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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>MTGS: Multi-Traversal Gaussian Splatting</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <base href="/html/2503.12552v2/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#S1" title="In MTGS: Multi-Traversal Gaussian Splatting">1 Introduction</a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#S2" title="In MTGS: Multi-Traversal Gaussian Splatting">2 Related Work</a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#S3" title="In MTGS: Multi-Traversal Gaussian Splatting">3 Background</a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#S3.SS1" title="In 3 Background ‣ MTGS: Multi-Traversal Gaussian Splatting">3.1 Preliminary on 3DGS</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#S3.SS2" title="In 3 Background ‣ MTGS: Multi-Traversal Gaussian Splatting">3.2 Problem Formulation</a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#S4" title="In MTGS: Multi-Traversal Gaussian Splatting">4 Multi-Traversal Gaussian Splatting</a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#S4.SS1" title="In 4 Multi-Traversal Gaussian Splatting ‣ MTGS: Multi-Traversal Gaussian Splatting">4.1 Multi-Traversal Scene Graph</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#S4.SS2" title="In 4 Multi-Traversal Gaussian Splatting ‣ MTGS: Multi-Traversal Gaussian Splatting">4.2 Appearance Modeling</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#S4.SS3" title="In 4 Multi-Traversal Gaussian Splatting ‣ MTGS: Multi-Traversal Gaussian Splatting">4.3 Regularization and Training</a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#S5" title="In MTGS: Multi-Traversal Gaussian Splatting">5 Experiment</a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#S5.SS1" title="In 5 Experiment ‣ MTGS: Multi-Traversal Gaussian Splatting">5.1 Setup and Protocols</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#S5.SS2" title="In 5 Experiment ‣ MTGS: Multi-Traversal Gaussian Splatting">5.2 Main Results</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#S5.SS3" title="In 5 Experiment ‣ MTGS: Multi-Traversal Gaussian Splatting">5.3 Ablation Study</a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#S6" title="In MTGS: Multi-Traversal Gaussian Splatting">6 Conclusion and Outlook</a></li> <li class="ltx_tocentry ltx_tocentry_appendix"> <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#A1" title="In MTGS: Multi-Traversal Gaussian Splatting">A Implementation Details</a> <ol class="ltx_toclist ltx_toclist_appendix"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#A1.SS1" title="In Appendix A Implementation Details ‣ MTGS: Multi-Traversal Gaussian Splatting">A.1 Dataset</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#A1.SS2" title="In Appendix A Implementation Details ‣ MTGS: Multi-Traversal Gaussian Splatting">A.2 MTGS</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#A1.SS3" title="In Appendix A Implementation Details ‣ MTGS: Multi-Traversal Gaussian Splatting">A.3 Reproduction of baselines</a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#A2" title="In MTGS: Multi-Traversal Gaussian Splatting">B Experiments</a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line ltx_pruned_first"> <h1 class="ltx_title ltx_title_document">MTGS: Multi-Traversal Gaussian Splatting</h1> <div class="ltx_authors"> Tianyu Li1,2∗ Yihang Qiu2∗ Zhenhua Wu1∗ Carl Lindström4 Peng Su2 Matthias Nießner3 Hongyang Li2 1Shanghai Innovation Institute 2OpenDriveLab and MMLab, The University of Hong Kong 3Technical University of Munich 4Chalmers University of Technology </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> Multi-traversal data, commonly collected through daily commutes or by self-driving fleets, provides multiple viewpoints for scene reconstruction within a road block. This data offers significant potential for high-quality novel view synthesis, which is crucial for applications such as autonomous vehicle simulators. However, inherent challenges in multi-traversal data often result in suboptimal reconstruction quality, including variations in appearance and the presence of dynamic objects. To address these issues, we propose Multi-Traversal Gaussian Splatting (MTGS), a novel approach that reconstructs high-quality driving scenes from arbitrarily collected multi-traversal data by modeling a shared static geometry while separately handling dynamic elements and appearance variations. Our method employs a multi-traversal dynamic scene graph with a shared static node and traversal-specific dynamic nodes, complemented by color correction nodes with learnable spherical harmonics coefficient residuals. This approach enables high-fidelity novel view synthesis and provides flexibility to navigate any viewpoint. We conduct extensive experiments on a large-scale driving dataset, nuPlan, with multi-traversal data. Our results demonstrate that MTGS improves LPIPS by 23.5% and geometry accuracy by 46.3% compared to single-traversal baselines. The code and data would be available to the public. </div> <div class="ltx_logical-block" id="id13"> <div class="ltx_para" id="id13.p1"> <img alt="[Uncaptioned image]" class="ltx_graphics ltx_centering ltx_img_landscape" height="516" id="id12.g1" src="x1.png" width="932"/> </div> <figure class="ltx_figure ltx_align_center" id="S0.F1"> <figcaption class="ltx_caption">Figure 1: Multi-Traversal Gaussian Splatting (MTGS) could reconstruct high-fidelity driving scenes from multi-traversal data. All images are rendered from a MTGS model of the same road block. (a) This approach preferably handles variations in lighting and shadows, rendering views conditioned on the traversal index (Trv #). (b) The extrapolation quality of MTGS is showcased. It maintains high visual quality, even with lateral shifts of 8 meters (i.e., two lanes). For clarity, we mark a fixed reference point across traversals with a red pin. </figcaption> </figure> </div> ††∗Equal contribution. Primary contact: tianyu@opendrivelab.com <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> 1 Introduction</h2> <div class="ltx_para" id="S1.p1"> Building photorealistic simulators is crucial for developing safe and robust autonomous vehicles (AVs), which could be adopted to create digital twins for testing autonomous systems <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib8" title="">8</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib24" title="">24</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib52" title="">52</a>]</cite>, or generate diverse data for training end-to-end planning algorithms <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib12" title="">12</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib2" title="">2</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib9" title="">9</a>]</cite>. To achieve this goal, the fundamental requirement is to synthesize high-fidelity renderings from arbitrary viewpoints while accurately preserving dynamic elements of the driving environment. </div> <div class="ltx_para" id="S1.p2"> Scene reconstruction from recorded sensor data of AVs has gained popularity in recent years for this purpose <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib47" title="">47</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib35" title="">35</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib45" title="">45</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib3" title="">3</a>]</cite>. However, methods that rely on single traversal logs often suffer from poor view extrapolation quality <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib14" title="">14</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib10" title="">10</a>]</cite>. In contrast, multi-traversal data covers a wide range of views. Intuitively, reconstruction using multi-traversal data improves quality for viewpoints that deviate from the original sequence. This is because views distributed across multiple lanes provide richer geometric constraints, potentially enabling view interpolation across the entire drivable area. </div> <div class="ltx_para" id="S1.p3"> Nonetheless, reconstructing a high-fidelity scene across multi-traversals is non-trivial. One characteristic of multi-traversal data is that it represents the same shared space, but the collection could span over a large time period. This indicates that effective interpolation across traversals applies to the spatial aspects of the scene only, corresponding to static 3D geometry, while temporal variations, such as scene dynamics and appearance, remain challenging to interpolate. In particular, the sunlight and weather can be mixed, resulting in different exposure, tone, white balance, and shadow. Furthermore, scene dynamics include both moving and parked vehicles, which are also time-variant. As a consequence, naive reconstruction approaches often struggle to model these inconsistencies, leading to blurred outputs or severe artifacts <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib30" title="">30</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib10" title="">10</a>]</cite>. </div> <div class="ltx_para" id="S1.p4"> To this end, we propose Multi-Traversal Gaussian Splatting (MTGS), the first approach able to reconstruct multi-traversal dynamic scenes and thus synthesize photorealistic extrapolated views, as depicted in <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#S0.F1" title="In MTGS: Multi-Traversal Gaussian Splatting">Fig. 1</a>. Our approach leverages images from multi-traversal sequences to reconstruct a shared static geometry while separately modeling scene dynamics and appearance variations across different traversals. Specifically, we propose a multi-traversal scene graph that builds a shared static node, and dynamic nodes within sub-graphs corresponding to each traversal. This design enables dynamic objects across traversals to be modeled in parallel. In addition, a LiDAR-guided exposure alignment module is introduced to ensure consistent appearance within individual traversal images. We further integrate an appearance node into each traversal subgraph to capture appearance variations in the form of the residual spherical harmonics coefficient. Finally, multiple regularization losses are developed to enhance the geometric alignment between traversals. </div> <div class="ltx_para" id="S1.p5"> To measure the view extrapolation performance of MTGS with prior works fairly, a dedicated benchmark on the public driving dataset, nuPlan <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib16" title="">16</a>]</cite>, is constructed. We select road blocks with multi-traversal data distributed across multiple lanes and evaluate one isolated traversal with minimal spatial overlap with others. Compared to single-traversal reconstruction, our multi-traversal approach consistently improves performance as additional traversals are incorporated, achieving up to an 18.5% improvement on the pixel-level metric (SSIM), 23.5% on the feature-level metric (LPIPS), and 46.3% on the geometry-level metric (absolute depth relative error). Our method also outperforms state-of-the-art approaches across all evaluation metrics. </div> <div class="ltx_para" id="S1.p6"> The contributions are summarized as follows: <ul class="ltx_itemize" id="S1.I1"> <li class="ltx_item" id="S1.I1.i1" style="list-style-type:none;"> • <div class="ltx_para" id="S1.I1.i1.p1"> We propose MTGS with a novel multi-traversal scene graph, including a shared static node that represents background geometry, an appearance node to model various appearances, and a transient node to preserve dynamic information. </div> </li> <li class="ltx_item" id="S1.I1.i2" style="list-style-type:none;"> • <div class="ltx_para" id="S1.I1.i2.p1"> MTGS enables high-fidelity reconstruction with extraordinary view extrapolation quality. We demonstrate that the MTGS achieves state-of-the-art performance in driving scene extrapolated view synthesis. It outperforms previous SOTA by 17.6% on SSIM, 42.4% on LPIPS and 35% on AbsRel. </div> </li> </ul> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> 2 Related Work</h2> <div class="ltx_para ltx_noindent" id="S2.p1"> Driving Scene Reconstruction. Recent approaches on driving scene reconstruction can be categorized into two paradigms: neural radiance fields (NeRF) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib27" title="">27</a>]</cite> and 3D Gaussian splatting (3DGS) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib17" title="">17</a>]</cite> based methods. NeRF-based methods <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib47" title="">47</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib39" title="">39</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib35" title="">35</a>]</cite> have shown remarkable success in reconstructing static backgrounds and dynamic agents via neural feature grids. Recent advancements in 3DGS provide a more efficient solution. DrivingGaussian <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib54" title="">54</a>]</cite>, HUGS <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib53" title="">53</a>]</cite>, and Street Gaussians <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib45" title="">45</a>]</cite> initialize dynamic objects using 3D bounding boxes and utilize the scene graph design that separates static backgrounds and dynamic objects to reconstruct driving scenes. Building upon this foundation, OmniRe <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib3" title="">3</a>]</cite> models cyclists and pedestrians using Deformable Gaussian <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib15" title="">15</a>]</cite> nodes and SMPL <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib25" title="">25</a>]</cite> nodes. SplatAD <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib11" title="">11</a>]</cite> explores LiDAR rasterization and solves the rolling shutter effect on both image and LiDAR to achieve better results. Yet, existing methods focus on the single traversal setting mainly, i.e., training and evaluating the original video sequence in a view interpolation manner. This work extends the dynamic scene graph design to a multi-traversal setting and evaluates an extrapolated view of unseen traversal. </div> <div class="ltx_para ltx_noindent" id="S2.p2"> Novel View Synthesis in Autonomous Driving. It emphasizes the extrapolation ability in reconstruction models. This topic follows two technical paradigms primarily: regularization-guided and generative-prior-guided. Among regularization-based methods, AutoSplat <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib18" title="">18</a>]</cite> introduces planar assumptions on the geometry of the road and sky while exploiting the symmetry of foreground objects to reconstruct unseen parts. Vid2Sim <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib41" title="">41</a>]</cite> enforces patch-normalized depth consistency and adjacent pixel normal vector alignment. Recent research <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib49" title="">49</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib14" title="">14</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib46" title="">46</a>]</cite> generates novel views with diffusion models conditioned on different features, e.g., images, depth, or LiDAR, to supplement training 3DGS. FreeSim <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib7" title="">7</a>]</cite> extends the coverage limit of LiDAR and adopts a hybrid generative reconstruction method to add generated views to the reconstruction process progressively. StreetUnveiler <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib43" title="">43</a>]</cite> removes parking cars, reconstructs occluded background with an inpainting diffusion model, and designs a near-to-far sampling strategy to improve temporal consistency. While these methods achieve photorealistic synthesis, they exhibit prohibitive computational costs and are limited by the quality of generation models. They also fall short of flexibility when inpainting unseen or occluded parts. Our work addresses these gaps through multi-traversal images collected from the real world, and utilizes regularization to achieve better geometry consistency. </div> <div class="ltx_para ltx_noindent" id="S2.p3"> Appearance Modeling. It has been a long-standing challenge in neural scene reconstruction. NeRF in the wild <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib26" title="">26</a>]</cite> pioneered appearance modeling for unstructured photo collections by presenting learnable per-image appearance embeddings. Block-NeRF <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib33" title="">33</a>]</cite> utilizes camera exposure parameters to optimize per-image appearance embeddings, enabling city-scale reconstruction. Recent works <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib50" title="">50</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib21" title="">21</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib42" title="">42</a>]</cite> in 3DGS explore appearance modeling similarly. For instance, <cite class="ltx_cite ltx_citemacro_citet">Kulhanek et al. [<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib19" title="">19</a>]</cite> propose to combine per-Gaussian and per-image appearance embeddings to model appearance variation. These methods aim to solve per-camera appearance alignment with large overlapped regions, or with additional information input. However, they often treat the transient as a distraction and use a semantic mask or uncertainty optimization to remove dynamic objects from the scene. We address the challenge of appearance modeling in multi-traversal AV sensor datasets, characterized by unbounded, non-object-centric, and dynamic scenes. Our approach leverages the inherent appearance consistency within individual traversals and the variations observed across multi-traversals to achieve improved appearance modeling. Moreover, we retain dynamic information to facilitate downstream applications. </div> <div class="ltx_para ltx_noindent" id="S2.p4"> Multi-traversal Street Reconstruction. It builds the scalable and robust 3D environmental representations for autonomous driving. <cite class="ltx_cite ltx_citemacro_citet">Qin et al. [<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib30" title="">30</a>]</cite> employ semantic segmentation to mask transient objects out and learn per-traversal appearance embeddings. 3DGM <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib20" title="">20</a>]</cite> proposes a self-supervised scene decomposition and mapping framework that leverages repeated traversals and pre-trained vision features to identify static backgrounds. The EUVS benchmark <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib10" title="">10</a>]</cite> is designed to evaluate view extrapolation quality using multi-traversal data. It also includes a baseline that trains on multi-traversal data confined to a single lane. Existing methods tend to produce blurred synthesized outputs, primarily due to their simplistic modeling of scene dynamics and appearance variations. They also filter out all dynamic objects during reconstruction, while we contend that preserving them is essential for achieving a comprehensive reconstruction and enabling downstream applications. </div> <figure class="ltx_figure" id="S2.F2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="493" id="S2.F2.g1" src="x2.png" width="904"/> <figcaption class="ltx_caption ltx_centering">Figure 2: Overview. MTGS reconstructs a scene graph from multi-traversal sensor sequences. The scene graph consists of three types of nodes. (a) The rendering of a traversal subgraph starts with a shared static node, representing the time-invariant part of the scene. (b) This is followed by an appearance node that applies traversal-specific appearance effects, such as lighting and shadows. (c) Finally, transient nodes are placed in the background. (d) We align exposure using the overlapping LiDAR point cloud to ensure lighting consistency within the subgraph. (e) Photometric loss and multiple geometric losses are applied to bootstrap the reconstruction fidelity. </figcaption> </figure> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> 3 Background</h2> <section class="ltx_subsection" id="S3.SS1"> <h3 class="ltx_title ltx_title_subsection"> 3.1 Preliminary on 3DGS</h3> <div class="ltx_para" id="S3.SS1.p1"> 3D Gaussian Splatting (3DGS), first proposed in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib17" title="">17</a>]</cite>, effectively reconstructs a scene with a set of 3D Gaussians <math alttext="\mathcal{G}=\left\{G_{i}\mid i=1,2,\cdots,N\right\}" class="ltx_Math" display="inline" id="S3.SS1.p1.1.m1.6"><semantics id="S3.SS1.p1.1.m1.6a"><mrow id="S3.SS1.p1.1.m1.6.6" xref="S3.SS1.p1.1.m1.6.6.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p1.1.m1.6.6.4" xref="S3.SS1.p1.1.m1.6.6.4.cmml">𝒢</mi><mo id="S3.SS1.p1.1.m1.6.6.3" xref="S3.SS1.p1.1.m1.6.6.3.cmml">=</mo><mrow id="S3.SS1.p1.1.m1.6.6.2.2" xref="S3.SS1.p1.1.m1.6.6.2.3.cmml"><mo id="S3.SS1.p1.1.m1.6.6.2.2.3" xref="S3.SS1.p1.1.m1.6.6.2.3.1.cmml">{</mo><msub id="S3.SS1.p1.1.m1.5.5.1.1.1" xref="S3.SS1.p1.1.m1.5.5.1.1.1.cmml"><mi id="S3.SS1.p1.1.m1.5.5.1.1.1.2" xref="S3.SS1.p1.1.m1.5.5.1.1.1.2.cmml">G</mi><mi id="S3.SS1.p1.1.m1.5.5.1.1.1.3" xref="S3.SS1.p1.1.m1.5.5.1.1.1.3.cmml">i</mi></msub><mo fence="true" id="S3.SS1.p1.1.m1.6.6.2.2.4" lspace="0em" rspace="0em" xref="S3.SS1.p1.1.m1.6.6.2.3.1.cmml">∣</mo><mrow id="S3.SS1.p1.1.m1.6.6.2.2.2" xref="S3.SS1.p1.1.m1.6.6.2.2.2.cmml"><mi id="S3.SS1.p1.1.m1.6.6.2.2.2.2" xref="S3.SS1.p1.1.m1.6.6.2.2.2.2.cmml">i</mi><mo id="S3.SS1.p1.1.m1.6.6.2.2.2.1" xref="S3.SS1.p1.1.m1.6.6.2.2.2.1.cmml">=</mo><mrow id="S3.SS1.p1.1.m1.6.6.2.2.2.3.2" xref="S3.SS1.p1.1.m1.6.6.2.2.2.3.1.cmml"><mn id="S3.SS1.p1.1.m1.1.1" xref="S3.SS1.p1.1.m1.1.1.cmml">1</mn><mo id="S3.SS1.p1.1.m1.6.6.2.2.2.3.2.1" xref="S3.SS1.p1.1.m1.6.6.2.2.2.3.1.cmml">,</mo><mn id="S3.SS1.p1.1.m1.2.2" xref="S3.SS1.p1.1.m1.2.2.cmml">2</mn><mo id="S3.SS1.p1.1.m1.6.6.2.2.2.3.2.2" xref="S3.SS1.p1.1.m1.6.6.2.2.2.3.1.cmml">,</mo><mi id="S3.SS1.p1.1.m1.3.3" mathvariant="normal" xref="S3.SS1.p1.1.m1.3.3.cmml">⋯</mi><mo id="S3.SS1.p1.1.m1.6.6.2.2.2.3.2.3" xref="S3.SS1.p1.1.m1.6.6.2.2.2.3.1.cmml">,</mo><mi id="S3.SS1.p1.1.m1.4.4" xref="S3.SS1.p1.1.m1.4.4.cmml">N</mi></mrow></mrow><mo id="S3.SS1.p1.1.m1.6.6.2.2.5" xref="S3.SS1.p1.1.m1.6.6.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.1.m1.6b"><apply id="S3.SS1.p1.1.m1.6.6.cmml" xref="S3.SS1.p1.1.m1.6.6"><eq id="S3.SS1.p1.1.m1.6.6.3.cmml" xref="S3.SS1.p1.1.m1.6.6.3"></eq><ci id="S3.SS1.p1.1.m1.6.6.4.cmml" xref="S3.SS1.p1.1.m1.6.6.4">𝒢</ci><apply id="S3.SS1.p1.1.m1.6.6.2.3.cmml" xref="S3.SS1.p1.1.m1.6.6.2.2"><csymbol cd="latexml" id="S3.SS1.p1.1.m1.6.6.2.3.1.cmml" xref="S3.SS1.p1.1.m1.6.6.2.2.3">conditional-set</csymbol><apply id="S3.SS1.p1.1.m1.5.5.1.1.1.cmml" xref="S3.SS1.p1.1.m1.5.5.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.1.m1.5.5.1.1.1.1.cmml" xref="S3.SS1.p1.1.m1.5.5.1.1.1">subscript</csymbol><ci id="S3.SS1.p1.1.m1.5.5.1.1.1.2.cmml" xref="S3.SS1.p1.1.m1.5.5.1.1.1.2">𝐺</ci><ci id="S3.SS1.p1.1.m1.5.5.1.1.1.3.cmml" xref="S3.SS1.p1.1.m1.5.5.1.1.1.3">𝑖</ci></apply><apply id="S3.SS1.p1.1.m1.6.6.2.2.2.cmml" xref="S3.SS1.p1.1.m1.6.6.2.2.2"><eq id="S3.SS1.p1.1.m1.6.6.2.2.2.1.cmml" xref="S3.SS1.p1.1.m1.6.6.2.2.2.1"></eq><ci id="S3.SS1.p1.1.m1.6.6.2.2.2.2.cmml" xref="S3.SS1.p1.1.m1.6.6.2.2.2.2">𝑖</ci><list id="S3.SS1.p1.1.m1.6.6.2.2.2.3.1.cmml" xref="S3.SS1.p1.1.m1.6.6.2.2.2.3.2"><cn id="S3.SS1.p1.1.m1.1.1.cmml" type="integer" xref="S3.SS1.p1.1.m1.1.1">1</cn><cn id="S3.SS1.p1.1.m1.2.2.cmml" type="integer" xref="S3.SS1.p1.1.m1.2.2">2</cn><ci id="S3.SS1.p1.1.m1.3.3.cmml" xref="S3.SS1.p1.1.m1.3.3">⋯</ci><ci id="S3.SS1.p1.1.m1.4.4.cmml" xref="S3.SS1.p1.1.m1.4.4">𝑁</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.1.m1.6c">\mathcal{G}=\left\{G_{i}\mid i=1,2,\cdots,N\right\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.1.m1.6d">caligraphic_G = { italic_G start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∣ italic_i = 1 , 2 , ⋯ , italic_N }</annotation></semantics></math>, where <math alttext="N" class="ltx_Math" display="inline" id="S3.SS1.p1.2.m2.1"><semantics id="S3.SS1.p1.2.m2.1a"><mi id="S3.SS1.p1.2.m2.1.1" xref="S3.SS1.p1.2.m2.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.2.m2.1b"><ci id="S3.SS1.p1.2.m2.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.2.m2.1c">N</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.2.m2.1d">italic_N</annotation></semantics></math> is the number of Gaussians. Each <math alttext="G_{i}(x)" class="ltx_Math" display="inline" id="S3.SS1.p1.3.m3.1"><semantics id="S3.SS1.p1.3.m3.1a"><mrow id="S3.SS1.p1.3.m3.1.2" xref="S3.SS1.p1.3.m3.1.2.cmml"><msub id="S3.SS1.p1.3.m3.1.2.2" xref="S3.SS1.p1.3.m3.1.2.2.cmml"><mi id="S3.SS1.p1.3.m3.1.2.2.2" xref="S3.SS1.p1.3.m3.1.2.2.2.cmml">G</mi><mi id="S3.SS1.p1.3.m3.1.2.2.3" xref="S3.SS1.p1.3.m3.1.2.2.3.cmml">i</mi></msub><mo id="S3.SS1.p1.3.m3.1.2.1" xref="S3.SS1.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.p1.3.m3.1.2.3.2" xref="S3.SS1.p1.3.m3.1.2.cmml"><mo id="S3.SS1.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S3.SS1.p1.3.m3.1.2.cmml">(</mo><mi id="S3.SS1.p1.3.m3.1.1" xref="S3.SS1.p1.3.m3.1.1.cmml">x</mi><mo id="S3.SS1.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S3.SS1.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.3.m3.1b"><apply id="S3.SS1.p1.3.m3.1.2.cmml" xref="S3.SS1.p1.3.m3.1.2"><times id="S3.SS1.p1.3.m3.1.2.1.cmml" xref="S3.SS1.p1.3.m3.1.2.1"></times><apply id="S3.SS1.p1.3.m3.1.2.2.cmml" xref="S3.SS1.p1.3.m3.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.p1.3.m3.1.2.2.1.cmml" xref="S3.SS1.p1.3.m3.1.2.2">subscript</csymbol><ci id="S3.SS1.p1.3.m3.1.2.2.2.cmml" xref="S3.SS1.p1.3.m3.1.2.2.2">𝐺</ci><ci id="S3.SS1.p1.3.m3.1.2.2.3.cmml" xref="S3.SS1.p1.3.m3.1.2.2.3">𝑖</ci></apply><ci id="S3.SS1.p1.3.m3.1.1.cmml" xref="S3.SS1.p1.3.m3.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.3.m3.1c">G_{i}(x)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.3.m3.1d">italic_G start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math> is a Gaussian distribution: <table class="ltx_equation ltx_eqn_table" id="S3.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="G_{i}(x)=\text{{exp}}\left[{-\frac{1}{2}(x-\mathbf{x}_{i})^{\top}\mathbf{% \Sigma}_{i}^{-1}(x-\mathbf{x}_{i})}\right]," class="ltx_Math" display="block" id="S3.E1.m1.2"><semantics id="S3.E1.m1.2a"><mrow id="S3.E1.m1.2.2.1" xref="S3.E1.m1.2.2.1.1.cmml"><mrow id="S3.E1.m1.2.2.1.1" xref="S3.E1.m1.2.2.1.1.cmml"><mrow id="S3.E1.m1.2.2.1.1.3" xref="S3.E1.m1.2.2.1.1.3.cmml"><msub id="S3.E1.m1.2.2.1.1.3.2" xref="S3.E1.m1.2.2.1.1.3.2.cmml"><mi id="S3.E1.m1.2.2.1.1.3.2.2" xref="S3.E1.m1.2.2.1.1.3.2.2.cmml">G</mi><mi id="S3.E1.m1.2.2.1.1.3.2.3" xref="S3.E1.m1.2.2.1.1.3.2.3.cmml">i</mi></msub><mo id="S3.E1.m1.2.2.1.1.3.1" xref="S3.E1.m1.2.2.1.1.3.1.cmml">⁢</mo><mrow id="S3.E1.m1.2.2.1.1.3.3.2" xref="S3.E1.m1.2.2.1.1.3.cmml"><mo id="S3.E1.m1.2.2.1.1.3.3.2.1" stretchy="false" xref="S3.E1.m1.2.2.1.1.3.cmml">(</mo><mi id="S3.E1.m1.1.1" xref="S3.E1.m1.1.1.cmml">x</mi><mo id="S3.E1.m1.2.2.1.1.3.3.2.2" stretchy="false" xref="S3.E1.m1.2.2.1.1.3.cmml">)</mo></mrow></mrow><mo id="S3.E1.m1.2.2.1.1.2" xref="S3.E1.m1.2.2.1.1.2.cmml">=</mo><mrow id="S3.E1.m1.2.2.1.1.1" 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xref="S3.SS1.p1.4.m1.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.p1.4.m1.2.2.2.2.1.cmml" xref="S3.SS1.p1.4.m1.2.2.2.2">subscript</csymbol><ci id="S3.SS1.p1.4.m1.2.2.2.2.2.cmml" xref="S3.SS1.p1.4.m1.2.2.2.2.2">𝐪</ci><ci id="S3.SS1.p1.4.m1.2.2.2.2.3.cmml" xref="S3.SS1.p1.4.m1.2.2.2.2.3">𝑖</ci></apply><apply id="S3.SS1.p1.4.m1.3.3.3.3.cmml" xref="S3.SS1.p1.4.m1.3.3.3.3"><csymbol cd="ambiguous" id="S3.SS1.p1.4.m1.3.3.3.3.1.cmml" xref="S3.SS1.p1.4.m1.3.3.3.3">subscript</csymbol><ci id="S3.SS1.p1.4.m1.3.3.3.3.2.cmml" xref="S3.SS1.p1.4.m1.3.3.3.3.2">𝐬</ci><ci id="S3.SS1.p1.4.m1.3.3.3.3.3.cmml" xref="S3.SS1.p1.4.m1.3.3.3.3.3">𝑖</ci></apply><apply id="S3.SS1.p1.4.m1.4.4.4.4.cmml" xref="S3.SS1.p1.4.m1.4.4.4.4"><csymbol cd="ambiguous" id="S3.SS1.p1.4.m1.4.4.4.4.1.cmml" xref="S3.SS1.p1.4.m1.4.4.4.4">subscript</csymbol><ci id="S3.SS1.p1.4.m1.4.4.4.4.2.cmml" xref="S3.SS1.p1.4.m1.4.4.4.4.2">𝛼</ci><ci id="S3.SS1.p1.4.m1.4.4.4.4.3.cmml" xref="S3.SS1.p1.4.m1.4.4.4.4.3">𝑖</ci></apply><apply id="S3.SS1.p1.4.m1.5.5.5.5.cmml" xref="S3.SS1.p1.4.m1.5.5.5.5"><csymbol cd="ambiguous" id="S3.SS1.p1.4.m1.5.5.5.5.1.cmml" xref="S3.SS1.p1.4.m1.5.5.5.5">subscript</csymbol><ci id="S3.SS1.p1.4.m1.5.5.5.5.2.cmml" xref="S3.SS1.p1.4.m1.5.5.5.5.2">𝜷</ci><ci id="S3.SS1.p1.4.m1.5.5.5.5.3.cmml" xref="S3.SS1.p1.4.m1.5.5.5.5.3">𝑖</ci></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.4.m1.5c">\left\{\mathbf{x}_{i},\mathbf{q}_{i},\mathbf{s}_{i},\alpha_{i},\boldsymbol{% \beta}_{i}\right\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.4.m1.5d">{ bold_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_β start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT }</annotation></semantics></math>. Here <math alttext="\mathbf{x}_{i}\in\mathbb{R}^{3}" class="ltx_Math" display="inline" id="S3.SS1.p1.5.m2.1"><semantics id="S3.SS1.p1.5.m2.1a"><mrow id="S3.SS1.p1.5.m2.1.1" xref="S3.SS1.p1.5.m2.1.1.cmml"><msub id="S3.SS1.p1.5.m2.1.1.2" xref="S3.SS1.p1.5.m2.1.1.2.cmml"><mi id="S3.SS1.p1.5.m2.1.1.2.2" xref="S3.SS1.p1.5.m2.1.1.2.2.cmml">𝐱</mi><mi id="S3.SS1.p1.5.m2.1.1.2.3" xref="S3.SS1.p1.5.m2.1.1.2.3.cmml">i</mi></msub><mo id="S3.SS1.p1.5.m2.1.1.1" xref="S3.SS1.p1.5.m2.1.1.1.cmml">∈</mo><msup id="S3.SS1.p1.5.m2.1.1.3" xref="S3.SS1.p1.5.m2.1.1.3.cmml"><mi id="S3.SS1.p1.5.m2.1.1.3.2" xref="S3.SS1.p1.5.m2.1.1.3.2.cmml">ℝ</mi><mn id="S3.SS1.p1.5.m2.1.1.3.3" xref="S3.SS1.p1.5.m2.1.1.3.3.cmml">3</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.5.m2.1b"><apply id="S3.SS1.p1.5.m2.1.1.cmml" xref="S3.SS1.p1.5.m2.1.1"><in id="S3.SS1.p1.5.m2.1.1.1.cmml" xref="S3.SS1.p1.5.m2.1.1.1"></in><apply id="S3.SS1.p1.5.m2.1.1.2.cmml" xref="S3.SS1.p1.5.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p1.5.m2.1.1.2.1.cmml" xref="S3.SS1.p1.5.m2.1.1.2">subscript</csymbol><ci id="S3.SS1.p1.5.m2.1.1.2.2.cmml" xref="S3.SS1.p1.5.m2.1.1.2.2">𝐱</ci><ci id="S3.SS1.p1.5.m2.1.1.2.3.cmml" xref="S3.SS1.p1.5.m2.1.1.2.3">𝑖</ci></apply><apply id="S3.SS1.p1.5.m2.1.1.3.cmml" xref="S3.SS1.p1.5.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p1.5.m2.1.1.3.1.cmml" xref="S3.SS1.p1.5.m2.1.1.3">superscript</csymbol><ci id="S3.SS1.p1.5.m2.1.1.3.2.cmml" xref="S3.SS1.p1.5.m2.1.1.3.2">ℝ</ci><cn id="S3.SS1.p1.5.m2.1.1.3.3.cmml" type="integer" xref="S3.SS1.p1.5.m2.1.1.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.5.m2.1c">\mathbf{x}_{i}\in\mathbb{R}^{3}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.5.m2.1d">bold_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\alpha_{i}\in\mathbb{R}" class="ltx_Math" display="inline" id="S3.SS1.p1.6.m3.1"><semantics id="S3.SS1.p1.6.m3.1a"><mrow id="S3.SS1.p1.6.m3.1.1" xref="S3.SS1.p1.6.m3.1.1.cmml"><msub id="S3.SS1.p1.6.m3.1.1.2" xref="S3.SS1.p1.6.m3.1.1.2.cmml"><mi id="S3.SS1.p1.6.m3.1.1.2.2" xref="S3.SS1.p1.6.m3.1.1.2.2.cmml">α</mi><mi id="S3.SS1.p1.6.m3.1.1.2.3" xref="S3.SS1.p1.6.m3.1.1.2.3.cmml">i</mi></msub><mo id="S3.SS1.p1.6.m3.1.1.1" xref="S3.SS1.p1.6.m3.1.1.1.cmml">∈</mo><mi id="S3.SS1.p1.6.m3.1.1.3" xref="S3.SS1.p1.6.m3.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.6.m3.1b"><apply id="S3.SS1.p1.6.m3.1.1.cmml" xref="S3.SS1.p1.6.m3.1.1"><in id="S3.SS1.p1.6.m3.1.1.1.cmml" xref="S3.SS1.p1.6.m3.1.1.1"></in><apply id="S3.SS1.p1.6.m3.1.1.2.cmml" xref="S3.SS1.p1.6.m3.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p1.6.m3.1.1.2.1.cmml" xref="S3.SS1.p1.6.m3.1.1.2">subscript</csymbol><ci id="S3.SS1.p1.6.m3.1.1.2.2.cmml" xref="S3.SS1.p1.6.m3.1.1.2.2">𝛼</ci><ci id="S3.SS1.p1.6.m3.1.1.2.3.cmml" xref="S3.SS1.p1.6.m3.1.1.2.3">𝑖</ci></apply><ci id="S3.SS1.p1.6.m3.1.1.3.cmml" xref="S3.SS1.p1.6.m3.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.6.m3.1c">\alpha_{i}\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.6.m3.1d">italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ blackboard_R</annotation></semantics></math> define the position and opacity of the Gaussian. The quaternion <math alttext="\mathbf{q}_{i}\in\mathbb{R}^{4}" class="ltx_Math" display="inline" id="S3.SS1.p1.7.m4.1"><semantics id="S3.SS1.p1.7.m4.1a"><mrow id="S3.SS1.p1.7.m4.1.1" xref="S3.SS1.p1.7.m4.1.1.cmml"><msub id="S3.SS1.p1.7.m4.1.1.2" xref="S3.SS1.p1.7.m4.1.1.2.cmml"><mi id="S3.SS1.p1.7.m4.1.1.2.2" xref="S3.SS1.p1.7.m4.1.1.2.2.cmml">𝐪</mi><mi id="S3.SS1.p1.7.m4.1.1.2.3" xref="S3.SS1.p1.7.m4.1.1.2.3.cmml">i</mi></msub><mo id="S3.SS1.p1.7.m4.1.1.1" xref="S3.SS1.p1.7.m4.1.1.1.cmml">∈</mo><msup id="S3.SS1.p1.7.m4.1.1.3" xref="S3.SS1.p1.7.m4.1.1.3.cmml"><mi id="S3.SS1.p1.7.m4.1.1.3.2" xref="S3.SS1.p1.7.m4.1.1.3.2.cmml">ℝ</mi><mn id="S3.SS1.p1.7.m4.1.1.3.3" xref="S3.SS1.p1.7.m4.1.1.3.3.cmml">4</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.7.m4.1b"><apply id="S3.SS1.p1.7.m4.1.1.cmml" xref="S3.SS1.p1.7.m4.1.1"><in id="S3.SS1.p1.7.m4.1.1.1.cmml" xref="S3.SS1.p1.7.m4.1.1.1"></in><apply id="S3.SS1.p1.7.m4.1.1.2.cmml" xref="S3.SS1.p1.7.m4.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p1.7.m4.1.1.2.1.cmml" xref="S3.SS1.p1.7.m4.1.1.2">subscript</csymbol><ci id="S3.SS1.p1.7.m4.1.1.2.2.cmml" xref="S3.SS1.p1.7.m4.1.1.2.2">𝐪</ci><ci id="S3.SS1.p1.7.m4.1.1.2.3.cmml" xref="S3.SS1.p1.7.m4.1.1.2.3">𝑖</ci></apply><apply id="S3.SS1.p1.7.m4.1.1.3.cmml" xref="S3.SS1.p1.7.m4.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p1.7.m4.1.1.3.1.cmml" xref="S3.SS1.p1.7.m4.1.1.3">superscript</csymbol><ci id="S3.SS1.p1.7.m4.1.1.3.2.cmml" xref="S3.SS1.p1.7.m4.1.1.3.2">ℝ</ci><cn id="S3.SS1.p1.7.m4.1.1.3.3.cmml" type="integer" xref="S3.SS1.p1.7.m4.1.1.3.3">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.7.m4.1c">\mathbf{q}_{i}\in\mathbb{R}^{4}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.7.m4.1d">bold_q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT</annotation></semantics></math> can be converted into a rotation matrix <math alttext="\mathbf{R}\in\mathbb{R}^{3\times 3}" class="ltx_Math" display="inline" id="S3.SS1.p1.8.m5.1"><semantics id="S3.SS1.p1.8.m5.1a"><mrow id="S3.SS1.p1.8.m5.1.1" xref="S3.SS1.p1.8.m5.1.1.cmml"><mi id="S3.SS1.p1.8.m5.1.1.2" xref="S3.SS1.p1.8.m5.1.1.2.cmml">𝐑</mi><mo id="S3.SS1.p1.8.m5.1.1.1" xref="S3.SS1.p1.8.m5.1.1.1.cmml">∈</mo><msup id="S3.SS1.p1.8.m5.1.1.3" xref="S3.SS1.p1.8.m5.1.1.3.cmml"><mi id="S3.SS1.p1.8.m5.1.1.3.2" xref="S3.SS1.p1.8.m5.1.1.3.2.cmml">ℝ</mi><mrow id="S3.SS1.p1.8.m5.1.1.3.3" xref="S3.SS1.p1.8.m5.1.1.3.3.cmml"><mn id="S3.SS1.p1.8.m5.1.1.3.3.2" xref="S3.SS1.p1.8.m5.1.1.3.3.2.cmml">3</mn><mo id="S3.SS1.p1.8.m5.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S3.SS1.p1.8.m5.1.1.3.3.1.cmml">×</mo><mn id="S3.SS1.p1.8.m5.1.1.3.3.3" xref="S3.SS1.p1.8.m5.1.1.3.3.3.cmml">3</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.8.m5.1b"><apply id="S3.SS1.p1.8.m5.1.1.cmml" xref="S3.SS1.p1.8.m5.1.1"><in id="S3.SS1.p1.8.m5.1.1.1.cmml" xref="S3.SS1.p1.8.m5.1.1.1"></in><ci id="S3.SS1.p1.8.m5.1.1.2.cmml" xref="S3.SS1.p1.8.m5.1.1.2">𝐑</ci><apply id="S3.SS1.p1.8.m5.1.1.3.cmml" xref="S3.SS1.p1.8.m5.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p1.8.m5.1.1.3.1.cmml" xref="S3.SS1.p1.8.m5.1.1.3">superscript</csymbol><ci id="S3.SS1.p1.8.m5.1.1.3.2.cmml" xref="S3.SS1.p1.8.m5.1.1.3.2">ℝ</ci><apply id="S3.SS1.p1.8.m5.1.1.3.3.cmml" xref="S3.SS1.p1.8.m5.1.1.3.3"><times id="S3.SS1.p1.8.m5.1.1.3.3.1.cmml" xref="S3.SS1.p1.8.m5.1.1.3.3.1"></times><cn id="S3.SS1.p1.8.m5.1.1.3.3.2.cmml" type="integer" xref="S3.SS1.p1.8.m5.1.1.3.3.2">3</cn><cn id="S3.SS1.p1.8.m5.1.1.3.3.3.cmml" type="integer" xref="S3.SS1.p1.8.m5.1.1.3.3.3">3</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.8.m5.1c">\mathbf{R}\in\mathbb{R}^{3\times 3}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.8.m5.1d">bold_R ∈ blackboard_R start_POSTSUPERSCRIPT 3 × 3 end_POSTSUPERSCRIPT</annotation></semantics></math>, which, along with the scale <math alttext="\mathbf{s}_{i}\in\mathbb{R}^{3}" class="ltx_Math" display="inline" id="S3.SS1.p1.9.m6.1"><semantics id="S3.SS1.p1.9.m6.1a"><mrow id="S3.SS1.p1.9.m6.1.1" xref="S3.SS1.p1.9.m6.1.1.cmml"><msub id="S3.SS1.p1.9.m6.1.1.2" xref="S3.SS1.p1.9.m6.1.1.2.cmml"><mi id="S3.SS1.p1.9.m6.1.1.2.2" xref="S3.SS1.p1.9.m6.1.1.2.2.cmml">𝐬</mi><mi id="S3.SS1.p1.9.m6.1.1.2.3" xref="S3.SS1.p1.9.m6.1.1.2.3.cmml">i</mi></msub><mo id="S3.SS1.p1.9.m6.1.1.1" xref="S3.SS1.p1.9.m6.1.1.1.cmml">∈</mo><msup id="S3.SS1.p1.9.m6.1.1.3" xref="S3.SS1.p1.9.m6.1.1.3.cmml"><mi id="S3.SS1.p1.9.m6.1.1.3.2" xref="S3.SS1.p1.9.m6.1.1.3.2.cmml">ℝ</mi><mn id="S3.SS1.p1.9.m6.1.1.3.3" xref="S3.SS1.p1.9.m6.1.1.3.3.cmml">3</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.9.m6.1b"><apply id="S3.SS1.p1.9.m6.1.1.cmml" xref="S3.SS1.p1.9.m6.1.1"><in id="S3.SS1.p1.9.m6.1.1.1.cmml" xref="S3.SS1.p1.9.m6.1.1.1"></in><apply id="S3.SS1.p1.9.m6.1.1.2.cmml" xref="S3.SS1.p1.9.m6.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p1.9.m6.1.1.2.1.cmml" xref="S3.SS1.p1.9.m6.1.1.2">subscript</csymbol><ci id="S3.SS1.p1.9.m6.1.1.2.2.cmml" xref="S3.SS1.p1.9.m6.1.1.2.2">𝐬</ci><ci id="S3.SS1.p1.9.m6.1.1.2.3.cmml" xref="S3.SS1.p1.9.m6.1.1.2.3">𝑖</ci></apply><apply id="S3.SS1.p1.9.m6.1.1.3.cmml" xref="S3.SS1.p1.9.m6.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p1.9.m6.1.1.3.1.cmml" xref="S3.SS1.p1.9.m6.1.1.3">superscript</csymbol><ci id="S3.SS1.p1.9.m6.1.1.3.2.cmml" xref="S3.SS1.p1.9.m6.1.1.3.2">ℝ</ci><cn id="S3.SS1.p1.9.m6.1.1.3.3.cmml" type="integer" xref="S3.SS1.p1.9.m6.1.1.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.9.m6.1c">\mathbf{s}_{i}\in\mathbb{R}^{3}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.9.m6.1d">bold_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT</annotation></semantics></math>, determines the covariance matrix <math alttext="\mathbf{\Sigma}_{i}" class="ltx_Math" display="inline" id="S3.SS1.p1.10.m7.1"><semantics id="S3.SS1.p1.10.m7.1a"><msub id="S3.SS1.p1.10.m7.1.1" xref="S3.SS1.p1.10.m7.1.1.cmml"><mi id="S3.SS1.p1.10.m7.1.1.2" xref="S3.SS1.p1.10.m7.1.1.2.cmml">𝚺</mi><mi id="S3.SS1.p1.10.m7.1.1.3" xref="S3.SS1.p1.10.m7.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.10.m7.1b"><apply id="S3.SS1.p1.10.m7.1.1.cmml" xref="S3.SS1.p1.10.m7.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.10.m7.1.1.1.cmml" xref="S3.SS1.p1.10.m7.1.1">subscript</csymbol><ci id="S3.SS1.p1.10.m7.1.1.2.cmml" xref="S3.SS1.p1.10.m7.1.1.2">𝚺</ci><ci id="S3.SS1.p1.10.m7.1.1.3.cmml" xref="S3.SS1.p1.10.m7.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.10.m7.1c">\mathbf{\Sigma}_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.10.m7.1d">bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> of the Gaussian, i.e., <table class="ltx_equation ltx_eqn_table" id="S3.E2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathbf{\Sigma}_{i}=\mathbf{R}\mathbf{S}\mathbf{S}^{\top}\mathbf{R}^{\top}% \text{, where }\mathbf{S}=\text{{dialog}}(\mathbf{s}_{i})." class="ltx_Math" display="block" id="S3.E2.m1.1"><semantics id="S3.E2.m1.1a"><mrow id="S3.E2.m1.1.1.1" xref="S3.E2.m1.1.1.1.1.cmml"><mrow id="S3.E2.m1.1.1.1.1" xref="S3.E2.m1.1.1.1.1.cmml"><msub id="S3.E2.m1.1.1.1.1.3" xref="S3.E2.m1.1.1.1.1.3.cmml"><mi id="S3.E2.m1.1.1.1.1.3.2" xref="S3.E2.m1.1.1.1.1.3.2.cmml">𝚺</mi><mi id="S3.E2.m1.1.1.1.1.3.3" xref="S3.E2.m1.1.1.1.1.3.3.cmml">i</mi></msub><mo id="S3.E2.m1.1.1.1.1.4" xref="S3.E2.m1.1.1.1.1.4.cmml">=</mo><mrow id="S3.E2.m1.1.1.1.1.5" xref="S3.E2.m1.1.1.1.1.5.cmml"><msup id="S3.E2.m1.1.1.1.1.5.2" xref="S3.E2.m1.1.1.1.1.5.2.cmml"><mi id="S3.E2.m1.1.1.1.1.5.2.2" xref="S3.E2.m1.1.1.1.1.5.2.2.cmml">𝐑𝐒𝐒</mi><mo id="S3.E2.m1.1.1.1.1.5.2.3" xref="S3.E2.m1.1.1.1.1.5.2.3.cmml">⊤</mo></msup><mo id="S3.E2.m1.1.1.1.1.5.1" xref="S3.E2.m1.1.1.1.1.5.1.cmml">⁢</mo><msup id="S3.E2.m1.1.1.1.1.5.3" xref="S3.E2.m1.1.1.1.1.5.3.cmml"><mi id="S3.E2.m1.1.1.1.1.5.3.2" xref="S3.E2.m1.1.1.1.1.5.3.2.cmml">𝐑</mi><mo id="S3.E2.m1.1.1.1.1.5.3.3" xref="S3.E2.m1.1.1.1.1.5.3.3.cmml">⊤</mo></msup><mo id="S3.E2.m1.1.1.1.1.5.1a" xref="S3.E2.m1.1.1.1.1.5.1.cmml">⁢</mo><mtext id="S3.E2.m1.1.1.1.1.5.4" xref="S3.E2.m1.1.1.1.1.5.4a.cmml">, where </mtext><mo id="S3.E2.m1.1.1.1.1.5.1b" xref="S3.E2.m1.1.1.1.1.5.1.cmml">⁢</mo><mi id="S3.E2.m1.1.1.1.1.5.5" xref="S3.E2.m1.1.1.1.1.5.5.cmml">𝐒</mi></mrow><mo id="S3.E2.m1.1.1.1.1.6" xref="S3.E2.m1.1.1.1.1.6.cmml">=</mo><mrow id="S3.E2.m1.1.1.1.1.1" xref="S3.E2.m1.1.1.1.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="S3.E2.m1.1.1.1.1.1.3" xref="S3.E2.m1.1.1.1.1.1.3a.cmml">dialog</mtext><mo id="S3.E2.m1.1.1.1.1.1.2" xref="S3.E2.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.E2.m1.1.1.1.1.1.1.1" xref="S3.E2.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.E2.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.E2.m1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.E2.m1.1.1.1.1.1.1.1.1" xref="S3.E2.m1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.E2.m1.1.1.1.1.1.1.1.1.2" xref="S3.E2.m1.1.1.1.1.1.1.1.1.2.cmml">𝐬</mi><mi id="S3.E2.m1.1.1.1.1.1.1.1.1.3" xref="S3.E2.m1.1.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.E2.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.E2.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S3.E2.m1.1.1.1.2" lspace="0em" xref="S3.E2.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E2.m1.1b"><apply id="S3.E2.m1.1.1.1.1.cmml" xref="S3.E2.m1.1.1.1"><and id="S3.E2.m1.1.1.1.1a.cmml" xref="S3.E2.m1.1.1.1"></and><apply id="S3.E2.m1.1.1.1.1b.cmml" xref="S3.E2.m1.1.1.1"><eq id="S3.E2.m1.1.1.1.1.4.cmml" xref="S3.E2.m1.1.1.1.1.4"></eq><apply id="S3.E2.m1.1.1.1.1.3.cmml" xref="S3.E2.m1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.E2.m1.1.1.1.1.3.1.cmml" xref="S3.E2.m1.1.1.1.1.3">subscript</csymbol><ci id="S3.E2.m1.1.1.1.1.3.2.cmml" xref="S3.E2.m1.1.1.1.1.3.2">𝚺</ci><ci id="S3.E2.m1.1.1.1.1.3.3.cmml" xref="S3.E2.m1.1.1.1.1.3.3">𝑖</ci></apply><apply id="S3.E2.m1.1.1.1.1.5.cmml" xref="S3.E2.m1.1.1.1.1.5"><times id="S3.E2.m1.1.1.1.1.5.1.cmml" xref="S3.E2.m1.1.1.1.1.5.1"></times><apply id="S3.E2.m1.1.1.1.1.5.2.cmml" xref="S3.E2.m1.1.1.1.1.5.2"><csymbol cd="ambiguous" id="S3.E2.m1.1.1.1.1.5.2.1.cmml" xref="S3.E2.m1.1.1.1.1.5.2">superscript</csymbol><ci id="S3.E2.m1.1.1.1.1.5.2.2.cmml" xref="S3.E2.m1.1.1.1.1.5.2.2">𝐑𝐒𝐒</ci><csymbol cd="latexml" id="S3.E2.m1.1.1.1.1.5.2.3.cmml" xref="S3.E2.m1.1.1.1.1.5.2.3">top</csymbol></apply><apply id="S3.E2.m1.1.1.1.1.5.3.cmml" xref="S3.E2.m1.1.1.1.1.5.3"><csymbol cd="ambiguous" id="S3.E2.m1.1.1.1.1.5.3.1.cmml" xref="S3.E2.m1.1.1.1.1.5.3">superscript</csymbol><ci id="S3.E2.m1.1.1.1.1.5.3.2.cmml" xref="S3.E2.m1.1.1.1.1.5.3.2">𝐑</ci><csymbol cd="latexml" id="S3.E2.m1.1.1.1.1.5.3.3.cmml" xref="S3.E2.m1.1.1.1.1.5.3.3">top</csymbol></apply><ci id="S3.E2.m1.1.1.1.1.5.4a.cmml" xref="S3.E2.m1.1.1.1.1.5.4"><mtext id="S3.E2.m1.1.1.1.1.5.4.cmml" xref="S3.E2.m1.1.1.1.1.5.4">, where </mtext></ci><ci id="S3.E2.m1.1.1.1.1.5.5.cmml" xref="S3.E2.m1.1.1.1.1.5.5">𝐒</ci></apply></apply><apply id="S3.E2.m1.1.1.1.1c.cmml" xref="S3.E2.m1.1.1.1"><eq id="S3.E2.m1.1.1.1.1.6.cmml" xref="S3.E2.m1.1.1.1.1.6"></eq><share href="https://arxiv.org/html/2503.12552v2#S3.E2.m1.1.1.1.1.5.cmml" id="S3.E2.m1.1.1.1.1d.cmml" xref="S3.E2.m1.1.1.1"></share><apply id="S3.E2.m1.1.1.1.1.1.cmml" xref="S3.E2.m1.1.1.1.1.1"><times id="S3.E2.m1.1.1.1.1.1.2.cmml" xref="S3.E2.m1.1.1.1.1.1.2"></times><ci id="S3.E2.m1.1.1.1.1.1.3a.cmml" xref="S3.E2.m1.1.1.1.1.1.3"><mtext class="ltx_mathvariant_monospace" id="S3.E2.m1.1.1.1.1.1.3.cmml" xref="S3.E2.m1.1.1.1.1.1.3">dialog</mtext></ci><apply id="S3.E2.m1.1.1.1.1.1.1.1.1.cmml" xref="S3.E2.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.E2.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.E2.m1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S3.E2.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S3.E2.m1.1.1.1.1.1.1.1.1.2">𝐬</ci><ci id="S3.E2.m1.1.1.1.1.1.1.1.1.3.cmml" xref="S3.E2.m1.1.1.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E2.m1.1c">\mathbf{\Sigma}_{i}=\mathbf{R}\mathbf{S}\mathbf{S}^{\top}\mathbf{R}^{\top}% \text{, where }\mathbf{S}=\text{{dialog}}(\mathbf{s}_{i}).</annotation><annotation encoding="application/x-llamapun" id="S3.E2.m1.1d">bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = bold_RSS start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_R start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT , where bold_S = dialog ( bold_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(2)</td> </tr></tbody> </table> In this way, <math alttext="\mathbf{\Sigma}_{i}" class="ltx_Math" display="inline" id="S3.SS1.p1.11.m1.1"><semantics id="S3.SS1.p1.11.m1.1a"><msub id="S3.SS1.p1.11.m1.1.1" xref="S3.SS1.p1.11.m1.1.1.cmml"><mi id="S3.SS1.p1.11.m1.1.1.2" xref="S3.SS1.p1.11.m1.1.1.2.cmml">𝚺</mi><mi id="S3.SS1.p1.11.m1.1.1.3" xref="S3.SS1.p1.11.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.11.m1.1b"><apply id="S3.SS1.p1.11.m1.1.1.cmml" xref="S3.SS1.p1.11.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.11.m1.1.1.1.cmml" xref="S3.SS1.p1.11.m1.1.1">subscript</csymbol><ci id="S3.SS1.p1.11.m1.1.1.2.cmml" xref="S3.SS1.p1.11.m1.1.1.2">𝚺</ci><ci id="S3.SS1.p1.11.m1.1.1.3.cmml" xref="S3.SS1.p1.11.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.11.m1.1c">\mathbf{\Sigma}_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.11.m1.1d">bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is guaranteed to be positive semi-definite. As for colors, <math alttext="\boldsymbol{\beta}_{i}=\left\{\boldsymbol{\beta}_{i,l,m}\right\}" class="ltx_Math" display="inline" id="S3.SS1.p1.12.m2.4"><semantics id="S3.SS1.p1.12.m2.4a"><mrow id="S3.SS1.p1.12.m2.4.4" xref="S3.SS1.p1.12.m2.4.4.cmml"><msub id="S3.SS1.p1.12.m2.4.4.3" xref="S3.SS1.p1.12.m2.4.4.3.cmml"><mi id="S3.SS1.p1.12.m2.4.4.3.2" xref="S3.SS1.p1.12.m2.4.4.3.2.cmml">𝜷</mi><mi id="S3.SS1.p1.12.m2.4.4.3.3" xref="S3.SS1.p1.12.m2.4.4.3.3.cmml">i</mi></msub><mo id="S3.SS1.p1.12.m2.4.4.2" xref="S3.SS1.p1.12.m2.4.4.2.cmml">=</mo><mrow id="S3.SS1.p1.12.m2.4.4.1.1" xref="S3.SS1.p1.12.m2.4.4.1.2.cmml"><mo id="S3.SS1.p1.12.m2.4.4.1.1.2" xref="S3.SS1.p1.12.m2.4.4.1.2.cmml">{</mo><msub id="S3.SS1.p1.12.m2.4.4.1.1.1" xref="S3.SS1.p1.12.m2.4.4.1.1.1.cmml"><mi id="S3.SS1.p1.12.m2.4.4.1.1.1.2" xref="S3.SS1.p1.12.m2.4.4.1.1.1.2.cmml">𝜷</mi><mrow id="S3.SS1.p1.12.m2.3.3.3.5" xref="S3.SS1.p1.12.m2.3.3.3.4.cmml"><mi id="S3.SS1.p1.12.m2.1.1.1.1" xref="S3.SS1.p1.12.m2.1.1.1.1.cmml">i</mi><mo id="S3.SS1.p1.12.m2.3.3.3.5.1" xref="S3.SS1.p1.12.m2.3.3.3.4.cmml">,</mo><mi id="S3.SS1.p1.12.m2.2.2.2.2" xref="S3.SS1.p1.12.m2.2.2.2.2.cmml">l</mi><mo id="S3.SS1.p1.12.m2.3.3.3.5.2" xref="S3.SS1.p1.12.m2.3.3.3.4.cmml">,</mo><mi id="S3.SS1.p1.12.m2.3.3.3.3" xref="S3.SS1.p1.12.m2.3.3.3.3.cmml">m</mi></mrow></msub><mo id="S3.SS1.p1.12.m2.4.4.1.1.3" xref="S3.SS1.p1.12.m2.4.4.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.12.m2.4b"><apply id="S3.SS1.p1.12.m2.4.4.cmml" xref="S3.SS1.p1.12.m2.4.4"><eq id="S3.SS1.p1.12.m2.4.4.2.cmml" xref="S3.SS1.p1.12.m2.4.4.2"></eq><apply id="S3.SS1.p1.12.m2.4.4.3.cmml" xref="S3.SS1.p1.12.m2.4.4.3"><csymbol cd="ambiguous" id="S3.SS1.p1.12.m2.4.4.3.1.cmml" xref="S3.SS1.p1.12.m2.4.4.3">subscript</csymbol><ci id="S3.SS1.p1.12.m2.4.4.3.2.cmml" xref="S3.SS1.p1.12.m2.4.4.3.2">𝜷</ci><ci id="S3.SS1.p1.12.m2.4.4.3.3.cmml" xref="S3.SS1.p1.12.m2.4.4.3.3">𝑖</ci></apply><set id="S3.SS1.p1.12.m2.4.4.1.2.cmml" xref="S3.SS1.p1.12.m2.4.4.1.1"><apply id="S3.SS1.p1.12.m2.4.4.1.1.1.cmml" xref="S3.SS1.p1.12.m2.4.4.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.12.m2.4.4.1.1.1.1.cmml" xref="S3.SS1.p1.12.m2.4.4.1.1.1">subscript</csymbol><ci id="S3.SS1.p1.12.m2.4.4.1.1.1.2.cmml" xref="S3.SS1.p1.12.m2.4.4.1.1.1.2">𝜷</ci><list id="S3.SS1.p1.12.m2.3.3.3.4.cmml" xref="S3.SS1.p1.12.m2.3.3.3.5"><ci id="S3.SS1.p1.12.m2.1.1.1.1.cmml" xref="S3.SS1.p1.12.m2.1.1.1.1">𝑖</ci><ci id="S3.SS1.p1.12.m2.2.2.2.2.cmml" xref="S3.SS1.p1.12.m2.2.2.2.2">𝑙</ci><ci id="S3.SS1.p1.12.m2.3.3.3.3.cmml" xref="S3.SS1.p1.12.m2.3.3.3.3">𝑚</ci></list></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.12.m2.4c">\boldsymbol{\beta}_{i}=\left\{\boldsymbol{\beta}_{i,l,m}\right\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.12.m2.4d">bold_italic_β start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = { bold_italic_β start_POSTSUBSCRIPT italic_i , italic_l , italic_m end_POSTSUBSCRIPT }</annotation></semantics></math> are coefficients for spherical harmonics <math alttext="\left\{Y_{l,m}\right\}_{0\leq l\leq l_{\mathtt{max}}}^{-l\leq m\leq l}" class="ltx_Math" display="inline" id="S3.SS1.p1.13.m3.3"><semantics id="S3.SS1.p1.13.m3.3a"><msubsup id="S3.SS1.p1.13.m3.3.3" xref="S3.SS1.p1.13.m3.3.3.cmml"><mrow id="S3.SS1.p1.13.m3.3.3.1.1.1" xref="S3.SS1.p1.13.m3.3.3.1.1.2.cmml"><mo id="S3.SS1.p1.13.m3.3.3.1.1.1.2" xref="S3.SS1.p1.13.m3.3.3.1.1.2.cmml">{</mo><msub id="S3.SS1.p1.13.m3.3.3.1.1.1.1" xref="S3.SS1.p1.13.m3.3.3.1.1.1.1.cmml"><mi id="S3.SS1.p1.13.m3.3.3.1.1.1.1.2" xref="S3.SS1.p1.13.m3.3.3.1.1.1.1.2.cmml">Y</mi><mrow id="S3.SS1.p1.13.m3.2.2.2.4" xref="S3.SS1.p1.13.m3.2.2.2.3.cmml"><mi id="S3.SS1.p1.13.m3.1.1.1.1" xref="S3.SS1.p1.13.m3.1.1.1.1.cmml">l</mi><mo id="S3.SS1.p1.13.m3.2.2.2.4.1" xref="S3.SS1.p1.13.m3.2.2.2.3.cmml">,</mo><mi id="S3.SS1.p1.13.m3.2.2.2.2" xref="S3.SS1.p1.13.m3.2.2.2.2.cmml">m</mi></mrow></msub><mo id="S3.SS1.p1.13.m3.3.3.1.1.1.3" xref="S3.SS1.p1.13.m3.3.3.1.1.2.cmml">}</mo></mrow><mrow id="S3.SS1.p1.13.m3.3.3.1.3" xref="S3.SS1.p1.13.m3.3.3.1.3.cmml"><mn id="S3.SS1.p1.13.m3.3.3.1.3.2" xref="S3.SS1.p1.13.m3.3.3.1.3.2.cmml">0</mn><mo id="S3.SS1.p1.13.m3.3.3.1.3.3" xref="S3.SS1.p1.13.m3.3.3.1.3.3.cmml">≤</mo><mi id="S3.SS1.p1.13.m3.3.3.1.3.4" xref="S3.SS1.p1.13.m3.3.3.1.3.4.cmml">l</mi><mo id="S3.SS1.p1.13.m3.3.3.1.3.5" xref="S3.SS1.p1.13.m3.3.3.1.3.5.cmml">≤</mo><msub id="S3.SS1.p1.13.m3.3.3.1.3.6" xref="S3.SS1.p1.13.m3.3.3.1.3.6.cmml"><mi id="S3.SS1.p1.13.m3.3.3.1.3.6.2" xref="S3.SS1.p1.13.m3.3.3.1.3.6.2.cmml">l</mi><mi id="S3.SS1.p1.13.m3.3.3.1.3.6.3" xref="S3.SS1.p1.13.m3.3.3.1.3.6.3.cmml">𝚖𝚊𝚡</mi></msub></mrow><mrow id="S3.SS1.p1.13.m3.3.3.3" xref="S3.SS1.p1.13.m3.3.3.3.cmml"><mrow id="S3.SS1.p1.13.m3.3.3.3.2" xref="S3.SS1.p1.13.m3.3.3.3.2.cmml"><mo id="S3.SS1.p1.13.m3.3.3.3.2a" xref="S3.SS1.p1.13.m3.3.3.3.2.cmml">−</mo><mi id="S3.SS1.p1.13.m3.3.3.3.2.2" xref="S3.SS1.p1.13.m3.3.3.3.2.2.cmml">l</mi></mrow><mo id="S3.SS1.p1.13.m3.3.3.3.3" xref="S3.SS1.p1.13.m3.3.3.3.3.cmml">≤</mo><mi id="S3.SS1.p1.13.m3.3.3.3.4" xref="S3.SS1.p1.13.m3.3.3.3.4.cmml">m</mi><mo id="S3.SS1.p1.13.m3.3.3.3.5" xref="S3.SS1.p1.13.m3.3.3.3.5.cmml">≤</mo><mi id="S3.SS1.p1.13.m3.3.3.3.6" xref="S3.SS1.p1.13.m3.3.3.3.6.cmml">l</mi></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.13.m3.3b"><apply id="S3.SS1.p1.13.m3.3.3.cmml" xref="S3.SS1.p1.13.m3.3.3"><csymbol cd="ambiguous" id="S3.SS1.p1.13.m3.3.3.2.cmml" xref="S3.SS1.p1.13.m3.3.3">superscript</csymbol><apply id="S3.SS1.p1.13.m3.3.3.1.cmml" xref="S3.SS1.p1.13.m3.3.3"><csymbol cd="ambiguous" id="S3.SS1.p1.13.m3.3.3.1.2.cmml" xref="S3.SS1.p1.13.m3.3.3">subscript</csymbol><set id="S3.SS1.p1.13.m3.3.3.1.1.2.cmml" xref="S3.SS1.p1.13.m3.3.3.1.1.1"><apply id="S3.SS1.p1.13.m3.3.3.1.1.1.1.cmml" xref="S3.SS1.p1.13.m3.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.13.m3.3.3.1.1.1.1.1.cmml" xref="S3.SS1.p1.13.m3.3.3.1.1.1.1">subscript</csymbol><ci id="S3.SS1.p1.13.m3.3.3.1.1.1.1.2.cmml" xref="S3.SS1.p1.13.m3.3.3.1.1.1.1.2">𝑌</ci><list id="S3.SS1.p1.13.m3.2.2.2.3.cmml" xref="S3.SS1.p1.13.m3.2.2.2.4"><ci id="S3.SS1.p1.13.m3.1.1.1.1.cmml" xref="S3.SS1.p1.13.m3.1.1.1.1">𝑙</ci><ci id="S3.SS1.p1.13.m3.2.2.2.2.cmml" xref="S3.SS1.p1.13.m3.2.2.2.2">𝑚</ci></list></apply></set><apply id="S3.SS1.p1.13.m3.3.3.1.3.cmml" xref="S3.SS1.p1.13.m3.3.3.1.3"><and id="S3.SS1.p1.13.m3.3.3.1.3a.cmml" xref="S3.SS1.p1.13.m3.3.3.1.3"></and><apply id="S3.SS1.p1.13.m3.3.3.1.3b.cmml" xref="S3.SS1.p1.13.m3.3.3.1.3"><leq id="S3.SS1.p1.13.m3.3.3.1.3.3.cmml" xref="S3.SS1.p1.13.m3.3.3.1.3.3"></leq><cn id="S3.SS1.p1.13.m3.3.3.1.3.2.cmml" type="integer" xref="S3.SS1.p1.13.m3.3.3.1.3.2">0</cn><ci id="S3.SS1.p1.13.m3.3.3.1.3.4.cmml" xref="S3.SS1.p1.13.m3.3.3.1.3.4">𝑙</ci></apply><apply id="S3.SS1.p1.13.m3.3.3.1.3c.cmml" xref="S3.SS1.p1.13.m3.3.3.1.3"><leq id="S3.SS1.p1.13.m3.3.3.1.3.5.cmml" xref="S3.SS1.p1.13.m3.3.3.1.3.5"></leq><share href="https://arxiv.org/html/2503.12552v2#S3.SS1.p1.13.m3.3.3.1.3.4.cmml" id="S3.SS1.p1.13.m3.3.3.1.3d.cmml" xref="S3.SS1.p1.13.m3.3.3.1.3"></share><apply id="S3.SS1.p1.13.m3.3.3.1.3.6.cmml" xref="S3.SS1.p1.13.m3.3.3.1.3.6"><csymbol cd="ambiguous" id="S3.SS1.p1.13.m3.3.3.1.3.6.1.cmml" xref="S3.SS1.p1.13.m3.3.3.1.3.6">subscript</csymbol><ci id="S3.SS1.p1.13.m3.3.3.1.3.6.2.cmml" xref="S3.SS1.p1.13.m3.3.3.1.3.6.2">𝑙</ci><ci id="S3.SS1.p1.13.m3.3.3.1.3.6.3.cmml" xref="S3.SS1.p1.13.m3.3.3.1.3.6.3">𝚖𝚊𝚡</ci></apply></apply></apply></apply><apply id="S3.SS1.p1.13.m3.3.3.3.cmml" xref="S3.SS1.p1.13.m3.3.3.3"><and id="S3.SS1.p1.13.m3.3.3.3a.cmml" xref="S3.SS1.p1.13.m3.3.3.3"></and><apply id="S3.SS1.p1.13.m3.3.3.3b.cmml" xref="S3.SS1.p1.13.m3.3.3.3"><leq id="S3.SS1.p1.13.m3.3.3.3.3.cmml" xref="S3.SS1.p1.13.m3.3.3.3.3"></leq><apply id="S3.SS1.p1.13.m3.3.3.3.2.cmml" xref="S3.SS1.p1.13.m3.3.3.3.2"><minus id="S3.SS1.p1.13.m3.3.3.3.2.1.cmml" xref="S3.SS1.p1.13.m3.3.3.3.2"></minus><ci id="S3.SS1.p1.13.m3.3.3.3.2.2.cmml" xref="S3.SS1.p1.13.m3.3.3.3.2.2">𝑙</ci></apply><ci id="S3.SS1.p1.13.m3.3.3.3.4.cmml" xref="S3.SS1.p1.13.m3.3.3.3.4">𝑚</ci></apply><apply id="S3.SS1.p1.13.m3.3.3.3c.cmml" xref="S3.SS1.p1.13.m3.3.3.3"><leq id="S3.SS1.p1.13.m3.3.3.3.5.cmml" xref="S3.SS1.p1.13.m3.3.3.3.5"></leq><share href="https://arxiv.org/html/2503.12552v2#S3.SS1.p1.13.m3.3.3.3.4.cmml" id="S3.SS1.p1.13.m3.3.3.3d.cmml" xref="S3.SS1.p1.13.m3.3.3.3"></share><ci id="S3.SS1.p1.13.m3.3.3.3.6.cmml" xref="S3.SS1.p1.13.m3.3.3.3.6">𝑙</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.13.m3.3c">\left\{Y_{l,m}\right\}_{0\leq l\leq l_{\mathtt{max}}}^{-l\leq m\leq l}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.13.m3.3d">{ italic_Y start_POSTSUBSCRIPT italic_l , italic_m end_POSTSUBSCRIPT } start_POSTSUBSCRIPT 0 ≤ italic_l ≤ italic_l start_POSTSUBSCRIPT typewriter_max end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - italic_l ≤ italic_m ≤ italic_l end_POSTSUPERSCRIPT</annotation></semantics></math>, where each coefficient <math alttext="\boldsymbol{\beta}_{i,m,l}\in\mathbb{R}^{3}" class="ltx_Math" display="inline" id="S3.SS1.p1.14.m4.3"><semantics id="S3.SS1.p1.14.m4.3a"><mrow id="S3.SS1.p1.14.m4.3.4" xref="S3.SS1.p1.14.m4.3.4.cmml"><msub id="S3.SS1.p1.14.m4.3.4.2" xref="S3.SS1.p1.14.m4.3.4.2.cmml"><mi id="S3.SS1.p1.14.m4.3.4.2.2" xref="S3.SS1.p1.14.m4.3.4.2.2.cmml">𝜷</mi><mrow id="S3.SS1.p1.14.m4.3.3.3.5" xref="S3.SS1.p1.14.m4.3.3.3.4.cmml"><mi id="S3.SS1.p1.14.m4.1.1.1.1" xref="S3.SS1.p1.14.m4.1.1.1.1.cmml">i</mi><mo id="S3.SS1.p1.14.m4.3.3.3.5.1" xref="S3.SS1.p1.14.m4.3.3.3.4.cmml">,</mo><mi id="S3.SS1.p1.14.m4.2.2.2.2" xref="S3.SS1.p1.14.m4.2.2.2.2.cmml">m</mi><mo id="S3.SS1.p1.14.m4.3.3.3.5.2" xref="S3.SS1.p1.14.m4.3.3.3.4.cmml">,</mo><mi id="S3.SS1.p1.14.m4.3.3.3.3" xref="S3.SS1.p1.14.m4.3.3.3.3.cmml">l</mi></mrow></msub><mo id="S3.SS1.p1.14.m4.3.4.1" xref="S3.SS1.p1.14.m4.3.4.1.cmml">∈</mo><msup id="S3.SS1.p1.14.m4.3.4.3" xref="S3.SS1.p1.14.m4.3.4.3.cmml"><mi id="S3.SS1.p1.14.m4.3.4.3.2" xref="S3.SS1.p1.14.m4.3.4.3.2.cmml">ℝ</mi><mn id="S3.SS1.p1.14.m4.3.4.3.3" xref="S3.SS1.p1.14.m4.3.4.3.3.cmml">3</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.14.m4.3b"><apply id="S3.SS1.p1.14.m4.3.4.cmml" xref="S3.SS1.p1.14.m4.3.4"><in id="S3.SS1.p1.14.m4.3.4.1.cmml" xref="S3.SS1.p1.14.m4.3.4.1"></in><apply id="S3.SS1.p1.14.m4.3.4.2.cmml" xref="S3.SS1.p1.14.m4.3.4.2"><csymbol cd="ambiguous" id="S3.SS1.p1.14.m4.3.4.2.1.cmml" xref="S3.SS1.p1.14.m4.3.4.2">subscript</csymbol><ci id="S3.SS1.p1.14.m4.3.4.2.2.cmml" xref="S3.SS1.p1.14.m4.3.4.2.2">𝜷</ci><list id="S3.SS1.p1.14.m4.3.3.3.4.cmml" xref="S3.SS1.p1.14.m4.3.3.3.5"><ci id="S3.SS1.p1.14.m4.1.1.1.1.cmml" xref="S3.SS1.p1.14.m4.1.1.1.1">𝑖</ci><ci id="S3.SS1.p1.14.m4.2.2.2.2.cmml" xref="S3.SS1.p1.14.m4.2.2.2.2">𝑚</ci><ci id="S3.SS1.p1.14.m4.3.3.3.3.cmml" xref="S3.SS1.p1.14.m4.3.3.3.3">𝑙</ci></list></apply><apply id="S3.SS1.p1.14.m4.3.4.3.cmml" xref="S3.SS1.p1.14.m4.3.4.3"><csymbol cd="ambiguous" id="S3.SS1.p1.14.m4.3.4.3.1.cmml" xref="S3.SS1.p1.14.m4.3.4.3">superscript</csymbol><ci id="S3.SS1.p1.14.m4.3.4.3.2.cmml" xref="S3.SS1.p1.14.m4.3.4.3.2">ℝ</ci><cn id="S3.SS1.p1.14.m4.3.4.3.3.cmml" type="integer" xref="S3.SS1.p1.14.m4.3.4.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.14.m4.3c">\boldsymbol{\beta}_{i,m,l}\in\mathbb{R}^{3}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.14.m4.3d">bold_italic_β start_POSTSUBSCRIPT italic_i , italic_m , italic_l end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT</annotation></semantics></math> corresponds to RGB channels. Since only <math alttext="Y_{0,0}" class="ltx_Math" display="inline" id="S3.SS1.p1.15.m5.2"><semantics id="S3.SS1.p1.15.m5.2a"><msub id="S3.SS1.p1.15.m5.2.3" xref="S3.SS1.p1.15.m5.2.3.cmml"><mi id="S3.SS1.p1.15.m5.2.3.2" xref="S3.SS1.p1.15.m5.2.3.2.cmml">Y</mi><mrow id="S3.SS1.p1.15.m5.2.2.2.4" xref="S3.SS1.p1.15.m5.2.2.2.3.cmml"><mn id="S3.SS1.p1.15.m5.1.1.1.1" xref="S3.SS1.p1.15.m5.1.1.1.1.cmml">0</mn><mo id="S3.SS1.p1.15.m5.2.2.2.4.1" xref="S3.SS1.p1.15.m5.2.2.2.3.cmml">,</mo><mn id="S3.SS1.p1.15.m5.2.2.2.2" xref="S3.SS1.p1.15.m5.2.2.2.2.cmml">0</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.15.m5.2b"><apply id="S3.SS1.p1.15.m5.2.3.cmml" xref="S3.SS1.p1.15.m5.2.3"><csymbol cd="ambiguous" id="S3.SS1.p1.15.m5.2.3.1.cmml" xref="S3.SS1.p1.15.m5.2.3">subscript</csymbol><ci id="S3.SS1.p1.15.m5.2.3.2.cmml" xref="S3.SS1.p1.15.m5.2.3.2">𝑌</ci><list id="S3.SS1.p1.15.m5.2.2.2.3.cmml" xref="S3.SS1.p1.15.m5.2.2.2.4"><cn id="S3.SS1.p1.15.m5.1.1.1.1.cmml" type="integer" xref="S3.SS1.p1.15.m5.1.1.1.1">0</cn><cn id="S3.SS1.p1.15.m5.2.2.2.2.cmml" type="integer" xref="S3.SS1.p1.15.m5.2.2.2.2">0</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.15.m5.2c">Y_{0,0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.15.m5.2d">italic_Y start_POSTSUBSCRIPT 0 , 0 end_POSTSUBSCRIPT</annotation></semantics></math> is rotation-invariant, the coefficient <math alttext="\boldsymbol{\beta}_{i,0,0}" class="ltx_Math" display="inline" id="S3.SS1.p1.16.m6.3"><semantics id="S3.SS1.p1.16.m6.3a"><msub id="S3.SS1.p1.16.m6.3.4" xref="S3.SS1.p1.16.m6.3.4.cmml"><mi id="S3.SS1.p1.16.m6.3.4.2" xref="S3.SS1.p1.16.m6.3.4.2.cmml">𝜷</mi><mrow id="S3.SS1.p1.16.m6.3.3.3.5" xref="S3.SS1.p1.16.m6.3.3.3.4.cmml"><mi id="S3.SS1.p1.16.m6.1.1.1.1" xref="S3.SS1.p1.16.m6.1.1.1.1.cmml">i</mi><mo id="S3.SS1.p1.16.m6.3.3.3.5.1" xref="S3.SS1.p1.16.m6.3.3.3.4.cmml">,</mo><mn id="S3.SS1.p1.16.m6.2.2.2.2" xref="S3.SS1.p1.16.m6.2.2.2.2.cmml">0</mn><mo id="S3.SS1.p1.16.m6.3.3.3.5.2" xref="S3.SS1.p1.16.m6.3.3.3.4.cmml">,</mo><mn id="S3.SS1.p1.16.m6.3.3.3.3" xref="S3.SS1.p1.16.m6.3.3.3.3.cmml">0</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.16.m6.3b"><apply id="S3.SS1.p1.16.m6.3.4.cmml" xref="S3.SS1.p1.16.m6.3.4"><csymbol cd="ambiguous" id="S3.SS1.p1.16.m6.3.4.1.cmml" xref="S3.SS1.p1.16.m6.3.4">subscript</csymbol><ci id="S3.SS1.p1.16.m6.3.4.2.cmml" xref="S3.SS1.p1.16.m6.3.4.2">𝜷</ci><list id="S3.SS1.p1.16.m6.3.3.3.4.cmml" xref="S3.SS1.p1.16.m6.3.3.3.5"><ci id="S3.SS1.p1.16.m6.1.1.1.1.cmml" xref="S3.SS1.p1.16.m6.1.1.1.1">𝑖</ci><cn id="S3.SS1.p1.16.m6.2.2.2.2.cmml" type="integer" xref="S3.SS1.p1.16.m6.2.2.2.2">0</cn><cn id="S3.SS1.p1.16.m6.3.3.3.3.cmml" type="integer" xref="S3.SS1.p1.16.m6.3.3.3.3">0</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.16.m6.3c">\boldsymbol{\beta}_{i,0,0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.16.m6.3d">bold_italic_β start_POSTSUBSCRIPT italic_i , 0 , 0 end_POSTSUBSCRIPT</annotation></semantics></math> defines the natural color of the Gaussian, while other coefficients control view-dependent effects like reflections and shading. </div> <div class="ltx_para" id="S3.SS1.p2"> Given a camera pose <math alttext="\boldsymbol{\xi}=\left\{\mathbf{W},\mathbf{K}\right\}" class="ltx_Math" display="inline" id="S3.SS1.p2.1.m1.2"><semantics id="S3.SS1.p2.1.m1.2a"><mrow id="S3.SS1.p2.1.m1.2.3" xref="S3.SS1.p2.1.m1.2.3.cmml"><mi id="S3.SS1.p2.1.m1.2.3.2" xref="S3.SS1.p2.1.m1.2.3.2.cmml">𝝃</mi><mo id="S3.SS1.p2.1.m1.2.3.1" xref="S3.SS1.p2.1.m1.2.3.1.cmml">=</mo><mrow id="S3.SS1.p2.1.m1.2.3.3.2" xref="S3.SS1.p2.1.m1.2.3.3.1.cmml"><mo id="S3.SS1.p2.1.m1.2.3.3.2.1" xref="S3.SS1.p2.1.m1.2.3.3.1.cmml">{</mo><mi id="S3.SS1.p2.1.m1.1.1" xref="S3.SS1.p2.1.m1.1.1.cmml">𝐖</mi><mo id="S3.SS1.p2.1.m1.2.3.3.2.2" xref="S3.SS1.p2.1.m1.2.3.3.1.cmml">,</mo><mi id="S3.SS1.p2.1.m1.2.2" xref="S3.SS1.p2.1.m1.2.2.cmml">𝐊</mi><mo id="S3.SS1.p2.1.m1.2.3.3.2.3" xref="S3.SS1.p2.1.m1.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.1.m1.2b"><apply id="S3.SS1.p2.1.m1.2.3.cmml" xref="S3.SS1.p2.1.m1.2.3"><eq id="S3.SS1.p2.1.m1.2.3.1.cmml" xref="S3.SS1.p2.1.m1.2.3.1"></eq><ci id="S3.SS1.p2.1.m1.2.3.2.cmml" xref="S3.SS1.p2.1.m1.2.3.2">𝝃</ci><set id="S3.SS1.p2.1.m1.2.3.3.1.cmml" xref="S3.SS1.p2.1.m1.2.3.3.2"><ci id="S3.SS1.p2.1.m1.1.1.cmml" xref="S3.SS1.p2.1.m1.1.1">𝐖</ci><ci id="S3.SS1.p2.1.m1.2.2.cmml" xref="S3.SS1.p2.1.m1.2.2">𝐊</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.1.m1.2c">\boldsymbol{\xi}=\left\{\mathbf{W},\mathbf{K}\right\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.1.m1.2d">bold_italic_ξ = { bold_W , bold_K }</annotation></semantics></math>, including viewing transformation <math alttext="\mathbf{W}" class="ltx_Math" display="inline" id="S3.SS1.p2.2.m2.1"><semantics id="S3.SS1.p2.2.m2.1a"><mi id="S3.SS1.p2.2.m2.1.1" xref="S3.SS1.p2.2.m2.1.1.cmml">𝐖</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.2.m2.1b"><ci id="S3.SS1.p2.2.m2.1.1.cmml" xref="S3.SS1.p2.2.m2.1.1">𝐖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.2.m2.1c">\mathbf{W}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.2.m2.1d">bold_W</annotation></semantics></math> from the world coordinates to the camera coordinates and the camera intrinsic <math alttext="\bf{K}" class="ltx_Math" display="inline" id="S3.SS1.p2.3.m3.1"><semantics id="S3.SS1.p2.3.m3.1a"><mi id="S3.SS1.p2.3.m3.1.1" xref="S3.SS1.p2.3.m3.1.1.cmml">𝐊</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.3.m3.1b"><ci id="S3.SS1.p2.3.m3.1.1.cmml" xref="S3.SS1.p2.3.m3.1.1">𝐊</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.3.m3.1c">\bf{K}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.3.m3.1d">bold_K</annotation></semantics></math>, a 3D Gaussian can be projected into a 2D one with means and covariance: <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A2.EGx1"> <tbody id="S3.E3"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathbf{x}^{\prime}_{i}=\mathbf{K}\mathbf{W}\mathbf{x}_{i},\quad% \mathbf{\Sigma}^{\prime}_{i}=\mathbf{J}\mathbf{W}\mathbf{\Sigma}_{i}\mathbf{W}% ^{\top}\mathbf{J}^{\top}," class="ltx_Math" display="inline" id="S3.E3.m1.1"><semantics id="S3.E3.m1.1a"><mrow id="S3.E3.m1.1.1.1"><mrow id="S3.E3.m1.1.1.1.1.2" xref="S3.E3.m1.1.1.1.1.3.cmml"><mrow id="S3.E3.m1.1.1.1.1.1.1" xref="S3.E3.m1.1.1.1.1.1.1.cmml"><msubsup id="S3.E3.m1.1.1.1.1.1.1.2" xref="S3.E3.m1.1.1.1.1.1.1.2.cmml"><mi id="S3.E3.m1.1.1.1.1.1.1.2.2.2" xref="S3.E3.m1.1.1.1.1.1.1.2.2.2.cmml">𝐱</mi><mi id="S3.E3.m1.1.1.1.1.1.1.2.3" xref="S3.E3.m1.1.1.1.1.1.1.2.3.cmml">i</mi><mo id="S3.E3.m1.1.1.1.1.1.1.2.2.3" xref="S3.E3.m1.1.1.1.1.1.1.2.2.3.cmml">′</mo></msubsup><mo id="S3.E3.m1.1.1.1.1.1.1.1" xref="S3.E3.m1.1.1.1.1.1.1.1.cmml">=</mo><msub id="S3.E3.m1.1.1.1.1.1.1.3" xref="S3.E3.m1.1.1.1.1.1.1.3.cmml"><mi id="S3.E3.m1.1.1.1.1.1.1.3.2" xref="S3.E3.m1.1.1.1.1.1.1.3.2.cmml">𝐊𝐖𝐱</mi><mi id="S3.E3.m1.1.1.1.1.1.1.3.3" xref="S3.E3.m1.1.1.1.1.1.1.3.3.cmml">i</mi></msub></mrow><mo id="S3.E3.m1.1.1.1.1.2.3" rspace="1.167em" xref="S3.E3.m1.1.1.1.1.3a.cmml">,</mo><mrow id="S3.E3.m1.1.1.1.1.2.2" xref="S3.E3.m1.1.1.1.1.2.2.cmml"><msubsup id="S3.E3.m1.1.1.1.1.2.2.2" xref="S3.E3.m1.1.1.1.1.2.2.2.cmml"><mi id="S3.E3.m1.1.1.1.1.2.2.2.2.2" xref="S3.E3.m1.1.1.1.1.2.2.2.2.2.cmml">𝚺</mi><mi id="S3.E3.m1.1.1.1.1.2.2.2.3" xref="S3.E3.m1.1.1.1.1.2.2.2.3.cmml">i</mi><mo id="S3.E3.m1.1.1.1.1.2.2.2.2.3" xref="S3.E3.m1.1.1.1.1.2.2.2.2.3.cmml">′</mo></msubsup><mo id="S3.E3.m1.1.1.1.1.2.2.1" xref="S3.E3.m1.1.1.1.1.2.2.1.cmml">=</mo><mrow id="S3.E3.m1.1.1.1.1.2.2.3" xref="S3.E3.m1.1.1.1.1.2.2.3.cmml"><mi id="S3.E3.m1.1.1.1.1.2.2.3.2" xref="S3.E3.m1.1.1.1.1.2.2.3.2.cmml">𝐉𝐖</mi><mo id="S3.E3.m1.1.1.1.1.2.2.3.1" xref="S3.E3.m1.1.1.1.1.2.2.3.1.cmml">⁢</mo><msub id="S3.E3.m1.1.1.1.1.2.2.3.3" xref="S3.E3.m1.1.1.1.1.2.2.3.3.cmml"><mi id="S3.E3.m1.1.1.1.1.2.2.3.3.2" xref="S3.E3.m1.1.1.1.1.2.2.3.3.2.cmml">𝚺</mi><mi id="S3.E3.m1.1.1.1.1.2.2.3.3.3" xref="S3.E3.m1.1.1.1.1.2.2.3.3.3.cmml">i</mi></msub><mo id="S3.E3.m1.1.1.1.1.2.2.3.1a" xref="S3.E3.m1.1.1.1.1.2.2.3.1.cmml">⁢</mo><msup id="S3.E3.m1.1.1.1.1.2.2.3.4" xref="S3.E3.m1.1.1.1.1.2.2.3.4.cmml"><mi id="S3.E3.m1.1.1.1.1.2.2.3.4.2" xref="S3.E3.m1.1.1.1.1.2.2.3.4.2.cmml">𝐖</mi><mo id="S3.E3.m1.1.1.1.1.2.2.3.4.3" xref="S3.E3.m1.1.1.1.1.2.2.3.4.3.cmml">⊤</mo></msup><mo id="S3.E3.m1.1.1.1.1.2.2.3.1b" xref="S3.E3.m1.1.1.1.1.2.2.3.1.cmml">⁢</mo><msup id="S3.E3.m1.1.1.1.1.2.2.3.5" xref="S3.E3.m1.1.1.1.1.2.2.3.5.cmml"><mi id="S3.E3.m1.1.1.1.1.2.2.3.5.2" xref="S3.E3.m1.1.1.1.1.2.2.3.5.2.cmml">𝐉</mi><mo id="S3.E3.m1.1.1.1.1.2.2.3.5.3" xref="S3.E3.m1.1.1.1.1.2.2.3.5.3.cmml">⊤</mo></msup></mrow></mrow></mrow><mo id="S3.E3.m1.1.1.1.2">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E3.m1.1b"><apply id="S3.E3.m1.1.1.1.1.3.cmml" xref="S3.E3.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.E3.m1.1.1.1.1.3a.cmml" xref="S3.E3.m1.1.1.1.1.2.3">formulae-sequence</csymbol><apply id="S3.E3.m1.1.1.1.1.1.1.cmml" xref="S3.E3.m1.1.1.1.1.1.1"><eq id="S3.E3.m1.1.1.1.1.1.1.1.cmml" xref="S3.E3.m1.1.1.1.1.1.1.1"></eq><apply id="S3.E3.m1.1.1.1.1.1.1.2.cmml" xref="S3.E3.m1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.E3.m1.1.1.1.1.1.1.2.1.cmml" xref="S3.E3.m1.1.1.1.1.1.1.2">subscript</csymbol><apply id="S3.E3.m1.1.1.1.1.1.1.2.2.cmml" xref="S3.E3.m1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.E3.m1.1.1.1.1.1.1.2.2.1.cmml" xref="S3.E3.m1.1.1.1.1.1.1.2">superscript</csymbol><ci id="S3.E3.m1.1.1.1.1.1.1.2.2.2.cmml" xref="S3.E3.m1.1.1.1.1.1.1.2.2.2">𝐱</ci><ci id="S3.E3.m1.1.1.1.1.1.1.2.2.3.cmml" xref="S3.E3.m1.1.1.1.1.1.1.2.2.3">′</ci></apply><ci id="S3.E3.m1.1.1.1.1.1.1.2.3.cmml" xref="S3.E3.m1.1.1.1.1.1.1.2.3">𝑖</ci></apply><apply id="S3.E3.m1.1.1.1.1.1.1.3.cmml" xref="S3.E3.m1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.E3.m1.1.1.1.1.1.1.3.1.cmml" xref="S3.E3.m1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S3.E3.m1.1.1.1.1.1.1.3.2.cmml" xref="S3.E3.m1.1.1.1.1.1.1.3.2">𝐊𝐖𝐱</ci><ci id="S3.E3.m1.1.1.1.1.1.1.3.3.cmml" xref="S3.E3.m1.1.1.1.1.1.1.3.3">𝑖</ci></apply></apply><apply id="S3.E3.m1.1.1.1.1.2.2.cmml" xref="S3.E3.m1.1.1.1.1.2.2"><eq id="S3.E3.m1.1.1.1.1.2.2.1.cmml" xref="S3.E3.m1.1.1.1.1.2.2.1"></eq><apply id="S3.E3.m1.1.1.1.1.2.2.2.cmml" xref="S3.E3.m1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="S3.E3.m1.1.1.1.1.2.2.2.1.cmml" xref="S3.E3.m1.1.1.1.1.2.2.2">subscript</csymbol><apply id="S3.E3.m1.1.1.1.1.2.2.2.2.cmml" xref="S3.E3.m1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="S3.E3.m1.1.1.1.1.2.2.2.2.1.cmml" xref="S3.E3.m1.1.1.1.1.2.2.2">superscript</csymbol><ci id="S3.E3.m1.1.1.1.1.2.2.2.2.2.cmml" xref="S3.E3.m1.1.1.1.1.2.2.2.2.2">𝚺</ci><ci id="S3.E3.m1.1.1.1.1.2.2.2.2.3.cmml" xref="S3.E3.m1.1.1.1.1.2.2.2.2.3">′</ci></apply><ci id="S3.E3.m1.1.1.1.1.2.2.2.3.cmml" xref="S3.E3.m1.1.1.1.1.2.2.2.3">𝑖</ci></apply><apply id="S3.E3.m1.1.1.1.1.2.2.3.cmml" xref="S3.E3.m1.1.1.1.1.2.2.3"><times id="S3.E3.m1.1.1.1.1.2.2.3.1.cmml" xref="S3.E3.m1.1.1.1.1.2.2.3.1"></times><ci id="S3.E3.m1.1.1.1.1.2.2.3.2.cmml" xref="S3.E3.m1.1.1.1.1.2.2.3.2">𝐉𝐖</ci><apply id="S3.E3.m1.1.1.1.1.2.2.3.3.cmml" xref="S3.E3.m1.1.1.1.1.2.2.3.3"><csymbol cd="ambiguous" id="S3.E3.m1.1.1.1.1.2.2.3.3.1.cmml" xref="S3.E3.m1.1.1.1.1.2.2.3.3">subscript</csymbol><ci id="S3.E3.m1.1.1.1.1.2.2.3.3.2.cmml" xref="S3.E3.m1.1.1.1.1.2.2.3.3.2">𝚺</ci><ci id="S3.E3.m1.1.1.1.1.2.2.3.3.3.cmml" xref="S3.E3.m1.1.1.1.1.2.2.3.3.3">𝑖</ci></apply><apply id="S3.E3.m1.1.1.1.1.2.2.3.4.cmml" xref="S3.E3.m1.1.1.1.1.2.2.3.4"><csymbol cd="ambiguous" id="S3.E3.m1.1.1.1.1.2.2.3.4.1.cmml" xref="S3.E3.m1.1.1.1.1.2.2.3.4">superscript</csymbol><ci id="S3.E3.m1.1.1.1.1.2.2.3.4.2.cmml" xref="S3.E3.m1.1.1.1.1.2.2.3.4.2">𝐖</ci><csymbol cd="latexml" id="S3.E3.m1.1.1.1.1.2.2.3.4.3.cmml" xref="S3.E3.m1.1.1.1.1.2.2.3.4.3">top</csymbol></apply><apply id="S3.E3.m1.1.1.1.1.2.2.3.5.cmml" xref="S3.E3.m1.1.1.1.1.2.2.3.5"><csymbol cd="ambiguous" id="S3.E3.m1.1.1.1.1.2.2.3.5.1.cmml" xref="S3.E3.m1.1.1.1.1.2.2.3.5">superscript</csymbol><ci id="S3.E3.m1.1.1.1.1.2.2.3.5.2.cmml" xref="S3.E3.m1.1.1.1.1.2.2.3.5.2">𝐉</ci><csymbol cd="latexml" id="S3.E3.m1.1.1.1.1.2.2.3.5.3.cmml" xref="S3.E3.m1.1.1.1.1.2.2.3.5.3">top</csymbol></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E3.m1.1c">\displaystyle\mathbf{x}^{\prime}_{i}=\mathbf{K}\mathbf{W}\mathbf{x}_{i},\quad% \mathbf{\Sigma}^{\prime}_{i}=\mathbf{J}\mathbf{W}\mathbf{\Sigma}_{i}\mathbf{W}% ^{\top}\mathbf{J}^{\top},</annotation><annotation encoding="application/x-llamapun" id="S3.E3.m1.1d">bold_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = bold_KWx start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_Σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = bold_JW bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT bold_W start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_J start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(3)</td> </tr></tbody> </table> where <math alttext="\mathbf{J}" class="ltx_Math" display="inline" id="S3.SS1.p2.4.m1.1"><semantics id="S3.SS1.p2.4.m1.1a"><mi id="S3.SS1.p2.4.m1.1.1" xref="S3.SS1.p2.4.m1.1.1.cmml">𝐉</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.4.m1.1b"><ci id="S3.SS1.p2.4.m1.1.1.cmml" xref="S3.SS1.p2.4.m1.1.1">𝐉</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.4.m1.1c">\mathbf{J}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.4.m1.1d">bold_J</annotation></semantics></math> is the Jacobian matrix of <math alttext="\mathbf{K}" class="ltx_Math" display="inline" id="S3.SS1.p2.5.m2.1"><semantics id="S3.SS1.p2.5.m2.1a"><mi id="S3.SS1.p2.5.m2.1.1" xref="S3.SS1.p2.5.m2.1.1.cmml">𝐊</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.5.m2.1b"><ci id="S3.SS1.p2.5.m2.1.1.cmml" xref="S3.SS1.p2.5.m2.1.1">𝐊</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.5.m2.1c">\mathbf{K}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.5.m2.1d">bold_K</annotation></semantics></math>. This 2D Gaussian projection gives the opacity of <math alttext="G_{i}" class="ltx_Math" display="inline" id="S3.SS1.p2.6.m3.1"><semantics id="S3.SS1.p2.6.m3.1a"><msub id="S3.SS1.p2.6.m3.1.1" xref="S3.SS1.p2.6.m3.1.1.cmml"><mi id="S3.SS1.p2.6.m3.1.1.2" xref="S3.SS1.p2.6.m3.1.1.2.cmml">G</mi><mi id="S3.SS1.p2.6.m3.1.1.3" xref="S3.SS1.p2.6.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.6.m3.1b"><apply id="S3.SS1.p2.6.m3.1.1.cmml" xref="S3.SS1.p2.6.m3.1.1"><csymbol cd="ambiguous" id="S3.SS1.p2.6.m3.1.1.1.cmml" xref="S3.SS1.p2.6.m3.1.1">subscript</csymbol><ci id="S3.SS1.p2.6.m3.1.1.2.cmml" xref="S3.SS1.p2.6.m3.1.1.2">𝐺</ci><ci id="S3.SS1.p2.6.m3.1.1.3.cmml" xref="S3.SS1.p2.6.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.6.m3.1c">G_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.6.m3.1d">italic_G start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> projected onto the pixel <math alttext="p" class="ltx_Math" display="inline" id="S3.SS1.p2.7.m4.1"><semantics id="S3.SS1.p2.7.m4.1a"><mi id="S3.SS1.p2.7.m4.1.1" xref="S3.SS1.p2.7.m4.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.7.m4.1b"><ci id="S3.SS1.p2.7.m4.1.1.cmml" xref="S3.SS1.p2.7.m4.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.7.m4.1c">p</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.7.m4.1d">italic_p</annotation></semantics></math>, denoted as <math alttext="\alpha_{i\to p}" class="ltx_Math" display="inline" id="S3.SS1.p2.8.m5.1"><semantics id="S3.SS1.p2.8.m5.1a"><msub id="S3.SS1.p2.8.m5.1.1" xref="S3.SS1.p2.8.m5.1.1.cmml"><mi id="S3.SS1.p2.8.m5.1.1.2" xref="S3.SS1.p2.8.m5.1.1.2.cmml">α</mi><mrow id="S3.SS1.p2.8.m5.1.1.3" xref="S3.SS1.p2.8.m5.1.1.3.cmml"><mi id="S3.SS1.p2.8.m5.1.1.3.2" xref="S3.SS1.p2.8.m5.1.1.3.2.cmml">i</mi><mo id="S3.SS1.p2.8.m5.1.1.3.1" stretchy="false" xref="S3.SS1.p2.8.m5.1.1.3.1.cmml">→</mo><mi id="S3.SS1.p2.8.m5.1.1.3.3" xref="S3.SS1.p2.8.m5.1.1.3.3.cmml">p</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.8.m5.1b"><apply id="S3.SS1.p2.8.m5.1.1.cmml" xref="S3.SS1.p2.8.m5.1.1"><csymbol cd="ambiguous" id="S3.SS1.p2.8.m5.1.1.1.cmml" xref="S3.SS1.p2.8.m5.1.1">subscript</csymbol><ci id="S3.SS1.p2.8.m5.1.1.2.cmml" xref="S3.SS1.p2.8.m5.1.1.2">𝛼</ci><apply id="S3.SS1.p2.8.m5.1.1.3.cmml" xref="S3.SS1.p2.8.m5.1.1.3"><ci id="S3.SS1.p2.8.m5.1.1.3.1.cmml" xref="S3.SS1.p2.8.m5.1.1.3.1">→</ci><ci id="S3.SS1.p2.8.m5.1.1.3.2.cmml" xref="S3.SS1.p2.8.m5.1.1.3.2">𝑖</ci><ci id="S3.SS1.p2.8.m5.1.1.3.3.cmml" xref="S3.SS1.p2.8.m5.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.8.m5.1c">\alpha_{i\to p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.8.m5.1d">italic_α start_POSTSUBSCRIPT italic_i → italic_p end_POSTSUBSCRIPT</annotation></semantics></math>, which yields the final opacity by multiplying <math alttext="\alpha_{i}" class="ltx_Math" display="inline" id="S3.SS1.p2.9.m6.1"><semantics id="S3.SS1.p2.9.m6.1a"><msub id="S3.SS1.p2.9.m6.1.1" xref="S3.SS1.p2.9.m6.1.1.cmml"><mi id="S3.SS1.p2.9.m6.1.1.2" xref="S3.SS1.p2.9.m6.1.1.2.cmml">α</mi><mi id="S3.SS1.p2.9.m6.1.1.3" xref="S3.SS1.p2.9.m6.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.9.m6.1b"><apply id="S3.SS1.p2.9.m6.1.1.cmml" xref="S3.SS1.p2.9.m6.1.1"><csymbol cd="ambiguous" id="S3.SS1.p2.9.m6.1.1.1.cmml" xref="S3.SS1.p2.9.m6.1.1">subscript</csymbol><ci id="S3.SS1.p2.9.m6.1.1.2.cmml" xref="S3.SS1.p2.9.m6.1.1.2">𝛼</ci><ci id="S3.SS1.p2.9.m6.1.1.3.cmml" xref="S3.SS1.p2.9.m6.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.9.m6.1c">\alpha_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.9.m6.1d">italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, the opacity of the Gaussian itself. Combined with the color <math alttext="\mathbf{c}_{i,p}" class="ltx_Math" display="inline" id="S3.SS1.p2.10.m7.2"><semantics id="S3.SS1.p2.10.m7.2a"><msub id="S3.SS1.p2.10.m7.2.3" xref="S3.SS1.p2.10.m7.2.3.cmml"><mi id="S3.SS1.p2.10.m7.2.3.2" xref="S3.SS1.p2.10.m7.2.3.2.cmml">𝐜</mi><mrow id="S3.SS1.p2.10.m7.2.2.2.4" xref="S3.SS1.p2.10.m7.2.2.2.3.cmml"><mi id="S3.SS1.p2.10.m7.1.1.1.1" xref="S3.SS1.p2.10.m7.1.1.1.1.cmml">i</mi><mo id="S3.SS1.p2.10.m7.2.2.2.4.1" xref="S3.SS1.p2.10.m7.2.2.2.3.cmml">,</mo><mi id="S3.SS1.p2.10.m7.2.2.2.2" xref="S3.SS1.p2.10.m7.2.2.2.2.cmml">p</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.10.m7.2b"><apply id="S3.SS1.p2.10.m7.2.3.cmml" xref="S3.SS1.p2.10.m7.2.3"><csymbol cd="ambiguous" id="S3.SS1.p2.10.m7.2.3.1.cmml" xref="S3.SS1.p2.10.m7.2.3">subscript</csymbol><ci id="S3.SS1.p2.10.m7.2.3.2.cmml" xref="S3.SS1.p2.10.m7.2.3.2">𝐜</ci><list id="S3.SS1.p2.10.m7.2.2.2.3.cmml" xref="S3.SS1.p2.10.m7.2.2.2.4"><ci id="S3.SS1.p2.10.m7.1.1.1.1.cmml" xref="S3.SS1.p2.10.m7.1.1.1.1">𝑖</ci><ci id="S3.SS1.p2.10.m7.2.2.2.2.cmml" xref="S3.SS1.p2.10.m7.2.2.2.2">𝑝</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.10.m7.2c">\mathbf{c}_{i,p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.10.m7.2d">bold_c start_POSTSUBSCRIPT italic_i , italic_p end_POSTSUBSCRIPT</annotation></semantics></math> of the Gaussian <math alttext="G_{i}" class="ltx_Math" display="inline" id="S3.SS1.p2.11.m8.1"><semantics id="S3.SS1.p2.11.m8.1a"><msub id="S3.SS1.p2.11.m8.1.1" xref="S3.SS1.p2.11.m8.1.1.cmml"><mi id="S3.SS1.p2.11.m8.1.1.2" xref="S3.SS1.p2.11.m8.1.1.2.cmml">G</mi><mi id="S3.SS1.p2.11.m8.1.1.3" xref="S3.SS1.p2.11.m8.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.11.m8.1b"><apply id="S3.SS1.p2.11.m8.1.1.cmml" xref="S3.SS1.p2.11.m8.1.1"><csymbol cd="ambiguous" id="S3.SS1.p2.11.m8.1.1.1.cmml" xref="S3.SS1.p2.11.m8.1.1">subscript</csymbol><ci id="S3.SS1.p2.11.m8.1.1.2.cmml" xref="S3.SS1.p2.11.m8.1.1.2">𝐺</ci><ci id="S3.SS1.p2.11.m8.1.1.3.cmml" xref="S3.SS1.p2.11.m8.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.11.m8.1c">G_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.11.m8.1d">italic_G start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> at pixel <math alttext="p" class="ltx_Math" display="inline" id="S3.SS1.p2.12.m9.1"><semantics id="S3.SS1.p2.12.m9.1a"><mi id="S3.SS1.p2.12.m9.1.1" xref="S3.SS1.p2.12.m9.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.12.m9.1b"><ci id="S3.SS1.p2.12.m9.1.1.cmml" xref="S3.SS1.p2.12.m9.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.12.m9.1c">p</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.12.m9.1d">italic_p</annotation></semantics></math> obtained from spherical harmonics, the color at pixel <math alttext="p" class="ltx_Math" display="inline" id="S3.SS1.p2.13.m10.1"><semantics id="S3.SS1.p2.13.m10.1a"><mi id="S3.SS1.p2.13.m10.1.1" xref="S3.SS1.p2.13.m10.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.13.m10.1b"><ci id="S3.SS1.p2.13.m10.1.1.cmml" xref="S3.SS1.p2.13.m10.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.13.m10.1c">p</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.13.m10.1d">italic_p</annotation></semantics></math> is determined via volumetric rendering, i.e., <table class="ltx_equation ltx_eqn_table" id="S3.E4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathbf{c}_{p}=\sum_{i=1}^{K}\mathbf{c}_{i,p}\alpha_{i,p}\prod_{j=1}^{i-1}% \alpha_{j,p},\text{ where }\alpha_{i,p}=\alpha_{i}\alpha_{i\to p}." 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xref="S3.E4.m1.9.9.1.1.1.1.3.2.3.2.cmml">α</mi><mrow id="S3.E4.m1.4.4.2.4" xref="S3.E4.m1.4.4.2.3.cmml"><mi id="S3.E4.m1.3.3.1.1" xref="S3.E4.m1.3.3.1.1.cmml">i</mi><mo id="S3.E4.m1.4.4.2.4.1" xref="S3.E4.m1.4.4.2.3.cmml">,</mo><mi id="S3.E4.m1.4.4.2.2" xref="S3.E4.m1.4.4.2.2.cmml">p</mi></mrow></msub><mo id="S3.E4.m1.9.9.1.1.1.1.3.2.1a" xref="S3.E4.m1.9.9.1.1.1.1.3.2.1.cmml">⁢</mo><mrow id="S3.E4.m1.9.9.1.1.1.1.3.2.4" xref="S3.E4.m1.9.9.1.1.1.1.3.2.4.cmml"><munderover id="S3.E4.m1.9.9.1.1.1.1.3.2.4.1" xref="S3.E4.m1.9.9.1.1.1.1.3.2.4.1.cmml"><mo id="S3.E4.m1.9.9.1.1.1.1.3.2.4.1.2.2" movablelimits="false" xref="S3.E4.m1.9.9.1.1.1.1.3.2.4.1.2.2.cmml">∏</mo><mrow id="S3.E4.m1.9.9.1.1.1.1.3.2.4.1.2.3" xref="S3.E4.m1.9.9.1.1.1.1.3.2.4.1.2.3.cmml"><mi id="S3.E4.m1.9.9.1.1.1.1.3.2.4.1.2.3.2" xref="S3.E4.m1.9.9.1.1.1.1.3.2.4.1.2.3.2.cmml">j</mi><mo id="S3.E4.m1.9.9.1.1.1.1.3.2.4.1.2.3.1" xref="S3.E4.m1.9.9.1.1.1.1.3.2.4.1.2.3.1.cmml">=</mo><mn id="S3.E4.m1.9.9.1.1.1.1.3.2.4.1.2.3.3" xref="S3.E4.m1.9.9.1.1.1.1.3.2.4.1.2.3.3.cmml">1</mn></mrow><mrow id="S3.E4.m1.9.9.1.1.1.1.3.2.4.1.3" xref="S3.E4.m1.9.9.1.1.1.1.3.2.4.1.3.cmml"><mi id="S3.E4.m1.9.9.1.1.1.1.3.2.4.1.3.2" xref="S3.E4.m1.9.9.1.1.1.1.3.2.4.1.3.2.cmml">i</mi><mo id="S3.E4.m1.9.9.1.1.1.1.3.2.4.1.3.1" xref="S3.E4.m1.9.9.1.1.1.1.3.2.4.1.3.1.cmml">−</mo><mn id="S3.E4.m1.9.9.1.1.1.1.3.2.4.1.3.3" xref="S3.E4.m1.9.9.1.1.1.1.3.2.4.1.3.3.cmml">1</mn></mrow></munderover><msub id="S3.E4.m1.9.9.1.1.1.1.3.2.4.2" xref="S3.E4.m1.9.9.1.1.1.1.3.2.4.2.cmml"><mi id="S3.E4.m1.9.9.1.1.1.1.3.2.4.2.2" xref="S3.E4.m1.9.9.1.1.1.1.3.2.4.2.2.cmml">α</mi><mrow id="S3.E4.m1.6.6.2.4" xref="S3.E4.m1.6.6.2.3.cmml"><mi id="S3.E4.m1.5.5.1.1" xref="S3.E4.m1.5.5.1.1.cmml">j</mi><mo id="S3.E4.m1.6.6.2.4.1" xref="S3.E4.m1.6.6.2.3.cmml">,</mo><mi id="S3.E4.m1.6.6.2.2" xref="S3.E4.m1.6.6.2.2.cmml">p</mi></mrow></msub></mrow></mrow></mrow></mrow><mo id="S3.E4.m1.9.9.1.1.2.3" xref="S3.E4.m1.9.9.1.1.3a.cmml">,</mo><mrow id="S3.E4.m1.9.9.1.1.2.2" xref="S3.E4.m1.9.9.1.1.2.2.cmml"><mrow id="S3.E4.m1.9.9.1.1.2.2.2" xref="S3.E4.m1.9.9.1.1.2.2.2.cmml"><mtext id="S3.E4.m1.9.9.1.1.2.2.2.2" xref="S3.E4.m1.9.9.1.1.2.2.2.2a.cmml"> where </mtext><mo id="S3.E4.m1.9.9.1.1.2.2.2.1" xref="S3.E4.m1.9.9.1.1.2.2.2.1.cmml">⁢</mo><msub id="S3.E4.m1.9.9.1.1.2.2.2.3" xref="S3.E4.m1.9.9.1.1.2.2.2.3.cmml"><mi id="S3.E4.m1.9.9.1.1.2.2.2.3.2" xref="S3.E4.m1.9.9.1.1.2.2.2.3.2.cmml">α</mi><mrow id="S3.E4.m1.8.8.2.4" xref="S3.E4.m1.8.8.2.3.cmml"><mi id="S3.E4.m1.7.7.1.1" xref="S3.E4.m1.7.7.1.1.cmml">i</mi><mo id="S3.E4.m1.8.8.2.4.1" xref="S3.E4.m1.8.8.2.3.cmml">,</mo><mi id="S3.E4.m1.8.8.2.2" xref="S3.E4.m1.8.8.2.2.cmml">p</mi></mrow></msub></mrow><mo id="S3.E4.m1.9.9.1.1.2.2.1" xref="S3.E4.m1.9.9.1.1.2.2.1.cmml">=</mo><mrow id="S3.E4.m1.9.9.1.1.2.2.3" xref="S3.E4.m1.9.9.1.1.2.2.3.cmml"><msub id="S3.E4.m1.9.9.1.1.2.2.3.2" xref="S3.E4.m1.9.9.1.1.2.2.3.2.cmml"><mi id="S3.E4.m1.9.9.1.1.2.2.3.2.2" xref="S3.E4.m1.9.9.1.1.2.2.3.2.2.cmml">α</mi><mi 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p}.</annotation><annotation encoding="application/x-llamapun" id="S3.E4.m1.9d">bold_c start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT bold_c start_POSTSUBSCRIPT italic_i , italic_p end_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT italic_i , italic_p end_POSTSUBSCRIPT ∏ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i - 1 end_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT italic_j , italic_p end_POSTSUBSCRIPT , where italic_α start_POSTSUBSCRIPT italic_i , italic_p end_POSTSUBSCRIPT = italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT italic_i → italic_p end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(4)</td> </tr></tbody> </table> The Gaussians are sorted by their depths from the viewpoint. By comparing the rendered image with the ground truth, we could optimize properties of 3D Gaussians to better the scene reconstruction. </div> </section> <section class="ltx_subsection" id="S3.SS2"> <h3 class="ltx_title ltx_title_subsection"> 3.2 Problem Formulation</h3> <div class="ltx_para ltx_noindent" id="S3.SS2.p1"> Inputs. The inputs for the task are videos captured in the same block but in different times. In other words, images <math alttext="\mathcal{I}=\left\{\mathbf{I}_{t,T}\in\mathbb{R}^{w\times h\times 3}\mid t=0,% \cdots,t_{T};T=1,\cdots,T_{\mathtt{all}}\right\}" class="ltx_Math" display="inline" id="S3.SS2.p1.1.m1.8"><semantics id="S3.SS2.p1.1.m1.8a"><mrow id="S3.SS2.p1.1.m1.8.8" xref="S3.SS2.p1.1.m1.8.8.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p1.1.m1.8.8.4" xref="S3.SS2.p1.1.m1.8.8.4.cmml">ℐ</mi><mo id="S3.SS2.p1.1.m1.8.8.3" xref="S3.SS2.p1.1.m1.8.8.3.cmml">=</mo><mrow id="S3.SS2.p1.1.m1.8.8.2.2" xref="S3.SS2.p1.1.m1.8.8.2.3.cmml"><mo id="S3.SS2.p1.1.m1.8.8.2.2.3" xref="S3.SS2.p1.1.m1.8.8.2.3.1.cmml">{</mo><mrow id="S3.SS2.p1.1.m1.7.7.1.1.1" xref="S3.SS2.p1.1.m1.7.7.1.1.1.cmml"><msub id="S3.SS2.p1.1.m1.7.7.1.1.1.2" xref="S3.SS2.p1.1.m1.7.7.1.1.1.2.cmml"><mi id="S3.SS2.p1.1.m1.7.7.1.1.1.2.2" xref="S3.SS2.p1.1.m1.7.7.1.1.1.2.2.cmml">𝐈</mi><mrow id="S3.SS2.p1.1.m1.2.2.2.4" xref="S3.SS2.p1.1.m1.2.2.2.3.cmml"><mi id="S3.SS2.p1.1.m1.1.1.1.1" 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xref="S3.SS2.p1.1.m1.7.7.1.1.1.3.3.4.cmml">3</mn></mrow></msup></mrow><mo fence="true" id="S3.SS2.p1.1.m1.8.8.2.2.4" lspace="0em" rspace="0em" xref="S3.SS2.p1.1.m1.8.8.2.3.1.cmml">∣</mo><mrow id="S3.SS2.p1.1.m1.8.8.2.2.2.2" xref="S3.SS2.p1.1.m1.8.8.2.2.2.3.cmml"><mrow id="S3.SS2.p1.1.m1.8.8.2.2.2.1.1" xref="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.cmml"><mi id="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.3" xref="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.3.cmml">t</mi><mo id="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.2" xref="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.2.cmml">=</mo><mrow id="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.1.1" xref="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.1.2.cmml"><mn id="S3.SS2.p1.1.m1.5.5" xref="S3.SS2.p1.1.m1.5.5.cmml">0</mn><mo id="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.1.1.2" xref="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.1.2.cmml">,</mo><mi id="S3.SS2.p1.1.m1.6.6" mathvariant="normal" xref="S3.SS2.p1.1.m1.6.6.cmml">⋯</mi><mo id="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.1.1.3" xref="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.1.2.cmml">,</mo><msub id="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.1.1.1" xref="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.1.1.1.cmml"><mi id="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.1.1.1.2" xref="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.1.1.1.2.cmml">t</mi><mi id="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.1.1.1.3" xref="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.1.1.1.3.cmml">T</mi></msub></mrow></mrow><mo id="S3.SS2.p1.1.m1.8.8.2.2.2.2.3" xref="S3.SS2.p1.1.m1.8.8.2.2.2.3a.cmml">;</mo><mrow id="S3.SS2.p1.1.m1.8.8.2.2.2.2.2" xref="S3.SS2.p1.1.m1.8.8.2.2.2.2.2.cmml"><mi id="S3.SS2.p1.1.m1.8.8.2.2.2.2.2.3" xref="S3.SS2.p1.1.m1.8.8.2.2.2.2.2.3.cmml">T</mi><mo id="S3.SS2.p1.1.m1.8.8.2.2.2.2.2.2" xref="S3.SS2.p1.1.m1.8.8.2.2.2.2.2.2.cmml">=</mo><mrow id="S3.SS2.p1.1.m1.8.8.2.2.2.2.2.1.1" xref="S3.SS2.p1.1.m1.8.8.2.2.2.2.2.1.2.cmml"><mn id="S3.SS2.p1.1.m1.3.3" xref="S3.SS2.p1.1.m1.3.3.cmml">1</mn><mo id="S3.SS2.p1.1.m1.8.8.2.2.2.2.2.1.1.2" xref="S3.SS2.p1.1.m1.8.8.2.2.2.2.2.1.2.cmml">,</mo><mi id="S3.SS2.p1.1.m1.4.4" mathvariant="normal" xref="S3.SS2.p1.1.m1.4.4.cmml">⋯</mi><mo id="S3.SS2.p1.1.m1.8.8.2.2.2.2.2.1.1.3" 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id="S3.SS2.p1.1.m1.7.7.1.1.1.1.cmml" xref="S3.SS2.p1.1.m1.7.7.1.1.1.1"></in><apply id="S3.SS2.p1.1.m1.7.7.1.1.1.2.cmml" xref="S3.SS2.p1.1.m1.7.7.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.p1.1.m1.7.7.1.1.1.2.1.cmml" xref="S3.SS2.p1.1.m1.7.7.1.1.1.2">subscript</csymbol><ci id="S3.SS2.p1.1.m1.7.7.1.1.1.2.2.cmml" xref="S3.SS2.p1.1.m1.7.7.1.1.1.2.2">𝐈</ci><list id="S3.SS2.p1.1.m1.2.2.2.3.cmml" xref="S3.SS2.p1.1.m1.2.2.2.4"><ci id="S3.SS2.p1.1.m1.1.1.1.1.cmml" xref="S3.SS2.p1.1.m1.1.1.1.1">𝑡</ci><ci id="S3.SS2.p1.1.m1.2.2.2.2.cmml" xref="S3.SS2.p1.1.m1.2.2.2.2">𝑇</ci></list></apply><apply id="S3.SS2.p1.1.m1.7.7.1.1.1.3.cmml" xref="S3.SS2.p1.1.m1.7.7.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p1.1.m1.7.7.1.1.1.3.1.cmml" xref="S3.SS2.p1.1.m1.7.7.1.1.1.3">superscript</csymbol><ci id="S3.SS2.p1.1.m1.7.7.1.1.1.3.2.cmml" xref="S3.SS2.p1.1.m1.7.7.1.1.1.3.2">ℝ</ci><apply id="S3.SS2.p1.1.m1.7.7.1.1.1.3.3.cmml" xref="S3.SS2.p1.1.m1.7.7.1.1.1.3.3"><times id="S3.SS2.p1.1.m1.7.7.1.1.1.3.3.1.cmml" xref="S3.SS2.p1.1.m1.7.7.1.1.1.3.3.1"></times><ci id="S3.SS2.p1.1.m1.7.7.1.1.1.3.3.2.cmml" xref="S3.SS2.p1.1.m1.7.7.1.1.1.3.3.2">𝑤</ci><ci id="S3.SS2.p1.1.m1.7.7.1.1.1.3.3.3.cmml" xref="S3.SS2.p1.1.m1.7.7.1.1.1.3.3.3">ℎ</ci><cn id="S3.SS2.p1.1.m1.7.7.1.1.1.3.3.4.cmml" type="integer" xref="S3.SS2.p1.1.m1.7.7.1.1.1.3.3.4">3</cn></apply></apply></apply><apply id="S3.SS2.p1.1.m1.8.8.2.2.2.3.cmml" xref="S3.SS2.p1.1.m1.8.8.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.p1.1.m1.8.8.2.2.2.3a.cmml" xref="S3.SS2.p1.1.m1.8.8.2.2.2.2.3">formulae-sequence</csymbol><apply id="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.cmml" xref="S3.SS2.p1.1.m1.8.8.2.2.2.1.1"><eq id="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.2.cmml" xref="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.2"></eq><ci id="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.3.cmml" xref="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.3">𝑡</ci><list id="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.1.2.cmml" xref="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.1.1"><cn id="S3.SS2.p1.1.m1.5.5.cmml" type="integer" xref="S3.SS2.p1.1.m1.5.5">0</cn><ci id="S3.SS2.p1.1.m1.6.6.cmml" xref="S3.SS2.p1.1.m1.6.6">⋯</ci><apply id="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.1.1.1.cmml" xref="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.1.1.1.1.cmml" xref="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.1.1.1">subscript</csymbol><ci id="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.1.1.1.2.cmml" xref="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.1.1.1.2">𝑡</ci><ci id="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.1.1.1.3.cmml" xref="S3.SS2.p1.1.m1.8.8.2.2.2.1.1.1.1.1.3">𝑇</ci></apply></list></apply><apply id="S3.SS2.p1.1.m1.8.8.2.2.2.2.2.cmml" xref="S3.SS2.p1.1.m1.8.8.2.2.2.2.2"><eq id="S3.SS2.p1.1.m1.8.8.2.2.2.2.2.2.cmml" xref="S3.SS2.p1.1.m1.8.8.2.2.2.2.2.2"></eq><ci id="S3.SS2.p1.1.m1.8.8.2.2.2.2.2.3.cmml" xref="S3.SS2.p1.1.m1.8.8.2.2.2.2.2.3">𝑇</ci><list id="S3.SS2.p1.1.m1.8.8.2.2.2.2.2.1.2.cmml" xref="S3.SS2.p1.1.m1.8.8.2.2.2.2.2.1.1"><cn id="S3.SS2.p1.1.m1.3.3.cmml" type="integer" xref="S3.SS2.p1.1.m1.3.3">1</cn><ci id="S3.SS2.p1.1.m1.4.4.cmml" xref="S3.SS2.p1.1.m1.4.4">⋯</ci><apply id="S3.SS2.p1.1.m1.8.8.2.2.2.2.2.1.1.1.cmml" xref="S3.SS2.p1.1.m1.8.8.2.2.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.1.m1.8.8.2.2.2.2.2.1.1.1.1.cmml" xref="S3.SS2.p1.1.m1.8.8.2.2.2.2.2.1.1.1">subscript</csymbol><ci id="S3.SS2.p1.1.m1.8.8.2.2.2.2.2.1.1.1.2.cmml" xref="S3.SS2.p1.1.m1.8.8.2.2.2.2.2.1.1.1.2">𝑇</ci><ci id="S3.SS2.p1.1.m1.8.8.2.2.2.2.2.1.1.1.3.cmml" xref="S3.SS2.p1.1.m1.8.8.2.2.2.2.2.1.1.1.3">𝚊𝚕𝚕</ci></apply></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.1.m1.8c">\mathcal{I}=\left\{\mathbf{I}_{t,T}\in\mathbb{R}^{w\times h\times 3}\mid t=0,% \cdots,t_{T};T=1,\cdots,T_{\mathtt{all}}\right\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.1.m1.8d">caligraphic_I = { bold_I start_POSTSUBSCRIPT italic_t , italic_T end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_w × italic_h × 3 end_POSTSUPERSCRIPT ∣ italic_t = 0 , ⋯ , italic_t start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ; italic_T = 1 , ⋯ , italic_T start_POSTSUBSCRIPT typewriter_all end_POSTSUBSCRIPT }</annotation></semantics></math> are given with corresponding camera poses <math alttext="\boldsymbol{\xi}_{t,T}=\left\{\mathbf{W}_{t,T},\mathbf{K}_{t,T}\right\}" class="ltx_Math" display="inline" id="S3.SS2.p1.2.m2.8"><semantics id="S3.SS2.p1.2.m2.8a"><mrow id="S3.SS2.p1.2.m2.8.8" xref="S3.SS2.p1.2.m2.8.8.cmml"><msub id="S3.SS2.p1.2.m2.8.8.4" xref="S3.SS2.p1.2.m2.8.8.4.cmml"><mi id="S3.SS2.p1.2.m2.8.8.4.2" xref="S3.SS2.p1.2.m2.8.8.4.2.cmml">𝝃</mi><mrow id="S3.SS2.p1.2.m2.2.2.2.4" xref="S3.SS2.p1.2.m2.2.2.2.3.cmml"><mi id="S3.SS2.p1.2.m2.1.1.1.1" xref="S3.SS2.p1.2.m2.1.1.1.1.cmml">t</mi><mo id="S3.SS2.p1.2.m2.2.2.2.4.1" xref="S3.SS2.p1.2.m2.2.2.2.3.cmml">,</mo><mi id="S3.SS2.p1.2.m2.2.2.2.2" xref="S3.SS2.p1.2.m2.2.2.2.2.cmml">T</mi></mrow></msub><mo id="S3.SS2.p1.2.m2.8.8.3" xref="S3.SS2.p1.2.m2.8.8.3.cmml">=</mo><mrow id="S3.SS2.p1.2.m2.8.8.2.2" 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encoding="application/x-tex" id="S3.SS2.p1.2.m2.8c">\boldsymbol{\xi}_{t,T}=\left\{\mathbf{W}_{t,T},\mathbf{K}_{t,T}\right\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.2.m2.8d">bold_italic_ξ start_POSTSUBSCRIPT italic_t , italic_T end_POSTSUBSCRIPT = { bold_W start_POSTSUBSCRIPT italic_t , italic_T end_POSTSUBSCRIPT , bold_K start_POSTSUBSCRIPT italic_t , italic_T end_POSTSUBSCRIPT }</annotation></semantics></math>, where <math alttext="t" class="ltx_Math" display="inline" id="S3.SS2.p1.3.m3.1"><semantics id="S3.SS2.p1.3.m3.1a"><mi id="S3.SS2.p1.3.m3.1.1" xref="S3.SS2.p1.3.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.3.m3.1b"><ci id="S3.SS2.p1.3.m3.1.1.cmml" xref="S3.SS2.p1.3.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.3.m3.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.3.m3.1d">italic_t</annotation></semantics></math> represents time and <math alttext="T" class="ltx_Math" display="inline" id="S3.SS2.p1.4.m4.1"><semantics id="S3.SS2.p1.4.m4.1a"><mi id="S3.SS2.p1.4.m4.1.1" xref="S3.SS2.p1.4.m4.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.4.m4.1b"><ci id="S3.SS2.p1.4.m4.1.1.cmml" xref="S3.SS2.p1.4.m4.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.4.m4.1c">T</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.4.m4.1d">italic_T</annotation></semantics></math> represents traversals. Colored LiDAR point clouds <math alttext="\mathbf{P}_{t,T}\in\mathbb{R}^{K\times 6}" class="ltx_Math" display="inline" id="S3.SS2.p1.5.m5.2"><semantics id="S3.SS2.p1.5.m5.2a"><mrow id="S3.SS2.p1.5.m5.2.3" xref="S3.SS2.p1.5.m5.2.3.cmml"><msub id="S3.SS2.p1.5.m5.2.3.2" xref="S3.SS2.p1.5.m5.2.3.2.cmml"><mi id="S3.SS2.p1.5.m5.2.3.2.2" xref="S3.SS2.p1.5.m5.2.3.2.2.cmml">𝐏</mi><mrow id="S3.SS2.p1.5.m5.2.2.2.4" xref="S3.SS2.p1.5.m5.2.2.2.3.cmml"><mi id="S3.SS2.p1.5.m5.1.1.1.1" xref="S3.SS2.p1.5.m5.1.1.1.1.cmml">t</mi><mo id="S3.SS2.p1.5.m5.2.2.2.4.1" xref="S3.SS2.p1.5.m5.2.2.2.3.cmml">,</mo><mi id="S3.SS2.p1.5.m5.2.2.2.2" xref="S3.SS2.p1.5.m5.2.2.2.2.cmml">T</mi></mrow></msub><mo id="S3.SS2.p1.5.m5.2.3.1" xref="S3.SS2.p1.5.m5.2.3.1.cmml">∈</mo><msup id="S3.SS2.p1.5.m5.2.3.3" xref="S3.SS2.p1.5.m5.2.3.3.cmml"><mi id="S3.SS2.p1.5.m5.2.3.3.2" xref="S3.SS2.p1.5.m5.2.3.3.2.cmml">ℝ</mi><mrow id="S3.SS2.p1.5.m5.2.3.3.3" xref="S3.SS2.p1.5.m5.2.3.3.3.cmml"><mi id="S3.SS2.p1.5.m5.2.3.3.3.2" xref="S3.SS2.p1.5.m5.2.3.3.3.2.cmml">K</mi><mo id="S3.SS2.p1.5.m5.2.3.3.3.1" lspace="0.222em" rspace="0.222em" xref="S3.SS2.p1.5.m5.2.3.3.3.1.cmml">×</mo><mn id="S3.SS2.p1.5.m5.2.3.3.3.3" xref="S3.SS2.p1.5.m5.2.3.3.3.3.cmml">6</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.5.m5.2b"><apply id="S3.SS2.p1.5.m5.2.3.cmml" xref="S3.SS2.p1.5.m5.2.3"><in id="S3.SS2.p1.5.m5.2.3.1.cmml" xref="S3.SS2.p1.5.m5.2.3.1"></in><apply id="S3.SS2.p1.5.m5.2.3.2.cmml" xref="S3.SS2.p1.5.m5.2.3.2"><csymbol cd="ambiguous" id="S3.SS2.p1.5.m5.2.3.2.1.cmml" xref="S3.SS2.p1.5.m5.2.3.2">subscript</csymbol><ci id="S3.SS2.p1.5.m5.2.3.2.2.cmml" xref="S3.SS2.p1.5.m5.2.3.2.2">𝐏</ci><list id="S3.SS2.p1.5.m5.2.2.2.3.cmml" xref="S3.SS2.p1.5.m5.2.2.2.4"><ci id="S3.SS2.p1.5.m5.1.1.1.1.cmml" xref="S3.SS2.p1.5.m5.1.1.1.1">𝑡</ci><ci id="S3.SS2.p1.5.m5.2.2.2.2.cmml" xref="S3.SS2.p1.5.m5.2.2.2.2">𝑇</ci></list></apply><apply id="S3.SS2.p1.5.m5.2.3.3.cmml" xref="S3.SS2.p1.5.m5.2.3.3"><csymbol cd="ambiguous" id="S3.SS2.p1.5.m5.2.3.3.1.cmml" xref="S3.SS2.p1.5.m5.2.3.3">superscript</csymbol><ci id="S3.SS2.p1.5.m5.2.3.3.2.cmml" xref="S3.SS2.p1.5.m5.2.3.3.2">ℝ</ci><apply id="S3.SS2.p1.5.m5.2.3.3.3.cmml" xref="S3.SS2.p1.5.m5.2.3.3.3"><times id="S3.SS2.p1.5.m5.2.3.3.3.1.cmml" xref="S3.SS2.p1.5.m5.2.3.3.3.1"></times><ci id="S3.SS2.p1.5.m5.2.3.3.3.2.cmml" xref="S3.SS2.p1.5.m5.2.3.3.3.2">𝐾</ci><cn id="S3.SS2.p1.5.m5.2.3.3.3.3.cmml" type="integer" xref="S3.SS2.p1.5.m5.2.3.3.3.3">6</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.5.m5.2c">\mathbf{P}_{t,T}\in\mathbb{R}^{K\times 6}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.5.m5.2d">bold_P start_POSTSUBSCRIPT italic_t , italic_T end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_K × 6 end_POSTSUPERSCRIPT</annotation></semantics></math> are also provided for initialization and sparse depth priors. </div> <div class="ltx_para ltx_noindent" id="S3.SS2.p2"> Assumptions. We assume that across multiple traversals in the same road block, the background remains largely consistent, i.e., sharing the same geometry despite variations in appearance. Meanwhile, foregrounds, such as moving vehicles and parked cars along the streets, are traversal-variant. </div> <div class="ltx_para ltx_noindent" id="S3.SS2.p3"> Outputs. The output of the problem is a scene representation <math alttext="f" class="ltx_Math" display="inline" id="S3.SS2.p3.1.m1.1"><semantics id="S3.SS2.p3.1.m1.1a"><mi id="S3.SS2.p3.1.m1.1.1" xref="S3.SS2.p3.1.m1.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.1.m1.1b"><ci id="S3.SS2.p3.1.m1.1.1.cmml" xref="S3.SS2.p3.1.m1.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.1.m1.1c">f</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.1.m1.1d">italic_f</annotation></semantics></math> that can render the result <math alttext="f(\boldsymbol{\xi},t,T)\in\mathbb{R}^{w\times h\times C}" class="ltx_Math" display="inline" id="S3.SS2.p3.2.m2.3"><semantics id="S3.SS2.p3.2.m2.3a"><mrow id="S3.SS2.p3.2.m2.3.4" xref="S3.SS2.p3.2.m2.3.4.cmml"><mrow id="S3.SS2.p3.2.m2.3.4.2" xref="S3.SS2.p3.2.m2.3.4.2.cmml"><mi id="S3.SS2.p3.2.m2.3.4.2.2" xref="S3.SS2.p3.2.m2.3.4.2.2.cmml">f</mi><mo id="S3.SS2.p3.2.m2.3.4.2.1" xref="S3.SS2.p3.2.m2.3.4.2.1.cmml">⁢</mo><mrow id="S3.SS2.p3.2.m2.3.4.2.3.2" xref="S3.SS2.p3.2.m2.3.4.2.3.1.cmml"><mo id="S3.SS2.p3.2.m2.3.4.2.3.2.1" stretchy="false" xref="S3.SS2.p3.2.m2.3.4.2.3.1.cmml">(</mo><mi id="S3.SS2.p3.2.m2.1.1" xref="S3.SS2.p3.2.m2.1.1.cmml">𝝃</mi><mo id="S3.SS2.p3.2.m2.3.4.2.3.2.2" xref="S3.SS2.p3.2.m2.3.4.2.3.1.cmml">,</mo><mi id="S3.SS2.p3.2.m2.2.2" xref="S3.SS2.p3.2.m2.2.2.cmml">t</mi><mo id="S3.SS2.p3.2.m2.3.4.2.3.2.3" xref="S3.SS2.p3.2.m2.3.4.2.3.1.cmml">,</mo><mi id="S3.SS2.p3.2.m2.3.3" xref="S3.SS2.p3.2.m2.3.3.cmml">T</mi><mo id="S3.SS2.p3.2.m2.3.4.2.3.2.4" stretchy="false" xref="S3.SS2.p3.2.m2.3.4.2.3.1.cmml">)</mo></mrow></mrow><mo id="S3.SS2.p3.2.m2.3.4.1" xref="S3.SS2.p3.2.m2.3.4.1.cmml">∈</mo><msup id="S3.SS2.p3.2.m2.3.4.3" xref="S3.SS2.p3.2.m2.3.4.3.cmml"><mi id="S3.SS2.p3.2.m2.3.4.3.2" xref="S3.SS2.p3.2.m2.3.4.3.2.cmml">ℝ</mi><mrow id="S3.SS2.p3.2.m2.3.4.3.3" xref="S3.SS2.p3.2.m2.3.4.3.3.cmml"><mi id="S3.SS2.p3.2.m2.3.4.3.3.2" 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id="S3.SS2.p3.2.m2.3d">italic_f ( bold_italic_ξ , italic_t , italic_T ) ∈ blackboard_R start_POSTSUPERSCRIPT italic_w × italic_h × italic_C end_POSTSUPERSCRIPT</annotation></semantics></math> captured by camera <math alttext="\boldsymbol{\xi}" class="ltx_Math" display="inline" id="S3.SS2.p3.3.m3.1"><semantics id="S3.SS2.p3.3.m3.1a"><mi id="S3.SS2.p3.3.m3.1.1" xref="S3.SS2.p3.3.m3.1.1.cmml">𝝃</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.3.m3.1b"><ci id="S3.SS2.p3.3.m3.1.1.cmml" xref="S3.SS2.p3.3.m3.1.1">𝝃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.3.m3.1c">\boldsymbol{\xi}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.3.m3.1d">bold_italic_ξ</annotation></semantics></math> at time <math alttext="t" class="ltx_Math" display="inline" id="S3.SS2.p3.4.m4.1"><semantics id="S3.SS2.p3.4.m4.1a"><mi id="S3.SS2.p3.4.m4.1.1" xref="S3.SS2.p3.4.m4.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.4.m4.1b"><ci id="S3.SS2.p3.4.m4.1.1.cmml" xref="S3.SS2.p3.4.m4.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.4.m4.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.4.m4.1d">italic_t</annotation></semantics></math> in traversal <math alttext="T" class="ltx_Math" display="inline" id="S3.SS2.p3.5.m5.1"><semantics id="S3.SS2.p3.5.m5.1a"><mi id="S3.SS2.p3.5.m5.1.1" xref="S3.SS2.p3.5.m5.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.5.m5.1b"><ci id="S3.SS2.p3.5.m5.1.1.cmml" xref="S3.SS2.p3.5.m5.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.5.m5.1c">T</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.5.m5.1d">italic_T</annotation></semantics></math>, where <math alttext="0\leq t\leq t_{T}" class="ltx_Math" display="inline" id="S3.SS2.p3.6.m6.1"><semantics id="S3.SS2.p3.6.m6.1a"><mrow id="S3.SS2.p3.6.m6.1.1" xref="S3.SS2.p3.6.m6.1.1.cmml"><mn id="S3.SS2.p3.6.m6.1.1.2" xref="S3.SS2.p3.6.m6.1.1.2.cmml">0</mn><mo id="S3.SS2.p3.6.m6.1.1.3" xref="S3.SS2.p3.6.m6.1.1.3.cmml">≤</mo><mi id="S3.SS2.p3.6.m6.1.1.4" xref="S3.SS2.p3.6.m6.1.1.4.cmml">t</mi><mo id="S3.SS2.p3.6.m6.1.1.5" xref="S3.SS2.p3.6.m6.1.1.5.cmml">≤</mo><msub id="S3.SS2.p3.6.m6.1.1.6" xref="S3.SS2.p3.6.m6.1.1.6.cmml"><mi id="S3.SS2.p3.6.m6.1.1.6.2" xref="S3.SS2.p3.6.m6.1.1.6.2.cmml">t</mi><mi id="S3.SS2.p3.6.m6.1.1.6.3" xref="S3.SS2.p3.6.m6.1.1.6.3.cmml">T</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.6.m6.1b"><apply id="S3.SS2.p3.6.m6.1.1.cmml" xref="S3.SS2.p3.6.m6.1.1"><and id="S3.SS2.p3.6.m6.1.1a.cmml" xref="S3.SS2.p3.6.m6.1.1"></and><apply id="S3.SS2.p3.6.m6.1.1b.cmml" xref="S3.SS2.p3.6.m6.1.1"><leq id="S3.SS2.p3.6.m6.1.1.3.cmml" xref="S3.SS2.p3.6.m6.1.1.3"></leq><cn id="S3.SS2.p3.6.m6.1.1.2.cmml" type="integer" xref="S3.SS2.p3.6.m6.1.1.2">0</cn><ci id="S3.SS2.p3.6.m6.1.1.4.cmml" xref="S3.SS2.p3.6.m6.1.1.4">𝑡</ci></apply><apply id="S3.SS2.p3.6.m6.1.1c.cmml" xref="S3.SS2.p3.6.m6.1.1"><leq id="S3.SS2.p3.6.m6.1.1.5.cmml" xref="S3.SS2.p3.6.m6.1.1.5"></leq><share href="https://arxiv.org/html/2503.12552v2#S3.SS2.p3.6.m6.1.1.4.cmml" id="S3.SS2.p3.6.m6.1.1d.cmml" xref="S3.SS2.p3.6.m6.1.1"></share><apply id="S3.SS2.p3.6.m6.1.1.6.cmml" xref="S3.SS2.p3.6.m6.1.1.6"><csymbol cd="ambiguous" id="S3.SS2.p3.6.m6.1.1.6.1.cmml" xref="S3.SS2.p3.6.m6.1.1.6">subscript</csymbol><ci id="S3.SS2.p3.6.m6.1.1.6.2.cmml" xref="S3.SS2.p3.6.m6.1.1.6.2">𝑡</ci><ci id="S3.SS2.p3.6.m6.1.1.6.3.cmml" xref="S3.SS2.p3.6.m6.1.1.6.3">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.6.m6.1c">0\leq t\leq t_{T}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.6.m6.1d">0 ≤ italic_t ≤ italic_t start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT</annotation></semantics></math>. <math alttext="C" class="ltx_Math" display="inline" id="S3.SS2.p3.7.m7.1"><semantics id="S3.SS2.p3.7.m7.1a"><mi id="S3.SS2.p3.7.m7.1.1" xref="S3.SS2.p3.7.m7.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.7.m7.1b"><ci id="S3.SS2.p3.7.m7.1.1.cmml" xref="S3.SS2.p3.7.m7.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.7.m7.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.7.m7.1d">italic_C</annotation></semantics></math> represents the number of expected channels, including RGB, depth, etc. Note that the representation should also be able to render results for unseen traversals. </div> <div class="ltx_para ltx_noindent" id="S3.SS2.p4"> Targets. The target is to optimize <math alttext="f" class="ltx_Math" display="inline" id="S3.SS2.p4.1.m1.1"><semantics id="S3.SS2.p4.1.m1.1a"><mi id="S3.SS2.p4.1.m1.1.1" xref="S3.SS2.p4.1.m1.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.1.m1.1b"><ci id="S3.SS2.p4.1.m1.1.1.cmml" xref="S3.SS2.p4.1.m1.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.1.m1.1c">f</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.1.m1.1d">italic_f</annotation></semantics></math> so that its rendered results <math alttext="f(\boldsymbol{\xi},t,T)" class="ltx_Math" display="inline" id="S3.SS2.p4.2.m2.3"><semantics id="S3.SS2.p4.2.m2.3a"><mrow id="S3.SS2.p4.2.m2.3.4" xref="S3.SS2.p4.2.m2.3.4.cmml"><mi id="S3.SS2.p4.2.m2.3.4.2" xref="S3.SS2.p4.2.m2.3.4.2.cmml">f</mi><mo id="S3.SS2.p4.2.m2.3.4.1" xref="S3.SS2.p4.2.m2.3.4.1.cmml">⁢</mo><mrow id="S3.SS2.p4.2.m2.3.4.3.2" xref="S3.SS2.p4.2.m2.3.4.3.1.cmml"><mo id="S3.SS2.p4.2.m2.3.4.3.2.1" stretchy="false" xref="S3.SS2.p4.2.m2.3.4.3.1.cmml">(</mo><mi id="S3.SS2.p4.2.m2.1.1" xref="S3.SS2.p4.2.m2.1.1.cmml">𝝃</mi><mo id="S3.SS2.p4.2.m2.3.4.3.2.2" xref="S3.SS2.p4.2.m2.3.4.3.1.cmml">,</mo><mi id="S3.SS2.p4.2.m2.2.2" xref="S3.SS2.p4.2.m2.2.2.cmml">t</mi><mo id="S3.SS2.p4.2.m2.3.4.3.2.3" xref="S3.SS2.p4.2.m2.3.4.3.1.cmml">,</mo><mi id="S3.SS2.p4.2.m2.3.3" xref="S3.SS2.p4.2.m2.3.3.cmml">T</mi><mo id="S3.SS2.p4.2.m2.3.4.3.2.4" stretchy="false" xref="S3.SS2.p4.2.m2.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.2.m2.3b"><apply id="S3.SS2.p4.2.m2.3.4.cmml" xref="S3.SS2.p4.2.m2.3.4"><times id="S3.SS2.p4.2.m2.3.4.1.cmml" xref="S3.SS2.p4.2.m2.3.4.1"></times><ci id="S3.SS2.p4.2.m2.3.4.2.cmml" xref="S3.SS2.p4.2.m2.3.4.2">𝑓</ci><vector id="S3.SS2.p4.2.m2.3.4.3.1.cmml" xref="S3.SS2.p4.2.m2.3.4.3.2"><ci id="S3.SS2.p4.2.m2.1.1.cmml" xref="S3.SS2.p4.2.m2.1.1">𝝃</ci><ci id="S3.SS2.p4.2.m2.2.2.cmml" xref="S3.SS2.p4.2.m2.2.2">𝑡</ci><ci id="S3.SS2.p4.2.m2.3.3.cmml" xref="S3.SS2.p4.2.m2.3.3">𝑇</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.2.m2.3c">f(\boldsymbol{\xi},t,T)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.2.m2.3d">italic_f ( bold_italic_ξ , italic_t , italic_T )</annotation></semantics></math> are as close as the ground truths of RGB, depth, etc., captured by camera <math alttext="\boldsymbol{\xi}" class="ltx_Math" display="inline" id="S3.SS2.p4.3.m3.1"><semantics id="S3.SS2.p4.3.m3.1a"><mi id="S3.SS2.p4.3.m3.1.1" xref="S3.SS2.p4.3.m3.1.1.cmml">𝝃</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.3.m3.1b"><ci id="S3.SS2.p4.3.m3.1.1.cmml" xref="S3.SS2.p4.3.m3.1.1">𝝃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.3.m3.1c">\boldsymbol{\xi}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.3.m3.1d">bold_italic_ξ</annotation></semantics></math> at time <math alttext="t" class="ltx_Math" display="inline" id="S3.SS2.p4.4.m4.1"><semantics id="S3.SS2.p4.4.m4.1a"><mi id="S3.SS2.p4.4.m4.1.1" xref="S3.SS2.p4.4.m4.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.4.m4.1b"><ci id="S3.SS2.p4.4.m4.1.1.cmml" xref="S3.SS2.p4.4.m4.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.4.m4.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.4.m4.1d">italic_t</annotation></semantics></math> in traversal <math alttext="T" class="ltx_Math" display="inline" id="S3.SS2.p4.5.m5.1"><semantics id="S3.SS2.p4.5.m5.1a"><mi id="S3.SS2.p4.5.m5.1.1" xref="S3.SS2.p4.5.m5.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.5.m5.1b"><ci id="S3.SS2.p4.5.m5.1.1.cmml" xref="S3.SS2.p4.5.m5.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.5.m5.1c">T</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.5.m5.1d">italic_T</annotation></semantics></math>. </div> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> 4 Multi-Traversal Gaussian Splatting</h2> <div class="ltx_para" id="S4.p1"> The overall pipeline of Multi-Traversal Gaussian splatting (MTGS) is depicted in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#S2.F2" title="Figure 2 ‣ 2 Related Work ‣ MTGS: Multi-Traversal Gaussian Splatting">2</a>. MTGS reconstructs a Multi-Traversal Scene Graph from a set of multi-traversal data, enabling the generation of high-fidelity images. In this section, we first introduce the design of the Multi-Traversal Scene Graph. Next, we describe our approach for tuning appearances across multiple traversals. Finally, we detail the geometric regularization techniques employed in MTGS and training objectives. </div> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> 4.1 Multi-Traversal Scene Graph</h3> <div class="ltx_para" id="S4.SS1.p1"> In multi-traversal settings, the state of the scene is determined by time <math alttext="t" class="ltx_Math" display="inline" id="S4.SS1.p1.1.m1.1"><semantics id="S4.SS1.p1.1.m1.1a"><mi id="S4.SS1.p1.1.m1.1.1" xref="S4.SS1.p1.1.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.1.m1.1b"><ci id="S4.SS1.p1.1.m1.1.1.cmml" xref="S4.SS1.p1.1.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.1.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.1.m1.1d">italic_t</annotation></semantics></math> and traversal <math alttext="T" class="ltx_Math" display="inline" id="S4.SS1.p1.2.m2.1"><semantics id="S4.SS1.p1.2.m2.1a"><mi id="S4.SS1.p1.2.m2.1.1" xref="S4.SS1.p1.2.m2.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.2.m2.1b"><ci id="S4.SS1.p1.2.m2.1.1.cmml" xref="S4.SS1.p1.2.m2.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.2.m2.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.2.m2.1d">italic_T</annotation></semantics></math>. To model transient objects and appearance changes, we represent the whole scene as a multi-traversal scene graph built upon 3DGS <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib17" title="">17</a>]</cite>, containing three types of nodes, one shared static node for backgrounds, <math alttext="\mathcal{G}^{\mathtt{static}}" class="ltx_Math" display="inline" id="S4.SS1.p1.3.m3.1"><semantics id="S4.SS1.p1.3.m3.1a"><msup id="S4.SS1.p1.3.m3.1.1" xref="S4.SS1.p1.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p1.3.m3.1.1.2" xref="S4.SS1.p1.3.m3.1.1.2.cmml">𝒢</mi><mi id="S4.SS1.p1.3.m3.1.1.3" xref="S4.SS1.p1.3.m3.1.1.3.cmml">𝚜𝚝𝚊𝚝𝚒𝚌</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.3.m3.1b"><apply id="S4.SS1.p1.3.m3.1.1.cmml" xref="S4.SS1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.3.m3.1.1.1.cmml" xref="S4.SS1.p1.3.m3.1.1">superscript</csymbol><ci id="S4.SS1.p1.3.m3.1.1.2.cmml" xref="S4.SS1.p1.3.m3.1.1.2">𝒢</ci><ci id="S4.SS1.p1.3.m3.1.1.3.cmml" xref="S4.SS1.p1.3.m3.1.1.3">𝚜𝚝𝚊𝚝𝚒𝚌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.3.m3.1c">\mathcal{G}^{\mathtt{static}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.3.m3.1d">caligraphic_G start_POSTSUPERSCRIPT typewriter_static end_POSTSUPERSCRIPT</annotation></semantics></math>, multiple appearance nodes for backgrounds, <math alttext="\mathcal{G}_{T}^{\mathtt{appr}}" class="ltx_Math" display="inline" id="S4.SS1.p1.4.m4.1"><semantics id="S4.SS1.p1.4.m4.1a"><msubsup id="S4.SS1.p1.4.m4.1.1" xref="S4.SS1.p1.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p1.4.m4.1.1.2.2" xref="S4.SS1.p1.4.m4.1.1.2.2.cmml">𝒢</mi><mi id="S4.SS1.p1.4.m4.1.1.2.3" xref="S4.SS1.p1.4.m4.1.1.2.3.cmml">T</mi><mi id="S4.SS1.p1.4.m4.1.1.3" xref="S4.SS1.p1.4.m4.1.1.3.cmml">𝚊𝚙𝚙𝚛</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.4.m4.1b"><apply id="S4.SS1.p1.4.m4.1.1.cmml" xref="S4.SS1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.4.m4.1.1.1.cmml" xref="S4.SS1.p1.4.m4.1.1">superscript</csymbol><apply id="S4.SS1.p1.4.m4.1.1.2.cmml" xref="S4.SS1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.4.m4.1.1.2.1.cmml" xref="S4.SS1.p1.4.m4.1.1">subscript</csymbol><ci id="S4.SS1.p1.4.m4.1.1.2.2.cmml" xref="S4.SS1.p1.4.m4.1.1.2.2">𝒢</ci><ci id="S4.SS1.p1.4.m4.1.1.2.3.cmml" xref="S4.SS1.p1.4.m4.1.1.2.3">𝑇</ci></apply><ci id="S4.SS1.p1.4.m4.1.1.3.cmml" xref="S4.SS1.p1.4.m4.1.1.3">𝚊𝚙𝚙𝚛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.4.m4.1c">\mathcal{G}_{T}^{\mathtt{appr}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.4.m4.1d">caligraphic_G start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT start_POSTSUPERSCRIPT typewriter_appr end_POSTSUPERSCRIPT</annotation></semantics></math> for traversal <math alttext="T" class="ltx_Math" display="inline" id="S4.SS1.p1.5.m5.1"><semantics id="S4.SS1.p1.5.m5.1a"><mi id="S4.SS1.p1.5.m5.1.1" xref="S4.SS1.p1.5.m5.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.5.m5.1b"><ci id="S4.SS1.p1.5.m5.1.1.cmml" xref="S4.SS1.p1.5.m5.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.5.m5.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.5.m5.1d">italic_T</annotation></semantics></math>, and multiple transient nodes that exist in exactly one traversal, <math alttext="\mathcal{G}_{T,k}^{\mathtt{tsnt}}" class="ltx_Math" display="inline" id="S4.SS1.p1.6.m6.2"><semantics id="S4.SS1.p1.6.m6.2a"><msubsup id="S4.SS1.p1.6.m6.2.3" xref="S4.SS1.p1.6.m6.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p1.6.m6.2.3.2.2" xref="S4.SS1.p1.6.m6.2.3.2.2.cmml">𝒢</mi><mrow id="S4.SS1.p1.6.m6.2.2.2.4" xref="S4.SS1.p1.6.m6.2.2.2.3.cmml"><mi id="S4.SS1.p1.6.m6.1.1.1.1" xref="S4.SS1.p1.6.m6.1.1.1.1.cmml">T</mi><mo id="S4.SS1.p1.6.m6.2.2.2.4.1" xref="S4.SS1.p1.6.m6.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p1.6.m6.2.2.2.2" xref="S4.SS1.p1.6.m6.2.2.2.2.cmml">k</mi></mrow><mi id="S4.SS1.p1.6.m6.2.3.3" xref="S4.SS1.p1.6.m6.2.3.3.cmml">𝚝𝚜𝚗𝚝</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.6.m6.2b"><apply id="S4.SS1.p1.6.m6.2.3.cmml" xref="S4.SS1.p1.6.m6.2.3"><csymbol cd="ambiguous" id="S4.SS1.p1.6.m6.2.3.1.cmml" xref="S4.SS1.p1.6.m6.2.3">superscript</csymbol><apply id="S4.SS1.p1.6.m6.2.3.2.cmml" xref="S4.SS1.p1.6.m6.2.3"><csymbol cd="ambiguous" id="S4.SS1.p1.6.m6.2.3.2.1.cmml" xref="S4.SS1.p1.6.m6.2.3">subscript</csymbol><ci id="S4.SS1.p1.6.m6.2.3.2.2.cmml" xref="S4.SS1.p1.6.m6.2.3.2.2">𝒢</ci><list id="S4.SS1.p1.6.m6.2.2.2.3.cmml" xref="S4.SS1.p1.6.m6.2.2.2.4"><ci id="S4.SS1.p1.6.m6.1.1.1.1.cmml" xref="S4.SS1.p1.6.m6.1.1.1.1">𝑇</ci><ci id="S4.SS1.p1.6.m6.2.2.2.2.cmml" xref="S4.SS1.p1.6.m6.2.2.2.2">𝑘</ci></list></apply><ci id="S4.SS1.p1.6.m6.2.3.3.cmml" xref="S4.SS1.p1.6.m6.2.3.3">𝚝𝚜𝚗𝚝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.6.m6.2c">\mathcal{G}_{T,k}^{\mathtt{tsnt}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.6.m6.2d">caligraphic_G start_POSTSUBSCRIPT italic_T , italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT typewriter_tsnt end_POSTSUPERSCRIPT</annotation></semantics></math> for the <math alttext="k" class="ltx_Math" display="inline" id="S4.SS1.p1.7.m7.1"><semantics id="S4.SS1.p1.7.m7.1a"><mi id="S4.SS1.p1.7.m7.1.1" xref="S4.SS1.p1.7.m7.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.7.m7.1b"><ci id="S4.SS1.p1.7.m7.1.1.cmml" xref="S4.SS1.p1.7.m7.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.7.m7.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.7.m7.1d">italic_k</annotation></semantics></math>-th node in traversal <math alttext="T" class="ltx_Math" display="inline" id="S4.SS1.p1.8.m8.1"><semantics id="S4.SS1.p1.8.m8.1a"><mi id="S4.SS1.p1.8.m8.1.1" xref="S4.SS1.p1.8.m8.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.8.m8.1b"><ci id="S4.SS1.p1.8.m8.1.1.cmml" xref="S4.SS1.p1.8.m8.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.8.m8.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.8.m8.1d">italic_T</annotation></semantics></math>. The scene in traversal <math alttext="T" class="ltx_Math" display="inline" id="S4.SS1.p1.9.m9.1"><semantics id="S4.SS1.p1.9.m9.1a"><mi id="S4.SS1.p1.9.m9.1.1" xref="S4.SS1.p1.9.m9.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.9.m9.1b"><ci id="S4.SS1.p1.9.m9.1.1.cmml" xref="S4.SS1.p1.9.m9.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.9.m9.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.9.m9.1d">italic_T</annotation></semantics></math> is thus a subgraph composed of the shared static node <math alttext="\mathcal{G}^{\mathtt{static}}" class="ltx_Math" display="inline" id="S4.SS1.p1.10.m10.1"><semantics id="S4.SS1.p1.10.m10.1a"><msup id="S4.SS1.p1.10.m10.1.1" xref="S4.SS1.p1.10.m10.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p1.10.m10.1.1.2" xref="S4.SS1.p1.10.m10.1.1.2.cmml">𝒢</mi><mi id="S4.SS1.p1.10.m10.1.1.3" xref="S4.SS1.p1.10.m10.1.1.3.cmml">𝚜𝚝𝚊𝚝𝚒𝚌</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.10.m10.1b"><apply id="S4.SS1.p1.10.m10.1.1.cmml" xref="S4.SS1.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.10.m10.1.1.1.cmml" xref="S4.SS1.p1.10.m10.1.1">superscript</csymbol><ci id="S4.SS1.p1.10.m10.1.1.2.cmml" xref="S4.SS1.p1.10.m10.1.1.2">𝒢</ci><ci id="S4.SS1.p1.10.m10.1.1.3.cmml" xref="S4.SS1.p1.10.m10.1.1.3">𝚜𝚝𝚊𝚝𝚒𝚌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.10.m10.1c">\mathcal{G}^{\mathtt{static}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.10.m10.1d">caligraphic_G start_POSTSUPERSCRIPT typewriter_static end_POSTSUPERSCRIPT</annotation></semantics></math>, one appearance node <math alttext="\mathcal{G}_{T}^{\mathtt{appr}}" class="ltx_Math" display="inline" id="S4.SS1.p1.11.m11.1"><semantics id="S4.SS1.p1.11.m11.1a"><msubsup id="S4.SS1.p1.11.m11.1.1" xref="S4.SS1.p1.11.m11.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p1.11.m11.1.1.2.2" xref="S4.SS1.p1.11.m11.1.1.2.2.cmml">𝒢</mi><mi id="S4.SS1.p1.11.m11.1.1.2.3" xref="S4.SS1.p1.11.m11.1.1.2.3.cmml">T</mi><mi id="S4.SS1.p1.11.m11.1.1.3" xref="S4.SS1.p1.11.m11.1.1.3.cmml">𝚊𝚙𝚙𝚛</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.11.m11.1b"><apply id="S4.SS1.p1.11.m11.1.1.cmml" xref="S4.SS1.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.11.m11.1.1.1.cmml" xref="S4.SS1.p1.11.m11.1.1">superscript</csymbol><apply id="S4.SS1.p1.11.m11.1.1.2.cmml" xref="S4.SS1.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.11.m11.1.1.2.1.cmml" xref="S4.SS1.p1.11.m11.1.1">subscript</csymbol><ci id="S4.SS1.p1.11.m11.1.1.2.2.cmml" xref="S4.SS1.p1.11.m11.1.1.2.2">𝒢</ci><ci id="S4.SS1.p1.11.m11.1.1.2.3.cmml" xref="S4.SS1.p1.11.m11.1.1.2.3">𝑇</ci></apply><ci id="S4.SS1.p1.11.m11.1.1.3.cmml" xref="S4.SS1.p1.11.m11.1.1.3">𝚊𝚙𝚙𝚛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.11.m11.1c">\mathcal{G}_{T}^{\mathtt{appr}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.11.m11.1d">caligraphic_G start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT start_POSTSUPERSCRIPT typewriter_appr end_POSTSUPERSCRIPT</annotation></semantics></math> and all transient nodes in the current traversal. </div> <div class="ltx_para ltx_noindent" id="S4.SS1.p2"> Static Node and Appearance Node. For <math alttext="G_{i}" class="ltx_Math" display="inline" id="S4.SS1.p2.1.m1.1"><semantics id="S4.SS1.p2.1.m1.1a"><msub id="S4.SS1.p2.1.m1.1.1" xref="S4.SS1.p2.1.m1.1.1.cmml"><mi id="S4.SS1.p2.1.m1.1.1.2" xref="S4.SS1.p2.1.m1.1.1.2.cmml">G</mi><mi id="S4.SS1.p2.1.m1.1.1.3" xref="S4.SS1.p2.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.1.m1.1b"><apply id="S4.SS1.p2.1.m1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.1.m1.1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1">subscript</csymbol><ci id="S4.SS1.p2.1.m1.1.1.2.cmml" xref="S4.SS1.p2.1.m1.1.1.2">𝐺</ci><ci id="S4.SS1.p2.1.m1.1.1.3.cmml" xref="S4.SS1.p2.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.1.m1.1c">G_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.1.m1.1d">italic_G start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> in the static backgrounds, <math alttext="\mathcal{G}^{\mathtt{static}}" class="ltx_Math" display="inline" id="S4.SS1.p2.2.m2.1"><semantics id="S4.SS1.p2.2.m2.1a"><msup id="S4.SS1.p2.2.m2.1.1" xref="S4.SS1.p2.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p2.2.m2.1.1.2" xref="S4.SS1.p2.2.m2.1.1.2.cmml">𝒢</mi><mi id="S4.SS1.p2.2.m2.1.1.3" xref="S4.SS1.p2.2.m2.1.1.3.cmml">𝚜𝚝𝚊𝚝𝚒𝚌</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.2.m2.1b"><apply id="S4.SS1.p2.2.m2.1.1.cmml" xref="S4.SS1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.2.m2.1.1.1.cmml" xref="S4.SS1.p2.2.m2.1.1">superscript</csymbol><ci id="S4.SS1.p2.2.m2.1.1.2.cmml" xref="S4.SS1.p2.2.m2.1.1.2">𝒢</ci><ci id="S4.SS1.p2.2.m2.1.1.3.cmml" xref="S4.SS1.p2.2.m2.1.1.3">𝚜𝚝𝚊𝚝𝚒𝚌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.2.m2.1c">\mathcal{G}^{\mathtt{static}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.2.m2.1d">caligraphic_G start_POSTSUPERSCRIPT typewriter_static end_POSTSUPERSCRIPT</annotation></semantics></math> provides traversal-invariant and time-invariant properties <math alttext="\left\{\mathbf{x}_{i},\mathbf{q}_{i},\mathbf{s}_{i},\alpha_{i},\boldsymbol{% \beta}^{\mathtt{base}}_{i}\right\}" class="ltx_Math" display="inline" id="S4.SS1.p2.3.m3.5"><semantics id="S4.SS1.p2.3.m3.5a"><mrow id="S4.SS1.p2.3.m3.5.5.5" xref="S4.SS1.p2.3.m3.5.5.6.cmml"><mo id="S4.SS1.p2.3.m3.5.5.5.6" xref="S4.SS1.p2.3.m3.5.5.6.cmml">{</mo><msub id="S4.SS1.p2.3.m3.1.1.1.1" xref="S4.SS1.p2.3.m3.1.1.1.1.cmml"><mi id="S4.SS1.p2.3.m3.1.1.1.1.2" xref="S4.SS1.p2.3.m3.1.1.1.1.2.cmml">𝐱</mi><mi id="S4.SS1.p2.3.m3.1.1.1.1.3" xref="S4.SS1.p2.3.m3.1.1.1.1.3.cmml">i</mi></msub><mo id="S4.SS1.p2.3.m3.5.5.5.7" xref="S4.SS1.p2.3.m3.5.5.6.cmml">,</mo><msub id="S4.SS1.p2.3.m3.2.2.2.2" xref="S4.SS1.p2.3.m3.2.2.2.2.cmml"><mi id="S4.SS1.p2.3.m3.2.2.2.2.2" xref="S4.SS1.p2.3.m3.2.2.2.2.2.cmml">𝐪</mi><mi id="S4.SS1.p2.3.m3.2.2.2.2.3" xref="S4.SS1.p2.3.m3.2.2.2.2.3.cmml">i</mi></msub><mo id="S4.SS1.p2.3.m3.5.5.5.8" xref="S4.SS1.p2.3.m3.5.5.6.cmml">,</mo><msub id="S4.SS1.p2.3.m3.3.3.3.3" xref="S4.SS1.p2.3.m3.3.3.3.3.cmml"><mi id="S4.SS1.p2.3.m3.3.3.3.3.2" xref="S4.SS1.p2.3.m3.3.3.3.3.2.cmml">𝐬</mi><mi id="S4.SS1.p2.3.m3.3.3.3.3.3" xref="S4.SS1.p2.3.m3.3.3.3.3.3.cmml">i</mi></msub><mo id="S4.SS1.p2.3.m3.5.5.5.9" xref="S4.SS1.p2.3.m3.5.5.6.cmml">,</mo><msub id="S4.SS1.p2.3.m3.4.4.4.4" xref="S4.SS1.p2.3.m3.4.4.4.4.cmml"><mi id="S4.SS1.p2.3.m3.4.4.4.4.2" xref="S4.SS1.p2.3.m3.4.4.4.4.2.cmml">α</mi><mi id="S4.SS1.p2.3.m3.4.4.4.4.3" xref="S4.SS1.p2.3.m3.4.4.4.4.3.cmml">i</mi></msub><mo id="S4.SS1.p2.3.m3.5.5.5.10" xref="S4.SS1.p2.3.m3.5.5.6.cmml">,</mo><msubsup id="S4.SS1.p2.3.m3.5.5.5.5" xref="S4.SS1.p2.3.m3.5.5.5.5.cmml"><mi id="S4.SS1.p2.3.m3.5.5.5.5.2.2" xref="S4.SS1.p2.3.m3.5.5.5.5.2.2.cmml">𝜷</mi><mi id="S4.SS1.p2.3.m3.5.5.5.5.3" xref="S4.SS1.p2.3.m3.5.5.5.5.3.cmml">i</mi><mi id="S4.SS1.p2.3.m3.5.5.5.5.2.3" xref="S4.SS1.p2.3.m3.5.5.5.5.2.3.cmml">𝚋𝚊𝚜𝚎</mi></msubsup><mo id="S4.SS1.p2.3.m3.5.5.5.11" xref="S4.SS1.p2.3.m3.5.5.6.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.3.m3.5b"><set id="S4.SS1.p2.3.m3.5.5.6.cmml" xref="S4.SS1.p2.3.m3.5.5.5"><apply id="S4.SS1.p2.3.m3.1.1.1.1.cmml" xref="S4.SS1.p2.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.3.m3.1.1.1.1.1.cmml" xref="S4.SS1.p2.3.m3.1.1.1.1">subscript</csymbol><ci id="S4.SS1.p2.3.m3.1.1.1.1.2.cmml" xref="S4.SS1.p2.3.m3.1.1.1.1.2">𝐱</ci><ci id="S4.SS1.p2.3.m3.1.1.1.1.3.cmml" xref="S4.SS1.p2.3.m3.1.1.1.1.3">𝑖</ci></apply><apply id="S4.SS1.p2.3.m3.2.2.2.2.cmml" xref="S4.SS1.p2.3.m3.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS1.p2.3.m3.2.2.2.2.1.cmml" xref="S4.SS1.p2.3.m3.2.2.2.2">subscript</csymbol><ci id="S4.SS1.p2.3.m3.2.2.2.2.2.cmml" xref="S4.SS1.p2.3.m3.2.2.2.2.2">𝐪</ci><ci id="S4.SS1.p2.3.m3.2.2.2.2.3.cmml" xref="S4.SS1.p2.3.m3.2.2.2.2.3">𝑖</ci></apply><apply id="S4.SS1.p2.3.m3.3.3.3.3.cmml" xref="S4.SS1.p2.3.m3.3.3.3.3"><csymbol cd="ambiguous" id="S4.SS1.p2.3.m3.3.3.3.3.1.cmml" xref="S4.SS1.p2.3.m3.3.3.3.3">subscript</csymbol><ci id="S4.SS1.p2.3.m3.3.3.3.3.2.cmml" xref="S4.SS1.p2.3.m3.3.3.3.3.2">𝐬</ci><ci id="S4.SS1.p2.3.m3.3.3.3.3.3.cmml" xref="S4.SS1.p2.3.m3.3.3.3.3.3">𝑖</ci></apply><apply id="S4.SS1.p2.3.m3.4.4.4.4.cmml" xref="S4.SS1.p2.3.m3.4.4.4.4"><csymbol cd="ambiguous" id="S4.SS1.p2.3.m3.4.4.4.4.1.cmml" xref="S4.SS1.p2.3.m3.4.4.4.4">subscript</csymbol><ci id="S4.SS1.p2.3.m3.4.4.4.4.2.cmml" xref="S4.SS1.p2.3.m3.4.4.4.4.2">𝛼</ci><ci id="S4.SS1.p2.3.m3.4.4.4.4.3.cmml" xref="S4.SS1.p2.3.m3.4.4.4.4.3">𝑖</ci></apply><apply id="S4.SS1.p2.3.m3.5.5.5.5.cmml" xref="S4.SS1.p2.3.m3.5.5.5.5"><csymbol cd="ambiguous" id="S4.SS1.p2.3.m3.5.5.5.5.1.cmml" xref="S4.SS1.p2.3.m3.5.5.5.5">subscript</csymbol><apply id="S4.SS1.p2.3.m3.5.5.5.5.2.cmml" xref="S4.SS1.p2.3.m3.5.5.5.5"><csymbol cd="ambiguous" id="S4.SS1.p2.3.m3.5.5.5.5.2.1.cmml" xref="S4.SS1.p2.3.m3.5.5.5.5">superscript</csymbol><ci id="S4.SS1.p2.3.m3.5.5.5.5.2.2.cmml" xref="S4.SS1.p2.3.m3.5.5.5.5.2.2">𝜷</ci><ci id="S4.SS1.p2.3.m3.5.5.5.5.2.3.cmml" xref="S4.SS1.p2.3.m3.5.5.5.5.2.3">𝚋𝚊𝚜𝚎</ci></apply><ci id="S4.SS1.p2.3.m3.5.5.5.5.3.cmml" xref="S4.SS1.p2.3.m3.5.5.5.5.3">𝑖</ci></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.3.m3.5c">\left\{\mathbf{x}_{i},\mathbf{q}_{i},\mathbf{s}_{i},\alpha_{i},\boldsymbol{% \beta}^{\mathtt{base}}_{i}\right\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.3.m3.5d">{ bold_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_β start_POSTSUPERSCRIPT typewriter_base end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT }</annotation></semantics></math> while the appearance node <math alttext="\mathcal{G}^{\mathtt{appr}}_{T}" class="ltx_Math" display="inline" id="S4.SS1.p2.4.m4.1"><semantics id="S4.SS1.p2.4.m4.1a"><msubsup id="S4.SS1.p2.4.m4.1.1" xref="S4.SS1.p2.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p2.4.m4.1.1.2.2" xref="S4.SS1.p2.4.m4.1.1.2.2.cmml">𝒢</mi><mi id="S4.SS1.p2.4.m4.1.1.3" xref="S4.SS1.p2.4.m4.1.1.3.cmml">T</mi><mi id="S4.SS1.p2.4.m4.1.1.2.3" xref="S4.SS1.p2.4.m4.1.1.2.3.cmml">𝚊𝚙𝚙𝚛</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.4.m4.1b"><apply id="S4.SS1.p2.4.m4.1.1.cmml" xref="S4.SS1.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.4.m4.1.1.1.cmml" xref="S4.SS1.p2.4.m4.1.1">subscript</csymbol><apply id="S4.SS1.p2.4.m4.1.1.2.cmml" xref="S4.SS1.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.4.m4.1.1.2.1.cmml" xref="S4.SS1.p2.4.m4.1.1">superscript</csymbol><ci id="S4.SS1.p2.4.m4.1.1.2.2.cmml" xref="S4.SS1.p2.4.m4.1.1.2.2">𝒢</ci><ci id="S4.SS1.p2.4.m4.1.1.2.3.cmml" xref="S4.SS1.p2.4.m4.1.1.2.3">𝚊𝚙𝚙𝚛</ci></apply><ci id="S4.SS1.p2.4.m4.1.1.3.cmml" xref="S4.SS1.p2.4.m4.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.4.m4.1c">\mathcal{G}^{\mathtt{appr}}_{T}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.4.m4.1d">caligraphic_G start_POSTSUPERSCRIPT typewriter_appr end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT</annotation></semantics></math> provides traversal-wise color residuals in traversal <math alttext="T" class="ltx_Math" display="inline" id="S4.SS1.p2.5.m5.1"><semantics id="S4.SS1.p2.5.m5.1a"><mi id="S4.SS1.p2.5.m5.1.1" xref="S4.SS1.p2.5.m5.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.5.m5.1b"><ci id="S4.SS1.p2.5.m5.1.1.cmml" xref="S4.SS1.p2.5.m5.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.5.m5.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.5.m5.1d">italic_T</annotation></semantics></math>, <math alttext="\left\{\boldsymbol{\beta}^{\mathtt{residual}}_{i,T}\right\}" class="ltx_Math" display="inline" id="S4.SS1.p2.6.m6.3"><semantics id="S4.SS1.p2.6.m6.3a"><mrow id="S4.SS1.p2.6.m6.3.3.1" xref="S4.SS1.p2.6.m6.3.3.2.cmml"><mo id="S4.SS1.p2.6.m6.3.3.1.2" xref="S4.SS1.p2.6.m6.3.3.2.cmml">{</mo><msubsup id="S4.SS1.p2.6.m6.3.3.1.1" xref="S4.SS1.p2.6.m6.3.3.1.1.cmml"><mi id="S4.SS1.p2.6.m6.3.3.1.1.2.2" xref="S4.SS1.p2.6.m6.3.3.1.1.2.2.cmml">𝜷</mi><mrow id="S4.SS1.p2.6.m6.2.2.2.4" xref="S4.SS1.p2.6.m6.2.2.2.3.cmml"><mi id="S4.SS1.p2.6.m6.1.1.1.1" xref="S4.SS1.p2.6.m6.1.1.1.1.cmml">i</mi><mo id="S4.SS1.p2.6.m6.2.2.2.4.1" xref="S4.SS1.p2.6.m6.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p2.6.m6.2.2.2.2" xref="S4.SS1.p2.6.m6.2.2.2.2.cmml">T</mi></mrow><mi id="S4.SS1.p2.6.m6.3.3.1.1.2.3" xref="S4.SS1.p2.6.m6.3.3.1.1.2.3.cmml">𝚛𝚎𝚜𝚒𝚍𝚞𝚊𝚕</mi></msubsup><mo id="S4.SS1.p2.6.m6.3.3.1.3" xref="S4.SS1.p2.6.m6.3.3.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.6.m6.3b"><set id="S4.SS1.p2.6.m6.3.3.2.cmml" xref="S4.SS1.p2.6.m6.3.3.1"><apply id="S4.SS1.p2.6.m6.3.3.1.1.cmml" xref="S4.SS1.p2.6.m6.3.3.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.6.m6.3.3.1.1.1.cmml" xref="S4.SS1.p2.6.m6.3.3.1.1">subscript</csymbol><apply id="S4.SS1.p2.6.m6.3.3.1.1.2.cmml" xref="S4.SS1.p2.6.m6.3.3.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.6.m6.3.3.1.1.2.1.cmml" xref="S4.SS1.p2.6.m6.3.3.1.1">superscript</csymbol><ci id="S4.SS1.p2.6.m6.3.3.1.1.2.2.cmml" xref="S4.SS1.p2.6.m6.3.3.1.1.2.2">𝜷</ci><ci id="S4.SS1.p2.6.m6.3.3.1.1.2.3.cmml" xref="S4.SS1.p2.6.m6.3.3.1.1.2.3">𝚛𝚎𝚜𝚒𝚍𝚞𝚊𝚕</ci></apply><list id="S4.SS1.p2.6.m6.2.2.2.3.cmml" xref="S4.SS1.p2.6.m6.2.2.2.4"><ci id="S4.SS1.p2.6.m6.1.1.1.1.cmml" xref="S4.SS1.p2.6.m6.1.1.1.1">𝑖</ci><ci id="S4.SS1.p2.6.m6.2.2.2.2.cmml" xref="S4.SS1.p2.6.m6.2.2.2.2">𝑇</ci></list></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.6.m6.3c">\left\{\boldsymbol{\beta}^{\mathtt{residual}}_{i,T}\right\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.6.m6.3d">{ bold_italic_β start_POSTSUPERSCRIPT typewriter_residual end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_T end_POSTSUBSCRIPT }</annotation></semantics></math>. Here, for <math alttext="G_{i}" class="ltx_Math" display="inline" id="S4.SS1.p2.7.m7.1"><semantics id="S4.SS1.p2.7.m7.1a"><msub id="S4.SS1.p2.7.m7.1.1" xref="S4.SS1.p2.7.m7.1.1.cmml"><mi id="S4.SS1.p2.7.m7.1.1.2" xref="S4.SS1.p2.7.m7.1.1.2.cmml">G</mi><mi id="S4.SS1.p2.7.m7.1.1.3" xref="S4.SS1.p2.7.m7.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.7.m7.1b"><apply id="S4.SS1.p2.7.m7.1.1.cmml" xref="S4.SS1.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.7.m7.1.1.1.cmml" xref="S4.SS1.p2.7.m7.1.1">subscript</csymbol><ci id="S4.SS1.p2.7.m7.1.1.2.cmml" xref="S4.SS1.p2.7.m7.1.1.2">𝐺</ci><ci id="S4.SS1.p2.7.m7.1.1.3.cmml" xref="S4.SS1.p2.7.m7.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.7.m7.1c">G_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.7.m7.1d">italic_G start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> in traversal <math alttext="T" class="ltx_Math" display="inline" id="S4.SS1.p2.8.m8.1"><semantics id="S4.SS1.p2.8.m8.1a"><mi id="S4.SS1.p2.8.m8.1.1" xref="S4.SS1.p2.8.m8.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.8.m8.1b"><ci id="S4.SS1.p2.8.m8.1.1.cmml" xref="S4.SS1.p2.8.m8.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.8.m8.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.8.m8.1d">italic_T</annotation></semantics></math>, <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A2.EGx2"> <tbody id="S4.E5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\boldsymbol{\beta}_{i,0,0}=\boldsymbol{\beta}_{i}^{\mathtt{base}}% ,\,\text{and }\left\{\boldsymbol{\beta}_{i,l,m}\right\}_{1\leq l\leq l_{% \mathtt{max}}}^{-l\leq m\leq l}=\boldsymbol{\beta}_{i,T}^{\mathtt{residual}}." 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id="S4.E5.m1.9c">\displaystyle\boldsymbol{\beta}_{i,0,0}=\boldsymbol{\beta}_{i}^{\mathtt{base}}% ,\,\text{and }\left\{\boldsymbol{\beta}_{i,l,m}\right\}_{1\leq l\leq l_{% \mathtt{max}}}^{-l\leq m\leq l}=\boldsymbol{\beta}_{i,T}^{\mathtt{residual}}.</annotation><annotation encoding="application/x-llamapun" id="S4.E5.m1.9d">bold_italic_β start_POSTSUBSCRIPT italic_i , 0 , 0 end_POSTSUBSCRIPT = bold_italic_β start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT typewriter_base end_POSTSUPERSCRIPT , and { bold_italic_β start_POSTSUBSCRIPT italic_i , italic_l , italic_m end_POSTSUBSCRIPT } start_POSTSUBSCRIPT 1 ≤ italic_l ≤ italic_l start_POSTSUBSCRIPT typewriter_max end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - italic_l ≤ italic_m ≤ italic_l end_POSTSUPERSCRIPT = bold_italic_β start_POSTSUBSCRIPT italic_i , italic_T end_POSTSUBSCRIPT start_POSTSUPERSCRIPT typewriter_residual end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(5)</td> </tr></tbody> </table> With such designs, only the coefficient for the rotation-invariant spherical harmonic (SH) <math alttext="Y_{0,0}" class="ltx_Math" display="inline" id="S4.SS1.p2.9.m1.2"><semantics id="S4.SS1.p2.9.m1.2a"><msub id="S4.SS1.p2.9.m1.2.3" xref="S4.SS1.p2.9.m1.2.3.cmml"><mi id="S4.SS1.p2.9.m1.2.3.2" xref="S4.SS1.p2.9.m1.2.3.2.cmml">Y</mi><mrow id="S4.SS1.p2.9.m1.2.2.2.4" xref="S4.SS1.p2.9.m1.2.2.2.3.cmml"><mn id="S4.SS1.p2.9.m1.1.1.1.1" xref="S4.SS1.p2.9.m1.1.1.1.1.cmml">0</mn><mo id="S4.SS1.p2.9.m1.2.2.2.4.1" xref="S4.SS1.p2.9.m1.2.2.2.3.cmml">,</mo><mn id="S4.SS1.p2.9.m1.2.2.2.2" xref="S4.SS1.p2.9.m1.2.2.2.2.cmml">0</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.9.m1.2b"><apply id="S4.SS1.p2.9.m1.2.3.cmml" xref="S4.SS1.p2.9.m1.2.3"><csymbol cd="ambiguous" id="S4.SS1.p2.9.m1.2.3.1.cmml" 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end_POSTSUBSCRIPT } start_POSTSUBSCRIPT 1 ≤ italic_l ≤ italic_l start_POSTSUBSCRIPT typewriter_max end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - italic_l ≤ italic_m ≤ italic_l end_POSTSUPERSCRIPT</annotation></semantics></math>. </div> <div class="ltx_para" id="S4.SS1.p3"> In contrast, if some coefficients in <math alttext="\boldsymbol{\beta}_{i,T}^{\mathtt{residual}}" class="ltx_Math" display="inline" id="S4.SS1.p3.1.m1.2"><semantics id="S4.SS1.p3.1.m1.2a"><msubsup id="S4.SS1.p3.1.m1.2.3" xref="S4.SS1.p3.1.m1.2.3.cmml"><mi id="S4.SS1.p3.1.m1.2.3.2.2" xref="S4.SS1.p3.1.m1.2.3.2.2.cmml">𝜷</mi><mrow id="S4.SS1.p3.1.m1.2.2.2.4" xref="S4.SS1.p3.1.m1.2.2.2.3.cmml"><mi id="S4.SS1.p3.1.m1.1.1.1.1" xref="S4.SS1.p3.1.m1.1.1.1.1.cmml">i</mi><mo id="S4.SS1.p3.1.m1.2.2.2.4.1" xref="S4.SS1.p3.1.m1.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p3.1.m1.2.2.2.2" xref="S4.SS1.p3.1.m1.2.2.2.2.cmml">T</mi></mrow><mi id="S4.SS1.p3.1.m1.2.3.3" xref="S4.SS1.p3.1.m1.2.3.3.cmml">𝚛𝚎𝚜𝚒𝚍𝚞𝚊𝚕</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.1.m1.2b"><apply id="S4.SS1.p3.1.m1.2.3.cmml" xref="S4.SS1.p3.1.m1.2.3"><csymbol cd="ambiguous" id="S4.SS1.p3.1.m1.2.3.1.cmml" xref="S4.SS1.p3.1.m1.2.3">superscript</csymbol><apply id="S4.SS1.p3.1.m1.2.3.2.cmml" xref="S4.SS1.p3.1.m1.2.3"><csymbol cd="ambiguous" id="S4.SS1.p3.1.m1.2.3.2.1.cmml" xref="S4.SS1.p3.1.m1.2.3">subscript</csymbol><ci id="S4.SS1.p3.1.m1.2.3.2.2.cmml" xref="S4.SS1.p3.1.m1.2.3.2.2">𝜷</ci><list id="S4.SS1.p3.1.m1.2.2.2.3.cmml" xref="S4.SS1.p3.1.m1.2.2.2.4"><ci id="S4.SS1.p3.1.m1.1.1.1.1.cmml" xref="S4.SS1.p3.1.m1.1.1.1.1">𝑖</ci><ci id="S4.SS1.p3.1.m1.2.2.2.2.cmml" xref="S4.SS1.p3.1.m1.2.2.2.2">𝑇</ci></list></apply><ci id="S4.SS1.p3.1.m1.2.3.3.cmml" xref="S4.SS1.p3.1.m1.2.3.3">𝚛𝚎𝚜𝚒𝚍𝚞𝚊𝚕</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.1.m1.2c">\boldsymbol{\beta}_{i,T}^{\mathtt{residual}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.1.m1.2d">bold_italic_β start_POSTSUBSCRIPT italic_i , italic_T end_POSTSUBSCRIPT start_POSTSUPERSCRIPT typewriter_residual end_POSTSUPERSCRIPT</annotation></semantics></math> are shared, changes caused by various traversals would be mistaken for those caused by various views. When no SH coefficients are shared, the geometry of backgrounds is not aligned, leading to undesired background deviations across traversals. </div> <div class="ltx_para ltx_noindent" id="S4.SS1.p4"> Transient Node. For Gaussians in <math alttext="\mathcal{G}_{T,k}^{\mathtt{tsnt}}" class="ltx_Math" display="inline" id="S4.SS1.p4.1.m1.2"><semantics id="S4.SS1.p4.1.m1.2a"><msubsup id="S4.SS1.p4.1.m1.2.3" xref="S4.SS1.p4.1.m1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p4.1.m1.2.3.2.2" xref="S4.SS1.p4.1.m1.2.3.2.2.cmml">𝒢</mi><mrow id="S4.SS1.p4.1.m1.2.2.2.4" xref="S4.SS1.p4.1.m1.2.2.2.3.cmml"><mi id="S4.SS1.p4.1.m1.1.1.1.1" xref="S4.SS1.p4.1.m1.1.1.1.1.cmml">T</mi><mo id="S4.SS1.p4.1.m1.2.2.2.4.1" xref="S4.SS1.p4.1.m1.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p4.1.m1.2.2.2.2" xref="S4.SS1.p4.1.m1.2.2.2.2.cmml">k</mi></mrow><mi id="S4.SS1.p4.1.m1.2.3.3" xref="S4.SS1.p4.1.m1.2.3.3.cmml">𝚝𝚜𝚗𝚝</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.1.m1.2b"><apply id="S4.SS1.p4.1.m1.2.3.cmml" xref="S4.SS1.p4.1.m1.2.3"><csymbol cd="ambiguous" id="S4.SS1.p4.1.m1.2.3.1.cmml" xref="S4.SS1.p4.1.m1.2.3">superscript</csymbol><apply id="S4.SS1.p4.1.m1.2.3.2.cmml" xref="S4.SS1.p4.1.m1.2.3"><csymbol cd="ambiguous" id="S4.SS1.p4.1.m1.2.3.2.1.cmml" xref="S4.SS1.p4.1.m1.2.3">subscript</csymbol><ci id="S4.SS1.p4.1.m1.2.3.2.2.cmml" xref="S4.SS1.p4.1.m1.2.3.2.2">𝒢</ci><list id="S4.SS1.p4.1.m1.2.2.2.3.cmml" xref="S4.SS1.p4.1.m1.2.2.2.4"><ci id="S4.SS1.p4.1.m1.1.1.1.1.cmml" xref="S4.SS1.p4.1.m1.1.1.1.1">𝑇</ci><ci id="S4.SS1.p4.1.m1.2.2.2.2.cmml" xref="S4.SS1.p4.1.m1.2.2.2.2">𝑘</ci></list></apply><ci id="S4.SS1.p4.1.m1.2.3.3.cmml" xref="S4.SS1.p4.1.m1.2.3.3">𝚝𝚜𝚗𝚝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.1.m1.2c">\mathcal{G}_{T,k}^{\mathtt{tsnt}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.1.m1.2d">caligraphic_G start_POSTSUBSCRIPT italic_T , italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT typewriter_tsnt end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\mathbf{x}_{i}" class="ltx_Math" display="inline" id="S4.SS1.p4.2.m2.1"><semantics id="S4.SS1.p4.2.m2.1a"><msub id="S4.SS1.p4.2.m2.1.1" xref="S4.SS1.p4.2.m2.1.1.cmml"><mi id="S4.SS1.p4.2.m2.1.1.2" xref="S4.SS1.p4.2.m2.1.1.2.cmml">𝐱</mi><mi id="S4.SS1.p4.2.m2.1.1.3" xref="S4.SS1.p4.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.2.m2.1b"><apply id="S4.SS1.p4.2.m2.1.1.cmml" xref="S4.SS1.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p4.2.m2.1.1.1.cmml" xref="S4.SS1.p4.2.m2.1.1">subscript</csymbol><ci id="S4.SS1.p4.2.m2.1.1.2.cmml" xref="S4.SS1.p4.2.m2.1.1.2">𝐱</ci><ci id="S4.SS1.p4.2.m2.1.1.3.cmml" xref="S4.SS1.p4.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.2.m2.1c">\mathbf{x}_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.2.m2.1d">bold_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mathbf{q}_{i}" class="ltx_Math" display="inline" id="S4.SS1.p4.3.m3.1"><semantics id="S4.SS1.p4.3.m3.1a"><msub id="S4.SS1.p4.3.m3.1.1" xref="S4.SS1.p4.3.m3.1.1.cmml"><mi id="S4.SS1.p4.3.m3.1.1.2" xref="S4.SS1.p4.3.m3.1.1.2.cmml">𝐪</mi><mi id="S4.SS1.p4.3.m3.1.1.3" xref="S4.SS1.p4.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.3.m3.1b"><apply id="S4.SS1.p4.3.m3.1.1.cmml" xref="S4.SS1.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS1.p4.3.m3.1.1.1.cmml" xref="S4.SS1.p4.3.m3.1.1">subscript</csymbol><ci id="S4.SS1.p4.3.m3.1.1.2.cmml" xref="S4.SS1.p4.3.m3.1.1.2">𝐪</ci><ci id="S4.SS1.p4.3.m3.1.1.3.cmml" xref="S4.SS1.p4.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.3.m3.1c">\mathbf{q}_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.3.m3.1d">bold_q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> are defined in local coordinates of the node, and can be transformed into world coordinates by <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A2.EGx3"> <tbody id="S4.E6"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathbf{x}^{\mathtt{world}}_{i}(t)" class="ltx_Math" display="inline" id="S4.E6.m1.1"><semantics id="S4.E6.m1.1a"><mrow id="S4.E6.m1.1.2" xref="S4.E6.m1.1.2.cmml"><msubsup id="S4.E6.m1.1.2.2" xref="S4.E6.m1.1.2.2.cmml"><mi id="S4.E6.m1.1.2.2.2.2" xref="S4.E6.m1.1.2.2.2.2.cmml">𝐱</mi><mi id="S4.E6.m1.1.2.2.3" xref="S4.E6.m1.1.2.2.3.cmml">i</mi><mi id="S4.E6.m1.1.2.2.2.3" xref="S4.E6.m1.1.2.2.2.3.cmml">𝚠𝚘𝚛𝚕𝚍</mi></msubsup><mo id="S4.E6.m1.1.2.1" xref="S4.E6.m1.1.2.1.cmml">⁢</mo><mrow id="S4.E6.m1.1.2.3.2" xref="S4.E6.m1.1.2.cmml"><mo id="S4.E6.m1.1.2.3.2.1" stretchy="false" xref="S4.E6.m1.1.2.cmml">(</mo><mi id="S4.E6.m1.1.1" xref="S4.E6.m1.1.1.cmml">t</mi><mo id="S4.E6.m1.1.2.3.2.2" stretchy="false" xref="S4.E6.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E6.m1.1b"><apply id="S4.E6.m1.1.2.cmml" xref="S4.E6.m1.1.2"><times id="S4.E6.m1.1.2.1.cmml" xref="S4.E6.m1.1.2.1"></times><apply id="S4.E6.m1.1.2.2.cmml" xref="S4.E6.m1.1.2.2"><csymbol cd="ambiguous" id="S4.E6.m1.1.2.2.1.cmml" xref="S4.E6.m1.1.2.2">subscript</csymbol><apply id="S4.E6.m1.1.2.2.2.cmml" xref="S4.E6.m1.1.2.2"><csymbol cd="ambiguous" id="S4.E6.m1.1.2.2.2.1.cmml" xref="S4.E6.m1.1.2.2">superscript</csymbol><ci id="S4.E6.m1.1.2.2.2.2.cmml" xref="S4.E6.m1.1.2.2.2.2">𝐱</ci><ci id="S4.E6.m1.1.2.2.2.3.cmml" xref="S4.E6.m1.1.2.2.2.3">𝚠𝚘𝚛𝚕𝚍</ci></apply><ci id="S4.E6.m1.1.2.2.3.cmml" xref="S4.E6.m1.1.2.2.3">𝑖</ci></apply><ci id="S4.E6.m1.1.1.cmml" xref="S4.E6.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E6.m1.1c">\displaystyle\mathbf{x}^{\mathtt{world}}_{i}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.E6.m1.1d">bold_x start_POSTSUPERSCRIPT typewriter_world end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\mathbf{R}_{T,k}(t)\mathbf{x}_{i}+\mathbf{T}_{T,k}(t)," class="ltx_Math" display="inline" id="S4.E6.m2.7"><semantics id="S4.E6.m2.7a"><mrow id="S4.E6.m2.7.7.1" xref="S4.E6.m2.7.7.1.1.cmml"><mrow id="S4.E6.m2.7.7.1.1" xref="S4.E6.m2.7.7.1.1.cmml"><mi id="S4.E6.m2.7.7.1.1.2" xref="S4.E6.m2.7.7.1.1.2.cmml"></mi><mo id="S4.E6.m2.7.7.1.1.1" xref="S4.E6.m2.7.7.1.1.1.cmml">=</mo><mrow id="S4.E6.m2.7.7.1.1.3" xref="S4.E6.m2.7.7.1.1.3.cmml"><mrow id="S4.E6.m2.7.7.1.1.3.2" xref="S4.E6.m2.7.7.1.1.3.2.cmml"><msub id="S4.E6.m2.7.7.1.1.3.2.2" xref="S4.E6.m2.7.7.1.1.3.2.2.cmml"><mi id="S4.E6.m2.7.7.1.1.3.2.2.2" xref="S4.E6.m2.7.7.1.1.3.2.2.2.cmml">𝐑</mi><mrow id="S4.E6.m2.2.2.2.4" xref="S4.E6.m2.2.2.2.3.cmml"><mi id="S4.E6.m2.1.1.1.1" xref="S4.E6.m2.1.1.1.1.cmml">T</mi><mo id="S4.E6.m2.2.2.2.4.1" xref="S4.E6.m2.2.2.2.3.cmml">,</mo><mi id="S4.E6.m2.2.2.2.2" xref="S4.E6.m2.2.2.2.2.cmml">k</mi></mrow></msub><mo id="S4.E6.m2.7.7.1.1.3.2.1" xref="S4.E6.m2.7.7.1.1.3.2.1.cmml">⁢</mo><mrow id="S4.E6.m2.7.7.1.1.3.2.3.2" xref="S4.E6.m2.7.7.1.1.3.2.cmml"><mo id="S4.E6.m2.7.7.1.1.3.2.3.2.1" stretchy="false" xref="S4.E6.m2.7.7.1.1.3.2.cmml">(</mo><mi id="S4.E6.m2.5.5" xref="S4.E6.m2.5.5.cmml">t</mi><mo id="S4.E6.m2.7.7.1.1.3.2.3.2.2" stretchy="false" xref="S4.E6.m2.7.7.1.1.3.2.cmml">)</mo></mrow><mo id="S4.E6.m2.7.7.1.1.3.2.1a" xref="S4.E6.m2.7.7.1.1.3.2.1.cmml">⁢</mo><msub id="S4.E6.m2.7.7.1.1.3.2.4" xref="S4.E6.m2.7.7.1.1.3.2.4.cmml"><mi id="S4.E6.m2.7.7.1.1.3.2.4.2" xref="S4.E6.m2.7.7.1.1.3.2.4.2.cmml">𝐱</mi><mi id="S4.E6.m2.7.7.1.1.3.2.4.3" xref="S4.E6.m2.7.7.1.1.3.2.4.3.cmml">i</mi></msub></mrow><mo id="S4.E6.m2.7.7.1.1.3.1" xref="S4.E6.m2.7.7.1.1.3.1.cmml">+</mo><mrow id="S4.E6.m2.7.7.1.1.3.3" xref="S4.E6.m2.7.7.1.1.3.3.cmml"><msub id="S4.E6.m2.7.7.1.1.3.3.2" xref="S4.E6.m2.7.7.1.1.3.3.2.cmml"><mi id="S4.E6.m2.7.7.1.1.3.3.2.2" xref="S4.E6.m2.7.7.1.1.3.3.2.2.cmml">𝐓</mi><mrow id="S4.E6.m2.4.4.2.4" xref="S4.E6.m2.4.4.2.3.cmml"><mi id="S4.E6.m2.3.3.1.1" xref="S4.E6.m2.3.3.1.1.cmml">T</mi><mo id="S4.E6.m2.4.4.2.4.1" xref="S4.E6.m2.4.4.2.3.cmml">,</mo><mi id="S4.E6.m2.4.4.2.2" xref="S4.E6.m2.4.4.2.2.cmml">k</mi></mrow></msub><mo id="S4.E6.m2.7.7.1.1.3.3.1" xref="S4.E6.m2.7.7.1.1.3.3.1.cmml">⁢</mo><mrow id="S4.E6.m2.7.7.1.1.3.3.3.2" xref="S4.E6.m2.7.7.1.1.3.3.cmml"><mo id="S4.E6.m2.7.7.1.1.3.3.3.2.1" stretchy="false" xref="S4.E6.m2.7.7.1.1.3.3.cmml">(</mo><mi id="S4.E6.m2.6.6" xref="S4.E6.m2.6.6.cmml">t</mi><mo id="S4.E6.m2.7.7.1.1.3.3.3.2.2" stretchy="false" xref="S4.E6.m2.7.7.1.1.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S4.E6.m2.7.7.1.2" xref="S4.E6.m2.7.7.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E6.m2.7b"><apply id="S4.E6.m2.7.7.1.1.cmml" xref="S4.E6.m2.7.7.1"><eq id="S4.E6.m2.7.7.1.1.1.cmml" xref="S4.E6.m2.7.7.1.1.1"></eq><csymbol cd="latexml" 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id="S4.E6.m2.7c">\displaystyle=\mathbf{R}_{T,k}(t)\mathbf{x}_{i}+\mathbf{T}_{T,k}(t),</annotation><annotation encoding="application/x-llamapun" id="S4.E6.m2.7d">= bold_R start_POSTSUBSCRIPT italic_T , italic_k end_POSTSUBSCRIPT ( italic_t ) bold_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + bold_T start_POSTSUBSCRIPT italic_T , italic_k end_POSTSUBSCRIPT ( italic_t ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(6)</td> </tr></tbody> <tbody id="S4.E7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathbf{q}^{\mathtt{world}}_{i}(t)" class="ltx_Math" display="inline" id="S4.E7.m1.1"><semantics id="S4.E7.m1.1a"><mrow id="S4.E7.m1.1.2" xref="S4.E7.m1.1.2.cmml"><msubsup id="S4.E7.m1.1.2.2" xref="S4.E7.m1.1.2.2.cmml"><mi id="S4.E7.m1.1.2.2.2.2" xref="S4.E7.m1.1.2.2.2.2.cmml">𝐪</mi><mi id="S4.E7.m1.1.2.2.3" xref="S4.E7.m1.1.2.2.3.cmml">i</mi><mi id="S4.E7.m1.1.2.2.2.3" xref="S4.E7.m1.1.2.2.2.3.cmml">𝚠𝚘𝚛𝚕𝚍</mi></msubsup><mo id="S4.E7.m1.1.2.1" xref="S4.E7.m1.1.2.1.cmml">⁢</mo><mrow id="S4.E7.m1.1.2.3.2" xref="S4.E7.m1.1.2.cmml"><mo id="S4.E7.m1.1.2.3.2.1" stretchy="false" xref="S4.E7.m1.1.2.cmml">(</mo><mi id="S4.E7.m1.1.1" xref="S4.E7.m1.1.1.cmml">t</mi><mo id="S4.E7.m1.1.2.3.2.2" stretchy="false" xref="S4.E7.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E7.m1.1b"><apply id="S4.E7.m1.1.2.cmml" xref="S4.E7.m1.1.2"><times id="S4.E7.m1.1.2.1.cmml" xref="S4.E7.m1.1.2.1"></times><apply id="S4.E7.m1.1.2.2.cmml" xref="S4.E7.m1.1.2.2"><csymbol cd="ambiguous" id="S4.E7.m1.1.2.2.1.cmml" xref="S4.E7.m1.1.2.2">subscript</csymbol><apply id="S4.E7.m1.1.2.2.2.cmml" xref="S4.E7.m1.1.2.2"><csymbol cd="ambiguous" id="S4.E7.m1.1.2.2.2.1.cmml" xref="S4.E7.m1.1.2.2">superscript</csymbol><ci id="S4.E7.m1.1.2.2.2.2.cmml" xref="S4.E7.m1.1.2.2.2.2">𝐪</ci><ci id="S4.E7.m1.1.2.2.2.3.cmml" xref="S4.E7.m1.1.2.2.2.3">𝚠𝚘𝚛𝚕𝚍</ci></apply><ci id="S4.E7.m1.1.2.2.3.cmml" xref="S4.E7.m1.1.2.2.3">𝑖</ci></apply><ci id="S4.E7.m1.1.1.cmml" xref="S4.E7.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E7.m1.1c">\displaystyle\mathbf{q}^{\mathtt{world}}_{i}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.E7.m1.1d">bold_q start_POSTSUPERSCRIPT typewriter_world end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\text{{RotToQuat}}(\mathbf{R}_{T,k}(t))\mathbf{q}_{i}," class="ltx_Math" display="inline" id="S4.E7.m2.4"><semantics id="S4.E7.m2.4a"><mrow id="S4.E7.m2.4.4.1" xref="S4.E7.m2.4.4.1.1.cmml"><mrow id="S4.E7.m2.4.4.1.1" 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id="S4.E7.m2.2.2.2.2" xref="S4.E7.m2.2.2.2.2.cmml">k</mi></mrow></msub><mo id="S4.E7.m2.4.4.1.1.1.1.1.1.1" xref="S4.E7.m2.4.4.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E7.m2.4.4.1.1.1.1.1.1.3.2" xref="S4.E7.m2.4.4.1.1.1.1.1.1.cmml"><mo id="S4.E7.m2.4.4.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S4.E7.m2.4.4.1.1.1.1.1.1.cmml">(</mo><mi id="S4.E7.m2.3.3" xref="S4.E7.m2.3.3.cmml">t</mi><mo id="S4.E7.m2.4.4.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S4.E7.m2.4.4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E7.m2.4.4.1.1.1.1.1.3" stretchy="false" xref="S4.E7.m2.4.4.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.E7.m2.4.4.1.1.1.2a" xref="S4.E7.m2.4.4.1.1.1.2.cmml">⁢</mo><msub id="S4.E7.m2.4.4.1.1.1.4" xref="S4.E7.m2.4.4.1.1.1.4.cmml"><mi id="S4.E7.m2.4.4.1.1.1.4.2" xref="S4.E7.m2.4.4.1.1.1.4.2.cmml">𝐪</mi><mi id="S4.E7.m2.4.4.1.1.1.4.3" xref="S4.E7.m2.4.4.1.1.1.4.3.cmml">i</mi></msub></mrow></mrow><mo id="S4.E7.m2.4.4.1.2" xref="S4.E7.m2.4.4.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E7.m2.4b"><apply id="S4.E7.m2.4.4.1.1.cmml" xref="S4.E7.m2.4.4.1"><eq id="S4.E7.m2.4.4.1.1.2.cmml" xref="S4.E7.m2.4.4.1.1.2"></eq><csymbol cd="latexml" id="S4.E7.m2.4.4.1.1.3.cmml" xref="S4.E7.m2.4.4.1.1.3">absent</csymbol><apply id="S4.E7.m2.4.4.1.1.1.cmml" xref="S4.E7.m2.4.4.1.1.1"><times id="S4.E7.m2.4.4.1.1.1.2.cmml" xref="S4.E7.m2.4.4.1.1.1.2"></times><ci id="S4.E7.m2.4.4.1.1.1.3a.cmml" xref="S4.E7.m2.4.4.1.1.1.3"><mtext class="ltx_mathvariant_monospace" id="S4.E7.m2.4.4.1.1.1.3.cmml" xref="S4.E7.m2.4.4.1.1.1.3">RotToQuat</mtext></ci><apply id="S4.E7.m2.4.4.1.1.1.1.1.1.cmml" xref="S4.E7.m2.4.4.1.1.1.1.1"><times id="S4.E7.m2.4.4.1.1.1.1.1.1.1.cmml" xref="S4.E7.m2.4.4.1.1.1.1.1.1.1"></times><apply id="S4.E7.m2.4.4.1.1.1.1.1.1.2.cmml" xref="S4.E7.m2.4.4.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.E7.m2.4.4.1.1.1.1.1.1.2.1.cmml" xref="S4.E7.m2.4.4.1.1.1.1.1.1.2">subscript</csymbol><ci id="S4.E7.m2.4.4.1.1.1.1.1.1.2.2.cmml" xref="S4.E7.m2.4.4.1.1.1.1.1.1.2.2">𝐑</ci><list id="S4.E7.m2.2.2.2.3.cmml" xref="S4.E7.m2.2.2.2.4"><ci id="S4.E7.m2.1.1.1.1.cmml" xref="S4.E7.m2.1.1.1.1">𝑇</ci><ci id="S4.E7.m2.2.2.2.2.cmml" xref="S4.E7.m2.2.2.2.2">𝑘</ci></list></apply><ci id="S4.E7.m2.3.3.cmml" xref="S4.E7.m2.3.3">𝑡</ci></apply><apply id="S4.E7.m2.4.4.1.1.1.4.cmml" xref="S4.E7.m2.4.4.1.1.1.4"><csymbol cd="ambiguous" id="S4.E7.m2.4.4.1.1.1.4.1.cmml" xref="S4.E7.m2.4.4.1.1.1.4">subscript</csymbol><ci id="S4.E7.m2.4.4.1.1.1.4.2.cmml" xref="S4.E7.m2.4.4.1.1.1.4.2">𝐪</ci><ci id="S4.E7.m2.4.4.1.1.1.4.3.cmml" xref="S4.E7.m2.4.4.1.1.1.4.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E7.m2.4c">\displaystyle=\text{{RotToQuat}}(\mathbf{R}_{T,k}(t))\mathbf{q}_{i},</annotation><annotation encoding="application/x-llamapun" id="S4.E7.m2.4d">= RotToQuat ( bold_R start_POSTSUBSCRIPT italic_T , italic_k end_POSTSUBSCRIPT ( italic_t ) ) bold_q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(7)</td> </tr></tbody> </table> where <math alttext="\mathbf{R}_{T,k}(t)" class="ltx_Math" display="inline" id="S4.SS1.p4.4.m1.3"><semantics id="S4.SS1.p4.4.m1.3a"><mrow id="S4.SS1.p4.4.m1.3.4" xref="S4.SS1.p4.4.m1.3.4.cmml"><msub id="S4.SS1.p4.4.m1.3.4.2" xref="S4.SS1.p4.4.m1.3.4.2.cmml"><mi id="S4.SS1.p4.4.m1.3.4.2.2" xref="S4.SS1.p4.4.m1.3.4.2.2.cmml">𝐑</mi><mrow id="S4.SS1.p4.4.m1.2.2.2.4" xref="S4.SS1.p4.4.m1.2.2.2.3.cmml"><mi id="S4.SS1.p4.4.m1.1.1.1.1" xref="S4.SS1.p4.4.m1.1.1.1.1.cmml">T</mi><mo id="S4.SS1.p4.4.m1.2.2.2.4.1" xref="S4.SS1.p4.4.m1.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p4.4.m1.2.2.2.2" xref="S4.SS1.p4.4.m1.2.2.2.2.cmml">k</mi></mrow></msub><mo id="S4.SS1.p4.4.m1.3.4.1" xref="S4.SS1.p4.4.m1.3.4.1.cmml">⁢</mo><mrow id="S4.SS1.p4.4.m1.3.4.3.2" xref="S4.SS1.p4.4.m1.3.4.cmml"><mo id="S4.SS1.p4.4.m1.3.4.3.2.1" stretchy="false" xref="S4.SS1.p4.4.m1.3.4.cmml">(</mo><mi id="S4.SS1.p4.4.m1.3.3" xref="S4.SS1.p4.4.m1.3.3.cmml">t</mi><mo id="S4.SS1.p4.4.m1.3.4.3.2.2" stretchy="false" xref="S4.SS1.p4.4.m1.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.4.m1.3b"><apply id="S4.SS1.p4.4.m1.3.4.cmml" xref="S4.SS1.p4.4.m1.3.4"><times id="S4.SS1.p4.4.m1.3.4.1.cmml" xref="S4.SS1.p4.4.m1.3.4.1"></times><apply id="S4.SS1.p4.4.m1.3.4.2.cmml" xref="S4.SS1.p4.4.m1.3.4.2"><csymbol cd="ambiguous" id="S4.SS1.p4.4.m1.3.4.2.1.cmml" xref="S4.SS1.p4.4.m1.3.4.2">subscript</csymbol><ci id="S4.SS1.p4.4.m1.3.4.2.2.cmml" xref="S4.SS1.p4.4.m1.3.4.2.2">𝐑</ci><list id="S4.SS1.p4.4.m1.2.2.2.3.cmml" xref="S4.SS1.p4.4.m1.2.2.2.4"><ci id="S4.SS1.p4.4.m1.1.1.1.1.cmml" xref="S4.SS1.p4.4.m1.1.1.1.1">𝑇</ci><ci id="S4.SS1.p4.4.m1.2.2.2.2.cmml" xref="S4.SS1.p4.4.m1.2.2.2.2">𝑘</ci></list></apply><ci id="S4.SS1.p4.4.m1.3.3.cmml" xref="S4.SS1.p4.4.m1.3.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.4.m1.3c">\mathbf{R}_{T,k}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.4.m1.3d">bold_R start_POSTSUBSCRIPT italic_T , italic_k end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> and <math alttext="\mathbf{T}_{T,k}(t)" class="ltx_Math" display="inline" id="S4.SS1.p4.5.m2.3"><semantics id="S4.SS1.p4.5.m2.3a"><mrow id="S4.SS1.p4.5.m2.3.4" xref="S4.SS1.p4.5.m2.3.4.cmml"><msub id="S4.SS1.p4.5.m2.3.4.2" xref="S4.SS1.p4.5.m2.3.4.2.cmml"><mi id="S4.SS1.p4.5.m2.3.4.2.2" xref="S4.SS1.p4.5.m2.3.4.2.2.cmml">𝐓</mi><mrow id="S4.SS1.p4.5.m2.2.2.2.4" xref="S4.SS1.p4.5.m2.2.2.2.3.cmml"><mi id="S4.SS1.p4.5.m2.1.1.1.1" xref="S4.SS1.p4.5.m2.1.1.1.1.cmml">T</mi><mo id="S4.SS1.p4.5.m2.2.2.2.4.1" xref="S4.SS1.p4.5.m2.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p4.5.m2.2.2.2.2" xref="S4.SS1.p4.5.m2.2.2.2.2.cmml">k</mi></mrow></msub><mo id="S4.SS1.p4.5.m2.3.4.1" xref="S4.SS1.p4.5.m2.3.4.1.cmml">⁢</mo><mrow id="S4.SS1.p4.5.m2.3.4.3.2" xref="S4.SS1.p4.5.m2.3.4.cmml"><mo id="S4.SS1.p4.5.m2.3.4.3.2.1" stretchy="false" xref="S4.SS1.p4.5.m2.3.4.cmml">(</mo><mi id="S4.SS1.p4.5.m2.3.3" xref="S4.SS1.p4.5.m2.3.3.cmml">t</mi><mo id="S4.SS1.p4.5.m2.3.4.3.2.2" stretchy="false" xref="S4.SS1.p4.5.m2.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.5.m2.3b"><apply id="S4.SS1.p4.5.m2.3.4.cmml" xref="S4.SS1.p4.5.m2.3.4"><times id="S4.SS1.p4.5.m2.3.4.1.cmml" xref="S4.SS1.p4.5.m2.3.4.1"></times><apply id="S4.SS1.p4.5.m2.3.4.2.cmml" xref="S4.SS1.p4.5.m2.3.4.2"><csymbol cd="ambiguous" id="S4.SS1.p4.5.m2.3.4.2.1.cmml" xref="S4.SS1.p4.5.m2.3.4.2">subscript</csymbol><ci id="S4.SS1.p4.5.m2.3.4.2.2.cmml" xref="S4.SS1.p4.5.m2.3.4.2.2">𝐓</ci><list id="S4.SS1.p4.5.m2.2.2.2.3.cmml" xref="S4.SS1.p4.5.m2.2.2.2.4"><ci id="S4.SS1.p4.5.m2.1.1.1.1.cmml" xref="S4.SS1.p4.5.m2.1.1.1.1">𝑇</ci><ci id="S4.SS1.p4.5.m2.2.2.2.2.cmml" xref="S4.SS1.p4.5.m2.2.2.2.2">𝑘</ci></list></apply><ci id="S4.SS1.p4.5.m2.3.3.cmml" xref="S4.SS1.p4.5.m2.3.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.5.m2.3c">\mathbf{T}_{T,k}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.5.m2.3d">bold_T start_POSTSUBSCRIPT italic_T , italic_k end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> are the rotation matrix and translation of the pose transform of the transient node over time, while <math alttext="\text{{RotToQuat}}(\cdot)" class="ltx_Math" display="inline" id="S4.SS1.p4.6.m3.1"><semantics id="S4.SS1.p4.6.m3.1a"><mrow id="S4.SS1.p4.6.m3.1.2" xref="S4.SS1.p4.6.m3.1.2.cmml"><mtext class="ltx_mathvariant_monospace" id="S4.SS1.p4.6.m3.1.2.2" xref="S4.SS1.p4.6.m3.1.2.2a.cmml">RotToQuat</mtext><mo id="S4.SS1.p4.6.m3.1.2.1" xref="S4.SS1.p4.6.m3.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.p4.6.m3.1.2.3.2" xref="S4.SS1.p4.6.m3.1.2.cmml"><mo id="S4.SS1.p4.6.m3.1.2.3.2.1" stretchy="false" xref="S4.SS1.p4.6.m3.1.2.cmml">(</mo><mo id="S4.SS1.p4.6.m3.1.1" lspace="0em" rspace="0em" xref="S4.SS1.p4.6.m3.1.1.cmml">⋅</mo><mo id="S4.SS1.p4.6.m3.1.2.3.2.2" stretchy="false" xref="S4.SS1.p4.6.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.6.m3.1b"><apply id="S4.SS1.p4.6.m3.1.2.cmml" xref="S4.SS1.p4.6.m3.1.2"><times id="S4.SS1.p4.6.m3.1.2.1.cmml" xref="S4.SS1.p4.6.m3.1.2.1"></times><ci id="S4.SS1.p4.6.m3.1.2.2a.cmml" xref="S4.SS1.p4.6.m3.1.2.2"><mtext class="ltx_mathvariant_monospace" id="S4.SS1.p4.6.m3.1.2.2.cmml" xref="S4.SS1.p4.6.m3.1.2.2">RotToQuat</mtext></ci><ci id="S4.SS1.p4.6.m3.1.1.cmml" xref="S4.SS1.p4.6.m3.1.1">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.6.m3.1c">\text{{RotToQuat}}(\cdot)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.6.m3.1d">RotToQuat ( ⋅ )</annotation></semantics></math> converts a rotation matrix into its corresponding quaternion. </div> <div class="ltx_para" id="S4.SS1.p5"> To prevent transient nodes from using floaters to overfit the backgrounds, an out-of-box loss is also introduced as: <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A2.EGx4"> <tbody id="S4.E8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{L}_{\text{oob}}=-\frac{1}{|\mathcal{G}^{\mathtt{oob}}_{T% ,k}|}\sum_{G_{i}\in\mathcal{G}_{k,T}^{\mathtt{oob}}}\log\left(1-\alpha_{i}% \right)," class="ltx_Math" display="inline" id="S4.E8.m1.7"><semantics id="S4.E8.m1.7a"><mrow id="S4.E8.m1.7.7.1" xref="S4.E8.m1.7.7.1.1.cmml"><mrow id="S4.E8.m1.7.7.1.1" xref="S4.E8.m1.7.7.1.1.cmml"><msub id="S4.E8.m1.7.7.1.1.3" xref="S4.E8.m1.7.7.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E8.m1.7.7.1.1.3.2" xref="S4.E8.m1.7.7.1.1.3.2.cmml">ℒ</mi><mtext id="S4.E8.m1.7.7.1.1.3.3" 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xref="S4.E8.m1.7.7.1.1.1.1.1.1.1.1.1.3.3">𝑖</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E8.m1.7c">\displaystyle\mathcal{L}_{\text{oob}}=-\frac{1}{|\mathcal{G}^{\mathtt{oob}}_{T% ,k}|}\sum_{G_{i}\in\mathcal{G}_{k,T}^{\mathtt{oob}}}\log\left(1-\alpha_{i}% \right),</annotation><annotation encoding="application/x-llamapun" id="S4.E8.m1.7d">caligraphic_L start_POSTSUBSCRIPT oob end_POSTSUBSCRIPT = - divide start_ARG 1 end_ARG start_ARG | caligraphic_G start_POSTSUPERSCRIPT typewriter_oob end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_T , italic_k end_POSTSUBSCRIPT | end_ARG ∑ start_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_G start_POSTSUBSCRIPT italic_k , italic_T end_POSTSUBSCRIPT start_POSTSUPERSCRIPT typewriter_oob end_POSTSUPERSCRIPT end_POSTSUBSCRIPT roman_log ( 1 - italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(8)</td> </tr></tbody> </table> where <math alttext="\mathcal{G}^{\mathtt{oob}}_{T,k}" class="ltx_Math" display="inline" id="S4.SS1.p5.1.m1.2"><semantics id="S4.SS1.p5.1.m1.2a"><msubsup id="S4.SS1.p5.1.m1.2.3" xref="S4.SS1.p5.1.m1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p5.1.m1.2.3.2.2" xref="S4.SS1.p5.1.m1.2.3.2.2.cmml">𝒢</mi><mrow id="S4.SS1.p5.1.m1.2.2.2.4" xref="S4.SS1.p5.1.m1.2.2.2.3.cmml"><mi id="S4.SS1.p5.1.m1.1.1.1.1" xref="S4.SS1.p5.1.m1.1.1.1.1.cmml">T</mi><mo id="S4.SS1.p5.1.m1.2.2.2.4.1" xref="S4.SS1.p5.1.m1.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p5.1.m1.2.2.2.2" xref="S4.SS1.p5.1.m1.2.2.2.2.cmml">k</mi></mrow><mi id="S4.SS1.p5.1.m1.2.3.2.3" xref="S4.SS1.p5.1.m1.2.3.2.3.cmml">𝚘𝚘𝚋</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.1.m1.2b"><apply id="S4.SS1.p5.1.m1.2.3.cmml" xref="S4.SS1.p5.1.m1.2.3"><csymbol 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is the set of Gaussians whose distance from the origin in the local coordinates is larger than <math alttext="\frac{1}{2}S_{T,k}^{\mathtt{tsnt}}+\theta_{\mathtt{tol}}" class="ltx_Math" display="inline" id="S4.SS1.p5.2.m2.2"><semantics id="S4.SS1.p5.2.m2.2a"><mrow id="S4.SS1.p5.2.m2.2.3" xref="S4.SS1.p5.2.m2.2.3.cmml"><mrow id="S4.SS1.p5.2.m2.2.3.2" xref="S4.SS1.p5.2.m2.2.3.2.cmml"><mfrac id="S4.SS1.p5.2.m2.2.3.2.2" xref="S4.SS1.p5.2.m2.2.3.2.2.cmml"><mn id="S4.SS1.p5.2.m2.2.3.2.2.2" xref="S4.SS1.p5.2.m2.2.3.2.2.2.cmml">1</mn><mn id="S4.SS1.p5.2.m2.2.3.2.2.3" xref="S4.SS1.p5.2.m2.2.3.2.2.3.cmml">2</mn></mfrac><mo id="S4.SS1.p5.2.m2.2.3.2.1" xref="S4.SS1.p5.2.m2.2.3.2.1.cmml">⁢</mo><msubsup id="S4.SS1.p5.2.m2.2.3.2.3" xref="S4.SS1.p5.2.m2.2.3.2.3.cmml"><mi id="S4.SS1.p5.2.m2.2.3.2.3.2.2" xref="S4.SS1.p5.2.m2.2.3.2.3.2.2.cmml">S</mi><mrow id="S4.SS1.p5.2.m2.2.2.2.4" xref="S4.SS1.p5.2.m2.2.2.2.3.cmml"><mi id="S4.SS1.p5.2.m2.1.1.1.1" xref="S4.SS1.p5.2.m2.1.1.1.1.cmml">T</mi><mo 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xref="S4.SS1.p5.2.m2.2.3.2.3.3">𝚝𝚜𝚗𝚝</ci></apply></apply><apply id="S4.SS1.p5.2.m2.2.3.3.cmml" xref="S4.SS1.p5.2.m2.2.3.3"><csymbol cd="ambiguous" id="S4.SS1.p5.2.m2.2.3.3.1.cmml" xref="S4.SS1.p5.2.m2.2.3.3">subscript</csymbol><ci id="S4.SS1.p5.2.m2.2.3.3.2.cmml" xref="S4.SS1.p5.2.m2.2.3.3.2">𝜃</ci><ci id="S4.SS1.p5.2.m2.2.3.3.3.cmml" xref="S4.SS1.p5.2.m2.2.3.3.3">𝚝𝚘𝚕</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.2.m2.2c">\frac{1}{2}S_{T,k}^{\mathtt{tsnt}}+\theta_{\mathtt{tol}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.2.m2.2d">divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_S start_POSTSUBSCRIPT italic_T , italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT typewriter_tsnt end_POSTSUPERSCRIPT + italic_θ start_POSTSUBSCRIPT typewriter_tol end_POSTSUBSCRIPT</annotation></semantics></math>. Here, <math alttext="\theta_{\mathtt{tol}}" class="ltx_Math" display="inline" id="S4.SS1.p5.3.m3.1"><semantics id="S4.SS1.p5.3.m3.1a"><msub id="S4.SS1.p5.3.m3.1.1" xref="S4.SS1.p5.3.m3.1.1.cmml"><mi id="S4.SS1.p5.3.m3.1.1.2" xref="S4.SS1.p5.3.m3.1.1.2.cmml">θ</mi><mi id="S4.SS1.p5.3.m3.1.1.3" xref="S4.SS1.p5.3.m3.1.1.3.cmml">𝚝𝚘𝚕</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.3.m3.1b"><apply id="S4.SS1.p5.3.m3.1.1.cmml" xref="S4.SS1.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS1.p5.3.m3.1.1.1.cmml" xref="S4.SS1.p5.3.m3.1.1">subscript</csymbol><ci id="S4.SS1.p5.3.m3.1.1.2.cmml" xref="S4.SS1.p5.3.m3.1.1.2">𝜃</ci><ci id="S4.SS1.p5.3.m3.1.1.3.cmml" xref="S4.SS1.p5.3.m3.1.1.3">𝚝𝚘𝚕</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.3.m3.1c">\theta_{\mathtt{tol}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.3.m3.1d">italic_θ start_POSTSUBSCRIPT typewriter_tol end_POSTSUBSCRIPT</annotation></semantics></math> acts as a tolerance threshold so that the shadow of the foreground is contained in the transient node. </div> <div class="ltx_para ltx_noindent" id="S4.SS1.p6"> Initialization. We initialize the scene graph structure with automatically labeled 3D bounding bounding boxes from the dataset <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib16" title="">16</a>]</cite>. From 3D boxes, we get transient nodes along with their sizes and transformations of poses over time. Gaussian points are initialized from aggregated LiDAR point clouds, with background and transient objects separated. Additionally, we employ point triangulation to initialize far-away Gaussians and randomly sample points on a semisphere to initialize Gaussians representing the sky. </div> <div class="ltx_para ltx_noindent" id="S4.SS1.p7"> Scene decomposition. We observe that reconstructing transient objects with such subgraph design, rather than simply masking them out, leads to better static reconstruction by preventing the background from overfitting on shadows of transients. Moreover, by this design, all transient objects, not just dynamic ones, can be decomposed from the background and are clearly reconstructed. For example, parked vehicles can be decoupled, as shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#S2.F2" title="Figure 2 ‣ 2 Related Work ‣ MTGS: Multi-Traversal Gaussian Splatting">2</a>. </div> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> 4.2 Appearance Modeling</h3> <div class="ltx_para" id="S4.SS2.p1"> In the multi-traversal setting, appearance modeling is two-fold, the alignment within a traversal and the appearance tuning across multiple traversals. For appearance tuning, we propose appearance nodes in the scene graph to adjust the appearance of backgrounds (See <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#S4.SS1" title="4.1 Multi-Traversal Scene Graph ‣ 4 Multi-Traversal Gaussian Splatting ‣ MTGS: Multi-Traversal Gaussian Splatting">Sec. 4.1</a>). For alignment within traversals, we introduce LiDAR-guided exposure alignment and learnable per-camera affine transforms. </div> <div class="ltx_para ltx_noindent" id="S4.SS2.p2"> LiDAR-Guided Exposure Alignment. Images might vary in exposure due to various lighting. To align the exposure within images taken by different cameras simultaneously at time <math alttext="t" class="ltx_Math" display="inline" id="S4.SS2.p2.1.m1.1"><semantics id="S4.SS2.p2.1.m1.1a"><mi id="S4.SS2.p2.1.m1.1.1" xref="S4.SS2.p2.1.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.1.m1.1b"><ci id="S4.SS2.p2.1.m1.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.1.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.1.m1.1d">italic_t</annotation></semantics></math>, we project colored LiDAR points at <math alttext="t" class="ltx_Math" display="inline" id="S4.SS2.p2.2.m2.1"><semantics id="S4.SS2.p2.2.m2.1a"><mi id="S4.SS2.p2.2.m2.1.1" xref="S4.SS2.p2.2.m2.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.2.m2.1b"><ci id="S4.SS2.p2.2.m2.1.1.cmml" xref="S4.SS2.p2.2.m2.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.2.m2.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.2.m2.1d">italic_t</annotation></semantics></math> into these images and adjust the exposure so that pixels corresponding to the same LiDAR point are of the same color. </div> <div class="ltx_para ltx_noindent" id="S4.SS2.p3"> Learnable Per-Camera Affine Transforms. To enhance consistency between images taken at different time within one traversal, a per-camera affine transform <math alttext="\text{{Aff}}(\cdot)" class="ltx_Math" display="inline" id="S4.SS2.p3.1.m1.1"><semantics id="S4.SS2.p3.1.m1.1a"><mrow id="S4.SS2.p3.1.m1.1.2" xref="S4.SS2.p3.1.m1.1.2.cmml"><mtext class="ltx_mathvariant_monospace" id="S4.SS2.p3.1.m1.1.2.2" xref="S4.SS2.p3.1.m1.1.2.2a.cmml">Aff</mtext><mo id="S4.SS2.p3.1.m1.1.2.1" xref="S4.SS2.p3.1.m1.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p3.1.m1.1.2.3.2" xref="S4.SS2.p3.1.m1.1.2.cmml"><mo id="S4.SS2.p3.1.m1.1.2.3.2.1" stretchy="false" xref="S4.SS2.p3.1.m1.1.2.cmml">(</mo><mo id="S4.SS2.p3.1.m1.1.1" lspace="0em" rspace="0em" xref="S4.SS2.p3.1.m1.1.1.cmml">⋅</mo><mo id="S4.SS2.p3.1.m1.1.2.3.2.2" stretchy="false" xref="S4.SS2.p3.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.1.m1.1b"><apply id="S4.SS2.p3.1.m1.1.2.cmml" xref="S4.SS2.p3.1.m1.1.2"><times id="S4.SS2.p3.1.m1.1.2.1.cmml" xref="S4.SS2.p3.1.m1.1.2.1"></times><ci id="S4.SS2.p3.1.m1.1.2.2a.cmml" xref="S4.SS2.p3.1.m1.1.2.2"><mtext class="ltx_mathvariant_monospace" id="S4.SS2.p3.1.m1.1.2.2.cmml" xref="S4.SS2.p3.1.m1.1.2.2">Aff</mtext></ci><ci id="S4.SS2.p3.1.m1.1.1.cmml" xref="S4.SS2.p3.1.m1.1.1">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.1.m1.1c">\text{{Aff}}(\cdot)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.1.m1.1d">Aff ( ⋅ )</annotation></semantics></math> is attached to refine the color tone, brightness, contrast, and exposure of image <math alttext="\mathbf{I}_{\text{idx}}\in\mathcal{I}" class="ltx_Math" display="inline" id="S4.SS2.p3.2.m2.1"><semantics id="S4.SS2.p3.2.m2.1a"><mrow id="S4.SS2.p3.2.m2.1.1" xref="S4.SS2.p3.2.m2.1.1.cmml"><msub id="S4.SS2.p3.2.m2.1.1.2" xref="S4.SS2.p3.2.m2.1.1.2.cmml"><mi id="S4.SS2.p3.2.m2.1.1.2.2" xref="S4.SS2.p3.2.m2.1.1.2.2.cmml">𝐈</mi><mtext id="S4.SS2.p3.2.m2.1.1.2.3" 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id="S4.SS2.p3.2.m2.1c">\mathbf{I}_{\text{idx}}\in\mathcal{I}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.2.m2.1d">bold_I start_POSTSUBSCRIPT idx end_POSTSUBSCRIPT ∈ caligraphic_I</annotation></semantics></math> by: <table class="ltx_equation ltx_eqn_table" id="S4.E9"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\text{{Aff}}(\mathbf{I})=\mathbf{W}_{\text{idx}}\mathbf{I}+\mathbf{b}_{\text{% idx}}." class="ltx_Math" display="block" id="S4.E9.m1.2"><semantics id="S4.E9.m1.2a"><mrow id="S4.E9.m1.2.2.1" xref="S4.E9.m1.2.2.1.1.cmml"><mrow id="S4.E9.m1.2.2.1.1" xref="S4.E9.m1.2.2.1.1.cmml"><mrow id="S4.E9.m1.2.2.1.1.2" xref="S4.E9.m1.2.2.1.1.2.cmml"><mtext class="ltx_mathvariant_monospace" id="S4.E9.m1.2.2.1.1.2.2" xref="S4.E9.m1.2.2.1.1.2.2a.cmml">Aff</mtext><mo id="S4.E9.m1.2.2.1.1.2.1" xref="S4.E9.m1.2.2.1.1.2.1.cmml">⁢</mo><mrow id="S4.E9.m1.2.2.1.1.2.3.2" xref="S4.E9.m1.2.2.1.1.2.cmml"><mo id="S4.E9.m1.2.2.1.1.2.3.2.1" stretchy="false" xref="S4.E9.m1.2.2.1.1.2.cmml">(</mo><mi id="S4.E9.m1.1.1" xref="S4.E9.m1.1.1.cmml">𝐈</mi><mo id="S4.E9.m1.2.2.1.1.2.3.2.2" stretchy="false" xref="S4.E9.m1.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.E9.m1.2.2.1.1.1" xref="S4.E9.m1.2.2.1.1.1.cmml">=</mo><mrow id="S4.E9.m1.2.2.1.1.3" xref="S4.E9.m1.2.2.1.1.3.cmml"><mrow id="S4.E9.m1.2.2.1.1.3.2" xref="S4.E9.m1.2.2.1.1.3.2.cmml"><msub id="S4.E9.m1.2.2.1.1.3.2.2" xref="S4.E9.m1.2.2.1.1.3.2.2.cmml"><mi id="S4.E9.m1.2.2.1.1.3.2.2.2" xref="S4.E9.m1.2.2.1.1.3.2.2.2.cmml">𝐖</mi><mtext id="S4.E9.m1.2.2.1.1.3.2.2.3" xref="S4.E9.m1.2.2.1.1.3.2.2.3a.cmml">idx</mtext></msub><mo id="S4.E9.m1.2.2.1.1.3.2.1" xref="S4.E9.m1.2.2.1.1.3.2.1.cmml">⁢</mo><mi id="S4.E9.m1.2.2.1.1.3.2.3" xref="S4.E9.m1.2.2.1.1.3.2.3.cmml">𝐈</mi></mrow><mo id="S4.E9.m1.2.2.1.1.3.1" xref="S4.E9.m1.2.2.1.1.3.1.cmml">+</mo><msub id="S4.E9.m1.2.2.1.1.3.3" 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id="S4.E9.m1.2.2.1.1.3.2.cmml" xref="S4.E9.m1.2.2.1.1.3.2"><times id="S4.E9.m1.2.2.1.1.3.2.1.cmml" xref="S4.E9.m1.2.2.1.1.3.2.1"></times><apply id="S4.E9.m1.2.2.1.1.3.2.2.cmml" xref="S4.E9.m1.2.2.1.1.3.2.2"><csymbol cd="ambiguous" id="S4.E9.m1.2.2.1.1.3.2.2.1.cmml" xref="S4.E9.m1.2.2.1.1.3.2.2">subscript</csymbol><ci id="S4.E9.m1.2.2.1.1.3.2.2.2.cmml" xref="S4.E9.m1.2.2.1.1.3.2.2.2">𝐖</ci><ci id="S4.E9.m1.2.2.1.1.3.2.2.3a.cmml" xref="S4.E9.m1.2.2.1.1.3.2.2.3"><mtext id="S4.E9.m1.2.2.1.1.3.2.2.3.cmml" mathsize="70%" xref="S4.E9.m1.2.2.1.1.3.2.2.3">idx</mtext></ci></apply><ci id="S4.E9.m1.2.2.1.1.3.2.3.cmml" xref="S4.E9.m1.2.2.1.1.3.2.3">𝐈</ci></apply><apply id="S4.E9.m1.2.2.1.1.3.3.cmml" xref="S4.E9.m1.2.2.1.1.3.3"><csymbol cd="ambiguous" id="S4.E9.m1.2.2.1.1.3.3.1.cmml" xref="S4.E9.m1.2.2.1.1.3.3">subscript</csymbol><ci id="S4.E9.m1.2.2.1.1.3.3.2.cmml" xref="S4.E9.m1.2.2.1.1.3.3.2">𝐛</ci><ci id="S4.E9.m1.2.2.1.1.3.3.3a.cmml" xref="S4.E9.m1.2.2.1.1.3.3.3"><mtext id="S4.E9.m1.2.2.1.1.3.3.3.cmml" mathsize="70%" xref="S4.E9.m1.2.2.1.1.3.3.3">idx</mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E9.m1.2c">\text{{Aff}}(\mathbf{I})=\mathbf{W}_{\text{idx}}\mathbf{I}+\mathbf{b}_{\text{% idx}}.</annotation><annotation encoding="application/x-llamapun" id="S4.E9.m1.2d">Aff ( bold_I ) = bold_W start_POSTSUBSCRIPT idx end_POSTSUBSCRIPT bold_I + bold_b start_POSTSUBSCRIPT idx end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(9)</td> </tr></tbody> </table> Note that learnable <math alttext="\mathbf{W}_{\text{idx}}\in\mathbb{R}^{3\times 3}" class="ltx_Math" display="inline" id="S4.SS2.p3.3.m1.1"><semantics id="S4.SS2.p3.3.m1.1a"><mrow id="S4.SS2.p3.3.m1.1.1" xref="S4.SS2.p3.3.m1.1.1.cmml"><msub id="S4.SS2.p3.3.m1.1.1.2" xref="S4.SS2.p3.3.m1.1.1.2.cmml"><mi id="S4.SS2.p3.3.m1.1.1.2.2" xref="S4.SS2.p3.3.m1.1.1.2.2.cmml">𝐖</mi><mtext id="S4.SS2.p3.3.m1.1.1.2.3" xref="S4.SS2.p3.3.m1.1.1.2.3a.cmml">idx</mtext></msub><mo id="S4.SS2.p3.3.m1.1.1.1" xref="S4.SS2.p3.3.m1.1.1.1.cmml">∈</mo><msup id="S4.SS2.p3.3.m1.1.1.3" xref="S4.SS2.p3.3.m1.1.1.3.cmml"><mi id="S4.SS2.p3.3.m1.1.1.3.2" xref="S4.SS2.p3.3.m1.1.1.3.2.cmml">ℝ</mi><mrow id="S4.SS2.p3.3.m1.1.1.3.3" xref="S4.SS2.p3.3.m1.1.1.3.3.cmml"><mn id="S4.SS2.p3.3.m1.1.1.3.3.2" xref="S4.SS2.p3.3.m1.1.1.3.3.2.cmml">3</mn><mo id="S4.SS2.p3.3.m1.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S4.SS2.p3.3.m1.1.1.3.3.1.cmml">×</mo><mn id="S4.SS2.p3.3.m1.1.1.3.3.3" xref="S4.SS2.p3.3.m1.1.1.3.3.3.cmml">3</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.3.m1.1b"><apply id="S4.SS2.p3.3.m1.1.1.cmml" xref="S4.SS2.p3.3.m1.1.1"><in id="S4.SS2.p3.3.m1.1.1.1.cmml" xref="S4.SS2.p3.3.m1.1.1.1"></in><apply id="S4.SS2.p3.3.m1.1.1.2.cmml" xref="S4.SS2.p3.3.m1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p3.3.m1.1.1.2.1.cmml" xref="S4.SS2.p3.3.m1.1.1.2">subscript</csymbol><ci id="S4.SS2.p3.3.m1.1.1.2.2.cmml" xref="S4.SS2.p3.3.m1.1.1.2.2">𝐖</ci><ci id="S4.SS2.p3.3.m1.1.1.2.3a.cmml" xref="S4.SS2.p3.3.m1.1.1.2.3"><mtext id="S4.SS2.p3.3.m1.1.1.2.3.cmml" mathsize="70%" xref="S4.SS2.p3.3.m1.1.1.2.3">idx</mtext></ci></apply><apply id="S4.SS2.p3.3.m1.1.1.3.cmml" xref="S4.SS2.p3.3.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p3.3.m1.1.1.3.1.cmml" xref="S4.SS2.p3.3.m1.1.1.3">superscript</csymbol><ci id="S4.SS2.p3.3.m1.1.1.3.2.cmml" xref="S4.SS2.p3.3.m1.1.1.3.2">ℝ</ci><apply id="S4.SS2.p3.3.m1.1.1.3.3.cmml" xref="S4.SS2.p3.3.m1.1.1.3.3"><times id="S4.SS2.p3.3.m1.1.1.3.3.1.cmml" xref="S4.SS2.p3.3.m1.1.1.3.3.1"></times><cn id="S4.SS2.p3.3.m1.1.1.3.3.2.cmml" type="integer" xref="S4.SS2.p3.3.m1.1.1.3.3.2">3</cn><cn id="S4.SS2.p3.3.m1.1.1.3.3.3.cmml" type="integer" xref="S4.SS2.p3.3.m1.1.1.3.3.3">3</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.3.m1.1c">\mathbf{W}_{\text{idx}}\in\mathbb{R}^{3\times 3}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.3.m1.1d">bold_W start_POSTSUBSCRIPT idx end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 3 × 3 end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\mathbf{b}_{\text{idx}}\in\mathbb{R}^{3}" class="ltx_Math" display="inline" id="S4.SS2.p3.4.m2.1"><semantics id="S4.SS2.p3.4.m2.1a"><mrow id="S4.SS2.p3.4.m2.1.1" xref="S4.SS2.p3.4.m2.1.1.cmml"><msub id="S4.SS2.p3.4.m2.1.1.2" xref="S4.SS2.p3.4.m2.1.1.2.cmml"><mi id="S4.SS2.p3.4.m2.1.1.2.2" xref="S4.SS2.p3.4.m2.1.1.2.2.cmml">𝐛</mi><mtext id="S4.SS2.p3.4.m2.1.1.2.3" xref="S4.SS2.p3.4.m2.1.1.2.3a.cmml">idx</mtext></msub><mo id="S4.SS2.p3.4.m2.1.1.1" xref="S4.SS2.p3.4.m2.1.1.1.cmml">∈</mo><msup id="S4.SS2.p3.4.m2.1.1.3" xref="S4.SS2.p3.4.m2.1.1.3.cmml"><mi id="S4.SS2.p3.4.m2.1.1.3.2" xref="S4.SS2.p3.4.m2.1.1.3.2.cmml">ℝ</mi><mn id="S4.SS2.p3.4.m2.1.1.3.3" xref="S4.SS2.p3.4.m2.1.1.3.3.cmml">3</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.4.m2.1b"><apply id="S4.SS2.p3.4.m2.1.1.cmml" xref="S4.SS2.p3.4.m2.1.1"><in id="S4.SS2.p3.4.m2.1.1.1.cmml" xref="S4.SS2.p3.4.m2.1.1.1"></in><apply id="S4.SS2.p3.4.m2.1.1.2.cmml" xref="S4.SS2.p3.4.m2.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p3.4.m2.1.1.2.1.cmml" xref="S4.SS2.p3.4.m2.1.1.2">subscript</csymbol><ci id="S4.SS2.p3.4.m2.1.1.2.2.cmml" xref="S4.SS2.p3.4.m2.1.1.2.2">𝐛</ci><ci id="S4.SS2.p3.4.m2.1.1.2.3a.cmml" xref="S4.SS2.p3.4.m2.1.1.2.3"><mtext id="S4.SS2.p3.4.m2.1.1.2.3.cmml" mathsize="70%" xref="S4.SS2.p3.4.m2.1.1.2.3">idx</mtext></ci></apply><apply id="S4.SS2.p3.4.m2.1.1.3.cmml" xref="S4.SS2.p3.4.m2.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p3.4.m2.1.1.3.1.cmml" xref="S4.SS2.p3.4.m2.1.1.3">superscript</csymbol><ci id="S4.SS2.p3.4.m2.1.1.3.2.cmml" xref="S4.SS2.p3.4.m2.1.1.3.2">ℝ</ci><cn id="S4.SS2.p3.4.m2.1.1.3.3.cmml" type="integer" xref="S4.SS2.p3.4.m2.1.1.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.4.m2.1c">\mathbf{b}_{\text{idx}}\in\mathbb{R}^{3}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.4.m2.1d">bold_b start_POSTSUBSCRIPT idx end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT</annotation></semantics></math> are image-wise, i.e., different parameters for different images. </div> </section> <section class="ltx_subsection" id="S4.SS3"> <h3 class="ltx_title ltx_title_subsection"> 4.3 Regularization and Training</h3> <div class="ltx_para" id="S4.SS3.p1"> To achieve high-quality 3D reconstruction and ensure consistency in geometry, we introduce two types of regularization: depth regularization and normal regularization. </div> <div class="ltx_para ltx_noindent" id="S4.SS3.p2"> Patch-wise LiDAR Depth Loss. The LiDAR depth loss contains an inverse L1 loss and a patch-wise normalized cross-correlation loss. We project sparse LiDAR points into the image plane to obtain sparse LiDAR depth as ground truth. The loss function for this regularization is defined as: <table class="ltx_equation ltx_eqn_table" id="S4.E10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{L}_{\text{depth}}=\left|\frac{1}{d_{\text{pred}}}-\frac{1}{d_{\text{% LiDAR}}}\right|," class="ltx_Math" display="block" id="S4.E10.m1.1"><semantics id="S4.E10.m1.1a"><mrow id="S4.E10.m1.1.1.1" xref="S4.E10.m1.1.1.1.1.cmml"><mrow id="S4.E10.m1.1.1.1.1" xref="S4.E10.m1.1.1.1.1.cmml"><msub id="S4.E10.m1.1.1.1.1.3" xref="S4.E10.m1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E10.m1.1.1.1.1.3.2" xref="S4.E10.m1.1.1.1.1.3.2.cmml">ℒ</mi><mtext id="S4.E10.m1.1.1.1.1.3.3" xref="S4.E10.m1.1.1.1.1.3.3a.cmml">depth</mtext></msub><mo id="S4.E10.m1.1.1.1.1.2" xref="S4.E10.m1.1.1.1.1.2.cmml">=</mo><mrow id="S4.E10.m1.1.1.1.1.1.1" xref="S4.E10.m1.1.1.1.1.1.2.cmml"><mo 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id="S4.E10.m1.1d">caligraphic_L start_POSTSUBSCRIPT depth end_POSTSUBSCRIPT = | divide start_ARG 1 end_ARG start_ARG italic_d start_POSTSUBSCRIPT pred end_POSTSUBSCRIPT end_ARG - divide start_ARG 1 end_ARG start_ARG italic_d start_POSTSUBSCRIPT LiDAR end_POSTSUBSCRIPT end_ARG | ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(10)</td> </tr></tbody> </table> where <math alttext="d_{\text{pred}}" class="ltx_Math" display="inline" id="S4.SS3.p2.1.m1.1"><semantics id="S4.SS3.p2.1.m1.1a"><msub id="S4.SS3.p2.1.m1.1.1" xref="S4.SS3.p2.1.m1.1.1.cmml"><mi id="S4.SS3.p2.1.m1.1.1.2" xref="S4.SS3.p2.1.m1.1.1.2.cmml">d</mi><mtext id="S4.SS3.p2.1.m1.1.1.3" xref="S4.SS3.p2.1.m1.1.1.3a.cmml">pred</mtext></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.1.m1.1b"><apply id="S4.SS3.p2.1.m1.1.1.cmml" xref="S4.SS3.p2.1.m1.1.1"><csymbol cd="ambiguous" 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To address this, we leverage a pre-trained dense depth estimator <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib29" title="">29</a>]</cite> and enforce a patch-based normalized cross-correlation (NCC) depth regularization <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib41" title="">41</a>]</cite>. NCC evaluates the similarity between scale-ambiguous pseudo depth and rendered depth patches, ensuring local consistency in depth rendering: <table class="ltx_equation ltx_eqn_table" id="S4.E11"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{L}_{\text{ncc}}=1-\frac{1}{|\Omega|}\sum_{p\in\Omega}\sum_{s=1}^{S^{2% }}\frac{\overline{D}_{p,s}D_{p,s}}{\overline{\sigma}_{p}\sigma_{p}}," class="ltx_Math" display="block" id="S4.E11.m1.6"><semantics id="S4.E11.m1.6a"><mrow id="S4.E11.m1.6.6.1" xref="S4.E11.m1.6.6.1.1.cmml"><mrow id="S4.E11.m1.6.6.1.1" xref="S4.E11.m1.6.6.1.1.cmml"><msub id="S4.E11.m1.6.6.1.1.2" xref="S4.E11.m1.6.6.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E11.m1.6.6.1.1.2.2" xref="S4.E11.m1.6.6.1.1.2.2.cmml">ℒ</mi><mtext id="S4.E11.m1.6.6.1.1.2.3" xref="S4.E11.m1.6.6.1.1.2.3a.cmml">ncc</mtext></msub><mo id="S4.E11.m1.6.6.1.1.1" 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id="S4.E11.m1.5.5.6.3.2.cmml" xref="S4.E11.m1.5.5.6.3.2">𝜎</ci><ci id="S4.E11.m1.5.5.6.3.3.cmml" xref="S4.E11.m1.5.5.6.3.3">𝑝</ci></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E11.m1.6c">\mathcal{L}_{\text{ncc}}=1-\frac{1}{|\Omega|}\sum_{p\in\Omega}\sum_{s=1}^{S^{2% }}\frac{\overline{D}_{p,s}D_{p,s}}{\overline{\sigma}_{p}\sigma_{p}},</annotation><annotation encoding="application/x-llamapun" id="S4.E11.m1.6d">caligraphic_L start_POSTSUBSCRIPT ncc end_POSTSUBSCRIPT = 1 - divide start_ARG 1 end_ARG start_ARG | roman_Ω | end_ARG ∑ start_POSTSUBSCRIPT italic_p ∈ roman_Ω end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_s = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT divide start_ARG over¯ start_ARG italic_D end_ARG start_POSTSUBSCRIPT italic_p , italic_s end_POSTSUBSCRIPT italic_D start_POSTSUBSCRIPT italic_p , italic_s end_POSTSUBSCRIPT end_ARG start_ARG over¯ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(11)</td> </tr></tbody> </table> where <math alttext="\Omega" class="ltx_Math" display="inline" id="S4.SS3.p3.1.m1.1"><semantics id="S4.SS3.p3.1.m1.1a"><mi id="S4.SS3.p3.1.m1.1.1" mathvariant="normal" xref="S4.SS3.p3.1.m1.1.1.cmml">Ω</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.1.m1.1b"><ci id="S4.SS3.p3.1.m1.1.1.cmml" xref="S4.SS3.p3.1.m1.1.1">Ω</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.1.m1.1c">\Omega</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.1.m1.1d">roman_Ω</annotation></semantics></math> is the patches set of depth map with size <math alttext="s\times s" class="ltx_Math" display="inline" id="S4.SS3.p3.2.m2.1"><semantics id="S4.SS3.p3.2.m2.1a"><mrow id="S4.SS3.p3.2.m2.1.1" xref="S4.SS3.p3.2.m2.1.1.cmml"><mi id="S4.SS3.p3.2.m2.1.1.2" xref="S4.SS3.p3.2.m2.1.1.2.cmml">s</mi><mo id="S4.SS3.p3.2.m2.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.SS3.p3.2.m2.1.1.1.cmml">×</mo><mi id="S4.SS3.p3.2.m2.1.1.3" xref="S4.SS3.p3.2.m2.1.1.3.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.2.m2.1b"><apply id="S4.SS3.p3.2.m2.1.1.cmml" xref="S4.SS3.p3.2.m2.1.1"><times id="S4.SS3.p3.2.m2.1.1.1.cmml" xref="S4.SS3.p3.2.m2.1.1.1"></times><ci id="S4.SS3.p3.2.m2.1.1.2.cmml" xref="S4.SS3.p3.2.m2.1.1.2">𝑠</ci><ci id="S4.SS3.p3.2.m2.1.1.3.cmml" xref="S4.SS3.p3.2.m2.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.2.m2.1c">s\times s</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.2.m2.1d">italic_s × italic_s</annotation></semantics></math> and stride <math alttext="k" class="ltx_Math" display="inline" id="S4.SS3.p3.3.m3.1"><semantics id="S4.SS3.p3.3.m3.1a"><mi id="S4.SS3.p3.3.m3.1.1" xref="S4.SS3.p3.3.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.3.m3.1b"><ci id="S4.SS3.p3.3.m3.1.1.cmml" xref="S4.SS3.p3.3.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.3.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.3.m3.1d">italic_k</annotation></semantics></math>. <math alttext="D_{p,s}" class="ltx_Math" display="inline" id="S4.SS3.p3.4.m4.2"><semantics id="S4.SS3.p3.4.m4.2a"><msub id="S4.SS3.p3.4.m4.2.3" xref="S4.SS3.p3.4.m4.2.3.cmml"><mi id="S4.SS3.p3.4.m4.2.3.2" xref="S4.SS3.p3.4.m4.2.3.2.cmml">D</mi><mrow id="S4.SS3.p3.4.m4.2.2.2.4" xref="S4.SS3.p3.4.m4.2.2.2.3.cmml"><mi id="S4.SS3.p3.4.m4.1.1.1.1" xref="S4.SS3.p3.4.m4.1.1.1.1.cmml">p</mi><mo id="S4.SS3.p3.4.m4.2.2.2.4.1" xref="S4.SS3.p3.4.m4.2.2.2.3.cmml">,</mo><mi id="S4.SS3.p3.4.m4.2.2.2.2" xref="S4.SS3.p3.4.m4.2.2.2.2.cmml">s</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.4.m4.2b"><apply id="S4.SS3.p3.4.m4.2.3.cmml" xref="S4.SS3.p3.4.m4.2.3"><csymbol cd="ambiguous" id="S4.SS3.p3.4.m4.2.3.1.cmml" xref="S4.SS3.p3.4.m4.2.3">subscript</csymbol><ci id="S4.SS3.p3.4.m4.2.3.2.cmml" xref="S4.SS3.p3.4.m4.2.3.2">𝐷</ci><list id="S4.SS3.p3.4.m4.2.2.2.3.cmml" xref="S4.SS3.p3.4.m4.2.2.2.4"><ci id="S4.SS3.p3.4.m4.1.1.1.1.cmml" xref="S4.SS3.p3.4.m4.1.1.1.1">𝑝</ci><ci id="S4.SS3.p3.4.m4.2.2.2.2.cmml" xref="S4.SS3.p3.4.m4.2.2.2.2">𝑠</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.4.m4.2c">D_{p,s}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.4.m4.2d">italic_D start_POSTSUBSCRIPT italic_p , italic_s end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\sigma_{p}" class="ltx_Math" display="inline" id="S4.SS3.p3.5.m5.1"><semantics id="S4.SS3.p3.5.m5.1a"><msub id="S4.SS3.p3.5.m5.1.1" xref="S4.SS3.p3.5.m5.1.1.cmml"><mi id="S4.SS3.p3.5.m5.1.1.2" xref="S4.SS3.p3.5.m5.1.1.2.cmml">σ</mi><mi id="S4.SS3.p3.5.m5.1.1.3" xref="S4.SS3.p3.5.m5.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.5.m5.1b"><apply id="S4.SS3.p3.5.m5.1.1.cmml" xref="S4.SS3.p3.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.5.m5.1.1.1.cmml" xref="S4.SS3.p3.5.m5.1.1">subscript</csymbol><ci id="S4.SS3.p3.5.m5.1.1.2.cmml" xref="S4.SS3.p3.5.m5.1.1.2">𝜎</ci><ci id="S4.SS3.p3.5.m5.1.1.3.cmml" xref="S4.SS3.p3.5.m5.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.5.m5.1c">\sigma_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.5.m5.1d">italic_σ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> represent a depth patch’s mean-centered values and standard deviations, respectively. </div> <div class="ltx_para ltx_noindent" id="S4.SS3.p4"> Normal Smooth Loss. To define the normal of a Gaussian, we first note that a Gaussian itself does not inherently possess a normal direction. However, we can derive a geometric normal based on its ellipsoidal shape. Specifically, the normal is defined as the direction of the smallest scaling axis of the Gaussian, which corresponds to its shortest axis in 3D space. Inspired by DN-Splatter <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib36" title="">36</a>]</cite>, for a Gaussian described by a rotation matrix <math alttext="\mathbf{R}\in\mathbb{R}^{3\times 3}" class="ltx_Math" display="inline" id="S4.SS3.p4.1.m1.1"><semantics id="S4.SS3.p4.1.m1.1a"><mrow id="S4.SS3.p4.1.m1.1.1" xref="S4.SS3.p4.1.m1.1.1.cmml"><mi id="S4.SS3.p4.1.m1.1.1.2" xref="S4.SS3.p4.1.m1.1.1.2.cmml">𝐑</mi><mo id="S4.SS3.p4.1.m1.1.1.1" xref="S4.SS3.p4.1.m1.1.1.1.cmml">∈</mo><msup id="S4.SS3.p4.1.m1.1.1.3" xref="S4.SS3.p4.1.m1.1.1.3.cmml"><mi id="S4.SS3.p4.1.m1.1.1.3.2" xref="S4.SS3.p4.1.m1.1.1.3.2.cmml">ℝ</mi><mrow id="S4.SS3.p4.1.m1.1.1.3.3" xref="S4.SS3.p4.1.m1.1.1.3.3.cmml"><mn id="S4.SS3.p4.1.m1.1.1.3.3.2" xref="S4.SS3.p4.1.m1.1.1.3.3.2.cmml">3</mn><mo id="S4.SS3.p4.1.m1.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S4.SS3.p4.1.m1.1.1.3.3.1.cmml">×</mo><mn id="S4.SS3.p4.1.m1.1.1.3.3.3" 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id="S4.SS3.p4.2.m2.5.5.1.1.cmml" xref="S4.SS3.p4.2.m2.5.5.1.1">𝑖</ci><cn id="S4.SS3.p4.2.m2.6.6.2.2.cmml" type="integer" xref="S4.SS3.p4.2.m2.6.6.2.2">2</cn></list></apply></list></apply><apply id="S4.SS3.p4.2.m2.9.9c.cmml" xref="S4.SS3.p4.2.m2.9.9"><in id="S4.SS3.p4.2.m2.9.9.7.cmml" xref="S4.SS3.p4.2.m2.9.9.7"></in><share href="https://arxiv.org/html/2503.12552v2#S4.SS3.p4.2.m2.9.9.3.cmml" id="S4.SS3.p4.2.m2.9.9d.cmml" xref="S4.SS3.p4.2.m2.9.9"></share><apply id="S4.SS3.p4.2.m2.9.9.8.cmml" xref="S4.SS3.p4.2.m2.9.9.8"><csymbol cd="ambiguous" id="S4.SS3.p4.2.m2.9.9.8.1.cmml" xref="S4.SS3.p4.2.m2.9.9.8">superscript</csymbol><ci id="S4.SS3.p4.2.m2.9.9.8.2.cmml" xref="S4.SS3.p4.2.m2.9.9.8.2">ℝ</ci><cn id="S4.SS3.p4.2.m2.9.9.8.3.cmml" type="integer" xref="S4.SS3.p4.2.m2.9.9.8.3">3</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.2.m2.9c">\mathbf{s}_{i}=\left[s_{i,0},s_{i,1},s_{i,2}\right]\in\mathbb{R}^{3}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.2.m2.9d">bold_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = [ italic_s start_POSTSUBSCRIPT italic_i , 0 end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT italic_i , 1 end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT italic_i , 2 end_POSTSUBSCRIPT ] ∈ blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT</annotation></semantics></math>, the normal is computed mathematically as: <table class="ltx_equation ltx_eqn_table" id="S4.E12"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathbf{\hat{n}}_{i,p}=\mathbf{R}\cdot\text{{OneHot}}\big{(}\operatorname{% argmin}(s_{i,0},s_{i,1},s_{i,2})\big{)}," class="ltx_Math" display="block" id="S4.E12.m1.10"><semantics id="S4.E12.m1.10a"><mrow id="S4.E12.m1.10.10.1" xref="S4.E12.m1.10.10.1.1.cmml"><mrow id="S4.E12.m1.10.10.1.1" xref="S4.E12.m1.10.10.1.1.cmml"><msub 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id="S4.E12.m1.10c">\mathbf{\hat{n}}_{i,p}=\mathbf{R}\cdot\text{{OneHot}}\big{(}\operatorname{% argmin}(s_{i,0},s_{i,1},s_{i,2})\big{)},</annotation><annotation encoding="application/x-llamapun" id="S4.E12.m1.10d">over^ start_ARG bold_n end_ARG start_POSTSUBSCRIPT italic_i , italic_p end_POSTSUBSCRIPT = bold_R ⋅ OneHot ( roman_argmin ( italic_s start_POSTSUBSCRIPT italic_i , 0 end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT italic_i , 1 end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT italic_i , 2 end_POSTSUBSCRIPT ) ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(12)</td> </tr></tbody> </table> where <math alttext="\text{{OneHot}}(\cdot)\in\mathbb{R}^{3}" class="ltx_Math" display="inline" id="S4.SS3.p4.3.m1.1"><semantics id="S4.SS3.p4.3.m1.1a"><mrow id="S4.SS3.p4.3.m1.1.2" xref="S4.SS3.p4.3.m1.1.2.cmml"><mrow id="S4.SS3.p4.3.m1.1.2.2" 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xref="S4.SS3.p4.3.m1.1.2"><in id="S4.SS3.p4.3.m1.1.2.1.cmml" xref="S4.SS3.p4.3.m1.1.2.1"></in><apply id="S4.SS3.p4.3.m1.1.2.2.cmml" xref="S4.SS3.p4.3.m1.1.2.2"><times id="S4.SS3.p4.3.m1.1.2.2.1.cmml" xref="S4.SS3.p4.3.m1.1.2.2.1"></times><ci id="S4.SS3.p4.3.m1.1.2.2.2a.cmml" xref="S4.SS3.p4.3.m1.1.2.2.2"><mtext class="ltx_mathvariant_monospace" id="S4.SS3.p4.3.m1.1.2.2.2.cmml" xref="S4.SS3.p4.3.m1.1.2.2.2">OneHot</mtext></ci><ci id="S4.SS3.p4.3.m1.1.1.cmml" xref="S4.SS3.p4.3.m1.1.1">⋅</ci></apply><apply id="S4.SS3.p4.3.m1.1.2.3.cmml" xref="S4.SS3.p4.3.m1.1.2.3"><csymbol cd="ambiguous" id="S4.SS3.p4.3.m1.1.2.3.1.cmml" xref="S4.SS3.p4.3.m1.1.2.3">superscript</csymbol><ci id="S4.SS3.p4.3.m1.1.2.3.2.cmml" xref="S4.SS3.p4.3.m1.1.2.3.2">ℝ</ci><cn id="S4.SS3.p4.3.m1.1.2.3.3.cmml" type="integer" xref="S4.SS3.p4.3.m1.1.2.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.3.m1.1c">\text{{OneHot}}(\cdot)\in\mathbb{R}^{3}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.3.m1.1d">OneHot ( ⋅ ) ∈ blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT</annotation></semantics></math> returns a unit vector with all zeros except at the position of minimum scaling. To generate per-pixel normal estimates, the corrected normals of 3D Gaussians are first transformed into camera space using the current camera transformation matrix. A per-pixel normal <math alttext="\hat{N}" class="ltx_Math" display="inline" id="S4.SS3.p4.4.m2.1"><semantics id="S4.SS3.p4.4.m2.1a"><mover accent="true" id="S4.SS3.p4.4.m2.1.1" xref="S4.SS3.p4.4.m2.1.1.cmml"><mi id="S4.SS3.p4.4.m2.1.1.2" xref="S4.SS3.p4.4.m2.1.1.2.cmml">N</mi><mo id="S4.SS3.p4.4.m2.1.1.1" xref="S4.SS3.p4.4.m2.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.4.m2.1b"><apply id="S4.SS3.p4.4.m2.1.1.cmml" xref="S4.SS3.p4.4.m2.1.1"><ci id="S4.SS3.p4.4.m2.1.1.1.cmml" xref="S4.SS3.p4.4.m2.1.1.1">^</ci><ci id="S4.SS3.p4.4.m2.1.1.2.cmml" xref="S4.SS3.p4.4.m2.1.1.2">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.4.m2.1c">\hat{N}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.4.m2.1d">over^ start_ARG italic_N end_ARG</annotation></semantics></math> is computed via alpha compositing. </div> <div class="ltx_para" id="S4.SS3.p5"> The pseudo normal <math alttext="N" class="ltx_Math" display="inline" id="S4.SS3.p5.1.m1.1"><semantics id="S4.SS3.p5.1.m1.1a"><mi id="S4.SS3.p5.1.m1.1.1" xref="S4.SS3.p5.1.m1.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p5.1.m1.1b"><ci id="S4.SS3.p5.1.m1.1.1.cmml" xref="S4.SS3.p5.1.m1.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p5.1.m1.1c">N</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p5.1.m1.1d">italic_N</annotation></semantics></math> is estimated from gradients of the pseudo-depth map, as in 2DGS <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib13" title="">13</a>]</cite>. To deal with noise in the pseudo normal, we introduce a total variation (TV) loss on the renderer normal. The normal regularization loss is: <table class="ltx_equation ltx_eqn_table" id="S4.E13"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{L}_{\text{normal}}=|\hat{N}-N|+\mathcal{L}_{\text{TV}}(\hat{N})." class="ltx_Math" display="block" id="S4.E13.m1.2"><semantics id="S4.E13.m1.2a"><mrow id="S4.E13.m1.2.2.1" xref="S4.E13.m1.2.2.1.1.cmml"><mrow id="S4.E13.m1.2.2.1.1" xref="S4.E13.m1.2.2.1.1.cmml"><msub id="S4.E13.m1.2.2.1.1.3" xref="S4.E13.m1.2.2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E13.m1.2.2.1.1.3.2" xref="S4.E13.m1.2.2.1.1.3.2.cmml">ℒ</mi><mtext id="S4.E13.m1.2.2.1.1.3.3" xref="S4.E13.m1.2.2.1.1.3.3a.cmml">normal</mtext></msub><mo id="S4.E13.m1.2.2.1.1.2" xref="S4.E13.m1.2.2.1.1.2.cmml">=</mo><mrow id="S4.E13.m1.2.2.1.1.1" xref="S4.E13.m1.2.2.1.1.1.cmml"><mrow id="S4.E13.m1.2.2.1.1.1.1.1" xref="S4.E13.m1.2.2.1.1.1.1.2.cmml"><mo id="S4.E13.m1.2.2.1.1.1.1.1.2" stretchy="false" xref="S4.E13.m1.2.2.1.1.1.1.2.1.cmml">|</mo><mrow id="S4.E13.m1.2.2.1.1.1.1.1.1" xref="S4.E13.m1.2.2.1.1.1.1.1.1.cmml"><mover accent="true" id="S4.E13.m1.2.2.1.1.1.1.1.1.2" xref="S4.E13.m1.2.2.1.1.1.1.1.1.2.cmml"><mi id="S4.E13.m1.2.2.1.1.1.1.1.1.2.2" xref="S4.E13.m1.2.2.1.1.1.1.1.1.2.2.cmml">N</mi><mo id="S4.E13.m1.2.2.1.1.1.1.1.1.2.1" xref="S4.E13.m1.2.2.1.1.1.1.1.1.2.1.cmml">^</mo></mover><mo id="S4.E13.m1.2.2.1.1.1.1.1.1.1" xref="S4.E13.m1.2.2.1.1.1.1.1.1.1.cmml">−</mo><mi id="S4.E13.m1.2.2.1.1.1.1.1.1.3" xref="S4.E13.m1.2.2.1.1.1.1.1.1.3.cmml">N</mi></mrow><mo id="S4.E13.m1.2.2.1.1.1.1.1.3" stretchy="false" xref="S4.E13.m1.2.2.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S4.E13.m1.2.2.1.1.1.2" xref="S4.E13.m1.2.2.1.1.1.2.cmml">+</mo><mrow id="S4.E13.m1.2.2.1.1.1.3" xref="S4.E13.m1.2.2.1.1.1.3.cmml"><msub id="S4.E13.m1.2.2.1.1.1.3.2" xref="S4.E13.m1.2.2.1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E13.m1.2.2.1.1.1.3.2.2" xref="S4.E13.m1.2.2.1.1.1.3.2.2.cmml">ℒ</mi><mtext id="S4.E13.m1.2.2.1.1.1.3.2.3" xref="S4.E13.m1.2.2.1.1.1.3.2.3a.cmml">TV</mtext></msub><mo id="S4.E13.m1.2.2.1.1.1.3.1" xref="S4.E13.m1.2.2.1.1.1.3.1.cmml">⁢</mo><mrow id="S4.E13.m1.2.2.1.1.1.3.3.2" xref="S4.E13.m1.1.1.cmml"><mo id="S4.E13.m1.2.2.1.1.1.3.3.2.1" stretchy="false" xref="S4.E13.m1.1.1.cmml">(</mo><mover accent="true" id="S4.E13.m1.1.1" xref="S4.E13.m1.1.1.cmml"><mi id="S4.E13.m1.1.1.2" xref="S4.E13.m1.1.1.2.cmml">N</mi><mo id="S4.E13.m1.1.1.1" xref="S4.E13.m1.1.1.1.cmml">^</mo></mover><mo id="S4.E13.m1.2.2.1.1.1.3.3.2.2" stretchy="false" xref="S4.E13.m1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S4.E13.m1.2.2.1.2" lspace="0em" xref="S4.E13.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E13.m1.2b"><apply id="S4.E13.m1.2.2.1.1.cmml" xref="S4.E13.m1.2.2.1"><eq id="S4.E13.m1.2.2.1.1.2.cmml" xref="S4.E13.m1.2.2.1.1.2"></eq><apply id="S4.E13.m1.2.2.1.1.3.cmml" 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xref="S4.E13.m1.1.1.2">𝑁</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E13.m1.2c">\mathcal{L}_{\text{normal}}=|\hat{N}-N|+\mathcal{L}_{\text{TV}}(\hat{N}).</annotation><annotation encoding="application/x-llamapun" id="S4.E13.m1.2d">caligraphic_L start_POSTSUBSCRIPT normal end_POSTSUBSCRIPT = | over^ start_ARG italic_N end_ARG - italic_N | + caligraphic_L start_POSTSUBSCRIPT TV end_POSTSUBSCRIPT ( over^ start_ARG italic_N end_ARG ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(13)</td> </tr></tbody> </table> To obtain a stable Gaussian normal, we add a Gaussian flatten regularization loss to regularize the ratio of the other two axes not exceeding <math alttext="r" class="ltx_Math" display="inline" id="S4.SS3.p5.2.m1.1"><semantics id="S4.SS3.p5.2.m1.1a"><mi id="S4.SS3.p5.2.m1.1.1" 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id="S4.E14.m1.7d">caligraphic_L start_POSTSUBSCRIPT flatten end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT max { divide start_ARG roman_max ( bold_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_ARG start_ARG roman_median ( bold_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_ARG , italic_r } - italic_r + roman_min ( bold_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(14)</td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS3.p6"> In the end, all components of MTGS are optimized jointly using the overall training loss: <table class="ltx_equationgroup ltx_eqn_table" id="S4.E15"> <tbody> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S4.E15X"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{L}=" class="ltx_Math" display="inline" id="S4.E15X.2.1.1.m1.1"><semantics id="S4.E15X.2.1.1.m1.1a"><mrow id="S4.E15X.2.1.1.m1.1.1" xref="S4.E15X.2.1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E15X.2.1.1.m1.1.1.2" xref="S4.E15X.2.1.1.m1.1.1.2.cmml">ℒ</mi><mo id="S4.E15X.2.1.1.m1.1.1.1" xref="S4.E15X.2.1.1.m1.1.1.1.cmml">=</mo><mi id="S4.E15X.2.1.1.m1.1.1.3" xref="S4.E15X.2.1.1.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="S4.E15X.2.1.1.m1.1b"><apply id="S4.E15X.2.1.1.m1.1.1.cmml" xref="S4.E15X.2.1.1.m1.1.1"><eq id="S4.E15X.2.1.1.m1.1.1.1.cmml" xref="S4.E15X.2.1.1.m1.1.1.1"></eq><ci id="S4.E15X.2.1.1.m1.1.1.2.cmml" xref="S4.E15X.2.1.1.m1.1.1.2">ℒ</ci><csymbol cd="latexml" id="S4.E15X.2.1.1.m1.1.1.3.cmml" xref="S4.E15X.2.1.1.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E15X.2.1.1.m1.1c">\displaystyle\mathcal{L}=</annotation><annotation 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id="S4.E15X.3.2.2.m1.1.1.1.1.1.1.3.3" xref="S4.E15X.3.2.2.m1.1.1.1.1.1.1.3.3.cmml">r</mi></msub></mrow><mo id="S4.E15X.3.2.2.m1.1.1.1.1.1.3" stretchy="false" xref="S4.E15X.3.2.2.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.E15X.3.2.2.m1.1.1.1.2" xref="S4.E15X.3.2.2.m1.1.1.1.2.cmml">⁢</mo><msub id="S4.E15X.3.2.2.m1.1.1.1.3" xref="S4.E15X.3.2.2.m1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E15X.3.2.2.m1.1.1.1.3.2" xref="S4.E15X.3.2.2.m1.1.1.1.3.2.cmml">ℒ</mi><mtext id="S4.E15X.3.2.2.m1.1.1.1.3.3" xref="S4.E15X.3.2.2.m1.1.1.1.3.3a.cmml">SSIM</mtext></msub></mrow><mo id="S4.E15X.3.2.2.m1.1.1.2a" xref="S4.E15X.3.2.2.m1.1.1.2.cmml">+</mo><mrow id="S4.E15X.3.2.2.m1.1.1.4" xref="S4.E15X.3.2.2.m1.1.1.4.cmml"><msub id="S4.E15X.3.2.2.m1.1.1.4.2" xref="S4.E15X.3.2.2.m1.1.1.4.2.cmml"><mi id="S4.E15X.3.2.2.m1.1.1.4.2.2" xref="S4.E15X.3.2.2.m1.1.1.4.2.2.cmml">λ</mi><mtext id="S4.E15X.3.2.2.m1.1.1.4.2.3" xref="S4.E15X.3.2.2.m1.1.1.4.2.3a.cmml">depth</mtext></msub><mo 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xref="S4.E15X.3.2.2.m1.1.1.1.1.1.1.3.2">𝜆</ci><ci id="S4.E15X.3.2.2.m1.1.1.1.1.1.1.3.3.cmml" xref="S4.E15X.3.2.2.m1.1.1.1.1.1.1.3.3">𝑟</ci></apply></apply><apply id="S4.E15X.3.2.2.m1.1.1.1.3.cmml" xref="S4.E15X.3.2.2.m1.1.1.1.3"><csymbol cd="ambiguous" id="S4.E15X.3.2.2.m1.1.1.1.3.1.cmml" xref="S4.E15X.3.2.2.m1.1.1.1.3">subscript</csymbol><ci id="S4.E15X.3.2.2.m1.1.1.1.3.2.cmml" xref="S4.E15X.3.2.2.m1.1.1.1.3.2">ℒ</ci><ci id="S4.E15X.3.2.2.m1.1.1.1.3.3a.cmml" xref="S4.E15X.3.2.2.m1.1.1.1.3.3"><mtext id="S4.E15X.3.2.2.m1.1.1.1.3.3.cmml" mathsize="70%" xref="S4.E15X.3.2.2.m1.1.1.1.3.3">SSIM</mtext></ci></apply></apply><apply id="S4.E15X.3.2.2.m1.1.1.4.cmml" xref="S4.E15X.3.2.2.m1.1.1.4"><times id="S4.E15X.3.2.2.m1.1.1.4.1.cmml" xref="S4.E15X.3.2.2.m1.1.1.4.1"></times><apply id="S4.E15X.3.2.2.m1.1.1.4.2.cmml" xref="S4.E15X.3.2.2.m1.1.1.4.2"><csymbol cd="ambiguous" id="S4.E15X.3.2.2.m1.1.1.4.2.1.cmml" xref="S4.E15X.3.2.2.m1.1.1.4.2">subscript</csymbol><ci id="S4.E15X.3.2.2.m1.1.1.4.2.2.cmml" xref="S4.E15X.3.2.2.m1.1.1.4.2.2">𝜆</ci><ci id="S4.E15X.3.2.2.m1.1.1.4.2.3a.cmml" xref="S4.E15X.3.2.2.m1.1.1.4.2.3"><mtext id="S4.E15X.3.2.2.m1.1.1.4.2.3.cmml" mathsize="70%" xref="S4.E15X.3.2.2.m1.1.1.4.2.3">depth</mtext></ci></apply><apply id="S4.E15X.3.2.2.m1.1.1.4.3.cmml" xref="S4.E15X.3.2.2.m1.1.1.4.3"><csymbol cd="ambiguous" id="S4.E15X.3.2.2.m1.1.1.4.3.1.cmml" xref="S4.E15X.3.2.2.m1.1.1.4.3">subscript</csymbol><ci id="S4.E15X.3.2.2.m1.1.1.4.3.2.cmml" xref="S4.E15X.3.2.2.m1.1.1.4.3.2">ℒ</ci><ci id="S4.E15X.3.2.2.m1.1.1.4.3.3a.cmml" xref="S4.E15X.3.2.2.m1.1.1.4.3.3"><mtext id="S4.E15X.3.2.2.m1.1.1.4.3.3.cmml" mathsize="70%" xref="S4.E15X.3.2.2.m1.1.1.4.3.3">depth</mtext></ci></apply></apply><apply id="S4.E15X.3.2.2.m1.1.1.5.cmml" xref="S4.E15X.3.2.2.m1.1.1.5"><times id="S4.E15X.3.2.2.m1.1.1.5.1.cmml" xref="S4.E15X.3.2.2.m1.1.1.5.1"></times><apply id="S4.E15X.3.2.2.m1.1.1.5.2.cmml" xref="S4.E15X.3.2.2.m1.1.1.5.2"><csymbol cd="ambiguous" id="S4.E15X.3.2.2.m1.1.1.5.2.1.cmml" xref="S4.E15X.3.2.2.m1.1.1.5.2">subscript</csymbol><ci id="S4.E15X.3.2.2.m1.1.1.5.2.2.cmml" xref="S4.E15X.3.2.2.m1.1.1.5.2.2">𝜆</ci><ci id="S4.E15X.3.2.2.m1.1.1.5.2.3a.cmml" xref="S4.E15X.3.2.2.m1.1.1.5.2.3"><mtext id="S4.E15X.3.2.2.m1.1.1.5.2.3.cmml" mathsize="70%" xref="S4.E15X.3.2.2.m1.1.1.5.2.3">ncc</mtext></ci></apply><apply id="S4.E15X.3.2.2.m1.1.1.5.3.cmml" xref="S4.E15X.3.2.2.m1.1.1.5.3"><csymbol cd="ambiguous" id="S4.E15X.3.2.2.m1.1.1.5.3.1.cmml" xref="S4.E15X.3.2.2.m1.1.1.5.3">subscript</csymbol><ci id="S4.E15X.3.2.2.m1.1.1.5.3.2.cmml" xref="S4.E15X.3.2.2.m1.1.1.5.3.2">ℒ</ci><ci id="S4.E15X.3.2.2.m1.1.1.5.3.3a.cmml" xref="S4.E15X.3.2.2.m1.1.1.5.3.3"><mtext id="S4.E15X.3.2.2.m1.1.1.5.3.3.cmml" mathsize="70%" xref="S4.E15X.3.2.2.m1.1.1.5.3.3">ncc</mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E15X.3.2.2.m1.1c">\displaystyle\lambda_{r}\mathcal{L}_{1}+(1-\lambda_{r})\mathcal{L}_{\text{SSIM% }}+\lambda_{\text{depth}}\mathcal{L}_{\text{depth}}+\lambda_{\text{ncc}}% \mathcal{L}_{\text{ncc}}</annotation><annotation encoding="application/x-llamapun" id="S4.E15X.3.2.2.m1.1d">italic_λ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT caligraphic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + ( 1 - italic_λ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) caligraphic_L start_POSTSUBSCRIPT SSIM end_POSTSUBSCRIPT + italic_λ start_POSTSUBSCRIPT depth end_POSTSUBSCRIPT caligraphic_L start_POSTSUBSCRIPT depth end_POSTSUBSCRIPT + italic_λ start_POSTSUBSCRIPT ncc end_POSTSUBSCRIPT caligraphic_L start_POSTSUBSCRIPT ncc end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="2">(15)</td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S4.E15Xa"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle+\lambda_{\text{normal}}\mathcal{L}_{\text{normal}}+\lambda_{% \text{flatten}}\mathcal{L}_{\text{flatten}}+\lambda_{\text{oob}}\mathcal{L}_{% \text{oob}}," class="ltx_Math" display="inline" id="S4.E15Xa.2.1.1.m1.1"><semantics id="S4.E15Xa.2.1.1.m1.1a"><mrow id="S4.E15Xa.2.1.1.m1.1.1.1" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.cmml"><mrow id="S4.E15Xa.2.1.1.m1.1.1.1.1" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.cmml"><mrow id="S4.E15Xa.2.1.1.m1.1.1.1.1.2" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.2.cmml"><mo id="S4.E15Xa.2.1.1.m1.1.1.1.1.2a" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.2.cmml">+</mo><mrow id="S4.E15Xa.2.1.1.m1.1.1.1.1.2.2" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.2.2.cmml"><msub id="S4.E15Xa.2.1.1.m1.1.1.1.1.2.2.2" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.2.2.2.cmml"><mi id="S4.E15Xa.2.1.1.m1.1.1.1.1.2.2.2.2" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.2.2.2.2.cmml">λ</mi><mtext id="S4.E15Xa.2.1.1.m1.1.1.1.1.2.2.2.3" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.2.2.2.3a.cmml">normal</mtext></msub><mo id="S4.E15Xa.2.1.1.m1.1.1.1.1.2.2.1" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.2.2.1.cmml">⁢</mo><msub id="S4.E15Xa.2.1.1.m1.1.1.1.1.2.2.3" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E15Xa.2.1.1.m1.1.1.1.1.2.2.3.2" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.2.2.3.2.cmml">ℒ</mi><mtext id="S4.E15Xa.2.1.1.m1.1.1.1.1.2.2.3.3" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.2.2.3.3a.cmml">normal</mtext></msub></mrow></mrow><mo id="S4.E15Xa.2.1.1.m1.1.1.1.1.1" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.1.cmml">+</mo><mrow id="S4.E15Xa.2.1.1.m1.1.1.1.1.3" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.3.cmml"><msub id="S4.E15Xa.2.1.1.m1.1.1.1.1.3.2" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.3.2.cmml"><mi id="S4.E15Xa.2.1.1.m1.1.1.1.1.3.2.2" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.3.2.2.cmml">λ</mi><mtext id="S4.E15Xa.2.1.1.m1.1.1.1.1.3.2.3" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.3.2.3a.cmml">flatten</mtext></msub><mo id="S4.E15Xa.2.1.1.m1.1.1.1.1.3.1" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.3.1.cmml">⁢</mo><msub id="S4.E15Xa.2.1.1.m1.1.1.1.1.3.3" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E15Xa.2.1.1.m1.1.1.1.1.3.3.2" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.3.3.2.cmml">ℒ</mi><mtext id="S4.E15Xa.2.1.1.m1.1.1.1.1.3.3.3" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.3.3.3a.cmml">flatten</mtext></msub></mrow><mo id="S4.E15Xa.2.1.1.m1.1.1.1.1.1a" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.1.cmml">+</mo><mrow id="S4.E15Xa.2.1.1.m1.1.1.1.1.4" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.4.cmml"><msub id="S4.E15Xa.2.1.1.m1.1.1.1.1.4.2" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.4.2.cmml"><mi id="S4.E15Xa.2.1.1.m1.1.1.1.1.4.2.2" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.4.2.2.cmml">λ</mi><mtext id="S4.E15Xa.2.1.1.m1.1.1.1.1.4.2.3" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.4.2.3a.cmml">oob</mtext></msub><mo id="S4.E15Xa.2.1.1.m1.1.1.1.1.4.1" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.4.1.cmml">⁢</mo><msub id="S4.E15Xa.2.1.1.m1.1.1.1.1.4.3" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.4.3.cmml"><mi 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id="S4.E15Xa.2.1.1.m1.1.1.1.1.3.cmml" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.3"><times id="S4.E15Xa.2.1.1.m1.1.1.1.1.3.1.cmml" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.3.1"></times><apply id="S4.E15Xa.2.1.1.m1.1.1.1.1.3.2.cmml" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="S4.E15Xa.2.1.1.m1.1.1.1.1.3.2.1.cmml" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.3.2">subscript</csymbol><ci id="S4.E15Xa.2.1.1.m1.1.1.1.1.3.2.2.cmml" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.3.2.2">𝜆</ci><ci id="S4.E15Xa.2.1.1.m1.1.1.1.1.3.2.3a.cmml" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.3.2.3"><mtext id="S4.E15Xa.2.1.1.m1.1.1.1.1.3.2.3.cmml" mathsize="70%" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.3.2.3">flatten</mtext></ci></apply><apply id="S4.E15Xa.2.1.1.m1.1.1.1.1.3.3.cmml" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S4.E15Xa.2.1.1.m1.1.1.1.1.3.3.1.cmml" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.3.3">subscript</csymbol><ci id="S4.E15Xa.2.1.1.m1.1.1.1.1.3.3.2.cmml" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.3.3.2">ℒ</ci><ci id="S4.E15Xa.2.1.1.m1.1.1.1.1.3.3.3a.cmml" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.3.3.3"><mtext id="S4.E15Xa.2.1.1.m1.1.1.1.1.3.3.3.cmml" mathsize="70%" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.3.3.3">flatten</mtext></ci></apply></apply><apply id="S4.E15Xa.2.1.1.m1.1.1.1.1.4.cmml" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.4"><times id="S4.E15Xa.2.1.1.m1.1.1.1.1.4.1.cmml" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.4.1"></times><apply id="S4.E15Xa.2.1.1.m1.1.1.1.1.4.2.cmml" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.4.2"><csymbol cd="ambiguous" id="S4.E15Xa.2.1.1.m1.1.1.1.1.4.2.1.cmml" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.4.2">subscript</csymbol><ci id="S4.E15Xa.2.1.1.m1.1.1.1.1.4.2.2.cmml" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.4.2.2">𝜆</ci><ci id="S4.E15Xa.2.1.1.m1.1.1.1.1.4.2.3a.cmml" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.4.2.3"><mtext id="S4.E15Xa.2.1.1.m1.1.1.1.1.4.2.3.cmml" mathsize="70%" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.4.2.3">oob</mtext></ci></apply><apply id="S4.E15Xa.2.1.1.m1.1.1.1.1.4.3.cmml" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.4.3"><csymbol cd="ambiguous" id="S4.E15Xa.2.1.1.m1.1.1.1.1.4.3.1.cmml" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.4.3">subscript</csymbol><ci id="S4.E15Xa.2.1.1.m1.1.1.1.1.4.3.2.cmml" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.4.3.2">ℒ</ci><ci id="S4.E15Xa.2.1.1.m1.1.1.1.1.4.3.3a.cmml" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.4.3.3"><mtext id="S4.E15Xa.2.1.1.m1.1.1.1.1.4.3.3.cmml" mathsize="70%" xref="S4.E15Xa.2.1.1.m1.1.1.1.1.4.3.3">oob</mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E15Xa.2.1.1.m1.1c">\displaystyle+\lambda_{\text{normal}}\mathcal{L}_{\text{normal}}+\lambda_{% \text{flatten}}\mathcal{L}_{\text{flatten}}+\lambda_{\text{oob}}\mathcal{L}_{% \text{oob}},</annotation><annotation encoding="application/x-llamapun" id="S4.E15Xa.2.1.1.m1.1d">+ italic_λ start_POSTSUBSCRIPT normal end_POSTSUBSCRIPT caligraphic_L start_POSTSUBSCRIPT normal end_POSTSUBSCRIPT + italic_λ start_POSTSUBSCRIPT flatten end_POSTSUBSCRIPT caligraphic_L start_POSTSUBSCRIPT flatten end_POSTSUBSCRIPT + italic_λ start_POSTSUBSCRIPT oob end_POSTSUBSCRIPT caligraphic_L start_POSTSUBSCRIPT oob end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> where <math alttext="\mathcal{L}_{1}" class="ltx_Math" display="inline" id="S4.SS3.p6.1.m1.1"><semantics id="S4.SS3.p6.1.m1.1a"><msub id="S4.SS3.p6.1.m1.1.1" xref="S4.SS3.p6.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p6.1.m1.1.1.2" xref="S4.SS3.p6.1.m1.1.1.2.cmml">ℒ</mi><mn id="S4.SS3.p6.1.m1.1.1.3" xref="S4.SS3.p6.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.1.m1.1b"><apply id="S4.SS3.p6.1.m1.1.1.cmml" xref="S4.SS3.p6.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.1.m1.1.1.1.cmml" xref="S4.SS3.p6.1.m1.1.1">subscript</csymbol><ci id="S4.SS3.p6.1.m1.1.1.2.cmml" xref="S4.SS3.p6.1.m1.1.1.2">ℒ</ci><cn id="S4.SS3.p6.1.m1.1.1.3.cmml" type="integer" xref="S4.SS3.p6.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.1.m1.1c">\mathcal{L}_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.1.m1.1d">caligraphic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mathcal{L}_{\text{SSIM}}" class="ltx_Math" display="inline" id="S4.SS3.p6.2.m2.1"><semantics id="S4.SS3.p6.2.m2.1a"><msub id="S4.SS3.p6.2.m2.1.1" xref="S4.SS3.p6.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p6.2.m2.1.1.2" xref="S4.SS3.p6.2.m2.1.1.2.cmml">ℒ</mi><mtext id="S4.SS3.p6.2.m2.1.1.3" xref="S4.SS3.p6.2.m2.1.1.3a.cmml">SSIM</mtext></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.2.m2.1b"><apply id="S4.SS3.p6.2.m2.1.1.cmml" xref="S4.SS3.p6.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.2.m2.1.1.1.cmml" xref="S4.SS3.p6.2.m2.1.1">subscript</csymbol><ci id="S4.SS3.p6.2.m2.1.1.2.cmml" xref="S4.SS3.p6.2.m2.1.1.2">ℒ</ci><ci id="S4.SS3.p6.2.m2.1.1.3a.cmml" xref="S4.SS3.p6.2.m2.1.1.3"><mtext id="S4.SS3.p6.2.m2.1.1.3.cmml" mathsize="70%" xref="S4.SS3.p6.2.m2.1.1.3">SSIM</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.2.m2.1c">\mathcal{L}_{\text{SSIM}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.2.m2.1d">caligraphic_L start_POSTSUBSCRIPT SSIM end_POSTSUBSCRIPT</annotation></semantics></math> are photometric losses between ground truth images and renderer images, <math alttext="\lambda_{r}" class="ltx_Math" display="inline" id="S4.SS3.p6.3.m3.1"><semantics id="S4.SS3.p6.3.m3.1a"><msub id="S4.SS3.p6.3.m3.1.1" xref="S4.SS3.p6.3.m3.1.1.cmml"><mi id="S4.SS3.p6.3.m3.1.1.2" xref="S4.SS3.p6.3.m3.1.1.2.cmml">λ</mi><mi id="S4.SS3.p6.3.m3.1.1.3" xref="S4.SS3.p6.3.m3.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.3.m3.1b"><apply id="S4.SS3.p6.3.m3.1.1.cmml" xref="S4.SS3.p6.3.m3.1.1"><csymbol cd="ambiguous" 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id="S4.SS3.p6.7.m7.1.1.2" xref="S4.SS3.p6.7.m7.1.1.2.cmml">λ</mi><mtext id="S4.SS3.p6.7.m7.1.1.3" xref="S4.SS3.p6.7.m7.1.1.3a.cmml">flatten</mtext></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.7.m7.1b"><apply id="S4.SS3.p6.7.m7.1.1.cmml" xref="S4.SS3.p6.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.7.m7.1.1.1.cmml" xref="S4.SS3.p6.7.m7.1.1">subscript</csymbol><ci id="S4.SS3.p6.7.m7.1.1.2.cmml" xref="S4.SS3.p6.7.m7.1.1.2">𝜆</ci><ci id="S4.SS3.p6.7.m7.1.1.3a.cmml" xref="S4.SS3.p6.7.m7.1.1.3"><mtext id="S4.SS3.p6.7.m7.1.1.3.cmml" mathsize="70%" xref="S4.SS3.p6.7.m7.1.1.3">flatten</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.7.m7.1c">\lambda_{\text{flatten}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.7.m7.1d">italic_λ start_POSTSUBSCRIPT flatten end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\lambda_{\text{oob}}" class="ltx_Math" display="inline" id="S4.SS3.p6.8.m8.1"><semantics id="S4.SS3.p6.8.m8.1a"><msub id="S4.SS3.p6.8.m8.1.1" xref="S4.SS3.p6.8.m8.1.1.cmml"><mi id="S4.SS3.p6.8.m8.1.1.2" xref="S4.SS3.p6.8.m8.1.1.2.cmml">λ</mi><mtext id="S4.SS3.p6.8.m8.1.1.3" xref="S4.SS3.p6.8.m8.1.1.3a.cmml">oob</mtext></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.8.m8.1b"><apply id="S4.SS3.p6.8.m8.1.1.cmml" xref="S4.SS3.p6.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.8.m8.1.1.1.cmml" xref="S4.SS3.p6.8.m8.1.1">subscript</csymbol><ci id="S4.SS3.p6.8.m8.1.1.2.cmml" xref="S4.SS3.p6.8.m8.1.1.2">𝜆</ci><ci id="S4.SS3.p6.8.m8.1.1.3a.cmml" xref="S4.SS3.p6.8.m8.1.1.3"><mtext id="S4.SS3.p6.8.m8.1.1.3.cmml" mathsize="70%" xref="S4.SS3.p6.8.m8.1.1.3">oob</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.8.m8.1c">\lambda_{\text{oob}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.8.m8.1d">italic_λ start_POSTSUBSCRIPT oob end_POSTSUBSCRIPT</annotation></semantics></math> are hyper-parameters. </div> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> 5 Experiment</h2> <section class="ltx_subsection" id="S5.SS1"> <h3 class="ltx_title ltx_title_subsection"> 5.1 Setup and Protocols</h3> <figure class="ltx_table" id="S5.T1"> <figcaption class="ltx_caption ltx_centering">Table 1: Comparison with SOTA. ‘ST’ denotes single-traversal reconstruction. ‘MT’ stands for multi-traversal reconstruction. For MT, results on training traversals are averaged and cannot be compared with those in ST directly. The evaluation for novel-view traversal is identical between ST and MT. <math alttext="\ast" class="ltx_Math" display="inline" id="S5.T1.3.1.m1.1"><semantics id="S5.T1.3.1.m1.1b"><mo id="S5.T1.3.1.m1.1.1" xref="S5.T1.3.1.m1.1.1.cmml">∗</mo><annotation-xml encoding="MathML-Content" id="S5.T1.3.1.m1.1c"><ci id="S5.T1.3.1.m1.1.1.cmml" xref="S5.T1.3.1.m1.1.1">∗</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.3.1.m1.1d">\ast</annotation><annotation encoding="application/x-llamapun" id="S5.T1.3.1.m1.1e">∗</annotation></semantics></math>: affine-aligned PSNR. <math alttext="\dagger" class="ltx_Math" display="inline" id="S5.T1.4.2.m2.1"><semantics id="S5.T1.4.2.m2.1b"><mo id="S5.T1.4.2.m2.1.1" xref="S5.T1.4.2.m2.1.1.cmml">†</mo><annotation-xml encoding="MathML-Content" id="S5.T1.4.2.m2.1c"><ci id="S5.T1.4.2.m2.1.1.cmml" xref="S5.T1.4.2.m2.1.1">†</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.4.2.m2.1d">\dagger</annotation><annotation encoding="application/x-llamapun" id="S5.T1.4.2.m2.1e">†</annotation></semantics></math>: adapted with multi-traversal transient nodes. First, second, third. </figcaption> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="S5.T1.16" style="width:496.9pt;height:142.4pt;vertical-align:-0.0pt;"> <table class="ltx_tabular ltx_align_middle" id="S5.T1.16.12"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S5.T1.16.12.13.1"> <td class="ltx_td ltx_border_tt" id="S5.T1.16.12.13.1.1"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_tt" id="S5.T1.16.12.13.1.2" rowspan="2">Method</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_tt" colspan="4" id="S5.T1.16.12.13.1.3">Training Traversal</td> <td class="ltx_td ltx_align_center ltx_border_tt" colspan="6" id="S5.T1.16.12.13.1.4">Novel-View Traversal</td> </tr> <tr class="ltx_tr" id="S5.T1.14.10.10"> <td class="ltx_td" id="S5.T1.14.10.10.11"></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.5.1.1.1" style="background-color:#FFFFFF;">PSNR <math alttext="\uparrow" class="ltx_Math" display="inline" id="S5.T1.5.1.1.1.1.1.m1.1" style="background-color:#FFFFFF;"><semantics id="S5.T1.5.1.1.1.1.1.m1.1a"><mo id="S5.T1.5.1.1.1.1.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="S5.T1.5.1.1.1.1.1.m1.1.1.cmml">↑</mo><annotation-xml encoding="MathML-Content" id="S5.T1.5.1.1.1.1.1.m1.1b"><ci id="S5.T1.5.1.1.1.1.1.m1.1.1.cmml" xref="S5.T1.5.1.1.1.1.1.m1.1.1">↑</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.5.1.1.1.1.1.m1.1c">\uparrow</annotation><annotation encoding="application/x-llamapun" id="S5.T1.5.1.1.1.1.1.m1.1d">↑</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.6.2.2.2" style="background-color:#FFFFFF;">SSIM <math alttext="\uparrow" class="ltx_Math" display="inline" id="S5.T1.6.2.2.2.1.1.m1.1" style="background-color:#FFFFFF;"><semantics id="S5.T1.6.2.2.2.1.1.m1.1a"><mo id="S5.T1.6.2.2.2.1.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="S5.T1.6.2.2.2.1.1.m1.1.1.cmml">↑</mo><annotation-xml encoding="MathML-Content" id="S5.T1.6.2.2.2.1.1.m1.1b"><ci id="S5.T1.6.2.2.2.1.1.m1.1.1.cmml" xref="S5.T1.6.2.2.2.1.1.m1.1.1">↑</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.6.2.2.2.1.1.m1.1c">\uparrow</annotation><annotation encoding="application/x-llamapun" id="S5.T1.6.2.2.2.1.1.m1.1d">↑</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.7.3.3.3" style="background-color:#FFFFFF;">LPIPS <math alttext="\downarrow" class="ltx_Math" display="inline" id="S5.T1.7.3.3.3.1.1.m1.1" style="background-color:#FFFFFF;"><semantics id="S5.T1.7.3.3.3.1.1.m1.1a"><mo id="S5.T1.7.3.3.3.1.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="S5.T1.7.3.3.3.1.1.m1.1.1.cmml">↓</mo><annotation-xml encoding="MathML-Content" id="S5.T1.7.3.3.3.1.1.m1.1b"><ci id="S5.T1.7.3.3.3.1.1.m1.1.1.cmml" xref="S5.T1.7.3.3.3.1.1.m1.1.1">↓</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.7.3.3.3.1.1.m1.1c">\downarrow</annotation><annotation encoding="application/x-llamapun" id="S5.T1.7.3.3.3.1.1.m1.1d">↓</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S5.T1.8.4.4.4" style="background-color:#FFFFFF;"> <table class="ltx_tabular ltx_align_middle" id="S5.T1.8.4.4.4.2" style="background-color:#FFFFFF;"> <tr class="ltx_tr" id="S5.T1.8.4.4.4.2.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T1.8.4.4.4.2.1.1">AbsRel</td> </tr> </table> <math alttext="\downarrow" class="ltx_Math" display="inline" id="S5.T1.8.4.4.4.1.m1.1" style="background-color:#FFFFFF;"><semantics id="S5.T1.8.4.4.4.1.m1.1a"><mo id="S5.T1.8.4.4.4.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="S5.T1.8.4.4.4.1.m1.1.1.cmml">↓</mo><annotation-xml encoding="MathML-Content" id="S5.T1.8.4.4.4.1.m1.1b"><ci id="S5.T1.8.4.4.4.1.m1.1.1.cmml" xref="S5.T1.8.4.4.4.1.m1.1.1">↓</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.8.4.4.4.1.m1.1c">\downarrow</annotation><annotation encoding="application/x-llamapun" id="S5.T1.8.4.4.4.1.m1.1d">↓</annotation></semantics></math> </td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.9.5.5.5" style="background-color:#FFFFFF;">PSNR*<math alttext="\uparrow" class="ltx_Math" display="inline" id="S5.T1.9.5.5.5.1.m1.1"><semantics id="S5.T1.9.5.5.5.1.m1.1a"><mo id="S5.T1.9.5.5.5.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="S5.T1.9.5.5.5.1.m1.1.1.cmml">↑</mo><annotation-xml encoding="MathML-Content" id="S5.T1.9.5.5.5.1.m1.1b"><ci id="S5.T1.9.5.5.5.1.m1.1.1.cmml" xref="S5.T1.9.5.5.5.1.m1.1.1">↑</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.9.5.5.5.1.m1.1c">\uparrow</annotation><annotation encoding="application/x-llamapun" id="S5.T1.9.5.5.5.1.m1.1d">↑</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.10.6.6.6" style="background-color:#FFFFFF;">SSIM <math alttext="\uparrow" class="ltx_Math" display="inline" id="S5.T1.10.6.6.6.1.1.m1.1" style="background-color:#FFFFFF;"><semantics id="S5.T1.10.6.6.6.1.1.m1.1a"><mo id="S5.T1.10.6.6.6.1.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="S5.T1.10.6.6.6.1.1.m1.1.1.cmml">↑</mo><annotation-xml encoding="MathML-Content" id="S5.T1.10.6.6.6.1.1.m1.1b"><ci id="S5.T1.10.6.6.6.1.1.m1.1.1.cmml" xref="S5.T1.10.6.6.6.1.1.m1.1.1">↑</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.10.6.6.6.1.1.m1.1c">\uparrow</annotation><annotation encoding="application/x-llamapun" id="S5.T1.10.6.6.6.1.1.m1.1d">↑</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.11.7.7.7" style="background-color:#FFFFFF;">LPIPS <math alttext="\downarrow" class="ltx_Math" display="inline" id="S5.T1.11.7.7.7.1.1.m1.1" style="background-color:#FFFFFF;"><semantics id="S5.T1.11.7.7.7.1.1.m1.1a"><mo id="S5.T1.11.7.7.7.1.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="S5.T1.11.7.7.7.1.1.m1.1.1.cmml">↓</mo><annotation-xml encoding="MathML-Content" id="S5.T1.11.7.7.7.1.1.m1.1b"><ci id="S5.T1.11.7.7.7.1.1.m1.1.1.cmml" xref="S5.T1.11.7.7.7.1.1.m1.1.1">↓</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.11.7.7.7.1.1.m1.1c">\downarrow</annotation><annotation encoding="application/x-llamapun" id="S5.T1.11.7.7.7.1.1.m1.1d">↓</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.12.8.8.8" style="background-color:#FFFFFF;"> <table class="ltx_tabular ltx_align_middle" id="S5.T1.12.8.8.8.1" style="background-color:#FFFFFF;"> <tr class="ltx_tr" id="S5.T1.12.8.8.8.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T1.12.8.8.8.1.1.1">Feat. Sim.</td> </tr> </table> <math alttext="\uparrow" class="ltx_Math" display="inline" id="S5.T1.12.8.8.8.m1.1" style="background-color:#FFFFFF;"><semantics id="S5.T1.12.8.8.8.m1.1a"><mo id="S5.T1.12.8.8.8.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="S5.T1.12.8.8.8.m1.1.1.cmml">↑</mo><annotation-xml encoding="MathML-Content" id="S5.T1.12.8.8.8.m1.1b"><ci id="S5.T1.12.8.8.8.m1.1.1.cmml" xref="S5.T1.12.8.8.8.m1.1.1">↑</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.12.8.8.8.m1.1c">\uparrow</annotation><annotation encoding="application/x-llamapun" id="S5.T1.12.8.8.8.m1.1d">↑</annotation></semantics></math> </td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.13.9.9.9" style="background-color:#FFFFFF;"> <table class="ltx_tabular ltx_align_middle" id="S5.T1.13.9.9.9.2" style="background-color:#FFFFFF;"> <tr class="ltx_tr" id="S5.T1.13.9.9.9.2.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T1.13.9.9.9.2.1.1">AbsRel</td> </tr> </table> <math alttext="\downarrow" class="ltx_Math" display="inline" id="S5.T1.13.9.9.9.1.m1.1" style="background-color:#FFFFFF;"><semantics id="S5.T1.13.9.9.9.1.m1.1a"><mo id="S5.T1.13.9.9.9.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="S5.T1.13.9.9.9.1.m1.1.1.cmml">↓</mo><annotation-xml encoding="MathML-Content" id="S5.T1.13.9.9.9.1.m1.1b"><ci id="S5.T1.13.9.9.9.1.m1.1.1.cmml" xref="S5.T1.13.9.9.9.1.m1.1.1">↓</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.13.9.9.9.1.m1.1c">\downarrow</annotation><annotation encoding="application/x-llamapun" id="S5.T1.13.9.9.9.1.m1.1d">↓</annotation></semantics></math> </td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.14.10.10.10" style="background-color:#FFFFFF;"> <table class="ltx_tabular ltx_align_middle" id="S5.T1.14.10.10.10.2" style="background-color:#FFFFFF;"> <tr class="ltx_tr" id="S5.T1.14.10.10.10.2.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T1.14.10.10.10.2.1.1">Delta1</td> </tr> </table> <math alttext="\uparrow" class="ltx_Math" display="inline" id="S5.T1.14.10.10.10.1.m1.1" style="background-color:#FFFFFF;"><semantics id="S5.T1.14.10.10.10.1.m1.1a"><mo id="S5.T1.14.10.10.10.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="S5.T1.14.10.10.10.1.m1.1.1.cmml">↑</mo><annotation-xml encoding="MathML-Content" id="S5.T1.14.10.10.10.1.m1.1b"><ci id="S5.T1.14.10.10.10.1.m1.1.1.cmml" xref="S5.T1.14.10.10.10.1.m1.1.1">↑</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.14.10.10.10.1.m1.1c">\uparrow</annotation><annotation encoding="application/x-llamapun" id="S5.T1.14.10.10.10.1.m1.1d">↑</annotation></semantics></math> </td> </tr> <tr class="ltx_tr" id="S5.T1.16.12.14.2"> <td class="ltx_td ltx_border_t" id="S5.T1.16.12.14.2.1"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S5.T1.16.12.14.2.2" style="background-color:#FFFFFF;">3DGS <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib17" title="">17</a>]</cite></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.16.12.14.2.3" style="background-color:#FFFFFF;">25.40</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.16.12.14.2.4" style="background-color:#FFFFFF;">0.775</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.16.12.14.2.5" style="background-color:#FFFFFF;">0.299</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S5.T1.16.12.14.2.6" style="background-color:#FFFFFF;">0.256</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.16.12.14.2.7" style="background-color:#FFFFFF;">19.15</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.16.12.14.2.8" style="background-color:#FFFFFF;">0.570</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.16.12.14.2.9" style="background-color:#FFFFFF;">0.414</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.16.12.14.2.10" style="background-color:#FFFFFF;">0.514</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.16.12.14.2.11" style="background-color:#FFFFFF;">0.285</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.16.12.14.2.12" style="background-color:#FFFFFF;">0.437</td> </tr> <tr class="ltx_tr" id="S5.T1.16.12.15.3"> <td class="ltx_td" id="S5.T1.16.12.15.3.1"></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T1.16.12.15.3.2" style="background-color:#FFFFFF;">StreetGS <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib45" title="">45</a>]</cite></td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.15.3.3" style="background-color:#FFFFFF;">23.32</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.15.3.4" style="background-color:#FFFFFF;">0.852</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.15.3.5" style="background-color:#FFFFFF;">0.304</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T1.16.12.15.3.6" style="background-color:#FFFFFF;">0.080</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.15.3.7" style="background-color:#FFFFFF;">17.39</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.15.3.8" style="background-color:#FFFFFF;">0.473</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.15.3.9" style="background-color:#FFFFFF;">0.479</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.15.3.10" style="background-color:#FFFFFF;">0.558</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.15.3.11" style="background-color:#FFFFFF;">0.157</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.15.3.12" style="background-color:#FFFFFF;">0.815</td> </tr> <tr class="ltx_tr" id="S5.T1.16.12.16.4"> <td class="ltx_td" id="S5.T1.16.12.16.4.1"></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T1.16.12.16.4.2" style="background-color:#FFFFFF;">OmniRe <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib3" title="">3</a>]</cite></td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.16.4.3" style="background-color:#FFFFFF;">23.64</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.16.4.4" style="background-color:#FFFFFF;">0.865</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.16.4.5" style="background-color:#FFFFFF;">0.283</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T1.16.12.16.4.6" style="background-color:#FFFFFF;">0.081</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.16.4.7" style="background-color:#FFFFFF;">17.34</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.16.4.8" style="background-color:#FFFFFF;">0.466</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.16.4.9" style="background-color:#FFFFFF;">0.474</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.16.4.10" style="background-color:#FFFFFF;">0.560</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.16.4.11" style="background-color:#FFFFFF;">0.162</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.16.4.12" style="background-color:#FFFFFF;">0.805</td> </tr> <tr class="ltx_tr" id="S5.T1.16.12.17.5"> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.17.5.1"> ST </td> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T1.16.12.17.5.2" style="background-color:#FFFFFF;">Ours</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.17.5.3" style="background-color:#FFFFFF;">29.43</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.17.5.4" style="background-color:#FFFFFF;">0.879</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.17.5.5" style="background-color:#FFFFFF;">0.150</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T1.16.12.17.5.6" style="background-color:#FFFFFF;">0.094</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.17.5.7" style="background-color:#FFFFFF;">20.11</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.17.5.8" style="background-color:#FFFFFF;">0.575</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.17.5.9" style="background-color:#FFFFB2;">0.313</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.17.5.10" style="background-color:#FFFFB2;">0.614</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.17.5.11" style="background-color:#FFFFFF;">0.145</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.17.5.12" style="background-color:#FFFFB2;">0.879</td> </tr> <tr class="ltx_tr" id="S5.T1.16.12.18.6"> <td class="ltx_td ltx_border_t" id="S5.T1.16.12.18.6.1"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S5.T1.16.12.18.6.2">3DGS <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib17" title="">17</a>]</cite> </td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.16.12.18.6.3" style="background-color:#FFFFB2;">22.04</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.16.12.18.6.4" style="background-color:#FFFFFF;">0.705</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.16.12.18.6.5" style="background-color:#FFFFB2;">0.390</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S5.T1.16.12.18.6.6" style="background-color:#FFFFFF;">0.332</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.16.12.18.6.7" style="background-color:#FFFFB2;">20.53</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.16.12.18.6.8" style="background-color:#FFFFB2;">0.614</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.16.12.18.6.9" style="background-color:#FFFFFF;">0.388</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.16.12.18.6.10" style="background-color:#FFFFFF;">0.557</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.16.12.18.6.11" style="background-color:#FFFFFF;">0.347</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.16.12.18.6.12" style="background-color:#FFFFFF;">0.312</td> </tr> <tr class="ltx_tr" id="S5.T1.15.11.11"> <td class="ltx_td" id="S5.T1.15.11.11.2"></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T1.15.11.11.1">StreetGS<math alttext="\dagger" class="ltx_Math" display="inline" id="S5.T1.15.11.11.1.m1.1"><semantics id="S5.T1.15.11.11.1.m1.1a"><mo id="S5.T1.15.11.11.1.m1.1.1" xref="S5.T1.15.11.11.1.m1.1.1.cmml">†</mo><annotation-xml encoding="MathML-Content" id="S5.T1.15.11.11.1.m1.1b"><ci id="S5.T1.15.11.11.1.m1.1.1.cmml" xref="S5.T1.15.11.11.1.m1.1.1">†</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.15.11.11.1.m1.1c">\dagger</annotation><annotation encoding="application/x-llamapun" id="S5.T1.15.11.11.1.m1.1d">†</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib45" title="">45</a>]</cite> </td> <td class="ltx_td ltx_align_center" id="S5.T1.15.11.11.3" style="background-color:#FFFFFF;">20.57</td> <td class="ltx_td ltx_align_center" id="S5.T1.15.11.11.4" style="background-color:#FFFFFF;">0.736</td> <td class="ltx_td ltx_align_center" id="S5.T1.15.11.11.5" style="background-color:#FFFFFF;">0.447</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T1.15.11.11.6" style="background-color:#FFFFB2;">0.097</td> <td class="ltx_td ltx_align_center" id="S5.T1.15.11.11.7" style="background-color:#FFFFFF;">18.18</td> <td class="ltx_td ltx_align_center" id="S5.T1.15.11.11.8" style="background-color:#FFFFFF;">0.527</td> <td class="ltx_td ltx_align_center" id="S5.T1.15.11.11.9" style="background-color:#FFFFFF;">0.488</td> <td class="ltx_td ltx_align_center" id="S5.T1.15.11.11.10" style="background-color:#FFFFFF;">0.577</td> <td class="ltx_td ltx_align_center" id="S5.T1.15.11.11.11" style="background-color:#FFFFFF;">0.148</td> <td class="ltx_td ltx_align_center" id="S5.T1.15.11.11.12" style="background-color:#FFFFFF;">0.826</td> </tr> <tr class="ltx_tr" id="S5.T1.16.12.12"> <td class="ltx_td" id="S5.T1.16.12.12.2"></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T1.16.12.12.1">OmniRe<math alttext="\dagger" class="ltx_Math" display="inline" id="S5.T1.16.12.12.1.m1.1"><semantics id="S5.T1.16.12.12.1.m1.1a"><mo id="S5.T1.16.12.12.1.m1.1.1" xref="S5.T1.16.12.12.1.m1.1.1.cmml">†</mo><annotation-xml encoding="MathML-Content" id="S5.T1.16.12.12.1.m1.1b"><ci id="S5.T1.16.12.12.1.m1.1.1.cmml" xref="S5.T1.16.12.12.1.m1.1.1">†</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.16.12.12.1.m1.1c">\dagger</annotation><annotation encoding="application/x-llamapun" id="S5.T1.16.12.12.1.m1.1d">†</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib3" title="">3</a>]</cite> </td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.12.3" style="background-color:#FFFFFF;">20.91</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.12.4" style="background-color:#FFFFB2;">0.755</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.12.5" style="background-color:#FFFFFF;">0.409</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T1.16.12.12.6" style="background-color:#FFB2B2;">0.092</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.12.7" style="background-color:#FFFFFF;">18.36</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.12.8" style="background-color:#FFFFFF;">0.527</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.12.9" style="background-color:#FFFFFF;">0.460</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.12.10" style="background-color:#FFFFFF;">0.594</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.12.11" style="background-color:#FFFFB2;">0.136</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.12.12" style="background-color:#FFFFFF;">0.859</td> </tr> <tr class="ltx_tr" id="S5.T1.16.12.19.7"> <td class="ltx_td" id="S5.T1.16.12.19.7.1"></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T1.16.12.19.7.2">Ours</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.19.7.3" style="background-color:#FFD9B2;">28.04</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.19.7.4" style="background-color:#FFD9B2;">0.848</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.19.7.5" style="background-color:#FFD9B2;">0.192</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T1.16.12.19.7.6" style="background-color:#FFD9B2;">0.094</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.19.7.7" style="background-color:#FFB2B2;">21.65</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.19.7.8" style="background-color:#FFB2B2;">0.628</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.19.7.9" style="background-color:#FFD9B2;">0.265</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.19.7.10" style="background-color:#FFD9B2;">0.670</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.19.7.11" style="background-color:#FFB2B2;">0.089</td> <td class="ltx_td ltx_align_center" id="S5.T1.16.12.19.7.12" style="background-color:#FFB2B2;">0.904</td> </tr> <tr class="ltx_tr" id="S5.T1.16.12.20.8"> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T1.16.12.20.8.1"> MT </td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_r" id="S5.T1.16.12.20.8.2">Ours (60k)</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T1.16.12.20.8.3" style="background-color:#FFB2B2;">28.73</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T1.16.12.20.8.4" style="background-color:#FFB2B2;">0.865</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T1.16.12.20.8.5" style="background-color:#FFB2B2;">0.169</td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_r" id="S5.T1.16.12.20.8.6" style="background-color:#FFD9B2;">0.094</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T1.16.12.20.8.7" style="background-color:#FFD9B2;">21.58</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T1.16.12.20.8.8" style="background-color:#FFD9B2;">0.620</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T1.16.12.20.8.9" style="background-color:#FFB2B2;">0.254</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T1.16.12.20.8.10" style="background-color:#FFB2B2;">0.676</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T1.16.12.20.8.11" style="background-color:#FFD9B2;">0.091</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T1.16.12.20.8.12" style="background-color:#FFD9B2;">0.902</td> </tr> </tbody> </table> </div> </figure> <figure class="ltx_figure" id="S5.F3"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="317" id="S5.F3.g1" src="x3.png" width="466"/> <figcaption class="ltx_caption">Figure 3: Novel-view performance when trained with more traversals. More traversals do not guarantee improvement on existing methods, while our design unleashes their significance. *: affine-aligned PSNR. </figcaption> </figure> <figure class="ltx_table" id="S5.T2"> <figcaption class="ltx_caption ltx_centering">Table 2: Ablation on appearance modeling. The full model in ID 5 validates the effectiveness of our designs, as well as outperforming existing methods in handling the appearance variations <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib19" title="">19</a>]</cite>. ‘CamAFF’ refers to per-camera affine, ‘LEA’ denotes LiDAR exposure alignment, and ‘Appr.Node’ represents the appearance node. First, second, third. </figcaption> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="S5.T2.10" style="width:496.9pt;height:117.7pt;vertical-align:-0.0pt;"> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="S5.T2.10.10"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S5.T2.10.10.11.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_tt" id="S5.T2.10.10.11.1.1" rowspan="2">ID</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_tt" colspan="3" id="S5.T2.10.10.11.1.2">Module</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_tt" colspan="4" id="S5.T2.10.10.11.1.3">Training Traversal</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" colspan="6" id="S5.T2.10.10.11.1.4">Novel-View Traversal</th> </tr> <tr class="ltx_tr" id="S5.T2.10.10.10"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.10.10.10.11" style="background-color:#FFFFFF;"> <table class="ltx_tabular ltx_align_middle" id="S5.T2.10.10.10.11.1" style="background-color:#FFFFFF;"> <tr class="ltx_tr" id="S5.T2.10.10.10.11.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T2.10.10.10.11.1.1.1">CamAFF</td> </tr> </table> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.10.10.10.12" style="background-color:#FFFFFF;"> <table class="ltx_tabular ltx_align_middle" id="S5.T2.10.10.10.12.1" style="background-color:#FFFFFF;"> <tr class="ltx_tr" id="S5.T2.10.10.10.12.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T2.10.10.10.12.1.1.1">LEA</td> </tr> </table> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S5.T2.10.10.10.13" style="background-color:#FFFFFF;">Appr. Node</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.1.1.1.1" style="background-color:#FFFFFF;">PSNR <math alttext="\uparrow" class="ltx_Math" display="inline" id="S5.T2.1.1.1.1.1.1.m1.1" style="background-color:#FFFFFF;"><semantics id="S5.T2.1.1.1.1.1.1.m1.1a"><mo id="S5.T2.1.1.1.1.1.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="S5.T2.1.1.1.1.1.1.m1.1.1.cmml">↑</mo><annotation-xml encoding="MathML-Content" id="S5.T2.1.1.1.1.1.1.m1.1b"><ci id="S5.T2.1.1.1.1.1.1.m1.1.1.cmml" xref="S5.T2.1.1.1.1.1.1.m1.1.1">↑</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.1.1.1.1.1.1.m1.1c">\uparrow</annotation><annotation encoding="application/x-llamapun" id="S5.T2.1.1.1.1.1.1.m1.1d">↑</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.2.2.2.2" style="background-color:#FFFFFF;">SSIM <math alttext="\uparrow" class="ltx_Math" display="inline" id="S5.T2.2.2.2.2.1.1.m1.1" style="background-color:#FFFFFF;"><semantics id="S5.T2.2.2.2.2.1.1.m1.1a"><mo id="S5.T2.2.2.2.2.1.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="S5.T2.2.2.2.2.1.1.m1.1.1.cmml">↑</mo><annotation-xml encoding="MathML-Content" id="S5.T2.2.2.2.2.1.1.m1.1b"><ci id="S5.T2.2.2.2.2.1.1.m1.1.1.cmml" xref="S5.T2.2.2.2.2.1.1.m1.1.1">↑</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.2.2.2.2.1.1.m1.1c">\uparrow</annotation><annotation encoding="application/x-llamapun" id="S5.T2.2.2.2.2.1.1.m1.1d">↑</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.3.3.3.3" style="background-color:#FFFFFF;">LPIPS <math alttext="\downarrow" class="ltx_Math" display="inline" id="S5.T2.3.3.3.3.1.1.m1.1" style="background-color:#FFFFFF;"><semantics id="S5.T2.3.3.3.3.1.1.m1.1a"><mo id="S5.T2.3.3.3.3.1.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="S5.T2.3.3.3.3.1.1.m1.1.1.cmml">↓</mo><annotation-xml encoding="MathML-Content" id="S5.T2.3.3.3.3.1.1.m1.1b"><ci id="S5.T2.3.3.3.3.1.1.m1.1.1.cmml" xref="S5.T2.3.3.3.3.1.1.m1.1.1">↓</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.3.3.3.3.1.1.m1.1c">\downarrow</annotation><annotation encoding="application/x-llamapun" id="S5.T2.3.3.3.3.1.1.m1.1d">↓</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S5.T2.4.4.4.4" style="background-color:#FFFFFF;"> <table class="ltx_tabular ltx_align_middle" id="S5.T2.4.4.4.4.2" style="background-color:#FFFFFF;"> <tr class="ltx_tr" id="S5.T2.4.4.4.4.2.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T2.4.4.4.4.2.1.1">AbsRel</td> </tr> </table> <math alttext="\downarrow" class="ltx_Math" display="inline" id="S5.T2.4.4.4.4.1.m1.1" style="background-color:#FFFFFF;"><semantics id="S5.T2.4.4.4.4.1.m1.1a"><mo id="S5.T2.4.4.4.4.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="S5.T2.4.4.4.4.1.m1.1.1.cmml">↓</mo><annotation-xml encoding="MathML-Content" id="S5.T2.4.4.4.4.1.m1.1b"><ci id="S5.T2.4.4.4.4.1.m1.1.1.cmml" xref="S5.T2.4.4.4.4.1.m1.1.1">↓</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.4.4.4.4.1.m1.1c">\downarrow</annotation><annotation encoding="application/x-llamapun" id="S5.T2.4.4.4.4.1.m1.1d">↓</annotation></semantics></math> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.5.5.5.5" style="background-color:#FFFFFF;">PSNR*<math alttext="\uparrow" class="ltx_Math" display="inline" id="S5.T2.5.5.5.5.1.m1.1"><semantics id="S5.T2.5.5.5.5.1.m1.1a"><mo id="S5.T2.5.5.5.5.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="S5.T2.5.5.5.5.1.m1.1.1.cmml">↑</mo><annotation-xml encoding="MathML-Content" id="S5.T2.5.5.5.5.1.m1.1b"><ci id="S5.T2.5.5.5.5.1.m1.1.1.cmml" xref="S5.T2.5.5.5.5.1.m1.1.1">↑</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.5.5.5.5.1.m1.1c">\uparrow</annotation><annotation encoding="application/x-llamapun" id="S5.T2.5.5.5.5.1.m1.1d">↑</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.6.6.6.6" style="background-color:#FFFFFF;">SSIM <math alttext="\uparrow" class="ltx_Math" display="inline" id="S5.T2.6.6.6.6.1.1.m1.1" style="background-color:#FFFFFF;"><semantics id="S5.T2.6.6.6.6.1.1.m1.1a"><mo id="S5.T2.6.6.6.6.1.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="S5.T2.6.6.6.6.1.1.m1.1.1.cmml">↑</mo><annotation-xml encoding="MathML-Content" id="S5.T2.6.6.6.6.1.1.m1.1b"><ci id="S5.T2.6.6.6.6.1.1.m1.1.1.cmml" xref="S5.T2.6.6.6.6.1.1.m1.1.1">↑</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.6.6.6.6.1.1.m1.1c">\uparrow</annotation><annotation encoding="application/x-llamapun" id="S5.T2.6.6.6.6.1.1.m1.1d">↑</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.7.7.7.7" style="background-color:#FFFFFF;">LPIPS <math alttext="\downarrow" class="ltx_Math" display="inline" id="S5.T2.7.7.7.7.1.1.m1.1" style="background-color:#FFFFFF;"><semantics id="S5.T2.7.7.7.7.1.1.m1.1a"><mo id="S5.T2.7.7.7.7.1.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="S5.T2.7.7.7.7.1.1.m1.1.1.cmml">↓</mo><annotation-xml encoding="MathML-Content" id="S5.T2.7.7.7.7.1.1.m1.1b"><ci id="S5.T2.7.7.7.7.1.1.m1.1.1.cmml" xref="S5.T2.7.7.7.7.1.1.m1.1.1">↓</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.7.7.7.7.1.1.m1.1c">\downarrow</annotation><annotation encoding="application/x-llamapun" id="S5.T2.7.7.7.7.1.1.m1.1d">↓</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.8.8.8.8" style="background-color:#FFFFFF;"> <table class="ltx_tabular ltx_align_middle" id="S5.T2.8.8.8.8.1" style="background-color:#FFFFFF;"> <tr class="ltx_tr" id="S5.T2.8.8.8.8.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T2.8.8.8.8.1.1.1">Feat. Sim.</td> </tr> </table> <math alttext="\uparrow" class="ltx_Math" display="inline" id="S5.T2.8.8.8.8.m1.1" style="background-color:#FFFFFF;"><semantics id="S5.T2.8.8.8.8.m1.1a"><mo id="S5.T2.8.8.8.8.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="S5.T2.8.8.8.8.m1.1.1.cmml">↑</mo><annotation-xml encoding="MathML-Content" id="S5.T2.8.8.8.8.m1.1b"><ci id="S5.T2.8.8.8.8.m1.1.1.cmml" xref="S5.T2.8.8.8.8.m1.1.1">↑</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.8.8.8.8.m1.1c">\uparrow</annotation><annotation encoding="application/x-llamapun" id="S5.T2.8.8.8.8.m1.1d">↑</annotation></semantics></math> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.9.9.9.9" style="background-color:#FFFFFF;"> <table class="ltx_tabular ltx_align_middle" id="S5.T2.9.9.9.9.2" style="background-color:#FFFFFF;"> <tr class="ltx_tr" id="S5.T2.9.9.9.9.2.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T2.9.9.9.9.2.1.1">AbsRel</td> </tr> </table> <math alttext="\downarrow" class="ltx_Math" display="inline" id="S5.T2.9.9.9.9.1.m1.1" style="background-color:#FFFFFF;"><semantics id="S5.T2.9.9.9.9.1.m1.1a"><mo id="S5.T2.9.9.9.9.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="S5.T2.9.9.9.9.1.m1.1.1.cmml">↓</mo><annotation-xml encoding="MathML-Content" id="S5.T2.9.9.9.9.1.m1.1b"><ci id="S5.T2.9.9.9.9.1.m1.1.1.cmml" xref="S5.T2.9.9.9.9.1.m1.1.1">↓</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.9.9.9.9.1.m1.1c">\downarrow</annotation><annotation encoding="application/x-llamapun" id="S5.T2.9.9.9.9.1.m1.1d">↓</annotation></semantics></math> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.10.10.10.10" style="background-color:#FFFFFF;"> <table class="ltx_tabular ltx_align_middle" id="S5.T2.10.10.10.10.2" style="background-color:#FFFFFF;"> <tr class="ltx_tr" id="S5.T2.10.10.10.10.2.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T2.10.10.10.10.2.1.1">Delta1</td> </tr> </table> <math alttext="\uparrow" class="ltx_Math" display="inline" id="S5.T2.10.10.10.10.1.m1.1" style="background-color:#FFFFFF;"><semantics id="S5.T2.10.10.10.10.1.m1.1a"><mo id="S5.T2.10.10.10.10.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="S5.T2.10.10.10.10.1.m1.1.1.cmml">↑</mo><annotation-xml encoding="MathML-Content" id="S5.T2.10.10.10.10.1.m1.1b"><ci id="S5.T2.10.10.10.10.1.m1.1.1.cmml" xref="S5.T2.10.10.10.10.1.m1.1.1">↑</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.10.10.10.10.1.m1.1c">\uparrow</annotation><annotation encoding="application/x-llamapun" id="S5.T2.10.10.10.10.1.m1.1d">↑</annotation></semantics></math> </th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S5.T2.10.10.12.1"> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S5.T2.10.10.12.1.1">0</td> <td class="ltx_td ltx_border_t" id="S5.T2.10.10.12.1.2"></td> <td class="ltx_td ltx_border_t" id="S5.T2.10.10.12.1.3"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S5.T2.10.10.12.1.4"></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T2.10.10.12.1.5" style="background-color:#FFFFFF;">23.46</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T2.10.10.12.1.6" style="background-color:#FFFFFF;">0.784</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T2.10.10.12.1.7" style="background-color:#FFFFFF;">0.249</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S5.T2.10.10.12.1.8" style="background-color:#FFFFFF;">0.105</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T2.10.10.12.1.9" style="background-color:#FFFFFF;">19.66</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T2.10.10.12.1.10" style="background-color:#FFFFFF;">0.582</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T2.10.10.12.1.11" style="background-color:#FFFFFF;">0.308</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T2.10.10.12.1.12" style="background-color:#FFFFFF;">0.612</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T2.10.10.12.1.13" style="background-color:#FFFFFF;">0.094</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T2.10.10.12.1.14" style="background-color:#FFFFFF;">0.901</td> </tr> <tr class="ltx_tr" id="S5.T2.10.10.13.2"> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.10.10.13.2.1">1</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.13.2.2" style="background-color:#FFFFFF;">✓</td> <td class="ltx_td" id="S5.T2.10.10.13.2.3"></td> <td class="ltx_td ltx_border_r" id="S5.T2.10.10.13.2.4"></td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.13.2.5" style="background-color:#FFFFFF;">24.95</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.13.2.6" style="background-color:#FFFFFF;">0.799</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.13.2.7" style="background-color:#FFFFFF;">0.229</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.10.10.13.2.8" style="background-color:#FFFFFF;">0.090</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.13.2.9" style="background-color:#FFFFFF;">20.05</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.13.2.10" style="background-color:#FFFFFF;">0.589</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.13.2.11" style="background-color:#FFFFFF;">0.293</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.13.2.12" style="background-color:#FFFFFF;">0.628</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.13.2.13" style="background-color:#FFFFB2;">0.082</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.13.2.14" style="background-color:#FFD9B2;">0.912</td> </tr> <tr class="ltx_tr" id="S5.T2.10.10.14.3"> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.10.10.14.3.1">2</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.14.3.2" style="background-color:#FFFFFF;">✓</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.14.3.3" style="background-color:#FFFFFF;">✓</td> <td class="ltx_td ltx_border_r" id="S5.T2.10.10.14.3.4"></td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.14.3.5" style="background-color:#FFFFFF;">26.16</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.14.3.6" style="background-color:#FFFFFF;">0.818</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.14.3.7" style="background-color:#FFFFFF;">0.219</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.10.10.14.3.8" style="background-color:#FFFFB2;">0.086</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.14.3.9" style="background-color:#FFB2B2;">20.85</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.14.3.10" style="background-color:#FFB2B2;">0.615</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.14.3.11" style="background-color:#FFFFB2;">0.281</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.14.3.12" style="background-color:#FFFFFF;">0.633</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.14.3.13" style="background-color:#FFD9B2;">0.078</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.14.3.14" style="background-color:#FFB2B2;">0.914</td> </tr> <tr class="ltx_tr" id="S5.T2.10.10.15.4"> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.10.10.15.4.1">3</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.15.4.2" style="background-color:#FFFFFF;">✓</td> <td class="ltx_td" id="S5.T2.10.10.15.4.3"></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.10.10.15.4.4" style="background-color:#FFFFFF;">✓</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.15.4.5" style="background-color:#FFFFB2;">27.08</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.15.4.6" style="background-color:#FFFFB2;">0.838</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.15.4.7" style="background-color:#FFFFB2;">0.199</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.10.10.15.4.8" style="background-color:#FFFFFF;">0.087</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.15.4.9" style="background-color:#FFFFFF;">20.02</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.15.4.10" style="background-color:#FFFFFF;">0.586</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.15.4.11" style="background-color:#FFFFFF;">0.288</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.15.4.12" style="background-color:#FFFFB2;">0.638</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.15.4.13" style="background-color:#FFD9B2;">0.078</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.15.4.14" style="background-color:#FFB2B2;">0.914</td> </tr> <tr class="ltx_tr" id="S5.T2.10.10.16.5"> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.10.10.16.5.1">4</td> <td class="ltx_td" id="S5.T2.10.10.16.5.2"></td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.16.5.3" style="background-color:#FFFFFF;">✓</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.10.10.16.5.4" style="background-color:#FFFFFF;">✓</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.16.5.5" style="background-color:#FFD9B2;">27.59</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.16.5.6" style="background-color:#FFD9B2;">0.853</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.16.5.7" style="background-color:#FFD9B2;">0.190</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.10.10.16.5.8" style="background-color:#FFB2B2;">0.084</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.16.5.9" style="background-color:#FFFFB2;">20.79</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.16.5.10" style="background-color:#FFD9B2;">0.612</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.16.5.11" style="background-color:#FFD9B2;">0.274</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.16.5.12" style="background-color:#FFD9B2;">0.641</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.16.5.13" style="background-color:#FFB2B2;">0.077</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.16.5.14" style="background-color:#FFB2B2;">0.914</td> </tr> <tr class="ltx_tr" id="S5.T2.10.10.17.6"> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.10.10.17.6.1">5</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.17.6.2" style="background-color:#FFFFFF;">✓</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.17.6.3" style="background-color:#FFFFFF;">✓</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.10.10.17.6.4" style="background-color:#FFFFFF;">✓</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.17.6.5" style="background-color:#FFB2B2;">28.51</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.17.6.6" style="background-color:#FFB2B2;">0.859</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.17.6.7" style="background-color:#FFB2B2;">0.179</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.10.10.17.6.8" style="background-color:#FFD9B2;">0.085</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.17.6.9" style="background-color:#FFD9B2;">20.83</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.17.6.10" style="background-color:#FFFFB2;">0.611</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.17.6.11" style="background-color:#FFB2B2;">0.271</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.17.6.12" style="background-color:#FFB2B2;">0.646</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.17.6.13" style="background-color:#FFD9B2;">0.078</td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.17.6.14" style="background-color:#FFB2B2;">0.914</td> </tr> <tr class="ltx_tr" id="S5.T2.10.10.18.7"> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t" id="S5.T2.10.10.18.7.1">6</td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t" colspan="3" id="S5.T2.10.10.18.7.2">WildGaussians <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib19" title="">19</a>]</cite> </td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="S5.T2.10.10.18.7.3" style="background-color:#FFFFFF;">25.20</td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="S5.T2.10.10.18.7.4" style="background-color:#FFFFFF;">0.805</td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="S5.T2.10.10.18.7.5" style="background-color:#FFFFFF;">0.229</td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t" id="S5.T2.10.10.18.7.6" style="background-color:#FFFFFF;">0.096</td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="S5.T2.10.10.18.7.7" style="background-color:#FFFFFF;">19.88</td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="S5.T2.10.10.18.7.8" style="background-color:#FFFFFF;">0.577</td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="S5.T2.10.10.18.7.9" style="background-color:#FFFFFF;">0.300</td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="S5.T2.10.10.18.7.10" style="background-color:#FFFFFF;">0.618</td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="S5.T2.10.10.18.7.11" style="background-color:#FFFFFF;">0.087</td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="S5.T2.10.10.18.7.12" style="background-color:#FFFFB2;">0.909</td> </tr> </tbody> </table> </div> </figure> <div class="ltx_para" id="S5.SS1.p1"> Dataset. The experiments are conducted on dedicated multi-traversal data extracted from nuPlan <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib16" title="">16</a>]</cite>. This large-scale driving dataset comprises over 100 hours of data, featuring eight surrounding-view images captured at 10 Hz and point clouds merged from 5 LiDAR sensors at 20 Hz. We use all eight views and LiDAR at 10 Hz, with the resolution of <math alttext="960\times 540" class="ltx_Math" display="inline" id="S5.SS1.p1.1.m1.1"><semantics id="S5.SS1.p1.1.m1.1a"><mrow id="S5.SS1.p1.1.m1.1.1" xref="S5.SS1.p1.1.m1.1.1.cmml"><mn id="S5.SS1.p1.1.m1.1.1.2" xref="S5.SS1.p1.1.m1.1.1.2.cmml">960</mn><mo id="S5.SS1.p1.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5.SS1.p1.1.m1.1.1.1.cmml">×</mo><mn id="S5.SS1.p1.1.m1.1.1.3" xref="S5.SS1.p1.1.m1.1.1.3.cmml">540</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.1.m1.1b"><apply id="S5.SS1.p1.1.m1.1.1.cmml" xref="S5.SS1.p1.1.m1.1.1"><times id="S5.SS1.p1.1.m1.1.1.1.cmml" xref="S5.SS1.p1.1.m1.1.1.1"></times><cn id="S5.SS1.p1.1.m1.1.1.2.cmml" type="integer" xref="S5.SS1.p1.1.m1.1.1.2">960</cn><cn id="S5.SS1.p1.1.m1.1.1.3.cmml" type="integer" xref="S5.SS1.p1.1.m1.1.1.3">540</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.1.m1.1c">960\times 540</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.1.m1.1d">960 × 540</annotation></semantics></math> for images across training and evaluation. </div> <div class="ltx_para" id="S5.SS1.p2"> We select six road blocks with multi-traversal data distributed across multiple lanes and evaluate one isolated traversal with minimal spatial overlap with others. During evaluation, non-rigid dynamics are ignored. All transient elements are masked when assessing novel-view traversals, as they are entirely unseen in training. </div> <div class="ltx_para ltx_noindent" id="S5.SS1.p3"> Implementation Details. Our method is implemented upon open-source repositories, nerfstudio and gsplat <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib34" title="">34</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib48" title="">48</a>]</cite>. As for unseen traversals, the appearance node of its nearest training traversal is used for appearance tuning. We select three baselines, 3DGS <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib17" title="">17</a>]</cite>, Street Gaussians <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib45" title="">45</a>]</cite>, and OmniRe <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib3" title="">3</a>]</cite>. The 3DGS baseline is implemented in gsplat, while other baselines are adapted from OmniRe’s codebase. By default, we train all methods with 30k steps using Adam optimizers. For details, please refer to the supplementary. </div> <div class="ltx_para ltx_noindent" id="S5.SS1.p4"> Metrics. We compute metrics on three aspects. All the metrics are adapted to support calculating with masks. <ul class="ltx_itemize" id="S5.I1"> <li class="ltx_item" id="S5.I1.i1" style="list-style-type:none;"> • <div class="ltx_para" id="S5.I1.i1.p1"> Pixel-level metrics. We use peak signal-to-noise ratio (PSNR), structural similarity index measure (SSIM) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib38" title="">38</a>]</cite>, and affine-aligned PSNR <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib1" title="">1</a>]</cite> for novel-view traversals. </div> </li> <li class="ltx_item" id="S5.I1.i2" style="list-style-type:none;"> • <div class="ltx_para" id="S5.I1.i2.p1"> Feature-level metrics. We employ learned perceptual image patch similarity (LPIPS) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib51" title="">51</a>]</cite> and DINOv2 <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib28" title="">28</a>]</cite> feature cosine similarity (Feat. Sim.), which matters more to the downstream visual models <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib22" title="">22</a>]</cite>. </div> </li> <li class="ltx_item" id="S5.I1.i3" style="list-style-type:none;"> • <div class="ltx_para" id="S5.I1.i3.p1"> Geometry-level metrics. We evaluate geometry accuracy with depth-related metrics, including the absolute relative error and <math alttext="\delta_{1.25}" class="ltx_Math" display="inline" id="S5.I1.i3.p1.1.m1.1"><semantics id="S5.I1.i3.p1.1.m1.1a"><msub id="S5.I1.i3.p1.1.m1.1.1" xref="S5.I1.i3.p1.1.m1.1.1.cmml"><mi id="S5.I1.i3.p1.1.m1.1.1.2" xref="S5.I1.i3.p1.1.m1.1.1.2.cmml">δ</mi><mn id="S5.I1.i3.p1.1.m1.1.1.3" xref="S5.I1.i3.p1.1.m1.1.1.3.cmml">1.25</mn></msub><annotation-xml encoding="MathML-Content" id="S5.I1.i3.p1.1.m1.1b"><apply id="S5.I1.i3.p1.1.m1.1.1.cmml" xref="S5.I1.i3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.I1.i3.p1.1.m1.1.1.1.cmml" xref="S5.I1.i3.p1.1.m1.1.1">subscript</csymbol><ci id="S5.I1.i3.p1.1.m1.1.1.2.cmml" xref="S5.I1.i3.p1.1.m1.1.1.2">𝛿</ci><cn id="S5.I1.i3.p1.1.m1.1.1.3.cmml" type="float" xref="S5.I1.i3.p1.1.m1.1.1.3">1.25</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i3.p1.1.m1.1c">\delta_{1.25}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i3.p1.1.m1.1d">italic_δ start_POSTSUBSCRIPT 1.25 end_POSTSUBSCRIPT</annotation></semantics></math> (delta 1), between the rendered depth and projected LiDAR depth within an 80-meter range. </div> </li> </ul> </div> </section> <section class="ltx_subsection" id="S5.SS2"> <h3 class="ltx_title ltx_title_subsection"> 5.2 Main Results</h3> <div class="ltx_para" id="S5.SS2.p1"> We show results in both single-traversal (ST) and multi-traversal (MT) settings in <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#S5.T1" title="In 5.1 Setup and Protocols ‣ 5 Experiment ‣ MTGS: Multi-Traversal Gaussian Splatting">Tab. 1</a>. In ST tests, our method outperforms others in image reconstruction, likely due to its effective inner-traversal appearance modeling. It also achieves the highest quality in novel-view synthesis across all metrics, especially at feature and geometry levels. </div> <div class="ltx_para" id="S5.SS2.p2"> In the MT setting, MTGS reconstructs a consistent scene across multiple traversals. Although OmniRe obtains good results on training traversals regarding SSIM and AbsRel, its severe overfitting leads to poor performance on novel-view traversals. In contrast, MTGS consistently delivers the best novel-view synthesis performance. Notably, with additional training iterations (60k), the feature-level metrics further improve, while the geometry metrics tend to converge. A qualitative comparison is shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#S5.F4" title="Figure 4 ‣ 5.2 Main Results ‣ 5 Experiment ‣ MTGS: Multi-Traversal Gaussian Splatting">4</a>. Our baselines produce blurry, artifact-prone images, whereas our method delivers clear, crisp results. </div> <figure class="ltx_figure" id="S5.F4"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="491" id="S5.F4.g1" src="x4.png" width="932"/> <figcaption class="ltx_caption ltx_centering">Figure 4: Visual comparison. Compared to OmniRE <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib3" title="">3</a>]</cite> and 3DGS <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib17" title="">17</a>]</cite>, MTGS produces images in higher fidelity, effectively handles appearance variations, and robustly extrapolates to novel views. Notably, our transient node accurately captures moving shadows (red box). </figcaption> </figure> <figure class="ltx_table" id="S5.T3"> <figcaption class="ltx_caption ltx_centering">Table 3: Ablation on modular designs. Nodes in the scene graph and regularization losses are all crucial for the final performance. ‘tsnt.’ stands for transient. First, second. </figcaption> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="S5.T3.6" style="width:238.5pt;height:63.1pt;vertical-align:-0.0pt;"> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="S5.T3.6.6"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S5.T3.6.6.7.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_tt" id="S5.T3.6.6.7.1.1" rowspan="2">ID</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_tt" id="S5.T3.6.6.7.1.2" rowspan="2">Exp. name</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" colspan="6" id="S5.T3.6.6.7.1.3">Novel-View Traversal</th> </tr> <tr class="ltx_tr" id="S5.T3.6.6.6"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T3.1.1.1.1" style="background-color:#FFFFFF;">PSNR*<math alttext="\uparrow" class="ltx_Math" display="inline" id="S5.T3.1.1.1.1.1.m1.1"><semantics id="S5.T3.1.1.1.1.1.m1.1a"><mo id="S5.T3.1.1.1.1.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="S5.T3.1.1.1.1.1.m1.1.1.cmml">↑</mo><annotation-xml encoding="MathML-Content" id="S5.T3.1.1.1.1.1.m1.1b"><ci id="S5.T3.1.1.1.1.1.m1.1.1.cmml" xref="S5.T3.1.1.1.1.1.m1.1.1">↑</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.1.1.1.1.1.m1.1c">\uparrow</annotation><annotation encoding="application/x-llamapun" id="S5.T3.1.1.1.1.1.m1.1d">↑</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T3.2.2.2.2" style="background-color:#FFFFFF;">SSIM <math alttext="\uparrow" class="ltx_Math" display="inline" id="S5.T3.2.2.2.2.1.1.m1.1" style="background-color:#FFFFFF;"><semantics id="S5.T3.2.2.2.2.1.1.m1.1a"><mo id="S5.T3.2.2.2.2.1.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="S5.T3.2.2.2.2.1.1.m1.1.1.cmml">↑</mo><annotation-xml encoding="MathML-Content" id="S5.T3.2.2.2.2.1.1.m1.1b"><ci id="S5.T3.2.2.2.2.1.1.m1.1.1.cmml" xref="S5.T3.2.2.2.2.1.1.m1.1.1">↑</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.2.2.2.2.1.1.m1.1c">\uparrow</annotation><annotation encoding="application/x-llamapun" id="S5.T3.2.2.2.2.1.1.m1.1d">↑</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T3.3.3.3.3" style="background-color:#FFFFFF;">LPIPS <math alttext="\downarrow" class="ltx_Math" display="inline" id="S5.T3.3.3.3.3.1.1.m1.1" style="background-color:#FFFFFF;"><semantics id="S5.T3.3.3.3.3.1.1.m1.1a"><mo id="S5.T3.3.3.3.3.1.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="S5.T3.3.3.3.3.1.1.m1.1.1.cmml">↓</mo><annotation-xml encoding="MathML-Content" id="S5.T3.3.3.3.3.1.1.m1.1b"><ci id="S5.T3.3.3.3.3.1.1.m1.1.1.cmml" xref="S5.T3.3.3.3.3.1.1.m1.1.1">↓</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.3.3.3.3.1.1.m1.1c">\downarrow</annotation><annotation encoding="application/x-llamapun" id="S5.T3.3.3.3.3.1.1.m1.1d">↓</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T3.4.4.4.4" style="background-color:#FFFFFF;"> <table class="ltx_tabular ltx_align_middle" id="S5.T3.4.4.4.4.1" style="background-color:#FFFFFF;"> <tr class="ltx_tr" id="S5.T3.4.4.4.4.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T3.4.4.4.4.1.1.1">Feat. Sim.</td> </tr> </table> <math alttext="\uparrow" class="ltx_Math" display="inline" id="S5.T3.4.4.4.4.m1.1" style="background-color:#FFFFFF;"><semantics id="S5.T3.4.4.4.4.m1.1a"><mo id="S5.T3.4.4.4.4.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="S5.T3.4.4.4.4.m1.1.1.cmml">↑</mo><annotation-xml encoding="MathML-Content" id="S5.T3.4.4.4.4.m1.1b"><ci id="S5.T3.4.4.4.4.m1.1.1.cmml" xref="S5.T3.4.4.4.4.m1.1.1">↑</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.4.4.4.4.m1.1c">\uparrow</annotation><annotation encoding="application/x-llamapun" id="S5.T3.4.4.4.4.m1.1d">↑</annotation></semantics></math> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T3.5.5.5.5" style="background-color:#FFFFFF;"> <table class="ltx_tabular ltx_align_middle" id="S5.T3.5.5.5.5.2" style="background-color:#FFFFFF;"> <tr class="ltx_tr" id="S5.T3.5.5.5.5.2.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T3.5.5.5.5.2.1.1">AbsRel</td> </tr> </table> <math alttext="\downarrow" class="ltx_Math" display="inline" id="S5.T3.5.5.5.5.1.m1.1" style="background-color:#FFFFFF;"><semantics id="S5.T3.5.5.5.5.1.m1.1a"><mo id="S5.T3.5.5.5.5.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="S5.T3.5.5.5.5.1.m1.1.1.cmml">↓</mo><annotation-xml encoding="MathML-Content" id="S5.T3.5.5.5.5.1.m1.1b"><ci id="S5.T3.5.5.5.5.1.m1.1.1.cmml" xref="S5.T3.5.5.5.5.1.m1.1.1">↓</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.5.5.5.5.1.m1.1c">\downarrow</annotation><annotation encoding="application/x-llamapun" id="S5.T3.5.5.5.5.1.m1.1d">↓</annotation></semantics></math> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T3.6.6.6.6" style="background-color:#FFFFFF;"> <table class="ltx_tabular ltx_align_middle" id="S5.T3.6.6.6.6.2" style="background-color:#FFFFFF;"> <tr class="ltx_tr" id="S5.T3.6.6.6.6.2.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T3.6.6.6.6.2.1.1">Delta1</td> </tr> </table> <math alttext="\uparrow" class="ltx_Math" display="inline" id="S5.T3.6.6.6.6.1.m1.1" style="background-color:#FFFFFF;"><semantics id="S5.T3.6.6.6.6.1.m1.1a"><mo id="S5.T3.6.6.6.6.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="S5.T3.6.6.6.6.1.m1.1.1.cmml">↑</mo><annotation-xml encoding="MathML-Content" id="S5.T3.6.6.6.6.1.m1.1b"><ci id="S5.T3.6.6.6.6.1.m1.1.1.cmml" xref="S5.T3.6.6.6.6.1.m1.1.1">↑</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.6.6.6.6.1.m1.1c">\uparrow</annotation><annotation encoding="application/x-llamapun" id="S5.T3.6.6.6.6.1.m1.1d">↑</annotation></semantics></math> </th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S5.T3.6.6.8.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S5.T3.6.6.8.1.1">0</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S5.T3.6.6.8.1.2">w/o tsnt. node</th> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T3.6.6.8.1.3" style="background-color:#FFFFFF;">20.62</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T3.6.6.8.1.4" style="background-color:#FFFFFF;">0.607</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T3.6.6.8.1.5" style="background-color:#FFFFFF;">0.276</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T3.6.6.8.1.6" style="background-color:#FFFFFF;">0.642</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T3.6.6.8.1.7" style="background-color:#FFFFFF;">0.086</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T3.6.6.8.1.8" style="background-color:#FFD9B2;">0.899</td> </tr> <tr class="ltx_tr" id="S5.T3.6.6.9.2"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r" id="S5.T3.6.6.9.2.1">1</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r" id="S5.T3.6.6.9.2.2">w/o normal loss</th> <td class="ltx_td ltx_align_center" id="S5.T3.6.6.9.2.3" style="background-color:#FFD9B2;">20.82</td> <td class="ltx_td ltx_align_center" id="S5.T3.6.6.9.2.4" style="background-color:#FFB2B2;">0.614</td> <td class="ltx_td ltx_align_center" id="S5.T3.6.6.9.2.5" style="background-color:#FFFFFF;">0.275</td> <td class="ltx_td ltx_align_center" id="S5.T3.6.6.9.2.6" style="background-color:#FFFFFF;">0.643</td> <td class="ltx_td ltx_align_center" id="S5.T3.6.6.9.2.7" style="background-color:#FFB2B2;">0.076</td> <td class="ltx_td ltx_align_center" id="S5.T3.6.6.9.2.8" style="background-color:#FFB2B2;">0.914</td> </tr> <tr class="ltx_tr" id="S5.T3.6.6.10.3"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r" id="S5.T3.6.6.10.3.1">2</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r" id="S5.T3.6.6.10.3.2">w/o depth loss</th> <td class="ltx_td ltx_align_center" id="S5.T3.6.6.10.3.3" style="background-color:#FFB2B2;">20.83</td> <td class="ltx_td ltx_align_center" id="S5.T3.6.6.10.3.4" style="background-color:#FFFFFF;">0.607</td> <td class="ltx_td ltx_align_center" id="S5.T3.6.6.10.3.5" style="background-color:#FFB2B2;">0.264</td> <td class="ltx_td ltx_align_center" id="S5.T3.6.6.10.3.6" style="background-color:#FFD9B2;">0.644</td> <td class="ltx_td ltx_align_center" id="S5.T3.6.6.10.3.7" style="background-color:#FFFFFF;">0.891</td> <td class="ltx_td ltx_align_center" id="S5.T3.6.6.10.3.8" style="background-color:#FFFFFF;">0.613</td> </tr> <tr class="ltx_tr" id="S5.T3.6.6.11.4"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_bb ltx_border_r" id="S5.T3.6.6.11.4.1">3</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_bb ltx_border_r" id="S5.T3.6.6.11.4.2">Ours (Full)</th> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T3.6.6.11.4.3" style="background-color:#FFB2B2;">20.83</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T3.6.6.11.4.4" style="background-color:#FFD9B2;">0.611</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T3.6.6.11.4.5" style="background-color:#FFD9B2;">0.271</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T3.6.6.11.4.6" style="background-color:#FFB2B2;">0.646</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T3.6.6.11.4.7" style="background-color:#FFD9B2;">0.078</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T3.6.6.11.4.8" style="background-color:#FFB2B2;">0.914</td> </tr> </tbody> </table> </div> </figure> </section> <section class="ltx_subsection" id="S5.SS3"> <h3 class="ltx_title ltx_title_subsection"> 5.3 Ablation Study</h3> <div class="ltx_para ltx_noindent" id="S5.SS3.p1"> Number of traversals. We conduct experiments on three road blocks with six traversals. Traversals in these blocks are not occluded by any buildings or obstructions on-road to ensure that the performance gain of multi-traversal is not simply from seeing the unseen part. As shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#S5.F3" title="Figure 3 ‣ 5.1 Setup and Protocols ‣ 5 Experiment ‣ MTGS: Multi-Traversal Gaussian Splatting">3</a>, our method enhances overall rendering quality on novel-view traversals as more traversals are incorporated. In contrast, OmniRe fails to maintain consistent geometry with the increased data. This demonstrates that MTGS effectively manages the appearance and dynamic variation across multiple traversals, resulting in a more accurate reconstruction of the shared static node. Full results are in the Supplement. </div> <div class="ltx_para ltx_noindent" id="S5.SS3.p2"> Multi-traversal appearance modeling. In <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#S5.T2" title="In 5.1 Setup and Protocols ‣ 5 Experiment ‣ MTGS: Multi-Traversal Gaussian Splatting">Tab. 2</a>, we demonstrate the effectiveness of our proposed appearance modeling designs by selecting a challenging subset of four road blocks, each containing three training traversals with various appearances. Removing modules from the final design (ID 2-4, compared to ID 5) leads to a performance drop, while incrementally adding modules to the baseline (ID 0-2, and 5) yields significant gains. These findings validate that our strategy effectively captures and reconstructs the diverse appearances across multiple traversals, thereby enhancing both image reconstruction and novel-view synthesis. We also compare our approach with a state-of-the-art Gaussian-based in-the-wild method, WildGaussians <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib19" title="">19</a>]</cite>. We re-implement its per-camera and per-gaussian appearance embeddings within our pipeline. The results reveal that modeling per-camera appearance in a multi-traversal setting is insufficient, as the limited overlapping regions between cameras complicate the optimization process. </div> <div class="ltx_para ltx_noindent" id="S5.SS3.p3"> Modular design. We further evaluate additional design choices in MTGS. As shown in Tab. <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#S5.T3" title="Table 3 ‣ 5.2 Main Results ‣ 5 Experiment ‣ MTGS: Multi-Traversal Gaussian Splatting">3</a>, removing the transient node (ID 0) degrades geometry accuracy, likely due to overfitting on the shadows cast by dynamic objects. These results demonstrate that preserving and modeling dynamic information can help multi-traversal reconstruction performance. Removing the normal smooth loss (ID 1) adversely affects the feature-level metrics, while removing the depth loss (ID 2) significantly harms learning the geometry. </div> </section> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> 6 Conclusion and Outlook</h2> <div class="ltx_para" id="S6.p1"> In this work, we propose Multi-Traversal Gaussian Splatting (MTGS), the first method capable of reconstructing multi-traversal dynamic scenes with high fidelity. By introducing a novel Multi-Traversal Scene Graph, our approach effectively captures a shared static background while separately modeling dynamic objects and appearance variations across multiple traversals. Extensive evaluations demonstrate that MTGS achieves both high-quality image reconstruction and robust novel-view synthesis, outperforming existing state-of-the-art methods. With its potential to serve as a foundation for photorealistic autonomous driving simulators, MTGS promises to enhance the safety and reliability of autonomous vehicle testing and development. </div> <div class="ltx_para" id="S6.p2"> Limitation and future works. In the current version, non-rigid objects, such as bicycles or pedestrians, are not reconstructed in the scene. However, our method can seamlessly support them by integrating deformable Gaussians or SMPL modeling into the transient node, as demonstrated in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib3" title="">3</a>]</cite>. In the shared background, there might be floating artifacts in regions not observed during training traversals, such as the space below parked cars. We also observe that rolling shutter effects, particularly in inverted traversals, introduce misalignment in the shared geometry. The static geometry of the scene is assumed to remain unchanged. Modeling and reconstruction of unlabeled transient objects and map changes are left for future works. Future endeavors may include simultaneous camera and LiDAR simulation, e.g. modeling the appearance diversity of LiDAR intensity and drop rate. </div> </section> <section class="ltx_section" id="Sx1"> <h2 class="ltx_title ltx_title_section">Acknowledgements</h2> <div class="ltx_para" id="Sx1.p1"> We extend our gratitude to Li Chen, Chonghao Sima, and Adam Tonderski for their profound discussions. </div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography" style="font-size:90%;">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> Barron et al. 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In CVPR, 2024c. </li> </ul> </section> <div class="ltx_pagination ltx_role_newpage"></div> <section class="ltx_appendix" id="Ax1"> <h2 class="ltx_title ltx_font_italic ltx_title_appendix" style="font-size:144%;">Appendix</h2> </section> <section class="ltx_appendix" id="A1"> <h2 class="ltx_title ltx_title_appendix"> Appendix A Implementation Details</h2> <div class="ltx_para" id="A1.p1"> We provide key implementation details on datasets, models, and experiments in the supplementary material. To encourage and facilitate further research, we will openly release the whole suite of code and models. </div> <section class="ltx_subsection" id="A1.SS1"> <h3 class="ltx_title ltx_title_subsection"> A.1 Dataset</h3> <figure class="ltx_table" id="A1.T4"> <figcaption class="ltx_caption ltx_centering">Table 4: Details of selected road blocks. 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start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT</annotation></semantics></math> in UTM coordinate. </figcaption> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="A1.T4.6"> <thead class="ltx_thead"> <tr class="ltx_tr" id="A1.T4.6.1.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="A1.T4.6.1.1.1">ID</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="A1.T4.6.1.1.2">City</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="A1.T4.6.1.1.3">Road Block Coordinate</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="A1.T4.6.1.1.4"># Traversal</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="A1.T4.6.2.1"> <td class="ltx_td ltx_align_center ltx_border_t" id="A1.T4.6.2.1.1">0</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A1.T4.6.2.1.2">us-ma-boston</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A1.T4.6.2.1.3">331220, 4690660, 331190, 4690710</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A1.T4.6.2.1.4">6</td> </tr> <tr class="ltx_tr" id="A1.T4.6.3.2"> <td class="ltx_td ltx_align_center" id="A1.T4.6.3.2.1">1</td> <td class="ltx_td ltx_align_center" id="A1.T4.6.3.2.2">sg-one-north</td> <td class="ltx_td ltx_align_center" id="A1.T4.6.3.2.3">365000, 144000, 365100, 144080</td> <td class="ltx_td ltx_align_center" id="A1.T4.6.3.2.4">3</td> </tr> <tr class="ltx_tr" id="A1.T4.6.4.3"> <td class="ltx_td ltx_align_center" id="A1.T4.6.4.3.1">2</td> <td class="ltx_td ltx_align_center" id="A1.T4.6.4.3.2">sg-one-north</td> <td class="ltx_td ltx_align_center" id="A1.T4.6.4.3.3">365530, 143960, 365630, 144060</td> <td class="ltx_td ltx_align_center" id="A1.T4.6.4.3.4">4</td> </tr> <tr class="ltx_tr" id="A1.T4.6.5.4"> <td class="ltx_td ltx_align_center" id="A1.T4.6.5.4.1">3</td> <td class="ltx_td ltx_align_center" id="A1.T4.6.5.4.2">us-pa-pittsburgh-hazelwood</td> <td class="ltx_td ltx_align_center" id="A1.T4.6.5.4.3">587400, 4475700, 587480, 4475800</td> <td class="ltx_td ltx_align_center" id="A1.T4.6.5.4.4">4</td> </tr> <tr class="ltx_tr" id="A1.T4.6.6.5"> <td class="ltx_td ltx_align_center" id="A1.T4.6.6.5.1">4</td> <td class="ltx_td ltx_align_center" id="A1.T4.6.6.5.2">us-pa-pittsburgh-hazelwood</td> <td class="ltx_td ltx_align_center" id="A1.T4.6.6.5.3">587640, 4475600, 587710, 4475660</td> <td class="ltx_td ltx_align_center" id="A1.T4.6.6.5.4">6</td> </tr> <tr class="ltx_tr" id="A1.T4.6.7.6"> <td class="ltx_td ltx_align_center ltx_border_bb" id="A1.T4.6.7.6.1">5</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="A1.T4.6.7.6.2">us-pa-pittsburgh-hazelwood</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="A1.T4.6.7.6.3">587860, 4475510, 587910, 4475570</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="A1.T4.6.7.6.4">6</td> </tr> </tbody> </table> </figure> <figure class="ltx_table ltx_figure_panel" id="A1.T5"> <figcaption class="ltx_caption">Table 5: Details of training hyperparameters </figcaption> <div class="ltx_inline-block ltx_figure_panel ltx_align_center ltx_transformed_outer" id="A1.T5.5" style="width:248.4pt;height:223.7pt;vertical-align:-0.0pt;"> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="A1.T5.5.1"> <thead class="ltx_thead"> <tr class="ltx_tr" id="A1.T5.5.1.1.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="A1.T5.5.1.1.1.1">Parameters</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="A1.T5.5.1.1.1.2">Initial LR</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="A1.T5.5.1.1.1.3">Final LR</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="A1.T5.5.1.1.1.4">Warm-up Steps</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="A1.T5.5.1.2.1"> <td class="ltx_td ltx_align_center ltx_border_t" id="A1.T5.5.1.2.1.1">means</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A1.T5.5.1.2.1.2">8e-4</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A1.T5.5.1.2.1.3">8e-6</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A1.T5.5.1.2.1.4">0</td> </tr> <tr class="ltx_tr" id="A1.T5.5.1.3.2"> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.3.2.1">static.features_dc</td> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.3.2.2">1.25e-4</td> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.3.2.3">1.25e-4</td> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.3.2.4">0</td> </tr> <tr class="ltx_tr" id="A1.T5.5.1.4.3"> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.4.3.1">appearance.features_rest</td> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.4.3.2">1.25e-4</td> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.4.3.3">1.25e-4</td> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.4.3.4">0</td> </tr> <tr class="ltx_tr" id="A1.T5.5.1.5.4"> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.5.4.1">transient.features_dc</td> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.5.4.2">2.5e-3</td> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.5.4.3">2.5e-3</td> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.5.4.4">0</td> </tr> <tr class="ltx_tr" id="A1.T5.5.1.6.5"> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.6.5.1">transient.feature_rest</td> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.6.5.2">1.25e-4</td> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.6.5.3">1.25e-4</td> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.6.5.4">0</td> </tr> <tr class="ltx_tr" id="A1.T5.5.1.7.6"> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.7.6.1">opacities</td> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.7.6.2">5e-2</td> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.7.6.3">5e-2</td> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.7.6.4">0</td> </tr> <tr class="ltx_tr" id="A1.T5.5.1.8.7"> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.8.7.1">scales</td> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.8.7.2">5e-3</td> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.8.7.3">5e-3</td> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.8.7.4">0</td> </tr> <tr class="ltx_tr" id="A1.T5.5.1.9.8"> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.9.8.1">quats</td> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.9.8.2">1e-3</td> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.9.8.3">1e-3</td> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.9.8.4">0</td> </tr> <tr class="ltx_tr" id="A1.T5.5.1.10.9"> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.10.9.1">camera_pose_opt</td> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.10.9.2">1e-4</td> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.10.9.3">5e-7</td> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.10.9.4">1500</td> </tr> <tr class="ltx_tr" id="A1.T5.5.1.11.10"> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.11.10.1">camera_affine</td> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.11.10.2">1e-3</td> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.11.10.3">1e-4</td> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.11.10.4">5000</td> </tr> <tr class="ltx_tr" id="A1.T5.5.1.12.11"> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.12.11.1">ins_rotation</td> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.12.11.2">1e-5</td> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.12.11.3">1e-6</td> <td class="ltx_td ltx_align_center" id="A1.T5.5.1.12.11.4">0</td> </tr> <tr class="ltx_tr" id="A1.T5.5.1.13.12"> <td class="ltx_td ltx_align_center ltx_border_bb" id="A1.T5.5.1.13.12.1">ins_translation</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="A1.T5.5.1.13.12.2">5e-4</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="A1.T5.5.1.13.12.3">1e-4</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="A1.T5.5.1.13.12.4">0</td> </tr> </tbody> </table> </div> </figure> <div class="ltx_para" id="A1.SS1.p1"> We conduct experiments on customized data from the nuPlan dataset <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib16" title="">16</a>]</cite>. We use all eight views and LiDAR at 10 Hz, with the resolution of <math alttext="960\times 540" class="ltx_Math" display="inline" id="A1.SS1.p1.1.m1.1"><semantics id="A1.SS1.p1.1.m1.1a"><mrow id="A1.SS1.p1.1.m1.1.1" xref="A1.SS1.p1.1.m1.1.1.cmml"><mn id="A1.SS1.p1.1.m1.1.1.2" xref="A1.SS1.p1.1.m1.1.1.2.cmml">960</mn><mo id="A1.SS1.p1.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="A1.SS1.p1.1.m1.1.1.1.cmml">×</mo><mn id="A1.SS1.p1.1.m1.1.1.3" xref="A1.SS1.p1.1.m1.1.1.3.cmml">540</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.p1.1.m1.1b"><apply id="A1.SS1.p1.1.m1.1.1.cmml" xref="A1.SS1.p1.1.m1.1.1"><times id="A1.SS1.p1.1.m1.1.1.1.cmml" xref="A1.SS1.p1.1.m1.1.1.1"></times><cn id="A1.SS1.p1.1.m1.1.1.2.cmml" type="integer" xref="A1.SS1.p1.1.m1.1.1.2">960</cn><cn id="A1.SS1.p1.1.m1.1.1.3.cmml" type="integer" xref="A1.SS1.p1.1.m1.1.1.3">540</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p1.1.m1.1c">960\times 540</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p1.1.m1.1d">960 × 540</annotation></semantics></math> for images across training and evaluation. </div> <div class="ltx_para ltx_noindent" id="A1.SS1.p2"> Handling of inaccurate pose alignment. Since the localization across multiple traversals in nuPlan is imprecise, we employ a LiDAR registration method <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib37" title="">37</a>]</cite> to align the multi-traversal poses accurately. The camera extrinsic is pre-calibrated but not perfectly synced with LiDAR, causing a pose shift when the car moves. To fix this problem, we composite the motion to the camera extrinsic by interpolation. We further use a camera pose optimizer <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib44" title="">44</a>]</cite> to handle this misalignment. </div> <div class="ltx_para ltx_noindent" id="A1.SS1.p3"> Handling of large image distortion. We also note that the camera distortion in nuPlan is severe and could cause bad outputs as in the raw implementation of OmniRe <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib3" title="">3</a>]</cite>. We undistort the images with OpenCV at optimal mode to preserve the field of view. To alleviate the inaccurate camera intrinsics in nuPlan, we employ several rounds of bundle adjustment of COLMAP <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib31" title="">31</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib32" title="">32</a>]</cite> to calibrate them. </div> <div class="ltx_para ltx_noindent" id="A1.SS1.p4"> Use of pre-trained models. The pseudo depth used during training is obtained from UniDepth <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib29" title="">29</a>]</cite> with a ViT-L <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib6" title="">6</a>]</cite> backbone. We input the undistorted images and the optimized focal length to UniDepth. Although it generates depth on a metric scale, the depth RMSE is still over 20 meters, which motivates us to apply the NCC loss in our model. To extract semantic masks, Mask2Former <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib4" title="">4</a>]</cite> with a Swin-L <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib23" title="">23</a>]</cite> backbone trained on Cityscapes <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib5" title="">5</a>]</cite> is adopted. </div> <div class="ltx_para ltx_noindent" id="A1.SS1.p5"> Benchmark. We list the road blocks used in experiments in Tab. <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#A1.T4" title="Table 4 ‣ A.1 Dataset ‣ Appendix A Implementation Details ‣ MTGS: Multi-Traversal Gaussian Splatting">4</a>. The traversals within road blocks are about 100 meters in length. The main comparison is based on all six traversals. The ablation on the number of traversals is based on traversals 0, 1, and 2. The rest of the ablations are based on traversals 0, 1, 2, and 5 with three training traversals. The principle of selecting traversals is shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#A1.F5" title="Figure 5 ‣ A.1 Dataset ‣ Appendix A Implementation Details ‣ MTGS: Multi-Traversal Gaussian Splatting">5</a>. </div> <figure class="ltx_figure" id="A1.F5"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="228" id="A1.F5.g1" src="extracted/6295770/figures/gfx/figure_data.jpg" width="316"/> <figcaption class="ltx_caption ltx_centering">Figure 5: Illustration of training and test traversals. We select traversals distributed across multiple lanes and choose the isolated traversal with minimum overlaps. </figcaption> </figure> </section> <section class="ltx_subsection" id="A1.SS2"> <h3 class="ltx_title ltx_title_subsection"> A.2 MTGS</h3> <div class="ltx_para ltx_noindent" id="A1.SS2.p1"> Transient Node. The initial poses for each transient node are derived from 3D bounding box annotations provided in the nuPlan dataset, which are generated by a pre-trained LiDAR 3D detector and tend to be inaccurate. Therefore, we treat these poses as learnable parameters, following the approach of Street Gaussians and OmniRe <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib45" title="">45</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib3" title="">3</a>]</cite>, without applying a smoothness loss. The poses of static objects are kept the same across frames. Object with movements of less than 3 meters is considered static. </div> <div class="ltx_para ltx_noindent" id="A1.SS2.p2"> Optimization. For the optimization process, we employ the Adam optimizer to train our model over 30,000 iterations. All the corresponding hyperparameters are explicitly outlined in <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#A1.T5" title="In A.1 Dataset ‣ Appendix A Implementation Details ‣ MTGS: Multi-Traversal Gaussian Splatting">Tab. 5</a>. For Gaussian density control, we keep most of the hyperparameters as in the original 3DGS. Since we train the scene on the metric scale without normalization, we adjust the scale threshold of densify to 0.2 meters and the scale threshold of culling to 0.5 meters. 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xref="A1.SS2.p4.5.m5.1.1.2.3.7">𝑒</ci><ci id="A1.SS2.p4.5.m5.1.1.2.3.8.cmml" xref="A1.SS2.p4.5.m5.1.1.2.3.8">𝑛</ci></apply></apply><cn id="A1.SS2.p4.5.m5.1.1.3.cmml" type="float" xref="A1.SS2.p4.5.m5.1.1.3">1.0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p4.5.m5.1c">\lambda_{flatten}=1.0</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p4.5.m5.1d">italic_λ start_POSTSUBSCRIPT italic_f italic_l italic_a italic_t italic_t italic_e italic_n end_POSTSUBSCRIPT = 1.0</annotation></semantics></math> and <math alttext="\lambda_{obb}=1.0" class="ltx_Math" display="inline" id="A1.SS2.p4.6.m6.1"><semantics id="A1.SS2.p4.6.m6.1a"><mrow id="A1.SS2.p4.6.m6.1.1" xref="A1.SS2.p4.6.m6.1.1.cmml"><msub id="A1.SS2.p4.6.m6.1.1.2" xref="A1.SS2.p4.6.m6.1.1.2.cmml"><mi id="A1.SS2.p4.6.m6.1.1.2.2" xref="A1.SS2.p4.6.m6.1.1.2.2.cmml">λ</mi><mrow id="A1.SS2.p4.6.m6.1.1.2.3" xref="A1.SS2.p4.6.m6.1.1.2.3.cmml"><mi id="A1.SS2.p4.6.m6.1.1.2.3.2" xref="A1.SS2.p4.6.m6.1.1.2.3.2.cmml">o</mi><mo id="A1.SS2.p4.6.m6.1.1.2.3.1" xref="A1.SS2.p4.6.m6.1.1.2.3.1.cmml">⁢</mo><mi id="A1.SS2.p4.6.m6.1.1.2.3.3" xref="A1.SS2.p4.6.m6.1.1.2.3.3.cmml">b</mi><mo id="A1.SS2.p4.6.m6.1.1.2.3.1a" xref="A1.SS2.p4.6.m6.1.1.2.3.1.cmml">⁢</mo><mi id="A1.SS2.p4.6.m6.1.1.2.3.4" xref="A1.SS2.p4.6.m6.1.1.2.3.4.cmml">b</mi></mrow></msub><mo id="A1.SS2.p4.6.m6.1.1.1" xref="A1.SS2.p4.6.m6.1.1.1.cmml">=</mo><mn id="A1.SS2.p4.6.m6.1.1.3" xref="A1.SS2.p4.6.m6.1.1.3.cmml">1.0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.p4.6.m6.1b"><apply id="A1.SS2.p4.6.m6.1.1.cmml" xref="A1.SS2.p4.6.m6.1.1"><eq id="A1.SS2.p4.6.m6.1.1.1.cmml" xref="A1.SS2.p4.6.m6.1.1.1"></eq><apply id="A1.SS2.p4.6.m6.1.1.2.cmml" xref="A1.SS2.p4.6.m6.1.1.2"><csymbol cd="ambiguous" id="A1.SS2.p4.6.m6.1.1.2.1.cmml" xref="A1.SS2.p4.6.m6.1.1.2">subscript</csymbol><ci id="A1.SS2.p4.6.m6.1.1.2.2.cmml" xref="A1.SS2.p4.6.m6.1.1.2.2">𝜆</ci><apply id="A1.SS2.p4.6.m6.1.1.2.3.cmml" xref="A1.SS2.p4.6.m6.1.1.2.3"><times id="A1.SS2.p4.6.m6.1.1.2.3.1.cmml" xref="A1.SS2.p4.6.m6.1.1.2.3.1"></times><ci id="A1.SS2.p4.6.m6.1.1.2.3.2.cmml" xref="A1.SS2.p4.6.m6.1.1.2.3.2">𝑜</ci><ci id="A1.SS2.p4.6.m6.1.1.2.3.3.cmml" xref="A1.SS2.p4.6.m6.1.1.2.3.3">𝑏</ci><ci id="A1.SS2.p4.6.m6.1.1.2.3.4.cmml" xref="A1.SS2.p4.6.m6.1.1.2.3.4">𝑏</ci></apply></apply><cn id="A1.SS2.p4.6.m6.1.1.3.cmml" type="float" xref="A1.SS2.p4.6.m6.1.1.3">1.0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p4.6.m6.1c">\lambda_{obb}=1.0</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p4.6.m6.1d">italic_λ start_POSTSUBSCRIPT italic_o italic_b italic_b end_POSTSUBSCRIPT = 1.0</annotation></semantics></math>. In the NCC loss, patch size <math alttext="s" class="ltx_Math" display="inline" id="A1.SS2.p4.7.m7.1"><semantics id="A1.SS2.p4.7.m7.1a"><mi id="A1.SS2.p4.7.m7.1.1" xref="A1.SS2.p4.7.m7.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.p4.7.m7.1b"><ci id="A1.SS2.p4.7.m7.1.1.cmml" xref="A1.SS2.p4.7.m7.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p4.7.m7.1c">s</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p4.7.m7.1d">italic_s</annotation></semantics></math> is set to 32, and <math alttext="k" class="ltx_Math" display="inline" id="A1.SS2.p4.8.m8.1"><semantics id="A1.SS2.p4.8.m8.1a"><mi id="A1.SS2.p4.8.m8.1.1" xref="A1.SS2.p4.8.m8.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.p4.8.m8.1b"><ci id="A1.SS2.p4.8.m8.1.1.cmml" xref="A1.SS2.p4.8.m8.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p4.8.m8.1c">k</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p4.8.m8.1d">italic_k</annotation></semantics></math> is set to 16. In the Gaussian flatten loss, <math alttext="r" class="ltx_Math" display="inline" id="A1.SS2.p4.9.m9.1"><semantics id="A1.SS2.p4.9.m9.1a"><mi id="A1.SS2.p4.9.m9.1.1" xref="A1.SS2.p4.9.m9.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.p4.9.m9.1b"><ci id="A1.SS2.p4.9.m9.1.1.cmml" xref="A1.SS2.p4.9.m9.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p4.9.m9.1c">r</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p4.9.m9.1d">italic_r</annotation></semantics></math> is set to 10 and is applied every 10 steps following <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib40" title="">40</a>]</cite>. </div> </section> <section class="ltx_subsection" id="A1.SS3"> <h3 class="ltx_title ltx_title_subsection"> A.3 Reproduction of baselines</h3> <div class="ltx_para" id="A1.SS3.p1"> 3DGS <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib17" title="">17</a>]</cite>. We reproduce 3DGS based on gsplat <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib48" title="">48</a>]</cite>. We set all the hyperparameters based on the original papers. The scene and the initialized point clouds are normalized with scale factor 5e-3, which corresponds to 200 meters scene extent. </div> <div class="ltx_para ltx_noindent" id="A1.SS3.p2"> OmniRe <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib3" title="">3</a>]</cite>. We adopt its official implementation with default hyperparameters. We perform equivalent data preprocessing steps, including LiDAR registration, bundle adjustment, and distortion correction, which are consistent with our method. Notably, as we do not assess human body reconstruction, we omit the SMPL node component from OmniRe’s pipeline and excluded pedestrians and bicycles during evaluation to ensure fairness. </div> <div class="ltx_para ltx_noindent" id="A1.SS3.p3"> Street Gaussians <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#bib.bib45" title="">45</a>]</cite>. For Street Gaussians, we employ the implementation in the OmniRe repository and the default parameters while maintaining identical data processing protocols. </div> </section> </section> <section class="ltx_appendix" id="A2"> <h2 class="ltx_title ltx_title_appendix"> Appendix B Experiments</h2> <div class="ltx_para ltx_noindent" id="A2.p1"> Ablation on the extrinsic calibration. As shown in Tab. <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#A2.T6" title="Table 6 ‣ Appendix B Experiments ‣ MTGS: Multi-Traversal Gaussian Splatting">6</a>, proper pose alignment significantly boosts both reconstruction and novel-view synthesis performance. Overfitting on inaccurate camera poses degrades view extrapolation. Since our pose alignment process is not fully optimized, improving multi-traversal localization represents a promising direction for enhanced reconstruction. </div> <figure class="ltx_table" id="A2.T6"> <figcaption class="ltx_caption ltx_centering">Table 6: Ablation on extrinsic calibration. </figcaption> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="A2.T6.10" style="width:496.9pt;height:67.1pt;vertical-align:-0.0pt;"> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="A2.T6.10.10"> <thead class="ltx_thead"> <tr class="ltx_tr" id="A2.T6.10.10.11.1"> <th class="ltx_td ltx_th ltx_th_column ltx_border_r ltx_border_tt" id="A2.T6.10.10.11.1.1"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_tt" colspan="2" id="A2.T6.10.10.11.1.2">Module</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" colspan="4" id="A2.T6.10.10.11.1.3">Training Traversal</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" colspan="6" id="A2.T6.10.10.11.1.4">Novel-View Traversal</th> </tr> <tr class="ltx_tr" id="A2.T6.10.10.10"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r" id="A2.T6.10.10.10.11">ID</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="A2.T6.10.10.10.12" style="background-color:#FFFFFF;">LiDAR Registration</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="A2.T6.10.10.10.13" style="background-color:#FFFFFF;">CamOptim</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="A2.T6.1.1.1.1" style="background-color:#FFFFFF;">PSNR <math alttext="\uparrow" class="ltx_Math" display="inline" id="A2.T6.1.1.1.1.1.1.m1.1" style="background-color:#FFFFFF;"><semantics id="A2.T6.1.1.1.1.1.1.m1.1a"><mo id="A2.T6.1.1.1.1.1.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="A2.T6.1.1.1.1.1.1.m1.1.1.cmml">↑</mo><annotation-xml encoding="MathML-Content" id="A2.T6.1.1.1.1.1.1.m1.1b"><ci id="A2.T6.1.1.1.1.1.1.m1.1.1.cmml" xref="A2.T6.1.1.1.1.1.1.m1.1.1">↑</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.T6.1.1.1.1.1.1.m1.1c">\uparrow</annotation><annotation encoding="application/x-llamapun" id="A2.T6.1.1.1.1.1.1.m1.1d">↑</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="A2.T6.2.2.2.2" style="background-color:#FFFFFF;">SSIM <math alttext="\uparrow" class="ltx_Math" display="inline" id="A2.T6.2.2.2.2.1.1.m1.1" style="background-color:#FFFFFF;"><semantics id="A2.T6.2.2.2.2.1.1.m1.1a"><mo id="A2.T6.2.2.2.2.1.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="A2.T6.2.2.2.2.1.1.m1.1.1.cmml">↑</mo><annotation-xml encoding="MathML-Content" id="A2.T6.2.2.2.2.1.1.m1.1b"><ci id="A2.T6.2.2.2.2.1.1.m1.1.1.cmml" xref="A2.T6.2.2.2.2.1.1.m1.1.1">↑</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.T6.2.2.2.2.1.1.m1.1c">\uparrow</annotation><annotation encoding="application/x-llamapun" id="A2.T6.2.2.2.2.1.1.m1.1d">↑</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="A2.T6.3.3.3.3" style="background-color:#FFFFFF;">LPIPS <math alttext="\downarrow" class="ltx_Math" display="inline" id="A2.T6.3.3.3.3.1.1.m1.1" style="background-color:#FFFFFF;"><semantics id="A2.T6.3.3.3.3.1.1.m1.1a"><mo id="A2.T6.3.3.3.3.1.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="A2.T6.3.3.3.3.1.1.m1.1.1.cmml">↓</mo><annotation-xml encoding="MathML-Content" id="A2.T6.3.3.3.3.1.1.m1.1b"><ci id="A2.T6.3.3.3.3.1.1.m1.1.1.cmml" xref="A2.T6.3.3.3.3.1.1.m1.1.1">↓</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.T6.3.3.3.3.1.1.m1.1c">\downarrow</annotation><annotation encoding="application/x-llamapun" id="A2.T6.3.3.3.3.1.1.m1.1d">↓</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="A2.T6.4.4.4.4" style="background-color:#FFFFFF;"> <table class="ltx_tabular ltx_align_middle" id="A2.T6.4.4.4.4.2" style="background-color:#FFFFFF;"> <tr class="ltx_tr" id="A2.T6.4.4.4.4.2.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="A2.T6.4.4.4.4.2.1.1">AbsRel</td> </tr> </table> <math alttext="\downarrow" class="ltx_Math" display="inline" id="A2.T6.4.4.4.4.1.m1.1" style="background-color:#FFFFFF;"><semantics id="A2.T6.4.4.4.4.1.m1.1a"><mo id="A2.T6.4.4.4.4.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="A2.T6.4.4.4.4.1.m1.1.1.cmml">↓</mo><annotation-xml encoding="MathML-Content" id="A2.T6.4.4.4.4.1.m1.1b"><ci id="A2.T6.4.4.4.4.1.m1.1.1.cmml" xref="A2.T6.4.4.4.4.1.m1.1.1">↓</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.T6.4.4.4.4.1.m1.1c">\downarrow</annotation><annotation encoding="application/x-llamapun" id="A2.T6.4.4.4.4.1.m1.1d">↓</annotation></semantics></math> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="A2.T6.5.5.5.5" style="background-color:#FFFFFF;">PSNR*<math alttext="\uparrow" class="ltx_Math" display="inline" id="A2.T6.5.5.5.5.1.m1.1"><semantics id="A2.T6.5.5.5.5.1.m1.1a"><mo id="A2.T6.5.5.5.5.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="A2.T6.5.5.5.5.1.m1.1.1.cmml">↑</mo><annotation-xml encoding="MathML-Content" id="A2.T6.5.5.5.5.1.m1.1b"><ci id="A2.T6.5.5.5.5.1.m1.1.1.cmml" xref="A2.T6.5.5.5.5.1.m1.1.1">↑</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.T6.5.5.5.5.1.m1.1c">\uparrow</annotation><annotation encoding="application/x-llamapun" id="A2.T6.5.5.5.5.1.m1.1d">↑</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="A2.T6.6.6.6.6" style="background-color:#FFFFFF;">SSIM <math alttext="\uparrow" class="ltx_Math" display="inline" id="A2.T6.6.6.6.6.1.1.m1.1" style="background-color:#FFFFFF;"><semantics id="A2.T6.6.6.6.6.1.1.m1.1a"><mo id="A2.T6.6.6.6.6.1.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="A2.T6.6.6.6.6.1.1.m1.1.1.cmml">↑</mo><annotation-xml encoding="MathML-Content" id="A2.T6.6.6.6.6.1.1.m1.1b"><ci id="A2.T6.6.6.6.6.1.1.m1.1.1.cmml" xref="A2.T6.6.6.6.6.1.1.m1.1.1">↑</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.T6.6.6.6.6.1.1.m1.1c">\uparrow</annotation><annotation encoding="application/x-llamapun" id="A2.T6.6.6.6.6.1.1.m1.1d">↑</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="A2.T6.7.7.7.7" style="background-color:#FFFFFF;">LPIPS <math alttext="\downarrow" class="ltx_Math" display="inline" id="A2.T6.7.7.7.7.1.1.m1.1" style="background-color:#FFFFFF;"><semantics id="A2.T6.7.7.7.7.1.1.m1.1a"><mo id="A2.T6.7.7.7.7.1.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="A2.T6.7.7.7.7.1.1.m1.1.1.cmml">↓</mo><annotation-xml encoding="MathML-Content" id="A2.T6.7.7.7.7.1.1.m1.1b"><ci id="A2.T6.7.7.7.7.1.1.m1.1.1.cmml" xref="A2.T6.7.7.7.7.1.1.m1.1.1">↓</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.T6.7.7.7.7.1.1.m1.1c">\downarrow</annotation><annotation encoding="application/x-llamapun" id="A2.T6.7.7.7.7.1.1.m1.1d">↓</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="A2.T6.8.8.8.8" style="background-color:#FFFFFF;"> <table class="ltx_tabular ltx_align_middle" id="A2.T6.8.8.8.8.1" style="background-color:#FFFFFF;"> <tr class="ltx_tr" id="A2.T6.8.8.8.8.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="A2.T6.8.8.8.8.1.1.1">Feat. Sim.</td> </tr> </table> <math alttext="\uparrow" class="ltx_Math" display="inline" id="A2.T6.8.8.8.8.m1.1" style="background-color:#FFFFFF;"><semantics id="A2.T6.8.8.8.8.m1.1a"><mo id="A2.T6.8.8.8.8.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="A2.T6.8.8.8.8.m1.1.1.cmml">↑</mo><annotation-xml encoding="MathML-Content" id="A2.T6.8.8.8.8.m1.1b"><ci id="A2.T6.8.8.8.8.m1.1.1.cmml" xref="A2.T6.8.8.8.8.m1.1.1">↑</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.T6.8.8.8.8.m1.1c">\uparrow</annotation><annotation encoding="application/x-llamapun" id="A2.T6.8.8.8.8.m1.1d">↑</annotation></semantics></math> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="A2.T6.9.9.9.9" style="background-color:#FFFFFF;"> <table class="ltx_tabular ltx_align_middle" id="A2.T6.9.9.9.9.2" style="background-color:#FFFFFF;"> <tr class="ltx_tr" id="A2.T6.9.9.9.9.2.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="A2.T6.9.9.9.9.2.1.1">AbsRel</td> </tr> </table> <math alttext="\downarrow" class="ltx_Math" display="inline" id="A2.T6.9.9.9.9.1.m1.1" style="background-color:#FFFFFF;"><semantics id="A2.T6.9.9.9.9.1.m1.1a"><mo id="A2.T6.9.9.9.9.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="A2.T6.9.9.9.9.1.m1.1.1.cmml">↓</mo><annotation-xml encoding="MathML-Content" id="A2.T6.9.9.9.9.1.m1.1b"><ci id="A2.T6.9.9.9.9.1.m1.1.1.cmml" xref="A2.T6.9.9.9.9.1.m1.1.1">↓</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.T6.9.9.9.9.1.m1.1c">\downarrow</annotation><annotation encoding="application/x-llamapun" id="A2.T6.9.9.9.9.1.m1.1d">↓</annotation></semantics></math> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="A2.T6.10.10.10.10" style="background-color:#FFFFFF;"> <table class="ltx_tabular ltx_align_middle" id="A2.T6.10.10.10.10.2" style="background-color:#FFFFFF;"> <tr class="ltx_tr" id="A2.T6.10.10.10.10.2.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="A2.T6.10.10.10.10.2.1.1">Delta1</td> </tr> </table> <math alttext="\uparrow" class="ltx_Math" display="inline" id="A2.T6.10.10.10.10.1.m1.1" style="background-color:#FFFFFF;"><semantics id="A2.T6.10.10.10.10.1.m1.1a"><mo id="A2.T6.10.10.10.10.1.m1.1.1" mathbackground="#FFFFFF" stretchy="false" xref="A2.T6.10.10.10.10.1.m1.1.1.cmml">↑</mo><annotation-xml encoding="MathML-Content" id="A2.T6.10.10.10.10.1.m1.1b"><ci id="A2.T6.10.10.10.10.1.m1.1.1.cmml" xref="A2.T6.10.10.10.10.1.m1.1.1">↑</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.T6.10.10.10.10.1.m1.1c">\uparrow</annotation><annotation encoding="application/x-llamapun" id="A2.T6.10.10.10.10.1.m1.1d">↑</annotation></semantics></math> </th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="A2.T6.10.10.12.1"> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A2.T6.10.10.12.1.1">0</td> <td class="ltx_td ltx_border_t" id="A2.T6.10.10.12.1.2"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="A2.T6.10.10.12.1.3"></td> <td class="ltx_td ltx_align_center ltx_border_t" id="A2.T6.10.10.12.1.4" style="background-color:#FFFFFF;">25.45</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A2.T6.10.10.12.1.5" style="background-color:#FFFFFF;">0.776</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A2.T6.10.10.12.1.6" style="background-color:#FFFFFF;">0.298</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A2.T6.10.10.12.1.7" style="background-color:#FFFFFF;">0.131</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A2.T6.10.10.12.1.8" style="background-color:#FFFFFF;">19.43</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A2.T6.10.10.12.1.9" style="background-color:#FFFFFF;">0.572</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A2.T6.10.10.12.1.10" style="background-color:#FFFFFF;">0.374</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A2.T6.10.10.12.1.11" style="background-color:#FFFFFF;">0.519</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A2.T6.10.10.12.1.12" style="background-color:#FFFFFF;">0.177</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A2.T6.10.10.12.1.13" style="background-color:#FFFFFF;">0.700</td> </tr> <tr class="ltx_tr" id="A2.T6.10.10.13.2"> <td class="ltx_td ltx_align_center ltx_border_r" id="A2.T6.10.10.13.2.1">1</td> <td class="ltx_td ltx_align_center" id="A2.T6.10.10.13.2.2" style="background-color:#FFFFFF;">✓</td> <td class="ltx_td ltx_border_r" id="A2.T6.10.10.13.2.3"></td> <td class="ltx_td ltx_align_center" id="A2.T6.10.10.13.2.4" style="background-color:#FFFFFF;">27.70</td> <td class="ltx_td ltx_align_center" id="A2.T6.10.10.13.2.5" style="background-color:#FFFFFF;">0.837</td> <td class="ltx_td ltx_align_center" id="A2.T6.10.10.13.2.6" style="background-color:#FFFFFF;">0.199</td> <td class="ltx_td ltx_align_center ltx_border_r" id="A2.T6.10.10.13.2.7" style="background-color:#FFFFFF;">0.082</td> <td class="ltx_td ltx_align_center" id="A2.T6.10.10.13.2.8" style="background-color:#FFFFFF;">20.79</td> <td class="ltx_td ltx_align_center" id="A2.T6.10.10.13.2.9" style="background-color:#FFFFFF;">0.613</td> <td class="ltx_td ltx_align_center" id="A2.T6.10.10.13.2.10" style="background-color:#FFFFFF;">0.281</td> <td class="ltx_td ltx_align_center" id="A2.T6.10.10.13.2.11" style="background-color:#FFFFFF;">0.639</td> <td class="ltx_td ltx_align_center" id="A2.T6.10.10.13.2.12" style="background-color:#FFFFFF;">0.078</td> <td class="ltx_td ltx_align_center" id="A2.T6.10.10.13.2.13" style="background-color:#FFFFFF;">0.914</td> </tr> <tr class="ltx_tr" id="A2.T6.10.10.14.3"> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_r" id="A2.T6.10.10.14.3.1">2</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="A2.T6.10.10.14.3.2" style="background-color:#FFFFFF;">✓</td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_r" id="A2.T6.10.10.14.3.3" style="background-color:#FFFFFF;">✓</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="A2.T6.10.10.14.3.4" style="background-color:#FFFFFF;">28.51</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="A2.T6.10.10.14.3.5" style="background-color:#FFFFFF;">0.859</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="A2.T6.10.10.14.3.6" style="background-color:#FFFFFF;">0.179</td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_r" id="A2.T6.10.10.14.3.7" style="background-color:#FFFFFF;">0.085</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="A2.T6.10.10.14.3.8" style="background-color:#FFFFFF;">20.83</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="A2.T6.10.10.14.3.9" style="background-color:#FFFFFF;">0.611</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="A2.T6.10.10.14.3.10" style="background-color:#FFFFFF;">0.271</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="A2.T6.10.10.14.3.11" style="background-color:#FFFFFF;">0.646</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="A2.T6.10.10.14.3.12" style="background-color:#FFFFFF;">0.078</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="A2.T6.10.10.14.3.13" style="background-color:#FFFFFF;">0.914</td> </tr> </tbody> </table> </div> </figure> <figure class="ltx_figure" id="A2.F6"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="218" id="A2.F6.g1" src="extracted/6295770/figures/gfx/intersection.png" width="316"/> <figcaption class="ltx_caption ltx_centering">Figure 6: An intersection with occluded areas. MTGS can also reconstruct big intersections with occlusions, e.g., buildings and road medians. </figcaption> </figure> <figure class="ltx_figure" id="A2.F7"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="693" id="A2.F7.g1" src="x5.png" width="951"/> <figcaption class="ltx_caption ltx_centering">Figure 7: More visualization on extrapolated views. MTGS consistently generates high-quality view extrapolations. However, since all transient nodes are removed when rendering unseen trajectories, floating artifacts appear over car parking areas.</figcaption> </figure> <div class="ltx_para ltx_noindent" id="A2.p2"> Reconstruction on big intersections. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#A2.F6" title="Figure 6 ‣ Appendix B Experiments ‣ MTGS: Multi-Traversal Gaussian Splatting">6</a> shows that MTGS can reconstruction big intersections with occlusions. We exclude such data from our evaluation to ensure that performance gains are not simply due to seeing the unseen regions. </div> <div class="ltx_para ltx_noindent" id="A2.p3"> More visualization. As shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#A2.F7" title="Figure 7 ‣ Appendix B Experiments ‣ MTGS: Multi-Traversal Gaussian Splatting">7</a>, we show more visualization on extrapolated views of our blocks. The visualization results of each block are arranged sequentially from left to right according to the temporal order of the traversal. </div> <div class="ltx_para ltx_noindent" id="A2.p4"> Ablation on the number of traversals. Results of all <math alttext="7" class="ltx_Math" display="inline" id="A2.p4.1.m1.1"><semantics id="A2.p4.1.m1.1a"><mn id="A2.p4.1.m1.1.1" xref="A2.p4.1.m1.1.1.cmml">7</mn><annotation-xml encoding="MathML-Content" id="A2.p4.1.m1.1b"><cn id="A2.p4.1.m1.1.1.cmml" type="integer" xref="A2.p4.1.m1.1.1">7</cn></annotation-xml><annotation encoding="application/x-tex" id="A2.p4.1.m1.1c">7</annotation><annotation encoding="application/x-llamapun" id="A2.p4.1.m1.1d">7</annotation></semantics></math> metrics on both training and novel-view traversals are shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.12552v2#A2.F8" title="Figure 8 ‣ Appendix B Experiments ‣ MTGS: Multi-Traversal Gaussian Splatting">8</a>. </div> <figure class="ltx_figure" id="A2.F8"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"> <div class="ltx_block ltx_figure_panel" id="A2.F8.2"> <figure class="ltx_figure ltx_align_center" id="A2.F8.sf1"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="482" id="A2.F8.sf1.g1" src="x6.png" width="428"/> <figcaption class="ltx_caption">(a) 3DGS.</figcaption> </figure> <figure class="ltx_figure ltx_align_center" id="A2.F8.sf2"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="482" id="A2.F8.sf2.g1" src="x7.png" width="428"/> <figcaption class="ltx_caption">(b) OmniRe.</figcaption> </figure> </div> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="A2.F8.sf3"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="482" id="A2.F8.sf3.g1" src="x8.png" width="428"/> <figcaption class="ltx_caption">(c) Ours.</figcaption> </figure> </div> </div> <figcaption class="ltx_caption ltx_centering">Figure 8: Performances of three methods when trained on more traversals. Note that outer rings represent better performance instead of larger scores. More traversals do not guarantee better performances for existing methods while our designs could continually benefit from more traversals used.</figcaption> </figure> </section> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Thu Mar 20 08:10:11 2025 by <a class="ltx_LaTeXML_logo" href="http://dlmf.nist.gov/LaTeXML/">LaTeXML<img alt="Mascot Sammy" src="data:image/png;base64,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"/></a> </div></footer> </div> </body> </html>