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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments</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/2406.16907v1/"/></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/2406.16907v1#S1" title="In RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">I </span><span class="ltx_text ltx_font_smallcaps">Introduction</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S2" title="In RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">II </span><span class="ltx_text ltx_font_smallcaps">Related Works</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S3" title="In RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">III </span><span class="ltx_text ltx_font_smallcaps">Neural Point Field for Wireless Channel</span></span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S3.SS1" title="In III Neural Point Field for Wireless Channel ‣ RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">III-A</span> </span><span class="ltx_text ltx_font_italic" style="color:#000000;">Data Preparation</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S3.SS2" title="In III Neural Point Field for Wireless Channel ‣ RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">III-B</span> </span><span class="ltx_text ltx_font_italic" style="color:#000000;">Point Cloud Feature Embedding</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S3.SS3" title="In III Neural Point Field for Wireless Channel ‣ RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">III-C</span> </span><span class="ltx_text ltx_font_italic" style="color:#000000;">Path Tracing with Light Probes and Point Clouds</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S3.SS4" title="In III Neural Point Field for Wireless Channel ‣ RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">III-D</span> </span><span class="ltx_text ltx_font_italic" style="color:#000000;">Receivers: Unveiling Ray Physics from Light Probes</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S3.SS5" title="In III Neural Point Field for Wireless Channel ‣ RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">III-E</span> </span><span class="ltx_text ltx_font_italic" style="color:#000000;">Spherical Harmonics-based Decoding of Ray Features</span></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S4" title="In RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">IV </span><span class="ltx_text ltx_font_smallcaps">Numerical Experiments</span></span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S4.SS1" title="In IV Numerical Experiments ‣ RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-A</span> </span><span class="ltx_text ltx_font_italic" style="color:#000000;">Experimental Setup and Training</span></span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S4.SS1.SSS1" title="In IV-A Experimental Setup and Training ‣ IV Numerical Experiments ‣ RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-A</span>1 </span>Data Collection</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S4.SS1.SSS2" title="In IV-A Experimental Setup and Training ‣ IV Numerical Experiments ‣ RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-A</span>2 </span>Training setups</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S4.SS1.SSS3" title="In IV-A Experimental Setup and Training ‣ IV Numerical Experiments ‣ RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-A</span>3 </span>Evaluation Metric</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S4.SS2" title="In IV Numerical Experiments ‣ RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-B</span> </span><span class="ltx_text ltx_font_italic" style="color:#000000;">Validation and Verification</span></span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S4.SS2.SSS1" title="In IV-B Validation and Verification ‣ IV Numerical Experiments ‣ RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-B</span>1 </span>Assessment in Learning EM Propagation Physics</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S4.SS2.SSS2" title="In IV-B Validation and Verification ‣ IV Numerical Experiments ‣ RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-B</span>2 </span>Comparison to Other Neural Surrogates</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S4.SS2.SSS3" title="In IV-B Validation and Verification ‣ IV Numerical Experiments ‣ RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-B</span>3 </span><span class="ltx_text">Verification through Ablation Experiment</span></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S4.SS3" title="In IV Numerical Experiments ‣ RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-C</span> </span><span class="ltx_text ltx_font_italic" style="color:#000000;">Evaluation in Large-scale, 3D Wireless Scenes</span></span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S4.SS3.SSS1" title="In IV-C Evaluation in Large-scale, 3D Wireless Scenes ‣ IV Numerical Experiments ‣ RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-C</span>1 </span>Large-scale Environment</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S4.SS3.SSS2" title="In IV-C Evaluation in Large-scale, 3D Wireless Scenes ‣ IV Numerical Experiments ‣ RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-C</span>2 </span>Antenna Radiation Pattern</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S4.SS3.SSS3" title="In IV-C Evaluation in Large-scale, 3D Wireless Scenes ‣ IV Numerical Experiments ‣ RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-C</span>3 </span>Quantitative Measurements</span></a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S5" title="In RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">V </span><span class="ltx_text ltx_font_smallcaps">Conclusion</span></span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_title_document">RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Ge Cao and Zhen Peng </span><span class="ltx_author_notes">G. Cao and Z. Peng are with the Center for Computational Electromagnetics, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA (e-mail: gecao2@illinois.edu; zvpeng@illinois.edu).</span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id1.id1">The radio wave propagation channel is central to the performance of wireless communication systems. In this paper, we introduce a novel machine learning-empowered methodology for wireless channel modeling. The key ingredients include a point-cloud-based neural network and a Spherical Harmonics encoder with light probes. Our approach offers several significant advantages, including the flexibility to adjust antenna radiation patterns and transmitter/receiver locations, the capability to predict radio power maps, and the scalability of large-scale wireless scenes. As a result, it lays the groundwork for an end-to-end pipeline for network planning and deployment optimization. The proposed work is validated in various outdoor and indoor radio environments.</p> </div> <div class="ltx_para" id="p1"> <span class="ltx_ERROR undefined" id="p1.3">{strip}</span> <div class="ltx_logical-block ltx_minipage ltx_align_middle" id="p1.2" style="width:433.6pt;"> <div class="ltx_para" id="p1.2.p1"> <img alt="[Uncaptioned image]" class="ltx_graphics ltx_img_landscape" height="142" id="p1.1.g1" src="extracted/5641587/figure/teaser.png" width="595"/> </div> <figure class="ltx_figure ltx_align_center" id="S0.F1"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S0.F1.2.1.1" style="font-size:90%;">Figure 1</span>: </span><span class="ltx_text" id="S0.F1.3.2" style="font-size:90%;"> The schematic illustrates the input, output, and application of our proposed neural point field network framework for predicting wireless radio channel properties in large-scale environments. </span></figcaption> </figure> </div> </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">I </span><span class="ltx_text ltx_font_smallcaps" id="S1.1.1">Introduction</span> </h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">Understanding and accurately modeling the characteristics of the propagation channel are essential for the design, deployment, and optimization of wireless communication networks <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib1" title="">1</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib2" title="">2</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib3" title="">3</a>]</cite>. Although Maxwell’s Equations govern the fundamental physics of wireless information transmission, obtaining full-wave solutions in large-scale environments is typically challenging and time-consuming <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib4" title="">4</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib5" title="">5</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib6" title="">6</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib7" title="">7</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib8" title="">8</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib9" title="">9</a>]</cite>. Ray tracing-based simulators are commonly employed for modeling wireless channel properties <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib10" title="">10</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib11" title="">11</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib12" title="">12</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib13" title="">13</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib14" title="">14</a>]</cite>. In the ray tracing process, electromagnetic (EM) rays are uniformly launched from the transmitter antenna, undergoing reflections, transmissions, and diffractions with various buildings and floors, ultimately reaching the receiver locations. These ray paths and interactions yield valuable wireless channel information such as channel gain, channel transfer function, and channel impulse response.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">While ray tracing has been a popular tool in wireless channel modeling, its computational complexity escalates with the number of ray-object interactions. Moreover, in wireless deployment and planning scenarios, frequent modifications to transmitter/receiver locations are common. Typically, a new ray tracing simulation is required for each configuration change. This exhibits a noticeable gap between the simulation time of ray-tracing simulators and the rapid time-to-solution demand of wireless network design and optimization. To address these needs, neural network-based forward surrogate models emerge as an attractive solution <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib15" title="">15</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib16" title="">16</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib17" title="">17</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib18" title="">18</a>]</cite>. Neural networks generally offer faster runtime compared to ray tracing algorithms, and their accuracy can be enhanced by refining the training dataset rather than increasing runtime.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1">The objective of this paper is to develop a neural network surrogate capable of predicting wireless channel properties across large-scale environments. The overview of the proposed framework is given in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S0.F1" title="Figure 1 ‣ RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_tag">1</span></a>. In our methodology, we train the neural surrogate using ray-tracing solutions corresponding to a finite set of transmitting locations within a specific radio environment. Once trained, the neural surrogate leverages its understanding of EM propagation physics to predict EM wave propagation for new transmitter/receiver locations and different antenna radiation patterns. This research emphasizes two key features: (1) the neural surrogate’s functionality to predict the spatial distribution of radiated power (i.e., the radio coverage or path loss map), and (2) its effective generalization to large-scale scenes in both outdoor and indoor environments.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1">In the realm of neural surrogate development for radio wave propagation, the learning of scene representations is an aspect that has received limited attention in previous works. Many existing approaches primarily focus on 2D image tasks, typically from a bird’s-eye view, and lack the incorporation of geometry information as input <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib19" title="">19</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib20" title="">20</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib21" title="">21</a>]</cite>. Another recent study <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib22" title="">22</a>]</cite> focuses on explicitly learning the meshed geometries, thereby limiting its generalizability to large outdoor scenes. In contrast, our proposed work offers a fresh perspective on neural scene representation. The 3D propagation environment (wireless scene) is rendered using point clouds, a representation well-known for its adaptability and intuitive scalability to large-scale scenes <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib23" title="">23</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.1">Moreover, we introduce the Neural Point Field framework to implicitly embed wireless channel state information into light probes <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib24" title="">24</a>]</cite>. Each light probe encapsulates EM ray properties, which are interpolated using a Spherical Harmonics encoder and decoder <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib25" title="">25</a>]</cite>. This facilitates the extraction of propagation information from queries in different ray directions. Conceptually, these light probes are designed to capture the site-specific EM ray propagation physics. Receivers can seamlessly extract path tracing and ray propagation from these probes, streamlining the process and enhancing overall efficiency.</p> </div> <div class="ltx_para" id="S1.p6"> <p class="ltx_p" id="S1.p6.1">Compared to existing neural ray tracing methods in the literature, the proposed work excels in scalability and flexibility, accommodating diverse levels of geometry complexity while maintaining high-quality channel prediction. We validate our proposed pipeline across small indoor, medium outdoor, and large city scenes. The results demonstrate the efficacy of our approach in predicting wireless channel properties across various scales of scenes.</p> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">II </span><span class="ltx_text ltx_font_smallcaps" id="S2.1.1">Related Works</span> </h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.1">In this section, we discuss related works from both the machine learning (ML) and wireless communication communities. Given the resemblance between rendering and wireless channel modeling algorithms, we particularly emphasize studies in neural rendering and computer graphics within the deep learning field. Additionally, since our pipeline design necessitates an implicit representation of geometry, we also introduce relevant works on geometry in neural networks.</p> </div> <div class="ltx_para" id="S2.p2"> <p class="ltx_p" id="S2.p2.1"><span class="ltx_text ltx_font_bold" id="S2.p2.1.1">Neural Rendering:</span> The ray tracing algorithm is widely used in the rendering process in 3D computer graphics. Leveraging this foundational understanding, our research explores valuable insights from advancements in neural rendering, enriching our approach to wireless channel modeling. Recently, advancements in 3D scene representation using neural networks have showcased their ability to render scenes quickly and flexibly. In these approaches, the radiance field is embedded within neural networks, such as Multi-Layer Perceptrons (MLPs), or at a higher level, within the volume space. This implies that the lighting information is typically fixed and cannot be modified. Despite the complexity of light sources in the rendering process <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib26" title="">26</a>]</cite>, several works have achieved relighting techniques <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib27" title="">27</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib28" title="">28</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib25" title="">25</a>]</cite>.</p> </div> <div class="ltx_para" id="S2.p3"> <p class="ltx_p" id="S2.p3.1">Since the publication of Kerbl et al.’s work on 3D Gaussian Splatting <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib29" title="">29</a>]</cite>, this new neural rendering technique has garnered significant attention. A Gaussian kernel is applied and learned to represent scene geometries in the format of point clouds. Subsequently, several related works have emerged, including research on relighting <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib30" title="">30</a>]</cite> and the reconstruction of human avatars <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib31" title="">31</a>]</cite>.</p> </div> <div class="ltx_para" id="S2.p4"> <p class="ltx_p" id="S2.p4.1">Before the development of 3D Gaussian Splatting, a strategy known as Neural Point Light Fields (NeuralPointLF) was introduced, demonstrating the potential of point cloud formats in the domain of neural rendering <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib23" title="">23</a>]</cite>. The distinction between NeuralPointLF and 3D Gaussian Splatting lies in the fact that NeuralPointLF does not necessitate a rasterization process in the pipeline. Since ray-tracing simulations in wireless channel modeling also do not require rasterization, our network draws inspiration from NeuralPointLF and incorporates attention techniques into the framework <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib32" title="">32</a>]</cite>. Furthermore, as NeuralPointLF lacks a relighting process, our pipeline incorporates relighting into its design. This addition addresses scenarios involving changing antenna locations or radiation patterns.</p> </div> <div class="ltx_para" id="S2.p5"> <p class="ltx_p" id="S2.p5.3"><span class="ltx_text ltx_font_bold" id="S2.p5.3.1">Geometry Representation:</span> The representation format of 3D geometry is crucial for all ML tasks involving three-dimensional data. The most common method for representing geometry is through mesh triangles, consisting of a set of vertices (<math alttext="\mathbb{V}" class="ltx_Math" display="inline" id="S2.p5.1.m1.1"><semantics id="S2.p5.1.m1.1a"><mi id="S2.p5.1.m1.1.1" xref="S2.p5.1.m1.1.1.cmml">𝕍</mi><annotation-xml encoding="MathML-Content" id="S2.p5.1.m1.1b"><ci id="S2.p5.1.m1.1.1.cmml" xref="S2.p5.1.m1.1.1">𝕍</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.1.m1.1c">\mathbb{V}</annotation><annotation encoding="application/x-llamapun" id="S2.p5.1.m1.1d">blackboard_V</annotation></semantics></math>), edges (<math alttext="\mathbb{E}" class="ltx_Math" display="inline" id="S2.p5.2.m2.1"><semantics id="S2.p5.2.m2.1a"><mi id="S2.p5.2.m2.1.1" xref="S2.p5.2.m2.1.1.cmml">𝔼</mi><annotation-xml encoding="MathML-Content" id="S2.p5.2.m2.1b"><ci id="S2.p5.2.m2.1.1.cmml" xref="S2.p5.2.m2.1.1">𝔼</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.2.m2.1c">\mathbb{E}</annotation><annotation encoding="application/x-llamapun" id="S2.p5.2.m2.1d">blackboard_E</annotation></semantics></math>), and faces (<math alttext="\mathbb{F}" class="ltx_Math" display="inline" id="S2.p5.3.m3.1"><semantics id="S2.p5.3.m3.1a"><mi id="S2.p5.3.m3.1.1" xref="S2.p5.3.m3.1.1.cmml">𝔽</mi><annotation-xml encoding="MathML-Content" id="S2.p5.3.m3.1b"><ci id="S2.p5.3.m3.1.1.cmml" xref="S2.p5.3.m3.1.1">𝔽</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.3.m3.1c">\mathbb{F}</annotation><annotation encoding="application/x-llamapun" id="S2.p5.3.m3.1d">blackboard_F</annotation></semantics></math>). While meshed geometries are widely utilized in computational science and engineering, their utilization in deep learning is limited due to the non-differentiability of triangle face indices. Although some researchers have attempted to apply statistical methods to make mesh triangles differentiable <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib33" title="">33</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib34" title="">34</a>]</cite>, these strategies are still computationally intensive for neural networks.</p> </div> <div class="ltx_para" id="S2.p6"> <p class="ltx_p" id="S2.p6.1">Point clouds have emerged as a preferred geometry representation format in neural network-based research. This representation is utilized across various tasks, including 3D surface reconstruction <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib35" title="">35</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib36" title="">36</a>]</cite>, geometry denoising <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib37" title="">37</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib38" title="">38</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib39" title="">39</a>]</cite>, and geometry completion <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib40" title="">40</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib41" title="">41</a>]</cite>. Leveraging the differentiability of point clouds, our work adopts the <em class="ltx_emph ltx_font_italic" id="S2.p6.1.1">PointNet</em> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib42" title="">42</a>]</cite> architecture for geometry representation. While mesh triangles and point clouds are prevalent, other representation formats exist, such as the multi-view model <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib43" title="">43</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib44" title="">44</a>]</cite> and surface random walk <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib45" title="">45</a>]</cite>.</p> </div> <div class="ltx_para" id="S2.p7"> <p class="ltx_p" id="S2.p7.1"><span class="ltx_text ltx_font_bold" id="S2.p7.1.1">Neural Radio Channel Modelling:</span> Physically-based simulation guided by neural networks is gaining popularity across various scientific domains, including fluid dynamics <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib46" title="">46</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib47" title="">47</a>]</cite>, soft body dynamics <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib48" title="">48</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib49" title="">49</a>]</cite>, and electrodynamics <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib50" title="">50</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib51" title="">51</a>]</cite>, etc. In the field of applied and computational electromagnetics, several approaches leveraging neural networks have been proposed <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib52" title="">52</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib53" title="">53</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib54" title="">54</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib55" title="">55</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib56" title="">56</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib57" title="">57</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib58" title="">58</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib59" title="">59</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib60" title="">60</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib61" title="">61</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib62" title="">62</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib63" title="">63</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib64" title="">64</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib65" title="">65</a>]</cite>. Many of these neural surrogates aim to learn the scattering process involving obstacles in free space. Given that wireless channel properties are governed by the propagation and scattering of EM waves, our work shares objectives related to those of these approaches. The emphasis of this work is to expand the application domain to encompass more complex scenarios, specifically extending into 3D environments featuring intricate obstacles like buildings.</p> </div> <div class="ltx_para" id="S2.p8"> <p class="ltx_p" id="S2.p8.1">Until now, there has been limited attention given to the task of wireless channel modeling in complex 3D environments. A recent work addressing this task is <em class="ltx_emph ltx_font_italic" id="S2.p8.1.1">WINERT</em> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib22" title="">22</a>]</cite>. In their approach, a complete ray tracing process is implemented, with a focus on learning the propagation properties (reflection, transmission, diffraction) of buildings. However, they did not implement the ray-triangle intersection process as differentiable, citing its non-differentiability. Furthermore, their pipeline is not suitable for handling large-scale and complicated scene geometries.</p> </div> <div class="ltx_para" id="S2.p9"> <p class="ltx_p" id="S2.p9.1">Several other works have also aimed to develop neural surrogates for predicting path loss map information. Nevertheless, most of these works focus on 2D tasks that do not explicitly require geometry representation. Instead, they rely on 2D bird’s-eye-view images (heatmaps) for training <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib19" title="">19</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib20" title="">20</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib21" title="">21</a>]</cite>. While this format simplifies the learning process and results in a faster pipeline, it may encounter difficulties in effectively capturing the complexities of 3D scenes in an end-to-end manner.</p> </div> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">III </span><span class="ltx_text ltx_font_smallcaps" id="S3.1.1">Neural Point Field for Wireless Channel</span> </h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.1">The proposed work aims to investigate a neural point field network to simulate the ray tracing process between transmitters and receivers within complex wireless scenes. At its core, this method relies on three fundamental elements: leveraging point clouds for the representation of geometric structures, integrating light probes to capture path tracing and ray propagation information, and utilizing spherical harmonic functions for the extraction of field data. An overview of the pipeline is illustrated in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S3.F2" title="Figure 2 ‣ III Neural Point Field for Wireless Channel ‣ RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_tag">2</span></a>, which we henceforth refer to as RayProNet. The detailed technical ingredients and underlying rationales are provided below.</p> </div> <figure class="ltx_figure" id="S3.F2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="127" id="S3.F2.g1" src="extracted/5641587/figure/pipeline.png" width="595"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S3.F2.2.1.1" style="font-size:90%;">Figure 2</span>: </span><span class="ltx_text" id="S3.F2.3.2" style="font-size:90%;">RayProNet: a neural point field framework for wireless channel modeling pipeline. (The symbols A - E represent the subsections in Section III.) </span></figcaption> </figure> <section class="ltx_subsection" id="S3.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S3.SS1.4.1.1">III-A</span> </span><span class="ltx_text ltx_font_italic" id="S3.SS1.5.2" style="color:#000000;">Data Preparation</span> </h3> <div class="ltx_para" id="S3.SS1.p1"> <p class="ltx_p" id="S3.SS1.p1.1">The RayProNet pipeline relies on two primary inputs: the locations of receivers and transmitters, alongside the 3D geometry of the environment, which is initially transformed into point clouds as the default format for representation. Point clouds offer an efficient means of encoding complex geometric features like obstacles, buildings, and terrain by sampling points in space to capture the characteristics of interacting objects within the environment. This approach allows for the effective encoding of interacting objects, with a particular emphasis on learning geometric features.</p> </div> <div class="ltx_para" id="S3.SS1.p2"> <p class="ltx_p" id="S3.SS1.p2.5">In addition, light probes are uniformly placed throughout the scene, capturing the propagation behavior of EM rays through space. Their integration into the pipeline allows the model to acquire essential insights into ray paths, reflections, and diffractions, thereby enhancing the accuracy and efficiency of the learning process. Light probes play a crucial role in encoding propagation information, particularly in environments characterized by sparse geometric structures, as elaborated in Section III.C. The data preparation stage proceeds as follows:</p> <ul class="ltx_itemize" id="S3.I1"> <li class="ltx_item" id="S3.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S3.I1.i1.p1"> <p class="ltx_p" id="S3.I1.i1.p1.1"><span class="ltx_text ltx_font_bold" id="S3.I1.i1.p1.1.1">Transmitter Setup:</span> Initially, we define the locations of transmitters and configure their antenna patterns. This process ensures an accurate representation of transmitter characteristics in the simulation.</p> </div> </li> <li class="ltx_item" id="S3.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S3.I1.i2.p1"> <p class="ltx_p" id="S3.I1.i2.p1.1"><span class="ltx_text ltx_font_bold" id="S3.I1.i2.p1.1.1">Receiver Setup:</span> Similarly, we specify the locations of receivers and configure their antenna patterns to accurately simulate receiver behavior in the wireless environment.</p> </div> </li> <li class="ltx_item" id="S3.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S3.I1.i3.p1"> <p class="ltx_p" id="S3.I1.i3.p1.2"><span class="ltx_text ltx_font_bold" id="S3.I1.i3.p1.1.1">Identify <math alttext="n" class="ltx_Math" display="inline" id="S3.I1.i3.p1.1.1.m1.1"><semantics id="S3.I1.i3.p1.1.1.m1.1a"><mi id="S3.I1.i3.p1.1.1.m1.1.1" xref="S3.I1.i3.p1.1.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i3.p1.1.1.m1.1b"><ci id="S3.I1.i3.p1.1.1.m1.1.1.cmml" xref="S3.I1.i3.p1.1.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i3.p1.1.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i3.p1.1.1.m1.1d">italic_n</annotation></semantics></math> Nearest Light Probes:</span> For each receiver, we identify the <math alttext="n" class="ltx_Math" display="inline" id="S3.I1.i3.p1.2.m1.1"><semantics id="S3.I1.i3.p1.2.m1.1a"><mi id="S3.I1.i3.p1.2.m1.1.1" xref="S3.I1.i3.p1.2.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i3.p1.2.m1.1b"><ci id="S3.I1.i3.p1.2.m1.1.1.cmml" xref="S3.I1.i3.p1.2.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i3.p1.2.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i3.p1.2.m1.1d">italic_n</annotation></semantics></math> nearest light probes and record ray directions, enabling the collection of electromagnetic field information from the surroundings (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S3.F3" title="Figure 3 ‣ III-C Path Tracing with Light Probes and Point Clouds ‣ III Neural Point Field for Wireless Channel ‣ RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_tag">3</span></a>).</p> </div> </li> <li class="ltx_item" id="S3.I1.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S3.I1.i4.p1"> <p class="ltx_p" id="S3.I1.i4.p1.3"><span class="ltx_text ltx_font_bold" id="S3.I1.i4.p1.1.1">Identify <math alttext="K" class="ltx_Math" display="inline" id="S3.I1.i4.p1.1.1.m1.1"><semantics id="S3.I1.i4.p1.1.1.m1.1a"><mi id="S3.I1.i4.p1.1.1.m1.1.1" xref="S3.I1.i4.p1.1.1.m1.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i4.p1.1.1.m1.1b"><ci id="S3.I1.i4.p1.1.1.m1.1.1.cmml" xref="S3.I1.i4.p1.1.1.m1.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i4.p1.1.1.m1.1c">K</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i4.p1.1.1.m1.1d">italic_K</annotation></semantics></math> Nearest Points:</span> Next, we determine the <math alttext="K" class="ltx_Math" display="inline" id="S3.I1.i4.p1.2.m1.1"><semantics id="S3.I1.i4.p1.2.m1.1a"><mi id="S3.I1.i4.p1.2.m1.1.1" xref="S3.I1.i4.p1.2.m1.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i4.p1.2.m1.1b"><ci id="S3.I1.i4.p1.2.m1.1.1.cmml" xref="S3.I1.i4.p1.2.m1.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i4.p1.2.m1.1c">K</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i4.p1.2.m1.1d">italic_K</annotation></semantics></math> nearest points for each light probe and record this as a <math alttext="K" class="ltx_Math" display="inline" id="S3.I1.i4.p1.3.m2.1"><semantics id="S3.I1.i4.p1.3.m2.1a"><mi id="S3.I1.i4.p1.3.m2.1.1" xref="S3.I1.i4.p1.3.m2.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i4.p1.3.m2.1b"><ci id="S3.I1.i4.p1.3.m2.1.1.cmml" xref="S3.I1.i4.p1.3.m2.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i4.p1.3.m2.1c">K</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i4.p1.3.m2.1d">italic_K</annotation></semantics></math>-closest direction attachment. This enables us to capture detailed geometric information about the scene (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S3.F4" title="Figure 4 ‣ III-C Path Tracing with Light Probes and Point Clouds ‣ III Neural Point Field for Wireless Channel ‣ RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_tag">4</span></a>).</p> </div> </li> </ul> <p class="ltx_p" id="S3.SS1.p2.4">The parameters <math alttext="n" class="ltx_Math" display="inline" id="S3.SS1.p2.1.m1.1"><semantics id="S3.SS1.p2.1.m1.1a"><mi id="S3.SS1.p2.1.m1.1.1" xref="S3.SS1.p2.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.1.m1.1b"><ci id="S3.SS1.p2.1.m1.1.1.cmml" xref="S3.SS1.p2.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.1.m1.1d">italic_n</annotation></semantics></math> and <math alttext="K" 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">K</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">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.2.m2.1c">K</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.2.m2.1d">italic_K</annotation></semantics></math> serve as hyperparameters that offer flexibility for customization based on scene complexity and application-specific consideration, allowing for tailored adjustments to the pipeline. For example, applications requiring highly fidelity predictions or precise localization may benefit from larger values of <math alttext="n" 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">n</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">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.3.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.3.m3.1d">italic_n</annotation></semantics></math> and <math alttext="K" class="ltx_Math" display="inline" id="S3.SS1.p2.4.m4.1"><semantics id="S3.SS1.p2.4.m4.1a"><mi id="S3.SS1.p2.4.m4.1.1" xref="S3.SS1.p2.4.m4.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.4.m4.1b"><ci id="S3.SS1.p2.4.m4.1.1.cmml" xref="S3.SS1.p2.4.m4.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.4.m4.1c">K</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.4.m4.1d">italic_K</annotation></semantics></math>.</p> </div> </section> <section class="ltx_subsection" id="S3.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S3.SS2.4.1.1">III-B</span> </span><span class="ltx_text ltx_font_italic" id="S3.SS2.5.2" style="color:#000000;">Point Cloud Feature Embedding</span> </h3> <div class="ltx_para" id="S3.SS2.p1"> <p class="ltx_p" id="S3.SS2.p1.5">Given our primary focus on wireless channel modeling rather than rendering, employing a multi-view model presents challenges due to the absence of a specific look-at direction in our task. Therefore, we adopt the PointNet model <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib42" title="">42</a>]</cite> for its effectiveness and robustness in learning various features of point clouds. PointNet is originally proposed for 3D recognition tasks such as object classification, part segmentation, and semantic segmentation. Unlike traditional convolutional neural networks (CNNs) that operate on grid-like data and images, PointNet can directly process point clouds without requiring any intermediate representation like voxelization. In our implementation, we begin by normalizing all scene point clouds to the range <math alttext="[-1,1]" class="ltx_Math" display="inline" id="S3.SS2.p1.1.m1.2"><semantics id="S3.SS2.p1.1.m1.2a"><mrow id="S3.SS2.p1.1.m1.2.2.1" xref="S3.SS2.p1.1.m1.2.2.2.cmml"><mo id="S3.SS2.p1.1.m1.2.2.1.2" stretchy="false" xref="S3.SS2.p1.1.m1.2.2.2.cmml">[</mo><mrow id="S3.SS2.p1.1.m1.2.2.1.1" xref="S3.SS2.p1.1.m1.2.2.1.1.cmml"><mo id="S3.SS2.p1.1.m1.2.2.1.1a" xref="S3.SS2.p1.1.m1.2.2.1.1.cmml">−</mo><mn id="S3.SS2.p1.1.m1.2.2.1.1.2" xref="S3.SS2.p1.1.m1.2.2.1.1.2.cmml">1</mn></mrow><mo id="S3.SS2.p1.1.m1.2.2.1.3" xref="S3.SS2.p1.1.m1.2.2.2.cmml">,</mo><mn id="S3.SS2.p1.1.m1.1.1" xref="S3.SS2.p1.1.m1.1.1.cmml">1</mn><mo id="S3.SS2.p1.1.m1.2.2.1.4" stretchy="false" xref="S3.SS2.p1.1.m1.2.2.2.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.1.m1.2b"><interval closure="closed" id="S3.SS2.p1.1.m1.2.2.2.cmml" xref="S3.SS2.p1.1.m1.2.2.1"><apply id="S3.SS2.p1.1.m1.2.2.1.1.cmml" xref="S3.SS2.p1.1.m1.2.2.1.1"><minus id="S3.SS2.p1.1.m1.2.2.1.1.1.cmml" xref="S3.SS2.p1.1.m1.2.2.1.1"></minus><cn id="S3.SS2.p1.1.m1.2.2.1.1.2.cmml" type="integer" xref="S3.SS2.p1.1.m1.2.2.1.1.2">1</cn></apply><cn id="S3.SS2.p1.1.m1.1.1.cmml" type="integer" xref="S3.SS2.p1.1.m1.1.1">1</cn></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.1.m1.2c">[-1,1]</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.1.m1.2d">[ - 1 , 1 ]</annotation></semantics></math>. We then utilize PointNet to generate a feature matrix <math alttext="\boldsymbol{l}_{j,k}\in\mathbb{R}^{n_{p}\times 128}" class="ltx_Math" display="inline" id="S3.SS2.p1.2.m2.2"><semantics id="S3.SS2.p1.2.m2.2a"><mrow id="S3.SS2.p1.2.m2.2.3" xref="S3.SS2.p1.2.m2.2.3.cmml"><msub id="S3.SS2.p1.2.m2.2.3.2" xref="S3.SS2.p1.2.m2.2.3.2.cmml"><mi id="S3.SS2.p1.2.m2.2.3.2.2" xref="S3.SS2.p1.2.m2.2.3.2.2.cmml">𝒍</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">j</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">k</mi></mrow></msub><mo id="S3.SS2.p1.2.m2.2.3.1" xref="S3.SS2.p1.2.m2.2.3.1.cmml">∈</mo><msup id="S3.SS2.p1.2.m2.2.3.3" xref="S3.SS2.p1.2.m2.2.3.3.cmml"><mi id="S3.SS2.p1.2.m2.2.3.3.2" xref="S3.SS2.p1.2.m2.2.3.3.2.cmml">ℝ</mi><mrow id="S3.SS2.p1.2.m2.2.3.3.3" xref="S3.SS2.p1.2.m2.2.3.3.3.cmml"><msub id="S3.SS2.p1.2.m2.2.3.3.3.2" xref="S3.SS2.p1.2.m2.2.3.3.3.2.cmml"><mi id="S3.SS2.p1.2.m2.2.3.3.3.2.2" xref="S3.SS2.p1.2.m2.2.3.3.3.2.2.cmml">n</mi><mi id="S3.SS2.p1.2.m2.2.3.3.3.2.3" xref="S3.SS2.p1.2.m2.2.3.3.3.2.3.cmml">p</mi></msub><mo id="S3.SS2.p1.2.m2.2.3.3.3.1" lspace="0.222em" rspace="0.222em" xref="S3.SS2.p1.2.m2.2.3.3.3.1.cmml">×</mo><mn id="S3.SS2.p1.2.m2.2.3.3.3.3" xref="S3.SS2.p1.2.m2.2.3.3.3.3.cmml">128</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.2.m2.2b"><apply id="S3.SS2.p1.2.m2.2.3.cmml" xref="S3.SS2.p1.2.m2.2.3"><in id="S3.SS2.p1.2.m2.2.3.1.cmml" xref="S3.SS2.p1.2.m2.2.3.1"></in><apply id="S3.SS2.p1.2.m2.2.3.2.cmml" xref="S3.SS2.p1.2.m2.2.3.2"><csymbol cd="ambiguous" id="S3.SS2.p1.2.m2.2.3.2.1.cmml" xref="S3.SS2.p1.2.m2.2.3.2">subscript</csymbol><ci id="S3.SS2.p1.2.m2.2.3.2.2.cmml" xref="S3.SS2.p1.2.m2.2.3.2.2">𝒍</ci><list id="S3.SS2.p1.2.m2.2.2.2.3.cmml" xref="S3.SS2.p1.2.m2.2.2.2.4"><ci id="S3.SS2.p1.2.m2.1.1.1.1.cmml" xref="S3.SS2.p1.2.m2.1.1.1.1">𝑗</ci><ci id="S3.SS2.p1.2.m2.2.2.2.2.cmml" xref="S3.SS2.p1.2.m2.2.2.2.2">𝑘</ci></list></apply><apply id="S3.SS2.p1.2.m2.2.3.3.cmml" xref="S3.SS2.p1.2.m2.2.3.3"><csymbol cd="ambiguous" id="S3.SS2.p1.2.m2.2.3.3.1.cmml" xref="S3.SS2.p1.2.m2.2.3.3">superscript</csymbol><ci id="S3.SS2.p1.2.m2.2.3.3.2.cmml" xref="S3.SS2.p1.2.m2.2.3.3.2">ℝ</ci><apply id="S3.SS2.p1.2.m2.2.3.3.3.cmml" xref="S3.SS2.p1.2.m2.2.3.3.3"><times id="S3.SS2.p1.2.m2.2.3.3.3.1.cmml" xref="S3.SS2.p1.2.m2.2.3.3.3.1"></times><apply id="S3.SS2.p1.2.m2.2.3.3.3.2.cmml" xref="S3.SS2.p1.2.m2.2.3.3.3.2"><csymbol cd="ambiguous" id="S3.SS2.p1.2.m2.2.3.3.3.2.1.cmml" xref="S3.SS2.p1.2.m2.2.3.3.3.2">subscript</csymbol><ci id="S3.SS2.p1.2.m2.2.3.3.3.2.2.cmml" xref="S3.SS2.p1.2.m2.2.3.3.3.2.2">𝑛</ci><ci id="S3.SS2.p1.2.m2.2.3.3.3.2.3.cmml" xref="S3.SS2.p1.2.m2.2.3.3.3.2.3">𝑝</ci></apply><cn id="S3.SS2.p1.2.m2.2.3.3.3.3.cmml" type="integer" xref="S3.SS2.p1.2.m2.2.3.3.3.3">128</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.2.m2.2c">\boldsymbol{l}_{j,k}\in\mathbb{R}^{n_{p}\times 128}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.2.m2.2d">bold_italic_l start_POSTSUBSCRIPT italic_j , italic_k end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT × 128 end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="n_{p}" class="ltx_Math" display="inline" id="S3.SS2.p1.3.m3.1"><semantics id="S3.SS2.p1.3.m3.1a"><msub id="S3.SS2.p1.3.m3.1.1" xref="S3.SS2.p1.3.m3.1.1.cmml"><mi id="S3.SS2.p1.3.m3.1.1.2" xref="S3.SS2.p1.3.m3.1.1.2.cmml">n</mi><mi id="S3.SS2.p1.3.m3.1.1.3" xref="S3.SS2.p1.3.m3.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.3.m3.1b"><apply id="S3.SS2.p1.3.m3.1.1.cmml" xref="S3.SS2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.3.m3.1.1.1.cmml" xref="S3.SS2.p1.3.m3.1.1">subscript</csymbol><ci id="S3.SS2.p1.3.m3.1.1.2.cmml" xref="S3.SS2.p1.3.m3.1.1.2">𝑛</ci><ci id="S3.SS2.p1.3.m3.1.1.3.cmml" xref="S3.SS2.p1.3.m3.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.3.m3.1c">n_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.3.m3.1d">italic_n start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> is the total number of point clouds. Afterwards, we split this matrix into <math alttext="\boldsymbol{l}_{j,k1}\in\mathbb{R}^{n_{p}\times 64}" class="ltx_Math" display="inline" id="S3.SS2.p1.4.m4.2"><semantics id="S3.SS2.p1.4.m4.2a"><mrow id="S3.SS2.p1.4.m4.2.3" xref="S3.SS2.p1.4.m4.2.3.cmml"><msub id="S3.SS2.p1.4.m4.2.3.2" xref="S3.SS2.p1.4.m4.2.3.2.cmml"><mi id="S3.SS2.p1.4.m4.2.3.2.2" xref="S3.SS2.p1.4.m4.2.3.2.2.cmml">𝒍</mi><mrow id="S3.SS2.p1.4.m4.2.2.2.2" xref="S3.SS2.p1.4.m4.2.2.2.3.cmml"><mi id="S3.SS2.p1.4.m4.1.1.1.1" xref="S3.SS2.p1.4.m4.1.1.1.1.cmml">j</mi><mo id="S3.SS2.p1.4.m4.2.2.2.2.2" xref="S3.SS2.p1.4.m4.2.2.2.3.cmml">,</mo><mrow id="S3.SS2.p1.4.m4.2.2.2.2.1" xref="S3.SS2.p1.4.m4.2.2.2.2.1.cmml"><mi id="S3.SS2.p1.4.m4.2.2.2.2.1.2" xref="S3.SS2.p1.4.m4.2.2.2.2.1.2.cmml">k</mi><mo id="S3.SS2.p1.4.m4.2.2.2.2.1.1" xref="S3.SS2.p1.4.m4.2.2.2.2.1.1.cmml">⁢</mo><mn id="S3.SS2.p1.4.m4.2.2.2.2.1.3" xref="S3.SS2.p1.4.m4.2.2.2.2.1.3.cmml">1</mn></mrow></mrow></msub><mo id="S3.SS2.p1.4.m4.2.3.1" xref="S3.SS2.p1.4.m4.2.3.1.cmml">∈</mo><msup id="S3.SS2.p1.4.m4.2.3.3" xref="S3.SS2.p1.4.m4.2.3.3.cmml"><mi id="S3.SS2.p1.4.m4.2.3.3.2" xref="S3.SS2.p1.4.m4.2.3.3.2.cmml">ℝ</mi><mrow id="S3.SS2.p1.4.m4.2.3.3.3" xref="S3.SS2.p1.4.m4.2.3.3.3.cmml"><msub id="S3.SS2.p1.4.m4.2.3.3.3.2" xref="S3.SS2.p1.4.m4.2.3.3.3.2.cmml"><mi id="S3.SS2.p1.4.m4.2.3.3.3.2.2" xref="S3.SS2.p1.4.m4.2.3.3.3.2.2.cmml">n</mi><mi id="S3.SS2.p1.4.m4.2.3.3.3.2.3" xref="S3.SS2.p1.4.m4.2.3.3.3.2.3.cmml">p</mi></msub><mo id="S3.SS2.p1.4.m4.2.3.3.3.1" lspace="0.222em" rspace="0.222em" xref="S3.SS2.p1.4.m4.2.3.3.3.1.cmml">×</mo><mn id="S3.SS2.p1.4.m4.2.3.3.3.3" xref="S3.SS2.p1.4.m4.2.3.3.3.3.cmml">64</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.4.m4.2b"><apply id="S3.SS2.p1.4.m4.2.3.cmml" xref="S3.SS2.p1.4.m4.2.3"><in id="S3.SS2.p1.4.m4.2.3.1.cmml" xref="S3.SS2.p1.4.m4.2.3.1"></in><apply id="S3.SS2.p1.4.m4.2.3.2.cmml" xref="S3.SS2.p1.4.m4.2.3.2"><csymbol cd="ambiguous" 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id="S3.SS2.p1.4.m4.2.3.3.3.1.cmml" xref="S3.SS2.p1.4.m4.2.3.3.3.1"></times><apply id="S3.SS2.p1.4.m4.2.3.3.3.2.cmml" xref="S3.SS2.p1.4.m4.2.3.3.3.2"><csymbol cd="ambiguous" id="S3.SS2.p1.4.m4.2.3.3.3.2.1.cmml" xref="S3.SS2.p1.4.m4.2.3.3.3.2">subscript</csymbol><ci id="S3.SS2.p1.4.m4.2.3.3.3.2.2.cmml" xref="S3.SS2.p1.4.m4.2.3.3.3.2.2">𝑛</ci><ci id="S3.SS2.p1.4.m4.2.3.3.3.2.3.cmml" xref="S3.SS2.p1.4.m4.2.3.3.3.2.3">𝑝</ci></apply><cn id="S3.SS2.p1.4.m4.2.3.3.3.3.cmml" type="integer" xref="S3.SS2.p1.4.m4.2.3.3.3.3">64</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.4.m4.2c">\boldsymbol{l}_{j,k1}\in\mathbb{R}^{n_{p}\times 64}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.4.m4.2d">bold_italic_l start_POSTSUBSCRIPT italic_j , italic_k 1 end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT × 64 end_POSTSUPERSCRIPT</annotation></semantics></math> and <math 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</div> </section> <section class="ltx_subsection" id="S3.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S3.SS3.4.1.1">III-C</span> </span><span class="ltx_text ltx_font_italic" id="S3.SS3.5.2" style="color:#000000;">Path Tracing with Light Probes and Point Clouds</span> </h3> <div class="ltx_para" id="S3.SS3.p1"> <p class="ltx_p" id="S3.SS3.p1.1">Integrating light probes into our pipeline stands as an important contribution to our pipeline. It represents a strategic solution to address the unique challenges encountered in wireless ray propagation scenarios. In environments characterized by sparse geometric structures, such as open landscapes or urban settings with tall buildings, a straightforward implementation of neural ray tracing may encounter limitations. When rays emitted from antennas fail to intersect with nearby point clouds, one has to extrapolate their trajectories into unobstructed space.</p> </div> <div class="ltx_para" id="S3.SS3.p2"> <p class="ltx_p" id="S3.SS3.p2.1">To mitigate potential inaccuracies arising from the absence of precise ray directions, the proposed work draws inspiration from the concept of light probes in computer graphics. Light probes serve as essential tools for capturing and simulating realistic lighting effects within virtual environments. These probes act as virtual cameras that record light information from different directions, allowing for the creation of dynamic and immersive lighting scenarios.</p> </div> <div class="ltx_para" id="S3.SS3.p3"> <p class="ltx_p" id="S3.SS3.p3.1">In our pipeline, we place a set of light probes (much fewer than the number of point clouds) throughout the scene. Each light probe serves as a virtual observation point, capturing and encoding surrounding ray propagation information. This encoded data allows nearby receivers to easily decode it using the ray direction and distance as queries. Essentially, individual light probes serve as a neural surrogate for baking the propagation information within their nearby space.</p> </div> <figure class="ltx_figure" id="S3.F3"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="207" id="S3.F3.g1" src="extracted/5641587/figure/etoilcenter_pts_with_probe_lines_marked.png" width="275"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S3.F3.5.2.1" style="font-size:90%;">Figure 3</span>: </span><span class="ltx_text ltx_font_bold" id="S3.F3.2.1" style="font-size:90%;">Identifying n-nearest light probes<span class="ltx_text ltx_font_medium" id="S3.F3.2.1.1">: Each receiver locates its <math alttext="n" class="ltx_Math" display="inline" id="S3.F3.2.1.1.m1.1"><semantics id="S3.F3.2.1.1.m1.1b"><mi id="S3.F3.2.1.1.m1.1.1" xref="S3.F3.2.1.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.F3.2.1.1.m1.1c"><ci id="S3.F3.2.1.1.m1.1.1.cmml" xref="S3.F3.2.1.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.2.1.1.m1.1d">n</annotation><annotation encoding="application/x-llamapun" id="S3.F3.2.1.1.m1.1e">italic_n</annotation></semantics></math> nearest light probes and retrieves radiance information from them.</span></span></figcaption> </figure> <figure class="ltx_figure" id="S3.F4"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="207" id="S3.F4.g1" src="extracted/5641587/figure/etoilcenter_pts_with_probe_framework_marked.png" width="275"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S3.F4.5.2.1" style="font-size:90%;">Figure 4</span>: </span><span class="ltx_text ltx_font_bold" id="S3.F4.2.1" style="font-size:90%;">Identifying K-nearest points<span class="ltx_text ltx_font_medium" id="S3.F4.2.1.1">: Each light probe finds its <math alttext="K" class="ltx_Math" display="inline" id="S3.F4.2.1.1.m1.1"><semantics id="S3.F4.2.1.1.m1.1b"><mi id="S3.F4.2.1.1.m1.1.1" xref="S3.F4.2.1.1.m1.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S3.F4.2.1.1.m1.1c"><ci id="S3.F4.2.1.1.m1.1.1.cmml" xref="S3.F4.2.1.1.m1.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.2.1.1.m1.1d">K</annotation><annotation encoding="application/x-llamapun" id="S3.F4.2.1.1.m1.1e">italic_K</annotation></semantics></math> closest points and encodes occlusion information.</span></span></figcaption> </figure> <div class="ltx_para" id="S3.SS3.p4"> <p class="ltx_p" id="S3.SS3.p4.3">Moreover, the introduction of light probes enables the extraction of EM field information from the embedded features of point clouds. 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This selection establishes a physical correspondence, where there is a Line of Sight (LOS) contribution from the transmitter (akin to direct illumination in rendering) and <math alttext="K" class="ltx_Math" display="inline" id="S3.SS3.p4.3.m3.1"><semantics id="S3.SS3.p4.3.m3.1a"><mi id="S3.SS3.p4.3.m3.1.1" xref="S3.SS3.p4.3.m3.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.3.m3.1b"><ci id="S3.SS3.p4.3.m3.1.1.cmml" xref="S3.SS3.p4.3.m3.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.3.m3.1c">K</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.3.m3.1d">italic_K</annotation></semantics></math> contributions (resembling wave physics of reflection, diffraction, and scattering) from point clouds.</p> </div> <div class="ltx_para" id="S3.SS3.p5"> <p class="ltx_p" id="S3.SS3.p5.8">In our design, an attention technique is employed for this extraction process (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S3.F5" title="Figure 5 ‣ III-C Path Tracing with Light Probes and Point Clouds ‣ III Neural Point Field for Wireless Channel ‣ RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_tag">5</span></a>), guided by the location information between light probes and transmitter (distance <math alttext="d_{t}" class="ltx_Math" display="inline" id="S3.SS3.p5.1.m1.1"><semantics id="S3.SS3.p5.1.m1.1a"><msub id="S3.SS3.p5.1.m1.1.1" xref="S3.SS3.p5.1.m1.1.1.cmml"><mi id="S3.SS3.p5.1.m1.1.1.2" xref="S3.SS3.p5.1.m1.1.1.2.cmml">d</mi><mi id="S3.SS3.p5.1.m1.1.1.3" xref="S3.SS3.p5.1.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.1.m1.1b"><apply id="S3.SS3.p5.1.m1.1.1.cmml" xref="S3.SS3.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p5.1.m1.1.1.1.cmml" xref="S3.SS3.p5.1.m1.1.1">subscript</csymbol><ci id="S3.SS3.p5.1.m1.1.1.2.cmml" 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xref="S3.SS3.p5.4.m4.1.1.2">𝑑</ci><ci id="S3.SS3.p5.4.m4.1.1.3.cmml" xref="S3.SS3.p5.4.m4.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.4.m4.1c">d_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.4.m4.1d">italic_d start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, elevation <math alttext="\theta_{j}" class="ltx_Math" display="inline" id="S3.SS3.p5.5.m5.1"><semantics id="S3.SS3.p5.5.m5.1a"><msub id="S3.SS3.p5.5.m5.1.1" xref="S3.SS3.p5.5.m5.1.1.cmml"><mi id="S3.SS3.p5.5.m5.1.1.2" xref="S3.SS3.p5.5.m5.1.1.2.cmml">θ</mi><mi id="S3.SS3.p5.5.m5.1.1.3" xref="S3.SS3.p5.5.m5.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.5.m5.1b"><apply id="S3.SS3.p5.5.m5.1.1.cmml" xref="S3.SS3.p5.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS3.p5.5.m5.1.1.1.cmml" xref="S3.SS3.p5.5.m5.1.1">subscript</csymbol><ci id="S3.SS3.p5.5.m5.1.1.2.cmml" xref="S3.SS3.p5.5.m5.1.1.2">𝜃</ci><ci id="S3.SS3.p5.5.m5.1.1.3.cmml" xref="S3.SS3.p5.5.m5.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.5.m5.1c">\theta_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.5.m5.1d">italic_θ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> and azimuth <math alttext="\phi_{j}" class="ltx_Math" display="inline" id="S3.SS3.p5.6.m6.1"><semantics id="S3.SS3.p5.6.m6.1a"><msub id="S3.SS3.p5.6.m6.1.1" xref="S3.SS3.p5.6.m6.1.1.cmml"><mi id="S3.SS3.p5.6.m6.1.1.2" xref="S3.SS3.p5.6.m6.1.1.2.cmml">ϕ</mi><mi id="S3.SS3.p5.6.m6.1.1.3" xref="S3.SS3.p5.6.m6.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.6.m6.1b"><apply id="S3.SS3.p5.6.m6.1.1.cmml" xref="S3.SS3.p5.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS3.p5.6.m6.1.1.1.cmml" xref="S3.SS3.p5.6.m6.1.1">subscript</csymbol><ci id="S3.SS3.p5.6.m6.1.1.2.cmml" xref="S3.SS3.p5.6.m6.1.1.2">italic-ϕ</ci><ci id="S3.SS3.p5.6.m6.1.1.3.cmml" xref="S3.SS3.p5.6.m6.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.6.m6.1c">\phi_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.6.m6.1d">italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>). Subsequently, we combine them with our previous embedded feature as <math alttext="\boldsymbol{K}_{j,k}\in\mathbb{R}^{K\times 67}" class="ltx_Math" display="inline" id="S3.SS3.p5.7.m7.2"><semantics id="S3.SS3.p5.7.m7.2a"><mrow id="S3.SS3.p5.7.m7.2.3" xref="S3.SS3.p5.7.m7.2.3.cmml"><msub id="S3.SS3.p5.7.m7.2.3.2" xref="S3.SS3.p5.7.m7.2.3.2.cmml"><mi id="S3.SS3.p5.7.m7.2.3.2.2" xref="S3.SS3.p5.7.m7.2.3.2.2.cmml">𝑲</mi><mrow id="S3.SS3.p5.7.m7.2.2.2.4" xref="S3.SS3.p5.7.m7.2.2.2.3.cmml"><mi id="S3.SS3.p5.7.m7.1.1.1.1" xref="S3.SS3.p5.7.m7.1.1.1.1.cmml">j</mi><mo id="S3.SS3.p5.7.m7.2.2.2.4.1" xref="S3.SS3.p5.7.m7.2.2.2.3.cmml">,</mo><mi id="S3.SS3.p5.7.m7.2.2.2.2" xref="S3.SS3.p5.7.m7.2.2.2.2.cmml">k</mi></mrow></msub><mo id="S3.SS3.p5.7.m7.2.3.1" xref="S3.SS3.p5.7.m7.2.3.1.cmml">∈</mo><msup id="S3.SS3.p5.7.m7.2.3.3" xref="S3.SS3.p5.7.m7.2.3.3.cmml"><mi id="S3.SS3.p5.7.m7.2.3.3.2" xref="S3.SS3.p5.7.m7.2.3.3.2.cmml">ℝ</mi><mrow id="S3.SS3.p5.7.m7.2.3.3.3" xref="S3.SS3.p5.7.m7.2.3.3.3.cmml"><mi id="S3.SS3.p5.7.m7.2.3.3.3.2" xref="S3.SS3.p5.7.m7.2.3.3.3.2.cmml">K</mi><mo id="S3.SS3.p5.7.m7.2.3.3.3.1" lspace="0.222em" rspace="0.222em" xref="S3.SS3.p5.7.m7.2.3.3.3.1.cmml">×</mo><mn id="S3.SS3.p5.7.m7.2.3.3.3.3" xref="S3.SS3.p5.7.m7.2.3.3.3.3.cmml">67</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.7.m7.2b"><apply id="S3.SS3.p5.7.m7.2.3.cmml" xref="S3.SS3.p5.7.m7.2.3"><in id="S3.SS3.p5.7.m7.2.3.1.cmml" xref="S3.SS3.p5.7.m7.2.3.1"></in><apply id="S3.SS3.p5.7.m7.2.3.2.cmml" xref="S3.SS3.p5.7.m7.2.3.2"><csymbol cd="ambiguous" id="S3.SS3.p5.7.m7.2.3.2.1.cmml" xref="S3.SS3.p5.7.m7.2.3.2">subscript</csymbol><ci id="S3.SS3.p5.7.m7.2.3.2.2.cmml" xref="S3.SS3.p5.7.m7.2.3.2.2">𝑲</ci><list id="S3.SS3.p5.7.m7.2.2.2.3.cmml" xref="S3.SS3.p5.7.m7.2.2.2.4"><ci id="S3.SS3.p5.7.m7.1.1.1.1.cmml" xref="S3.SS3.p5.7.m7.1.1.1.1">𝑗</ci><ci id="S3.SS3.p5.7.m7.2.2.2.2.cmml" xref="S3.SS3.p5.7.m7.2.2.2.2">𝑘</ci></list></apply><apply id="S3.SS3.p5.7.m7.2.3.3.cmml" xref="S3.SS3.p5.7.m7.2.3.3"><csymbol cd="ambiguous" id="S3.SS3.p5.7.m7.2.3.3.1.cmml" xref="S3.SS3.p5.7.m7.2.3.3">superscript</csymbol><ci id="S3.SS3.p5.7.m7.2.3.3.2.cmml" xref="S3.SS3.p5.7.m7.2.3.3.2">ℝ</ci><apply id="S3.SS3.p5.7.m7.2.3.3.3.cmml" xref="S3.SS3.p5.7.m7.2.3.3.3"><times id="S3.SS3.p5.7.m7.2.3.3.3.1.cmml" xref="S3.SS3.p5.7.m7.2.3.3.3.1"></times><ci id="S3.SS3.p5.7.m7.2.3.3.3.2.cmml" xref="S3.SS3.p5.7.m7.2.3.3.3.2">𝐾</ci><cn id="S3.SS3.p5.7.m7.2.3.3.3.3.cmml" type="integer" xref="S3.SS3.p5.7.m7.2.3.3.3.3">67</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.7.m7.2c">\boldsymbol{K}_{j,k}\in\mathbb{R}^{K\times 67}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.7.m7.2d">bold_italic_K start_POSTSUBSCRIPT italic_j , italic_k end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_K × 67 end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\boldsymbol{V}_{j,k}\in\mathbb{R}^{K\times 67}" class="ltx_Math" display="inline" id="S3.SS3.p5.8.m8.2"><semantics id="S3.SS3.p5.8.m8.2a"><mrow id="S3.SS3.p5.8.m8.2.3" xref="S3.SS3.p5.8.m8.2.3.cmml"><msub id="S3.SS3.p5.8.m8.2.3.2" xref="S3.SS3.p5.8.m8.2.3.2.cmml"><mi id="S3.SS3.p5.8.m8.2.3.2.2" xref="S3.SS3.p5.8.m8.2.3.2.2.cmml">𝑽</mi><mrow id="S3.SS3.p5.8.m8.2.2.2.4" xref="S3.SS3.p5.8.m8.2.2.2.3.cmml"><mi id="S3.SS3.p5.8.m8.1.1.1.1" xref="S3.SS3.p5.8.m8.1.1.1.1.cmml">j</mi><mo id="S3.SS3.p5.8.m8.2.2.2.4.1" xref="S3.SS3.p5.8.m8.2.2.2.3.cmml">,</mo><mi id="S3.SS3.p5.8.m8.2.2.2.2" xref="S3.SS3.p5.8.m8.2.2.2.2.cmml">k</mi></mrow></msub><mo id="S3.SS3.p5.8.m8.2.3.1" xref="S3.SS3.p5.8.m8.2.3.1.cmml">∈</mo><msup id="S3.SS3.p5.8.m8.2.3.3" xref="S3.SS3.p5.8.m8.2.3.3.cmml"><mi id="S3.SS3.p5.8.m8.2.3.3.2" xref="S3.SS3.p5.8.m8.2.3.3.2.cmml">ℝ</mi><mrow id="S3.SS3.p5.8.m8.2.3.3.3" xref="S3.SS3.p5.8.m8.2.3.3.3.cmml"><mi id="S3.SS3.p5.8.m8.2.3.3.3.2" xref="S3.SS3.p5.8.m8.2.3.3.3.2.cmml">K</mi><mo id="S3.SS3.p5.8.m8.2.3.3.3.1" lspace="0.222em" rspace="0.222em" xref="S3.SS3.p5.8.m8.2.3.3.3.1.cmml">×</mo><mn id="S3.SS3.p5.8.m8.2.3.3.3.3" xref="S3.SS3.p5.8.m8.2.3.3.3.3.cmml">67</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.8.m8.2b"><apply id="S3.SS3.p5.8.m8.2.3.cmml" xref="S3.SS3.p5.8.m8.2.3"><in id="S3.SS3.p5.8.m8.2.3.1.cmml" xref="S3.SS3.p5.8.m8.2.3.1"></in><apply id="S3.SS3.p5.8.m8.2.3.2.cmml" xref="S3.SS3.p5.8.m8.2.3.2"><csymbol cd="ambiguous" id="S3.SS3.p5.8.m8.2.3.2.1.cmml" xref="S3.SS3.p5.8.m8.2.3.2">subscript</csymbol><ci id="S3.SS3.p5.8.m8.2.3.2.2.cmml" xref="S3.SS3.p5.8.m8.2.3.2.2">𝑽</ci><list id="S3.SS3.p5.8.m8.2.2.2.3.cmml" xref="S3.SS3.p5.8.m8.2.2.2.4"><ci id="S3.SS3.p5.8.m8.1.1.1.1.cmml" xref="S3.SS3.p5.8.m8.1.1.1.1">𝑗</ci><ci id="S3.SS3.p5.8.m8.2.2.2.2.cmml" xref="S3.SS3.p5.8.m8.2.2.2.2">𝑘</ci></list></apply><apply id="S3.SS3.p5.8.m8.2.3.3.cmml" xref="S3.SS3.p5.8.m8.2.3.3"><csymbol cd="ambiguous" id="S3.SS3.p5.8.m8.2.3.3.1.cmml" xref="S3.SS3.p5.8.m8.2.3.3">superscript</csymbol><ci id="S3.SS3.p5.8.m8.2.3.3.2.cmml" xref="S3.SS3.p5.8.m8.2.3.3.2">ℝ</ci><apply id="S3.SS3.p5.8.m8.2.3.3.3.cmml" xref="S3.SS3.p5.8.m8.2.3.3.3"><times id="S3.SS3.p5.8.m8.2.3.3.3.1.cmml" xref="S3.SS3.p5.8.m8.2.3.3.3.1"></times><ci id="S3.SS3.p5.8.m8.2.3.3.3.2.cmml" xref="S3.SS3.p5.8.m8.2.3.3.3.2">𝐾</ci><cn id="S3.SS3.p5.8.m8.2.3.3.3.3.cmml" type="integer" xref="S3.SS3.p5.8.m8.2.3.3.3.3">67</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.8.m8.2c">\boldsymbol{V}_{j,k}\in\mathbb{R}^{K\times 67}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.8.m8.2d">bold_italic_V start_POSTSUBSCRIPT italic_j , italic_k end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_K × 67 end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS3.p6"> <table class="ltx_equationgroup ltx_eqn_table" id="S3.E1"> <tbody> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S3.E1X"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\centering\begin{cases}\boldsymbol{K}_{j,k}=\boldsymbol{l}_{j,k1}% \oplus\{d_{j},\theta_{j},\phi_{j}\}\\ \boldsymbol{V}_{j,k}=\boldsymbol{l}_{j,k2}\oplus\{d_{t},\theta_{t},\phi_{t}\}% \\ \end{cases}" class="ltx_Math" display="inline" id="S3.E1X.2.1.1.m1.1"><semantics id="S3.E1X.2.1.1.m1.1a"><mrow id="S3.E1.m1.2.2.2.2.2.2a" xref="S3.E1X.2.1.1.m1.1.1.1.cmml"><mo id="S3.E1.m1.2.2.2.2.2.2a.3" xref="S3.E1X.2.1.1.m1.1.1.1.1.cmml">{</mo><mtable columnspacing="5pt" id="S3.E1.m1.2.2.2.2.2.2.2a" rowspacing="0pt" xref="S3.E1X.2.1.1.m1.1.1.1.cmml"><mtr id="S3.E1.m1.2.2.2.2.2.2.2aa" xref="S3.E1X.2.1.1.m1.1.1.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S3.E1.m1.2.2.2.2.2.2.2ab" xref="S3.E1X.2.1.1.m1.1.1.1.cmml"><mrow id="S3.E1.m1.1.1.1.1.1.1.1.1.1.1.mf" 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start_POSTSUBSCRIPT italic_j , italic_k end_POSTSUBSCRIPT = bold_italic_l start_POSTSUBSCRIPT italic_j , italic_k 1 end_POSTSUBSCRIPT ⊕ { italic_d start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_θ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT } end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL bold_italic_V start_POSTSUBSCRIPT italic_j , italic_k end_POSTSUBSCRIPT = bold_italic_l start_POSTSUBSCRIPT italic_j , italic_k 2 end_POSTSUBSCRIPT ⊕ { italic_d start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_θ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_ϕ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT } end_CELL start_CELL end_CELL end_ROW</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"><span class="ltx_tag ltx_tag_equationgroup ltx_align_right">(1)</span></td> </tr> </tbody> </table> </div> <div class="ltx_para" id="S3.SS3.p7"> <p class="ltx_p" id="S3.SS3.p7.3">By adding a query into the pipeline, which is instructed by <math alttext="K+1" class="ltx_Math" display="inline" id="S3.SS3.p7.1.m1.1"><semantics id="S3.SS3.p7.1.m1.1a"><mrow id="S3.SS3.p7.1.m1.1.1" xref="S3.SS3.p7.1.m1.1.1.cmml"><mi id="S3.SS3.p7.1.m1.1.1.2" xref="S3.SS3.p7.1.m1.1.1.2.cmml">K</mi><mo id="S3.SS3.p7.1.m1.1.1.1" xref="S3.SS3.p7.1.m1.1.1.1.cmml">+</mo><mn id="S3.SS3.p7.1.m1.1.1.3" xref="S3.SS3.p7.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p7.1.m1.1b"><apply id="S3.SS3.p7.1.m1.1.1.cmml" xref="S3.SS3.p7.1.m1.1.1"><plus id="S3.SS3.p7.1.m1.1.1.1.cmml" xref="S3.SS3.p7.1.m1.1.1.1"></plus><ci id="S3.SS3.p7.1.m1.1.1.2.cmml" xref="S3.SS3.p7.1.m1.1.1.2">𝐾</ci><cn id="S3.SS3.p7.1.m1.1.1.3.cmml" type="integer" xref="S3.SS3.p7.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p7.1.m1.1c">K+1</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p7.1.m1.1d">italic_K + 1</annotation></semantics></math> closest direction <math alttext="\boldsymbol{d}_{j}" class="ltx_Math" display="inline" id="S3.SS3.p7.2.m2.1"><semantics id="S3.SS3.p7.2.m2.1a"><msub id="S3.SS3.p7.2.m2.1.1" xref="S3.SS3.p7.2.m2.1.1.cmml"><mi id="S3.SS3.p7.2.m2.1.1.2" xref="S3.SS3.p7.2.m2.1.1.2.cmml">𝒅</mi><mi id="S3.SS3.p7.2.m2.1.1.3" xref="S3.SS3.p7.2.m2.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p7.2.m2.1b"><apply id="S3.SS3.p7.2.m2.1.1.cmml" xref="S3.SS3.p7.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS3.p7.2.m2.1.1.1.cmml" xref="S3.SS3.p7.2.m2.1.1">subscript</csymbol><ci id="S3.SS3.p7.2.m2.1.1.2.cmml" xref="S3.SS3.p7.2.m2.1.1.2">𝒅</ci><ci id="S3.SS3.p7.2.m2.1.1.3.cmml" xref="S3.SS3.p7.2.m2.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p7.2.m2.1c">\boldsymbol{d}_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p7.2.m2.1d">bold_italic_d start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> from light probes to point clouds and transmitter, EM field profiles are effectively baked into our light probes. These closest directions are shot from light probes directly to point clouds, and positionally encoded by encoder <math alttext="\boldsymbol{F}_{\theta_{Q}}" class="ltx_Math" display="inline" id="S3.SS3.p7.3.m3.1"><semantics id="S3.SS3.p7.3.m3.1a"><msub id="S3.SS3.p7.3.m3.1.1" xref="S3.SS3.p7.3.m3.1.1.cmml"><mi id="S3.SS3.p7.3.m3.1.1.2" xref="S3.SS3.p7.3.m3.1.1.2.cmml">𝑭</mi><msub id="S3.SS3.p7.3.m3.1.1.3" xref="S3.SS3.p7.3.m3.1.1.3.cmml"><mi id="S3.SS3.p7.3.m3.1.1.3.2" xref="S3.SS3.p7.3.m3.1.1.3.2.cmml">θ</mi><mi id="S3.SS3.p7.3.m3.1.1.3.3" xref="S3.SS3.p7.3.m3.1.1.3.3.cmml">Q</mi></msub></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p7.3.m3.1b"><apply id="S3.SS3.p7.3.m3.1.1.cmml" xref="S3.SS3.p7.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS3.p7.3.m3.1.1.1.cmml" xref="S3.SS3.p7.3.m3.1.1">subscript</csymbol><ci id="S3.SS3.p7.3.m3.1.1.2.cmml" xref="S3.SS3.p7.3.m3.1.1.2">𝑭</ci><apply id="S3.SS3.p7.3.m3.1.1.3.cmml" xref="S3.SS3.p7.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.p7.3.m3.1.1.3.1.cmml" xref="S3.SS3.p7.3.m3.1.1.3">subscript</csymbol><ci id="S3.SS3.p7.3.m3.1.1.3.2.cmml" xref="S3.SS3.p7.3.m3.1.1.3.2">𝜃</ci><ci id="S3.SS3.p7.3.m3.1.1.3.3.cmml" xref="S3.SS3.p7.3.m3.1.1.3.3">𝑄</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p7.3.m3.1c">\boldsymbol{F}_{\theta_{Q}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p7.3.m3.1d">bold_italic_F start_POSTSUBSCRIPT italic_θ start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> (<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S3.E2" title="In III-C Path Tracing with Light Probes and Point Clouds ‣ III Neural Point Field for Wireless Channel ‣ RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_tag">2</span></a>).</p> </div> <div class="ltx_para" id="S3.SS3.p8"> <table class="ltx_equationgroup ltx_eqn_table" id="S3.E2"> <tbody> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S3.E2X"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\centering\boldsymbol{Q}_{j}=\boldsymbol{F}_{\theta_{Q}}(% \boldsymbol{d}_{j})" class="ltx_Math" display="inline" id="S3.E2X.2.1.1.m1.1"><semantics id="S3.E2X.2.1.1.m1.1a"><mrow id="S3.E2X.2.1.1.m1.1.1" xref="S3.E2X.2.1.1.m1.1.1.cmml"><msub id="S3.E2X.2.1.1.m1.1.1.3" xref="S3.E2X.2.1.1.m1.1.1.3.cmml"><mi id="S3.E2X.2.1.1.m1.1.1.3.2" xref="S3.E2X.2.1.1.m1.1.1.3.2.cmml">𝑸</mi><mi id="S3.E2X.2.1.1.m1.1.1.3.3" xref="S3.E2X.2.1.1.m1.1.1.3.3.cmml">j</mi></msub><mo id="S3.E2X.2.1.1.m1.1.1.2" xref="S3.E2X.2.1.1.m1.1.1.2.cmml">=</mo><mrow id="S3.E2X.2.1.1.m1.1.1.1" xref="S3.E2X.2.1.1.m1.1.1.1.cmml"><msub id="S3.E2X.2.1.1.m1.1.1.1.3" xref="S3.E2X.2.1.1.m1.1.1.1.3.cmml"><mi id="S3.E2X.2.1.1.m1.1.1.1.3.2" xref="S3.E2X.2.1.1.m1.1.1.1.3.2.cmml">𝑭</mi><msub 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start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_italic_d start_POSTSUBSCRIPT italic_j 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"><span class="ltx_tag ltx_tag_equationgroup ltx_align_right">(2)</span></td> </tr> </tbody> </table> </div> <div class="ltx_para" id="S3.SS3.p9"> <p class="ltx_p" id="S3.SS3.p9.3">Subsequently, we apply multi-head attention to learn the power information of light probes: Given key-value pair <math alttext="(\boldsymbol{k}_{j,k},\boldsymbol{V}_{j,k})" class="ltx_Math" display="inline" id="S3.SS3.p9.1.m1.6"><semantics id="S3.SS3.p9.1.m1.6a"><mrow id="S3.SS3.p9.1.m1.6.6.2" xref="S3.SS3.p9.1.m1.6.6.3.cmml"><mo id="S3.SS3.p9.1.m1.6.6.2.3" stretchy="false" xref="S3.SS3.p9.1.m1.6.6.3.cmml">(</mo><msub id="S3.SS3.p9.1.m1.5.5.1.1" xref="S3.SS3.p9.1.m1.5.5.1.1.cmml"><mi 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xref="S3.SS3.p9.1.m1.6.6.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p9.1.m1.6b"><interval closure="open" id="S3.SS3.p9.1.m1.6.6.3.cmml" xref="S3.SS3.p9.1.m1.6.6.2"><apply id="S3.SS3.p9.1.m1.5.5.1.1.cmml" xref="S3.SS3.p9.1.m1.5.5.1.1"><csymbol cd="ambiguous" id="S3.SS3.p9.1.m1.5.5.1.1.1.cmml" xref="S3.SS3.p9.1.m1.5.5.1.1">subscript</csymbol><ci id="S3.SS3.p9.1.m1.5.5.1.1.2.cmml" xref="S3.SS3.p9.1.m1.5.5.1.1.2">𝒌</ci><list id="S3.SS3.p9.1.m1.2.2.2.3.cmml" xref="S3.SS3.p9.1.m1.2.2.2.4"><ci id="S3.SS3.p9.1.m1.1.1.1.1.cmml" xref="S3.SS3.p9.1.m1.1.1.1.1">𝑗</ci><ci id="S3.SS3.p9.1.m1.2.2.2.2.cmml" xref="S3.SS3.p9.1.m1.2.2.2.2">𝑘</ci></list></apply><apply id="S3.SS3.p9.1.m1.6.6.2.2.cmml" xref="S3.SS3.p9.1.m1.6.6.2.2"><csymbol cd="ambiguous" id="S3.SS3.p9.1.m1.6.6.2.2.1.cmml" xref="S3.SS3.p9.1.m1.6.6.2.2">subscript</csymbol><ci id="S3.SS3.p9.1.m1.6.6.2.2.2.cmml" xref="S3.SS3.p9.1.m1.6.6.2.2.2">𝑽</ci><list id="S3.SS3.p9.1.m1.4.4.2.3.cmml" xref="S3.SS3.p9.1.m1.4.4.2.4"><ci id="S3.SS3.p9.1.m1.3.3.1.1.cmml" xref="S3.SS3.p9.1.m1.3.3.1.1">𝑗</ci><ci id="S3.SS3.p9.1.m1.4.4.2.2.cmml" xref="S3.SS3.p9.1.m1.4.4.2.2">𝑘</ci></list></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p9.1.m1.6c">(\boldsymbol{k}_{j,k},\boldsymbol{V}_{j,k})</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p9.1.m1.6d">( bold_italic_k start_POSTSUBSCRIPT italic_j , italic_k end_POSTSUBSCRIPT , bold_italic_V start_POSTSUBSCRIPT italic_j , italic_k end_POSTSUBSCRIPT )</annotation></semantics></math>, the task is to predict a weight corresponding to the query ray vector <math alttext="\boldsymbol{Q}_{j}" class="ltx_Math" display="inline" id="S3.SS3.p9.2.m2.1"><semantics id="S3.SS3.p9.2.m2.1a"><msub id="S3.SS3.p9.2.m2.1.1" xref="S3.SS3.p9.2.m2.1.1.cmml"><mi id="S3.SS3.p9.2.m2.1.1.2" xref="S3.SS3.p9.2.m2.1.1.2.cmml">𝑸</mi><mi id="S3.SS3.p9.2.m2.1.1.3" xref="S3.SS3.p9.2.m2.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p9.2.m2.1b"><apply id="S3.SS3.p9.2.m2.1.1.cmml" xref="S3.SS3.p9.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS3.p9.2.m2.1.1.1.cmml" xref="S3.SS3.p9.2.m2.1.1">subscript</csymbol><ci id="S3.SS3.p9.2.m2.1.1.2.cmml" xref="S3.SS3.p9.2.m2.1.1.2">𝑸</ci><ci id="S3.SS3.p9.2.m2.1.1.3.cmml" xref="S3.SS3.p9.2.m2.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p9.2.m2.1c">\boldsymbol{Q}_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p9.2.m2.1d">bold_italic_Q start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>. The output weight is then encoded into point cloud feature vector <math alttext="\boldsymbol{l}_{i,j}\in\mathbb{R}^{n\times 128}" class="ltx_Math" display="inline" id="S3.SS3.p9.3.m3.2"><semantics id="S3.SS3.p9.3.m3.2a"><mrow id="S3.SS3.p9.3.m3.2.3" xref="S3.SS3.p9.3.m3.2.3.cmml"><msub id="S3.SS3.p9.3.m3.2.3.2" xref="S3.SS3.p9.3.m3.2.3.2.cmml"><mi id="S3.SS3.p9.3.m3.2.3.2.2" xref="S3.SS3.p9.3.m3.2.3.2.2.cmml">𝒍</mi><mrow id="S3.SS3.p9.3.m3.2.2.2.4" xref="S3.SS3.p9.3.m3.2.2.2.3.cmml"><mi id="S3.SS3.p9.3.m3.1.1.1.1" xref="S3.SS3.p9.3.m3.1.1.1.1.cmml">i</mi><mo id="S3.SS3.p9.3.m3.2.2.2.4.1" xref="S3.SS3.p9.3.m3.2.2.2.3.cmml">,</mo><mi id="S3.SS3.p9.3.m3.2.2.2.2" xref="S3.SS3.p9.3.m3.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S3.SS3.p9.3.m3.2.3.1" xref="S3.SS3.p9.3.m3.2.3.1.cmml">∈</mo><msup id="S3.SS3.p9.3.m3.2.3.3" xref="S3.SS3.p9.3.m3.2.3.3.cmml"><mi id="S3.SS3.p9.3.m3.2.3.3.2" xref="S3.SS3.p9.3.m3.2.3.3.2.cmml">ℝ</mi><mrow id="S3.SS3.p9.3.m3.2.3.3.3" xref="S3.SS3.p9.3.m3.2.3.3.3.cmml"><mi id="S3.SS3.p9.3.m3.2.3.3.3.2" xref="S3.SS3.p9.3.m3.2.3.3.3.2.cmml">n</mi><mo id="S3.SS3.p9.3.m3.2.3.3.3.1" lspace="0.222em" rspace="0.222em" xref="S3.SS3.p9.3.m3.2.3.3.3.1.cmml">×</mo><mn id="S3.SS3.p9.3.m3.2.3.3.3.3" xref="S3.SS3.p9.3.m3.2.3.3.3.3.cmml">128</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p9.3.m3.2b"><apply id="S3.SS3.p9.3.m3.2.3.cmml" xref="S3.SS3.p9.3.m3.2.3"><in id="S3.SS3.p9.3.m3.2.3.1.cmml" xref="S3.SS3.p9.3.m3.2.3.1"></in><apply id="S3.SS3.p9.3.m3.2.3.2.cmml" xref="S3.SS3.p9.3.m3.2.3.2"><csymbol cd="ambiguous" id="S3.SS3.p9.3.m3.2.3.2.1.cmml" xref="S3.SS3.p9.3.m3.2.3.2">subscript</csymbol><ci id="S3.SS3.p9.3.m3.2.3.2.2.cmml" xref="S3.SS3.p9.3.m3.2.3.2.2">𝒍</ci><list id="S3.SS3.p9.3.m3.2.2.2.3.cmml" xref="S3.SS3.p9.3.m3.2.2.2.4"><ci id="S3.SS3.p9.3.m3.1.1.1.1.cmml" xref="S3.SS3.p9.3.m3.1.1.1.1">𝑖</ci><ci id="S3.SS3.p9.3.m3.2.2.2.2.cmml" xref="S3.SS3.p9.3.m3.2.2.2.2">𝑗</ci></list></apply><apply id="S3.SS3.p9.3.m3.2.3.3.cmml" xref="S3.SS3.p9.3.m3.2.3.3"><csymbol cd="ambiguous" id="S3.SS3.p9.3.m3.2.3.3.1.cmml" xref="S3.SS3.p9.3.m3.2.3.3">superscript</csymbol><ci id="S3.SS3.p9.3.m3.2.3.3.2.cmml" xref="S3.SS3.p9.3.m3.2.3.3.2">ℝ</ci><apply id="S3.SS3.p9.3.m3.2.3.3.3.cmml" xref="S3.SS3.p9.3.m3.2.3.3.3"><times id="S3.SS3.p9.3.m3.2.3.3.3.1.cmml" xref="S3.SS3.p9.3.m3.2.3.3.3.1"></times><ci id="S3.SS3.p9.3.m3.2.3.3.3.2.cmml" xref="S3.SS3.p9.3.m3.2.3.3.3.2">𝑛</ci><cn id="S3.SS3.p9.3.m3.2.3.3.3.3.cmml" type="integer" xref="S3.SS3.p9.3.m3.2.3.3.3.3">128</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p9.3.m3.2c">\boldsymbol{l}_{i,j}\in\mathbb{R}^{n\times 128}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p9.3.m3.2d">bold_italic_l start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_n × 128 end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS3.p10"> <table class="ltx_equationgroup ltx_eqn_table" id="S3.E3"> <tbody> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S3.E3X"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\centering\boldsymbol{l}_{i,j}=\boldsymbol{F}_{\theta_{atten}}(% \boldsymbol{K}_{j,k},\boldsymbol{V}_{j,k},\boldsymbol{Q}_{j})" class="ltx_Math" display="inline" id="S3.E3X.2.1.1.m1.9"><semantics id="S3.E3X.2.1.1.m1.9a"><mrow id="S3.E3X.2.1.1.m1.9.9" xref="S3.E3X.2.1.1.m1.9.9.cmml"><msub id="S3.E3X.2.1.1.m1.9.9.5" xref="S3.E3X.2.1.1.m1.9.9.5.cmml"><mi id="S3.E3X.2.1.1.m1.9.9.5.2" xref="S3.E3X.2.1.1.m1.9.9.5.2.cmml">𝒍</mi><mrow id="S3.E3X.2.1.1.m1.2.2.2.4" xref="S3.E3X.2.1.1.m1.2.2.2.3.cmml"><mi id="S3.E3X.2.1.1.m1.1.1.1.1" xref="S3.E3X.2.1.1.m1.1.1.1.1.cmml">i</mi><mo id="S3.E3X.2.1.1.m1.2.2.2.4.1" xref="S3.E3X.2.1.1.m1.2.2.2.3.cmml">,</mo><mi id="S3.E3X.2.1.1.m1.2.2.2.2" 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\boldsymbol{K}_{j,k},\boldsymbol{V}_{j,k},\boldsymbol{Q}_{j})</annotation><annotation encoding="application/x-llamapun" id="S3.E3X.2.1.1.m1.9d">bold_italic_l start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = bold_italic_F start_POSTSUBSCRIPT italic_θ start_POSTSUBSCRIPT italic_a italic_t italic_t italic_e italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_italic_K start_POSTSUBSCRIPT italic_j , italic_k end_POSTSUBSCRIPT , bold_italic_V start_POSTSUBSCRIPT italic_j , italic_k end_POSTSUBSCRIPT , bold_italic_Q start_POSTSUBSCRIPT italic_j 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"><span class="ltx_tag ltx_tag_equationgroup ltx_align_right">(3)</span></td> </tr> </tbody> </table> </div> <figure class="ltx_figure" id="S3.F5"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="193" id="S3.F5.g1" src="extracted/5641587/figure/Attention_module.png" width="287"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S3.F5.7.3.1" style="font-size:90%;">Figure 5</span>: </span><span class="ltx_text ltx_font_bold" id="S3.F5.4.2" style="font-size:90%;">Multi-head attention<span class="ltx_text ltx_font_medium" id="S3.F5.4.2.2">: In Section III.C, a multi-head attention module is employed to aggregate the point cloud feature vector <math alttext="l_{j,k}" class="ltx_Math" display="inline" id="S3.F5.3.1.1.m1.2"><semantics id="S3.F5.3.1.1.m1.2b"><msub id="S3.F5.3.1.1.m1.2.3" xref="S3.F5.3.1.1.m1.2.3.cmml"><mi id="S3.F5.3.1.1.m1.2.3.2" xref="S3.F5.3.1.1.m1.2.3.2.cmml">l</mi><mrow id="S3.F5.3.1.1.m1.2.2.2.4" xref="S3.F5.3.1.1.m1.2.2.2.3.cmml"><mi id="S3.F5.3.1.1.m1.1.1.1.1" xref="S3.F5.3.1.1.m1.1.1.1.1.cmml">j</mi><mo id="S3.F5.3.1.1.m1.2.2.2.4.1" xref="S3.F5.3.1.1.m1.2.2.2.3.cmml">,</mo><mi id="S3.F5.3.1.1.m1.2.2.2.2" xref="S3.F5.3.1.1.m1.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.F5.3.1.1.m1.2c"><apply id="S3.F5.3.1.1.m1.2.3.cmml" xref="S3.F5.3.1.1.m1.2.3"><csymbol cd="ambiguous" id="S3.F5.3.1.1.m1.2.3.1.cmml" xref="S3.F5.3.1.1.m1.2.3">subscript</csymbol><ci id="S3.F5.3.1.1.m1.2.3.2.cmml" xref="S3.F5.3.1.1.m1.2.3.2">𝑙</ci><list id="S3.F5.3.1.1.m1.2.2.2.3.cmml" xref="S3.F5.3.1.1.m1.2.2.2.4"><ci id="S3.F5.3.1.1.m1.1.1.1.1.cmml" xref="S3.F5.3.1.1.m1.1.1.1.1">𝑗</ci><ci id="S3.F5.3.1.1.m1.2.2.2.2.cmml" xref="S3.F5.3.1.1.m1.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.3.1.1.m1.2d">l_{j,k}</annotation><annotation encoding="application/x-llamapun" id="S3.F5.3.1.1.m1.2e">italic_l start_POSTSUBSCRIPT italic_j , italic_k end_POSTSUBSCRIPT</annotation></semantics></math> along with the K-closest direction <math alttext="j" class="ltx_Math" display="inline" id="S3.F5.4.2.2.m2.1"><semantics id="S3.F5.4.2.2.m2.1b"><mi id="S3.F5.4.2.2.m2.1.1" xref="S3.F5.4.2.2.m2.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S3.F5.4.2.2.m2.1c"><ci id="S3.F5.4.2.2.m2.1.1.cmml" xref="S3.F5.4.2.2.m2.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.4.2.2.m2.1d">j</annotation><annotation encoding="application/x-llamapun" id="S3.F5.4.2.2.m2.1e">italic_j</annotation></semantics></math>. This process generates a light probe feature. The attention module described in Section III.D follows a similar structure.</span></span></figcaption> </figure> </section> <section class="ltx_subsection" id="S3.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S3.SS4.4.1.1">III-D</span> </span><span class="ltx_text ltx_font_italic" id="S3.SS4.5.2" style="color:#000000;">Receivers: Unveiling Ray Physics from Light Probes</span> </h3> <div class="ltx_para" id="S3.SS4.p1"> <p class="ltx_p" id="S3.SS4.p1.4">Once the EM propagation information has been baked into light probes, the next step is to determine a format of interpolation for storing this data. Similar to the previous section, we employ another attention technique when receivers extract EM power from light probes. In this process, we select the <math alttext="n" class="ltx_Math" display="inline" id="S3.SS4.p1.1.m1.1"><semantics id="S3.SS4.p1.1.m1.1a"><mi id="S3.SS4.p1.1.m1.1.1" xref="S3.SS4.p1.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.1.m1.1b"><ci id="S3.SS4.p1.1.m1.1.1.cmml" xref="S3.SS4.p1.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.1.m1.1d">italic_n</annotation></semantics></math> closest light probes and generate a ray for each of them. The direction of these rays serves as the query for our attention block. Thereby, we design two instructions (key and value) with the combination of three variables: distance <math alttext="d_{i}" class="ltx_Math" display="inline" id="S3.SS4.p1.2.m2.1"><semantics id="S3.SS4.p1.2.m2.1a"><msub id="S3.SS4.p1.2.m2.1.1" xref="S3.SS4.p1.2.m2.1.1.cmml"><mi id="S3.SS4.p1.2.m2.1.1.2" xref="S3.SS4.p1.2.m2.1.1.2.cmml">d</mi><mi id="S3.SS4.p1.2.m2.1.1.3" xref="S3.SS4.p1.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.2.m2.1b"><apply id="S3.SS4.p1.2.m2.1.1.cmml" xref="S3.SS4.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS4.p1.2.m2.1.1.1.cmml" xref="S3.SS4.p1.2.m2.1.1">subscript</csymbol><ci id="S3.SS4.p1.2.m2.1.1.2.cmml" xref="S3.SS4.p1.2.m2.1.1.2">𝑑</ci><ci id="S3.SS4.p1.2.m2.1.1.3.cmml" xref="S3.SS4.p1.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.2.m2.1c">d_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.2.m2.1d">italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, elevation <math alttext="\theta_{i}" class="ltx_Math" display="inline" id="S3.SS4.p1.3.m3.1"><semantics id="S3.SS4.p1.3.m3.1a"><msub id="S3.SS4.p1.3.m3.1.1" xref="S3.SS4.p1.3.m3.1.1.cmml"><mi id="S3.SS4.p1.3.m3.1.1.2" xref="S3.SS4.p1.3.m3.1.1.2.cmml">θ</mi><mi id="S3.SS4.p1.3.m3.1.1.3" xref="S3.SS4.p1.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.3.m3.1b"><apply id="S3.SS4.p1.3.m3.1.1.cmml" xref="S3.SS4.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS4.p1.3.m3.1.1.1.cmml" xref="S3.SS4.p1.3.m3.1.1">subscript</csymbol><ci id="S3.SS4.p1.3.m3.1.1.2.cmml" xref="S3.SS4.p1.3.m3.1.1.2">𝜃</ci><ci id="S3.SS4.p1.3.m3.1.1.3.cmml" xref="S3.SS4.p1.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.3.m3.1c">\theta_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.3.m3.1d">italic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, and azimuth <math alttext="\phi_{i}" class="ltx_Math" display="inline" id="S3.SS4.p1.4.m4.1"><semantics id="S3.SS4.p1.4.m4.1a"><msub id="S3.SS4.p1.4.m4.1.1" xref="S3.SS4.p1.4.m4.1.1.cmml"><mi id="S3.SS4.p1.4.m4.1.1.2" xref="S3.SS4.p1.4.m4.1.1.2.cmml">ϕ</mi><mi id="S3.SS4.p1.4.m4.1.1.3" xref="S3.SS4.p1.4.m4.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.4.m4.1b"><apply id="S3.SS4.p1.4.m4.1.1.cmml" xref="S3.SS4.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS4.p1.4.m4.1.1.1.cmml" xref="S3.SS4.p1.4.m4.1.1">subscript</csymbol><ci id="S3.SS4.p1.4.m4.1.1.2.cmml" xref="S3.SS4.p1.4.m4.1.1.2">italic-ϕ</ci><ci id="S3.SS4.p1.4.m4.1.1.3.cmml" xref="S3.SS4.p1.4.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.4.m4.1c">\phi_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.4.m4.1d">italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. They are between point clouds and receivers (key), transmitters and receivers (value). This section is very similar to the previous part.</p> </div> <div class="ltx_para" id="S3.SS4.p2"> <table class="ltx_equationgroup ltx_eqn_table" id="S3.E4"> <tbody> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S3.E4X"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\centering\begin{cases}\boldsymbol{K}_{i,j}=\boldsymbol{l}_{i,j1}% \oplus\{d_{i},\theta_{i},\phi_{i}\}\\ \boldsymbol{V}_{i,j}=\boldsymbol{l}_{i,j2}\oplus\{d_{t},\theta_{t},\phi_{t}\}% \\ \end{cases}" class="ltx_Math" display="inline" id="S3.E4X.2.1.1.m1.1"><semantics id="S3.E4X.2.1.1.m1.1a"><mrow id="S3.E4.m1.2.2.2.2.2.2a" xref="S3.E4X.2.1.1.m1.1.1.1.cmml"><mo id="S3.E4.m1.2.2.2.2.2.2a.3" xref="S3.E4X.2.1.1.m1.1.1.1.1.cmml">{</mo><mtable columnspacing="5pt" id="S3.E4.m1.2.2.2.2.2.2.2a" rowspacing="0pt" xref="S3.E4X.2.1.1.m1.1.1.1.cmml"><mtr id="S3.E4.m1.2.2.2.2.2.2.2aa" xref="S3.E4X.2.1.1.m1.1.1.1.cmml"><mtd 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id="S3.E4X.2.1.1.m1.1c">\displaystyle\centering\begin{cases}\boldsymbol{K}_{i,j}=\boldsymbol{l}_{i,j1}% \oplus\{d_{i},\theta_{i},\phi_{i}\}\\ \boldsymbol{V}_{i,j}=\boldsymbol{l}_{i,j2}\oplus\{d_{t},\theta_{t},\phi_{t}\}% \\ \end{cases}</annotation><annotation encoding="application/x-llamapun" id="S3.E4X.2.1.1.m1.1d">{ start_ROW start_CELL bold_italic_K start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = bold_italic_l start_POSTSUBSCRIPT italic_i , italic_j 1 end_POSTSUBSCRIPT ⊕ { italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL bold_italic_V start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = bold_italic_l start_POSTSUBSCRIPT italic_i , italic_j 2 end_POSTSUBSCRIPT ⊕ { italic_d start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_θ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_ϕ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT } end_CELL start_CELL end_CELL end_ROW</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"><span class="ltx_tag ltx_tag_equationgroup ltx_align_right">(4)</span></td> </tr> </tbody> </table> </div> <div class="ltx_para" id="S3.SS4.p3"> <p class="ltx_p" id="S3.SS4.p3.7">Upon receiving a key-value pair, we encode <math alttext="n" class="ltx_Math" display="inline" id="S3.SS4.p3.1.m1.1"><semantics id="S3.SS4.p3.1.m1.1a"><mi id="S3.SS4.p3.1.m1.1.1" xref="S3.SS4.p3.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.p3.1.m1.1b"><ci id="S3.SS4.p3.1.m1.1.1.cmml" xref="S3.SS4.p3.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p3.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p3.1.m1.1d">italic_n</annotation></semantics></math> ray directions <math alttext="\boldsymbol{d}_{i}" class="ltx_Math" display="inline" id="S3.SS4.p3.2.m2.1"><semantics id="S3.SS4.p3.2.m2.1a"><msub id="S3.SS4.p3.2.m2.1.1" xref="S3.SS4.p3.2.m2.1.1.cmml"><mi id="S3.SS4.p3.2.m2.1.1.2" xref="S3.SS4.p3.2.m2.1.1.2.cmml">𝒅</mi><mi id="S3.SS4.p3.2.m2.1.1.3" xref="S3.SS4.p3.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p3.2.m2.1b"><apply id="S3.SS4.p3.2.m2.1.1.cmml" xref="S3.SS4.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS4.p3.2.m2.1.1.1.cmml" xref="S3.SS4.p3.2.m2.1.1">subscript</csymbol><ci id="S3.SS4.p3.2.m2.1.1.2.cmml" xref="S3.SS4.p3.2.m2.1.1.2">𝒅</ci><ci id="S3.SS4.p3.2.m2.1.1.3.cmml" xref="S3.SS4.p3.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p3.2.m2.1c">\boldsymbol{d}_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p3.2.m2.1d">bold_italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, which is shot from receivers to light probes. Following positional encoding, a query vector <math alttext="\boldsymbol{Q}_{i}" class="ltx_Math" display="inline" id="S3.SS4.p3.3.m3.1"><semantics id="S3.SS4.p3.3.m3.1a"><msub id="S3.SS4.p3.3.m3.1.1" xref="S3.SS4.p3.3.m3.1.1.cmml"><mi id="S3.SS4.p3.3.m3.1.1.2" xref="S3.SS4.p3.3.m3.1.1.2.cmml">𝑸</mi><mi id="S3.SS4.p3.3.m3.1.1.3" xref="S3.SS4.p3.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p3.3.m3.1b"><apply id="S3.SS4.p3.3.m3.1.1.cmml" xref="S3.SS4.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS4.p3.3.m3.1.1.1.cmml" xref="S3.SS4.p3.3.m3.1.1">subscript</csymbol><ci id="S3.SS4.p3.3.m3.1.1.2.cmml" xref="S3.SS4.p3.3.m3.1.1.2">𝑸</ci><ci id="S3.SS4.p3.3.m3.1.1.3.cmml" xref="S3.SS4.p3.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p3.3.m3.1c">\boldsymbol{Q}_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p3.3.m3.1d">bold_italic_Q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is generated, which is then applied to another Multi-head attention neural block for feature extraction. The resulting output is a ray feature vector <math alttext="\boldsymbol{l}_{i}\in\mathbb{R}^{n+1}" class="ltx_Math" display="inline" id="S3.SS4.p3.4.m4.1"><semantics id="S3.SS4.p3.4.m4.1a"><mrow id="S3.SS4.p3.4.m4.1.1" xref="S3.SS4.p3.4.m4.1.1.cmml"><msub id="S3.SS4.p3.4.m4.1.1.2" xref="S3.SS4.p3.4.m4.1.1.2.cmml"><mi id="S3.SS4.p3.4.m4.1.1.2.2" xref="S3.SS4.p3.4.m4.1.1.2.2.cmml">𝒍</mi><mi id="S3.SS4.p3.4.m4.1.1.2.3" xref="S3.SS4.p3.4.m4.1.1.2.3.cmml">i</mi></msub><mo id="S3.SS4.p3.4.m4.1.1.1" xref="S3.SS4.p3.4.m4.1.1.1.cmml">∈</mo><msup id="S3.SS4.p3.4.m4.1.1.3" xref="S3.SS4.p3.4.m4.1.1.3.cmml"><mi id="S3.SS4.p3.4.m4.1.1.3.2" xref="S3.SS4.p3.4.m4.1.1.3.2.cmml">ℝ</mi><mrow id="S3.SS4.p3.4.m4.1.1.3.3" xref="S3.SS4.p3.4.m4.1.1.3.3.cmml"><mi id="S3.SS4.p3.4.m4.1.1.3.3.2" xref="S3.SS4.p3.4.m4.1.1.3.3.2.cmml">n</mi><mo id="S3.SS4.p3.4.m4.1.1.3.3.1" xref="S3.SS4.p3.4.m4.1.1.3.3.1.cmml">+</mo><mn id="S3.SS4.p3.4.m4.1.1.3.3.3" xref="S3.SS4.p3.4.m4.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p3.4.m4.1b"><apply id="S3.SS4.p3.4.m4.1.1.cmml" xref="S3.SS4.p3.4.m4.1.1"><in id="S3.SS4.p3.4.m4.1.1.1.cmml" xref="S3.SS4.p3.4.m4.1.1.1"></in><apply id="S3.SS4.p3.4.m4.1.1.2.cmml" xref="S3.SS4.p3.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.p3.4.m4.1.1.2.1.cmml" xref="S3.SS4.p3.4.m4.1.1.2">subscript</csymbol><ci id="S3.SS4.p3.4.m4.1.1.2.2.cmml" xref="S3.SS4.p3.4.m4.1.1.2.2">𝒍</ci><ci id="S3.SS4.p3.4.m4.1.1.2.3.cmml" xref="S3.SS4.p3.4.m4.1.1.2.3">𝑖</ci></apply><apply id="S3.SS4.p3.4.m4.1.1.3.cmml" xref="S3.SS4.p3.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.p3.4.m4.1.1.3.1.cmml" xref="S3.SS4.p3.4.m4.1.1.3">superscript</csymbol><ci id="S3.SS4.p3.4.m4.1.1.3.2.cmml" xref="S3.SS4.p3.4.m4.1.1.3.2">ℝ</ci><apply id="S3.SS4.p3.4.m4.1.1.3.3.cmml" xref="S3.SS4.p3.4.m4.1.1.3.3"><plus id="S3.SS4.p3.4.m4.1.1.3.3.1.cmml" xref="S3.SS4.p3.4.m4.1.1.3.3.1"></plus><ci id="S3.SS4.p3.4.m4.1.1.3.3.2.cmml" xref="S3.SS4.p3.4.m4.1.1.3.3.2">𝑛</ci><cn id="S3.SS4.p3.4.m4.1.1.3.3.3.cmml" type="integer" xref="S3.SS4.p3.4.m4.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p3.4.m4.1c">\boldsymbol{l}_{i}\in\mathbb{R}^{n+1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p3.4.m4.1d">bold_italic_l start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="n" class="ltx_Math" display="inline" id="S3.SS4.p3.5.m5.1"><semantics id="S3.SS4.p3.5.m5.1a"><mi id="S3.SS4.p3.5.m5.1.1" xref="S3.SS4.p3.5.m5.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.p3.5.m5.1b"><ci id="S3.SS4.p3.5.m5.1.1.cmml" xref="S3.SS4.p3.5.m5.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p3.5.m5.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p3.5.m5.1d">italic_n</annotation></semantics></math> represents the number of rays. Notably, the inclusion of LoS necessitates the addition of another receiver-transmitter ray into our pipeline, thereby augmenting the final feature count to <math alttext="n+1" class="ltx_Math" display="inline" id="S3.SS4.p3.6.m6.1"><semantics id="S3.SS4.p3.6.m6.1a"><mrow id="S3.SS4.p3.6.m6.1.1" xref="S3.SS4.p3.6.m6.1.1.cmml"><mi id="S3.SS4.p3.6.m6.1.1.2" xref="S3.SS4.p3.6.m6.1.1.2.cmml">n</mi><mo id="S3.SS4.p3.6.m6.1.1.1" xref="S3.SS4.p3.6.m6.1.1.1.cmml">+</mo><mn id="S3.SS4.p3.6.m6.1.1.3" xref="S3.SS4.p3.6.m6.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p3.6.m6.1b"><apply id="S3.SS4.p3.6.m6.1.1.cmml" xref="S3.SS4.p3.6.m6.1.1"><plus id="S3.SS4.p3.6.m6.1.1.1.cmml" xref="S3.SS4.p3.6.m6.1.1.1"></plus><ci id="S3.SS4.p3.6.m6.1.1.2.cmml" xref="S3.SS4.p3.6.m6.1.1.2">𝑛</ci><cn id="S3.SS4.p3.6.m6.1.1.3.cmml" type="integer" xref="S3.SS4.p3.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p3.6.m6.1c">n+1</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p3.6.m6.1d">italic_n + 1</annotation></semantics></math> instead of <math alttext="n" class="ltx_Math" display="inline" id="S3.SS4.p3.7.m7.1"><semantics id="S3.SS4.p3.7.m7.1a"><mi id="S3.SS4.p3.7.m7.1.1" xref="S3.SS4.p3.7.m7.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.p3.7.m7.1b"><ci id="S3.SS4.p3.7.m7.1.1.cmml" xref="S3.SS4.p3.7.m7.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p3.7.m7.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p3.7.m7.1d">italic_n</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS4.p4"> <table class="ltx_equationgroup ltx_eqn_table" id="S3.E5"> <tbody> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S3.E5X"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\centering\boldsymbol{Q}_{i}=\boldsymbol{F}_{\theta_{Q}}(% \boldsymbol{d}_{i})" class="ltx_Math" display="inline" id="S3.E5X.2.1.1.m1.1"><semantics id="S3.E5X.2.1.1.m1.1a"><mrow id="S3.E5X.2.1.1.m1.1.1" xref="S3.E5X.2.1.1.m1.1.1.cmml"><msub id="S3.E5X.2.1.1.m1.1.1.3" xref="S3.E5X.2.1.1.m1.1.1.3.cmml"><mi id="S3.E5X.2.1.1.m1.1.1.3.2" xref="S3.E5X.2.1.1.m1.1.1.3.2.cmml">𝑸</mi><mi id="S3.E5X.2.1.1.m1.1.1.3.3" xref="S3.E5X.2.1.1.m1.1.1.3.3.cmml">i</mi></msub><mo id="S3.E5X.2.1.1.m1.1.1.2" xref="S3.E5X.2.1.1.m1.1.1.2.cmml">=</mo><mrow id="S3.E5X.2.1.1.m1.1.1.1" xref="S3.E5X.2.1.1.m1.1.1.1.cmml"><msub id="S3.E5X.2.1.1.m1.1.1.1.3" xref="S3.E5X.2.1.1.m1.1.1.1.3.cmml"><mi id="S3.E5X.2.1.1.m1.1.1.1.3.2" xref="S3.E5X.2.1.1.m1.1.1.1.3.2.cmml">𝑭</mi><msub id="S3.E5X.2.1.1.m1.1.1.1.3.3" xref="S3.E5X.2.1.1.m1.1.1.1.3.3.cmml"><mi id="S3.E5X.2.1.1.m1.1.1.1.3.3.2" xref="S3.E5X.2.1.1.m1.1.1.1.3.3.2.cmml">θ</mi><mi id="S3.E5X.2.1.1.m1.1.1.1.3.3.3" xref="S3.E5X.2.1.1.m1.1.1.1.3.3.3.cmml">Q</mi></msub></msub><mo id="S3.E5X.2.1.1.m1.1.1.1.2" xref="S3.E5X.2.1.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S3.E5X.2.1.1.m1.1.1.1.1.1" xref="S3.E5X.2.1.1.m1.1.1.1.1.1.1.cmml"><mo id="S3.E5X.2.1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S3.E5X.2.1.1.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.E5X.2.1.1.m1.1.1.1.1.1.1" xref="S3.E5X.2.1.1.m1.1.1.1.1.1.1.cmml"><mi id="S3.E5X.2.1.1.m1.1.1.1.1.1.1.2" xref="S3.E5X.2.1.1.m1.1.1.1.1.1.1.2.cmml">𝒅</mi><mi id="S3.E5X.2.1.1.m1.1.1.1.1.1.1.3" xref="S3.E5X.2.1.1.m1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.E5X.2.1.1.m1.1.1.1.1.1.3" stretchy="false" xref="S3.E5X.2.1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.E5X.2.1.1.m1.1b"><apply id="S3.E5X.2.1.1.m1.1.1.cmml" xref="S3.E5X.2.1.1.m1.1.1"><eq id="S3.E5X.2.1.1.m1.1.1.2.cmml" xref="S3.E5X.2.1.1.m1.1.1.2"></eq><apply id="S3.E5X.2.1.1.m1.1.1.3.cmml" xref="S3.E5X.2.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.E5X.2.1.1.m1.1.1.3.1.cmml" xref="S3.E5X.2.1.1.m1.1.1.3">subscript</csymbol><ci id="S3.E5X.2.1.1.m1.1.1.3.2.cmml" xref="S3.E5X.2.1.1.m1.1.1.3.2">𝑸</ci><ci id="S3.E5X.2.1.1.m1.1.1.3.3.cmml" xref="S3.E5X.2.1.1.m1.1.1.3.3">𝑖</ci></apply><apply id="S3.E5X.2.1.1.m1.1.1.1.cmml" xref="S3.E5X.2.1.1.m1.1.1.1"><times id="S3.E5X.2.1.1.m1.1.1.1.2.cmml" xref="S3.E5X.2.1.1.m1.1.1.1.2"></times><apply id="S3.E5X.2.1.1.m1.1.1.1.3.cmml" xref="S3.E5X.2.1.1.m1.1.1.1.3"><csymbol cd="ambiguous" id="S3.E5X.2.1.1.m1.1.1.1.3.1.cmml" xref="S3.E5X.2.1.1.m1.1.1.1.3">subscript</csymbol><ci id="S3.E5X.2.1.1.m1.1.1.1.3.2.cmml" xref="S3.E5X.2.1.1.m1.1.1.1.3.2">𝑭</ci><apply id="S3.E5X.2.1.1.m1.1.1.1.3.3.cmml" xref="S3.E5X.2.1.1.m1.1.1.1.3.3"><csymbol cd="ambiguous" id="S3.E5X.2.1.1.m1.1.1.1.3.3.1.cmml" xref="S3.E5X.2.1.1.m1.1.1.1.3.3">subscript</csymbol><ci id="S3.E5X.2.1.1.m1.1.1.1.3.3.2.cmml" xref="S3.E5X.2.1.1.m1.1.1.1.3.3.2">𝜃</ci><ci id="S3.E5X.2.1.1.m1.1.1.1.3.3.3.cmml" xref="S3.E5X.2.1.1.m1.1.1.1.3.3.3">𝑄</ci></apply></apply><apply id="S3.E5X.2.1.1.m1.1.1.1.1.1.1.cmml" xref="S3.E5X.2.1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.E5X.2.1.1.m1.1.1.1.1.1.1.1.cmml" xref="S3.E5X.2.1.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S3.E5X.2.1.1.m1.1.1.1.1.1.1.2.cmml" xref="S3.E5X.2.1.1.m1.1.1.1.1.1.1.2">𝒅</ci><ci id="S3.E5X.2.1.1.m1.1.1.1.1.1.1.3.cmml" xref="S3.E5X.2.1.1.m1.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E5X.2.1.1.m1.1c">\displaystyle\centering\boldsymbol{Q}_{i}=\boldsymbol{F}_{\theta_{Q}}(% \boldsymbol{d}_{i})</annotation><annotation encoding="application/x-llamapun" id="S3.E5X.2.1.1.m1.1d">bold_italic_Q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = bold_italic_F start_POSTSUBSCRIPT italic_θ start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_italic_d 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"><span class="ltx_tag ltx_tag_equationgroup ltx_align_right">(5)</span></td> </tr> </tbody> </table> </div> <div class="ltx_para" id="S3.SS4.p5"> <table class="ltx_equationgroup ltx_eqn_table" id="S3.E6"> <tbody> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S3.E6X"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\centering\boldsymbol{l}_{i}=\boldsymbol{F}_{\theta_{atten}}(% \boldsymbol{K}_{i,j},\boldsymbol{V}_{i,j},\boldsymbol{Q}_{i})" class="ltx_Math" display="inline" id="S3.E6X.2.1.1.m1.7"><semantics id="S3.E6X.2.1.1.m1.7a"><mrow id="S3.E6X.2.1.1.m1.7.7" xref="S3.E6X.2.1.1.m1.7.7.cmml"><msub id="S3.E6X.2.1.1.m1.7.7.5" xref="S3.E6X.2.1.1.m1.7.7.5.cmml"><mi id="S3.E6X.2.1.1.m1.7.7.5.2" xref="S3.E6X.2.1.1.m1.7.7.5.2.cmml">𝒍</mi><mi id="S3.E6X.2.1.1.m1.7.7.5.3" 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bold_italic_F start_POSTSUBSCRIPT italic_θ start_POSTSUBSCRIPT italic_a italic_t italic_t italic_e italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_italic_K start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT , bold_italic_V start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT , bold_italic_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"><span class="ltx_tag ltx_tag_equationgroup ltx_align_right">(6)</span></td> </tr> </tbody> </table> </div> </section> <section class="ltx_subsection" id="S3.SS5"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S3.SS5.4.1.1">III-E</span> </span><span class="ltx_text ltx_font_italic" id="S3.SS5.5.2" style="color:#000000;">Spherical Harmonics-based Decoding of Ray Features</span> </h3> <div class="ltx_para" id="S3.SS5.p1"> <p class="ltx_p" id="S3.SS5.p1.1">After decoding the ray features from our attention neural blocks, we represent this information as a set of spherical harmonics coefficients. Spherical harmonics are special functions defined on the surface of a sphere, widely utilized in various fields such as atomic and molecular physics, quantum mechanics, and computer graphics. These functions constitute an orthogonal and complete set of basis functions, particularly renowned for their utility in encoding or decoding directional information. A visualization of 3-order Spherical Harmonics is shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S3.F6" title="Figure 6 ‣ III-E Spherical Harmonics-based Decoding of Ray Features ‣ III Neural Point Field for Wireless Channel ‣ RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_tag">6</span></a>, where a higher order suggests enhanced performance in restoring higher frequency and directional information.</p> </div> <div class="ltx_para" id="S3.SS5.p2"> <p class="ltx_p" id="S3.SS5.p2.3">In the previous subsection, a ray feature <math alttext="\boldsymbol{l}_{i}\in\mathbb{R}^{n+1}" class="ltx_Math" display="inline" id="S3.SS5.p2.1.m1.1"><semantics id="S3.SS5.p2.1.m1.1a"><mrow id="S3.SS5.p2.1.m1.1.1" xref="S3.SS5.p2.1.m1.1.1.cmml"><msub id="S3.SS5.p2.1.m1.1.1.2" xref="S3.SS5.p2.1.m1.1.1.2.cmml"><mi id="S3.SS5.p2.1.m1.1.1.2.2" xref="S3.SS5.p2.1.m1.1.1.2.2.cmml">𝒍</mi><mi id="S3.SS5.p2.1.m1.1.1.2.3" xref="S3.SS5.p2.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S3.SS5.p2.1.m1.1.1.1" xref="S3.SS5.p2.1.m1.1.1.1.cmml">∈</mo><msup id="S3.SS5.p2.1.m1.1.1.3" xref="S3.SS5.p2.1.m1.1.1.3.cmml"><mi id="S3.SS5.p2.1.m1.1.1.3.2" xref="S3.SS5.p2.1.m1.1.1.3.2.cmml">ℝ</mi><mrow id="S3.SS5.p2.1.m1.1.1.3.3" xref="S3.SS5.p2.1.m1.1.1.3.3.cmml"><mi id="S3.SS5.p2.1.m1.1.1.3.3.2" xref="S3.SS5.p2.1.m1.1.1.3.3.2.cmml">n</mi><mo id="S3.SS5.p2.1.m1.1.1.3.3.1" xref="S3.SS5.p2.1.m1.1.1.3.3.1.cmml">+</mo><mn id="S3.SS5.p2.1.m1.1.1.3.3.3" xref="S3.SS5.p2.1.m1.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS5.p2.1.m1.1b"><apply id="S3.SS5.p2.1.m1.1.1.cmml" xref="S3.SS5.p2.1.m1.1.1"><in id="S3.SS5.p2.1.m1.1.1.1.cmml" xref="S3.SS5.p2.1.m1.1.1.1"></in><apply id="S3.SS5.p2.1.m1.1.1.2.cmml" xref="S3.SS5.p2.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.SS5.p2.1.m1.1.1.2.1.cmml" xref="S3.SS5.p2.1.m1.1.1.2">subscript</csymbol><ci id="S3.SS5.p2.1.m1.1.1.2.2.cmml" xref="S3.SS5.p2.1.m1.1.1.2.2">𝒍</ci><ci id="S3.SS5.p2.1.m1.1.1.2.3.cmml" xref="S3.SS5.p2.1.m1.1.1.2.3">𝑖</ci></apply><apply id="S3.SS5.p2.1.m1.1.1.3.cmml" xref="S3.SS5.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS5.p2.1.m1.1.1.3.1.cmml" xref="S3.SS5.p2.1.m1.1.1.3">superscript</csymbol><ci id="S3.SS5.p2.1.m1.1.1.3.2.cmml" xref="S3.SS5.p2.1.m1.1.1.3.2">ℝ</ci><apply id="S3.SS5.p2.1.m1.1.1.3.3.cmml" xref="S3.SS5.p2.1.m1.1.1.3.3"><plus id="S3.SS5.p2.1.m1.1.1.3.3.1.cmml" xref="S3.SS5.p2.1.m1.1.1.3.3.1"></plus><ci id="S3.SS5.p2.1.m1.1.1.3.3.2.cmml" xref="S3.SS5.p2.1.m1.1.1.3.3.2">𝑛</ci><cn id="S3.SS5.p2.1.m1.1.1.3.3.3.cmml" type="integer" xref="S3.SS5.p2.1.m1.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS5.p2.1.m1.1c">\boldsymbol{l}_{i}\in\mathbb{R}^{n+1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS5.p2.1.m1.1d">bold_italic_l start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT</annotation></semantics></math> is provided. Here, we employ an 8-layer multi-layer perceptron (MLP) with 256 channels. 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xref="S3.SS5.p2.2.m2.1.1.1.1.1.1.3">1</cn></apply><apply id="S3.SS5.p2.2.m2.1.1.1.3.cmml" xref="S3.SS5.p2.2.m2.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS5.p2.2.m2.1.1.1.3.1.cmml" xref="S3.SS5.p2.2.m2.1.1.1.3">subscript</csymbol><ci id="S3.SS5.p2.2.m2.1.1.1.3.2.cmml" xref="S3.SS5.p2.2.m2.1.1.1.3.2">𝑛</ci><ci id="S3.SS5.p2.2.m2.1.1.1.3.3.cmml" xref="S3.SS5.p2.2.m2.1.1.1.3.3">𝑐</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS5.p2.2.m2.1c">\boldsymbol{c}_{i}\in\mathbb{R}^{(n+1)\times n_{c}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS5.p2.2.m2.1d">bold_italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT ( italic_n + 1 ) × italic_n start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="n_{c}" class="ltx_Math" display="inline" id="S3.SS5.p2.3.m3.1"><semantics id="S3.SS5.p2.3.m3.1a"><msub id="S3.SS5.p2.3.m3.1.1" xref="S3.SS5.p2.3.m3.1.1.cmml"><mi id="S3.SS5.p2.3.m3.1.1.2" xref="S3.SS5.p2.3.m3.1.1.2.cmml">n</mi><mi id="S3.SS5.p2.3.m3.1.1.3" xref="S3.SS5.p2.3.m3.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS5.p2.3.m3.1b"><apply id="S3.SS5.p2.3.m3.1.1.cmml" xref="S3.SS5.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS5.p2.3.m3.1.1.1.cmml" xref="S3.SS5.p2.3.m3.1.1">subscript</csymbol><ci id="S3.SS5.p2.3.m3.1.1.2.cmml" xref="S3.SS5.p2.3.m3.1.1.2">𝑛</ci><ci id="S3.SS5.p2.3.m3.1.1.3.cmml" xref="S3.SS5.p2.3.m3.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS5.p2.3.m3.1c">n_{c}</annotation><annotation encoding="application/x-llamapun" id="S3.SS5.p2.3.m3.1d">italic_n start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math> is the output channel, typically set as the interpolation degree.</p> </div> <div class="ltx_para" id="S3.SS5.p3"> <table class="ltx_equationgroup ltx_eqn_table" 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xref="S3.E7X.2.1.1.m1.1.1.1.3.3.3.4">𝑃</ci></apply></apply></apply><apply id="S3.E7X.2.1.1.m1.1.1.1.1.1.1.cmml" xref="S3.E7X.2.1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.E7X.2.1.1.m1.1.1.1.1.1.1.1.cmml" xref="S3.E7X.2.1.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S3.E7X.2.1.1.m1.1.1.1.1.1.1.2.cmml" xref="S3.E7X.2.1.1.m1.1.1.1.1.1.1.2">𝒍</ci><ci id="S3.E7X.2.1.1.m1.1.1.1.1.1.1.3.cmml" xref="S3.E7X.2.1.1.m1.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E7X.2.1.1.m1.1c">\displaystyle\centering\boldsymbol{c}_{i}=\boldsymbol{F}_{\theta_{MLP}}(% \boldsymbol{l}_{i})</annotation><annotation encoding="application/x-llamapun" id="S3.E7X.2.1.1.m1.1d">bold_italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = bold_italic_F start_POSTSUBSCRIPT italic_θ start_POSTSUBSCRIPT italic_M italic_L italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_italic_l 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"><span class="ltx_tag ltx_tag_equationgroup ltx_align_right">(7)</span></td> </tr> </tbody> </table> </div> <div class="ltx_para" id="S3.SS5.p4"> <p class="ltx_p" id="S3.SS5.p4.3">Subsequently, we divide the output coefficient <math alttext="\boldsymbol{c}_{i}\in\mathbb{R}^{(n+1)\times n_{c}}" class="ltx_Math" display="inline" id="S3.SS5.p4.1.m1.1"><semantics id="S3.SS5.p4.1.m1.1a"><mrow id="S3.SS5.p4.1.m1.1.2" xref="S3.SS5.p4.1.m1.1.2.cmml"><msub id="S3.SS5.p4.1.m1.1.2.2" xref="S3.SS5.p4.1.m1.1.2.2.cmml"><mi id="S3.SS5.p4.1.m1.1.2.2.2" xref="S3.SS5.p4.1.m1.1.2.2.2.cmml">𝒄</mi><mi id="S3.SS5.p4.1.m1.1.2.2.3" xref="S3.SS5.p4.1.m1.1.2.2.3.cmml">i</mi></msub><mo id="S3.SS5.p4.1.m1.1.2.1" xref="S3.SS5.p4.1.m1.1.2.1.cmml">∈</mo><msup id="S3.SS5.p4.1.m1.1.2.3" xref="S3.SS5.p4.1.m1.1.2.3.cmml"><mi 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cd="ambiguous" id="S3.SS5.p4.3.m3.1.1.3.3.1.cmml" xref="S3.SS5.p4.3.m3.1.1.3.3">subscript</csymbol><ci id="S3.SS5.p4.3.m3.1.1.3.3.2.cmml" xref="S3.SS5.p4.3.m3.1.1.3.3.2">𝑛</ci><ci id="S3.SS5.p4.3.m3.1.1.3.3.3.cmml" xref="S3.SS5.p4.3.m3.1.1.3.3.3">𝑐</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS5.p4.3.m3.1c">\boldsymbol{c}_{i2}\in\mathbb{R}^{n_{c}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS5.p4.3.m3.1d">bold_italic_c start_POSTSUBSCRIPT italic_i 2 end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>. Finally, a Spherical Harmonics decoder is applied.</p> </div> <div class="ltx_para" id="S3.SS5.p5"> <table class="ltx_equationgroup ltx_eqn_table" id="S3.E8"> <tbody> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S3.E8X"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\centering\boldsymbol{o}_{i}=\boldsymbol{c}_{i1}^{T}SH(% \boldsymbol{d}_{i})+\boldsymbol{c}_{i2}" class="ltx_Math" display="inline" id="S3.E8X.2.1.1.m1.1"><semantics id="S3.E8X.2.1.1.m1.1a"><mrow id="S3.E8X.2.1.1.m1.1.1" xref="S3.E8X.2.1.1.m1.1.1.cmml"><msub id="S3.E8X.2.1.1.m1.1.1.3" xref="S3.E8X.2.1.1.m1.1.1.3.cmml"><mi id="S3.E8X.2.1.1.m1.1.1.3.2" xref="S3.E8X.2.1.1.m1.1.1.3.2.cmml">𝒐</mi><mi id="S3.E8X.2.1.1.m1.1.1.3.3" xref="S3.E8X.2.1.1.m1.1.1.3.3.cmml">i</mi></msub><mo id="S3.E8X.2.1.1.m1.1.1.2" xref="S3.E8X.2.1.1.m1.1.1.2.cmml">=</mo><mrow id="S3.E8X.2.1.1.m1.1.1.1" xref="S3.E8X.2.1.1.m1.1.1.1.cmml"><mrow id="S3.E8X.2.1.1.m1.1.1.1.1" xref="S3.E8X.2.1.1.m1.1.1.1.1.cmml"><msubsup id="S3.E8X.2.1.1.m1.1.1.1.1.3" xref="S3.E8X.2.1.1.m1.1.1.1.1.3.cmml"><mi id="S3.E8X.2.1.1.m1.1.1.1.1.3.2.2" xref="S3.E8X.2.1.1.m1.1.1.1.1.3.2.2.cmml">𝒄</mi><mrow id="S3.E8X.2.1.1.m1.1.1.1.1.3.2.3" xref="S3.E8X.2.1.1.m1.1.1.1.1.3.2.3.cmml"><mi id="S3.E8X.2.1.1.m1.1.1.1.1.3.2.3.2" xref="S3.E8X.2.1.1.m1.1.1.1.1.3.2.3.2.cmml">i</mi><mo id="S3.E8X.2.1.1.m1.1.1.1.1.3.2.3.1" xref="S3.E8X.2.1.1.m1.1.1.1.1.3.2.3.1.cmml">⁢</mo><mn id="S3.E8X.2.1.1.m1.1.1.1.1.3.2.3.3" xref="S3.E8X.2.1.1.m1.1.1.1.1.3.2.3.3.cmml">1</mn></mrow><mi id="S3.E8X.2.1.1.m1.1.1.1.1.3.3" xref="S3.E8X.2.1.1.m1.1.1.1.1.3.3.cmml">T</mi></msubsup><mo id="S3.E8X.2.1.1.m1.1.1.1.1.2" xref="S3.E8X.2.1.1.m1.1.1.1.1.2.cmml">⁢</mo><mi id="S3.E8X.2.1.1.m1.1.1.1.1.4" xref="S3.E8X.2.1.1.m1.1.1.1.1.4.cmml">S</mi><mo id="S3.E8X.2.1.1.m1.1.1.1.1.2a" xref="S3.E8X.2.1.1.m1.1.1.1.1.2.cmml">⁢</mo><mi id="S3.E8X.2.1.1.m1.1.1.1.1.5" 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id="S3.E8X.2.1.1.m1.1.1.1.3.3.cmml" xref="S3.E8X.2.1.1.m1.1.1.1.3.3"><times id="S3.E8X.2.1.1.m1.1.1.1.3.3.1.cmml" xref="S3.E8X.2.1.1.m1.1.1.1.3.3.1"></times><ci id="S3.E8X.2.1.1.m1.1.1.1.3.3.2.cmml" xref="S3.E8X.2.1.1.m1.1.1.1.3.3.2">𝑖</ci><cn id="S3.E8X.2.1.1.m1.1.1.1.3.3.3.cmml" type="integer" xref="S3.E8X.2.1.1.m1.1.1.1.3.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E8X.2.1.1.m1.1c">\displaystyle\centering\boldsymbol{o}_{i}=\boldsymbol{c}_{i1}^{T}SH(% \boldsymbol{d}_{i})+\boldsymbol{c}_{i2}</annotation><annotation encoding="application/x-llamapun" id="S3.E8X.2.1.1.m1.1d">bold_italic_o start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = bold_italic_c start_POSTSUBSCRIPT italic_i 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT italic_S italic_H ( bold_italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) + bold_italic_c 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"><span class="ltx_tag ltx_tag_equationgroup ltx_align_right">(8)</span></td> </tr> </tbody> </table> </div> <figure class="ltx_figure" id="S3.F6"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="216" id="S3.F6.g1" src="extracted/5641587/figure/SH.png" width="287"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S3.F6.11.5.1" style="font-size:90%;">Figure 6</span>: </span><span class="ltx_text ltx_font_bold" id="S3.F6.8.4" style="font-size:90%;">Spherical Harmonics<span class="ltx_text ltx_font_medium" id="S3.F6.8.4.4">: This figure visualizes <math alttext="3^{\textrm{rd}}" class="ltx_Math" display="inline" id="S3.F6.5.1.1.m1.1"><semantics id="S3.F6.5.1.1.m1.1b"><msup id="S3.F6.5.1.1.m1.1.1" xref="S3.F6.5.1.1.m1.1.1.cmml"><mn id="S3.F6.5.1.1.m1.1.1.2" xref="S3.F6.5.1.1.m1.1.1.2.cmml">3</mn><mtext id="S3.F6.5.1.1.m1.1.1.3" xref="S3.F6.5.1.1.m1.1.1.3a.cmml">rd</mtext></msup><annotation-xml encoding="MathML-Content" id="S3.F6.5.1.1.m1.1c"><apply id="S3.F6.5.1.1.m1.1.1.cmml" xref="S3.F6.5.1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.F6.5.1.1.m1.1.1.1.cmml" xref="S3.F6.5.1.1.m1.1.1">superscript</csymbol><cn id="S3.F6.5.1.1.m1.1.1.2.cmml" type="integer" xref="S3.F6.5.1.1.m1.1.1.2">3</cn><ci id="S3.F6.5.1.1.m1.1.1.3a.cmml" xref="S3.F6.5.1.1.m1.1.1.3"><mtext id="S3.F6.5.1.1.m1.1.1.3.cmml" mathsize="70%" xref="S3.F6.5.1.1.m1.1.1.3">rd</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F6.5.1.1.m1.1d">3^{\textrm{rd}}</annotation><annotation encoding="application/x-llamapun" id="S3.F6.5.1.1.m1.1e">3 start_POSTSUPERSCRIPT rd end_POSTSUPERSCRIPT</annotation></semantics></math>-order Spherical Harmonics, whose solution is a multiple of the associated Legendre polynomial <math alttext="P_{l}^{|m|}" class="ltx_Math" display="inline" id="S3.F6.6.2.2.m2.1"><semantics id="S3.F6.6.2.2.m2.1b"><msubsup id="S3.F6.6.2.2.m2.1.2" xref="S3.F6.6.2.2.m2.1.2.cmml"><mi id="S3.F6.6.2.2.m2.1.2.2.2" xref="S3.F6.6.2.2.m2.1.2.2.2.cmml">P</mi><mi id="S3.F6.6.2.2.m2.1.2.2.3" xref="S3.F6.6.2.2.m2.1.2.2.3.cmml">l</mi><mrow id="S3.F6.6.2.2.m2.1.1.1.3" xref="S3.F6.6.2.2.m2.1.1.1.2.cmml"><mo id="S3.F6.6.2.2.m2.1.1.1.3.1" stretchy="false" xref="S3.F6.6.2.2.m2.1.1.1.2.1.cmml">|</mo><mi id="S3.F6.6.2.2.m2.1.1.1.1" xref="S3.F6.6.2.2.m2.1.1.1.1.cmml">m</mi><mo id="S3.F6.6.2.2.m2.1.1.1.3.2" stretchy="false" xref="S3.F6.6.2.2.m2.1.1.1.2.1.cmml">|</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.F6.6.2.2.m2.1c"><apply id="S3.F6.6.2.2.m2.1.2.cmml" xref="S3.F6.6.2.2.m2.1.2"><csymbol cd="ambiguous" id="S3.F6.6.2.2.m2.1.2.1.cmml" xref="S3.F6.6.2.2.m2.1.2">superscript</csymbol><apply id="S3.F6.6.2.2.m2.1.2.2.cmml" xref="S3.F6.6.2.2.m2.1.2"><csymbol cd="ambiguous" id="S3.F6.6.2.2.m2.1.2.2.1.cmml" xref="S3.F6.6.2.2.m2.1.2">subscript</csymbol><ci id="S3.F6.6.2.2.m2.1.2.2.2.cmml" xref="S3.F6.6.2.2.m2.1.2.2.2">𝑃</ci><ci id="S3.F6.6.2.2.m2.1.2.2.3.cmml" xref="S3.F6.6.2.2.m2.1.2.2.3">𝑙</ci></apply><apply id="S3.F6.6.2.2.m2.1.1.1.2.cmml" xref="S3.F6.6.2.2.m2.1.1.1.3"><abs id="S3.F6.6.2.2.m2.1.1.1.2.1.cmml" xref="S3.F6.6.2.2.m2.1.1.1.3.1"></abs><ci id="S3.F6.6.2.2.m2.1.1.1.1.cmml" xref="S3.F6.6.2.2.m2.1.1.1.1">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F6.6.2.2.m2.1d">P_{l}^{|m|}</annotation><annotation encoding="application/x-llamapun" id="S3.F6.6.2.2.m2.1e">italic_P start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT start_POSTSUPERSCRIPT | italic_m | end_POSTSUPERSCRIPT</annotation></semantics></math> with input of azimuth <math alttext="\varphi" class="ltx_Math" display="inline" id="S3.F6.7.3.3.m3.1"><semantics id="S3.F6.7.3.3.m3.1b"><mi id="S3.F6.7.3.3.m3.1.1" xref="S3.F6.7.3.3.m3.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S3.F6.7.3.3.m3.1c"><ci id="S3.F6.7.3.3.m3.1.1.cmml" xref="S3.F6.7.3.3.m3.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F6.7.3.3.m3.1d">\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.F6.7.3.3.m3.1e">italic_φ</annotation></semantics></math> and elevation <math alttext="\theta" class="ltx_Math" display="inline" id="S3.F6.8.4.4.m4.1"><semantics id="S3.F6.8.4.4.m4.1b"><mi id="S3.F6.8.4.4.m4.1.1" xref="S3.F6.8.4.4.m4.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S3.F6.8.4.4.m4.1c"><ci id="S3.F6.8.4.4.m4.1.1.cmml" xref="S3.F6.8.4.4.m4.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F6.8.4.4.m4.1d">\theta</annotation><annotation encoding="application/x-llamapun" id="S3.F6.8.4.4.m4.1e">italic_θ</annotation></semantics></math>. In our methodology, Spherical Harmonics, which takes the ray direction as input, are employed for radiance encoding. </span></span></figcaption> </figure> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">IV </span><span class="ltx_text ltx_font_smallcaps" id="S4.1.1">Numerical Experiments</span> </h2> <figure class="ltx_figure" id="S4.F10.sf1"> </figure> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">In this section, we quantitatively validate our proposed method for predicting radio propagation characteristics across various wireless environments. The section is divided into three parts. Part <span class="ltx_text ltx_font_italic" id="S4.p1.1.1" style="color:#000000;">A</span> details the experimental setup, data collection, and the training process employed. Part <span class="ltx_text ltx_font_italic" id="S4.p1.1.2" style="color:#000000;">B</span> focuses on the validation and verification procedures. Finally, Part <span class="ltx_text ltx_font_italic" id="S4.p1.1.3" style="color:#000000;">C</span> demonstrates and evaluates the proposed methodology in large-scale, 3D wireless scenarios.</p> </div> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S4.SS1.4.1.1">IV-A</span> </span><span class="ltx_text ltx_font_italic" id="S4.SS1.5.2" style="color:#000000;">Experimental Setup and Training</span> </h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.1">Our evaluation aims to assess the effectiveness of the proposed model in accurately predicting signal strength and coverage within wireless environments. We focus on two key outcomes: path loss maps and received signal strength at designated locations. Path loss maps depict the attenuation of electromagnetic signals as they propagate through the wireless scenes, offering valuable insights into coverage areas. Meanwhile, the received signal strength at specific locations provides crucial information for tasks such as localization and connectivity assessment. Both outcomes are essential for network planning and optimization, informing decisions regarding antenna placement and transmission power levels to optimize network performance and reliability.</p> </div> <section class="ltx_subsubsection" id="S4.SS1.SSS1"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection"><span class="ltx_text" id="S4.SS1.SSS1.4.1.1">IV-A</span>1 </span>Data Collection</h4> <div class="ltx_para" id="S4.SS1.SSS1.p1"> <p class="ltx_p" id="S4.SS1.SSS1.p1.1">Our datasets are generated using an open-sourced ray-tracing simulator: <em class="ltx_emph ltx_font_italic" id="S4.SS1.SSS1.p1.1.1">Sionna</em> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib66" title="">66</a>]</cite>. We generate our dataset in various scales of scenes such as: <em class="ltx_emph ltx_font_italic" id="S4.SS1.SSS1.p1.1.2">wiindoor</em> (small indoor room scene), <em class="ltx_emph ltx_font_italic" id="S4.SS1.SSS1.p1.1.3">etoile</em> (Medium city block scene), and Munich (large urban city scene).</p> </div> <div class="ltx_para" id="S4.SS1.SSS1.p2"> <p class="ltx_p" id="S4.SS1.SSS1.p2.5">As described in Table <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S4.T1" title="TABLE I ‣ IV-A1 Data Collection ‣ IV-A Experimental Setup and Training ‣ IV Numerical Experiments ‣ RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_tag">I</span></a>, the dataset comprises <math alttext="175\sim 375" class="ltx_Math" display="inline" id="S4.SS1.SSS1.p2.1.m1.1"><semantics id="S4.SS1.SSS1.p2.1.m1.1a"><mrow id="S4.SS1.SSS1.p2.1.m1.1.1" xref="S4.SS1.SSS1.p2.1.m1.1.1.cmml"><mn id="S4.SS1.SSS1.p2.1.m1.1.1.2" xref="S4.SS1.SSS1.p2.1.m1.1.1.2.cmml">175</mn><mo id="S4.SS1.SSS1.p2.1.m1.1.1.1" xref="S4.SS1.SSS1.p2.1.m1.1.1.1.cmml">∼</mo><mn id="S4.SS1.SSS1.p2.1.m1.1.1.3" xref="S4.SS1.SSS1.p2.1.m1.1.1.3.cmml">375</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS1.p2.1.m1.1b"><apply id="S4.SS1.SSS1.p2.1.m1.1.1.cmml" xref="S4.SS1.SSS1.p2.1.m1.1.1"><csymbol cd="latexml" id="S4.SS1.SSS1.p2.1.m1.1.1.1.cmml" xref="S4.SS1.SSS1.p2.1.m1.1.1.1">similar-to</csymbol><cn id="S4.SS1.SSS1.p2.1.m1.1.1.2.cmml" type="integer" xref="S4.SS1.SSS1.p2.1.m1.1.1.2">175</cn><cn id="S4.SS1.SSS1.p2.1.m1.1.1.3.cmml" type="integer" xref="S4.SS1.SSS1.p2.1.m1.1.1.3">375</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS1.p2.1.m1.1c">175\sim 375</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS1.p2.1.m1.1d">175 ∼ 375</annotation></semantics></math> transmitter locations and approximately <math alttext="1,920\sim 35,816" class="ltx_Math" display="inline" id="S4.SS1.SSS1.p2.2.m2.4"><semantics id="S4.SS1.SSS1.p2.2.m2.4a"><mrow id="S4.SS1.SSS1.p2.2.m2.4.4.1" xref="S4.SS1.SSS1.p2.2.m2.4.4.2.cmml"><mrow id="S4.SS1.SSS1.p2.2.m2.4.4.1.1" xref="S4.SS1.SSS1.p2.2.m2.4.4.1.1.cmml"><mrow id="S4.SS1.SSS1.p2.2.m2.4.4.1.1.2.2" xref="S4.SS1.SSS1.p2.2.m2.4.4.1.1.2.1.cmml"><mn id="S4.SS1.SSS1.p2.2.m2.1.1" xref="S4.SS1.SSS1.p2.2.m2.1.1.cmml">1</mn><mo id="S4.SS1.SSS1.p2.2.m2.4.4.1.1.2.2.1" xref="S4.SS1.SSS1.p2.2.m2.4.4.1.1.2.1.cmml">,</mo><mn id="S4.SS1.SSS1.p2.2.m2.2.2" xref="S4.SS1.SSS1.p2.2.m2.2.2.cmml">920</mn></mrow><mo id="S4.SS1.SSS1.p2.2.m2.4.4.1.1.1" xref="S4.SS1.SSS1.p2.2.m2.4.4.1.1.1.cmml">∼</mo><mn id="S4.SS1.SSS1.p2.2.m2.4.4.1.1.3" xref="S4.SS1.SSS1.p2.2.m2.4.4.1.1.3.cmml">35</mn></mrow><mo id="S4.SS1.SSS1.p2.2.m2.4.4.1.2" xref="S4.SS1.SSS1.p2.2.m2.4.4.2a.cmml">,</mo><mn id="S4.SS1.SSS1.p2.2.m2.3.3" xref="S4.SS1.SSS1.p2.2.m2.3.3.cmml">816</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS1.p2.2.m2.4b"><apply id="S4.SS1.SSS1.p2.2.m2.4.4.2.cmml" xref="S4.SS1.SSS1.p2.2.m2.4.4.1"><csymbol cd="ambiguous" id="S4.SS1.SSS1.p2.2.m2.4.4.2a.cmml" xref="S4.SS1.SSS1.p2.2.m2.4.4.1.2">formulae-sequence</csymbol><apply id="S4.SS1.SSS1.p2.2.m2.4.4.1.1.cmml" xref="S4.SS1.SSS1.p2.2.m2.4.4.1.1"><csymbol cd="latexml" id="S4.SS1.SSS1.p2.2.m2.4.4.1.1.1.cmml" xref="S4.SS1.SSS1.p2.2.m2.4.4.1.1.1">similar-to</csymbol><list id="S4.SS1.SSS1.p2.2.m2.4.4.1.1.2.1.cmml" xref="S4.SS1.SSS1.p2.2.m2.4.4.1.1.2.2"><cn id="S4.SS1.SSS1.p2.2.m2.1.1.cmml" type="integer" xref="S4.SS1.SSS1.p2.2.m2.1.1">1</cn><cn id="S4.SS1.SSS1.p2.2.m2.2.2.cmml" type="integer" xref="S4.SS1.SSS1.p2.2.m2.2.2">920</cn></list><cn id="S4.SS1.SSS1.p2.2.m2.4.4.1.1.3.cmml" type="integer" xref="S4.SS1.SSS1.p2.2.m2.4.4.1.1.3">35</cn></apply><cn id="S4.SS1.SSS1.p2.2.m2.3.3.cmml" type="integer" xref="S4.SS1.SSS1.p2.2.m2.3.3">816</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS1.p2.2.m2.4c">1,920\sim 35,816</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS1.p2.2.m2.4d">1 , 920 ∼ 35 , 816</annotation></semantics></math> uniformly sampled receiver locations for each scene. About <math alttext="85\%" class="ltx_Math" display="inline" id="S4.SS1.SSS1.p2.3.m3.1"><semantics id="S4.SS1.SSS1.p2.3.m3.1a"><mrow id="S4.SS1.SSS1.p2.3.m3.1.1" xref="S4.SS1.SSS1.p2.3.m3.1.1.cmml"><mn id="S4.SS1.SSS1.p2.3.m3.1.1.2" xref="S4.SS1.SSS1.p2.3.m3.1.1.2.cmml">85</mn><mo id="S4.SS1.SSS1.p2.3.m3.1.1.1" xref="S4.SS1.SSS1.p2.3.m3.1.1.1.cmml">%</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS1.p2.3.m3.1b"><apply id="S4.SS1.SSS1.p2.3.m3.1.1.cmml" xref="S4.SS1.SSS1.p2.3.m3.1.1"><csymbol cd="latexml" id="S4.SS1.SSS1.p2.3.m3.1.1.1.cmml" xref="S4.SS1.SSS1.p2.3.m3.1.1.1">percent</csymbol><cn id="S4.SS1.SSS1.p2.3.m3.1.1.2.cmml" type="integer" xref="S4.SS1.SSS1.p2.3.m3.1.1.2">85</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS1.p2.3.m3.1c">85\%</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS1.p2.3.m3.1d">85 %</annotation></semantics></math> of them is used for training, with the remaining <math alttext="15\%" class="ltx_Math" display="inline" id="S4.SS1.SSS1.p2.4.m4.1"><semantics id="S4.SS1.SSS1.p2.4.m4.1a"><mrow id="S4.SS1.SSS1.p2.4.m4.1.1" xref="S4.SS1.SSS1.p2.4.m4.1.1.cmml"><mn id="S4.SS1.SSS1.p2.4.m4.1.1.2" xref="S4.SS1.SSS1.p2.4.m4.1.1.2.cmml">15</mn><mo id="S4.SS1.SSS1.p2.4.m4.1.1.1" xref="S4.SS1.SSS1.p2.4.m4.1.1.1.cmml">%</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS1.p2.4.m4.1b"><apply id="S4.SS1.SSS1.p2.4.m4.1.1.cmml" xref="S4.SS1.SSS1.p2.4.m4.1.1"><csymbol cd="latexml" id="S4.SS1.SSS1.p2.4.m4.1.1.1.cmml" xref="S4.SS1.SSS1.p2.4.m4.1.1.1">percent</csymbol><cn id="S4.SS1.SSS1.p2.4.m4.1.1.2.cmml" type="integer" xref="S4.SS1.SSS1.p2.4.m4.1.1.2">15</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS1.p2.4.m4.1c">15\%</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS1.p2.4.m4.1d">15 %</annotation></semantics></math> reserved for validation. The operating frequency is <math alttext="2.14\rm{GHz}" class="ltx_Math" display="inline" id="S4.SS1.SSS1.p2.5.m5.1"><semantics id="S4.SS1.SSS1.p2.5.m5.1a"><mrow id="S4.SS1.SSS1.p2.5.m5.1.1" xref="S4.SS1.SSS1.p2.5.m5.1.1.cmml"><mn id="S4.SS1.SSS1.p2.5.m5.1.1.2" xref="S4.SS1.SSS1.p2.5.m5.1.1.2.cmml">2.14</mn><mo id="S4.SS1.SSS1.p2.5.m5.1.1.1" xref="S4.SS1.SSS1.p2.5.m5.1.1.1.cmml">⁢</mo><mi id="S4.SS1.SSS1.p2.5.m5.1.1.3" xref="S4.SS1.SSS1.p2.5.m5.1.1.3.cmml">GHz</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS1.p2.5.m5.1b"><apply id="S4.SS1.SSS1.p2.5.m5.1.1.cmml" xref="S4.SS1.SSS1.p2.5.m5.1.1"><times id="S4.SS1.SSS1.p2.5.m5.1.1.1.cmml" xref="S4.SS1.SSS1.p2.5.m5.1.1.1"></times><cn id="S4.SS1.SSS1.p2.5.m5.1.1.2.cmml" type="float" xref="S4.SS1.SSS1.p2.5.m5.1.1.2">2.14</cn><ci id="S4.SS1.SSS1.p2.5.m5.1.1.3.cmml" xref="S4.SS1.SSS1.p2.5.m5.1.1.3">GHz</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS1.p2.5.m5.1c">2.14\rm{GHz}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS1.p2.5.m5.1d">2.14 roman_GHz</annotation></semantics></math>. After training, the model serves as a neural surrogate for wireless channel prediction.</p> </div> <figure class="ltx_table" id="S4.T1"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S4.T1.16.1.1" style="font-size:90%;">TABLE I</span>: </span><span class="ltx_text ltx_font_bold" id="S4.T1.17.2" style="font-size:90%;">Data collection<span class="ltx_text ltx_font_medium" id="S4.T1.17.2.1">: We validate our methodology across three different scene scales. This table provides the configuration details of our dataset.</span></span></figcaption> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S4.T1.13.13"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T1.13.13.14.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.13.13.14.1.1"><span class="ltx_text ltx_font_bold" id="S4.T1.13.13.14.1.1.1">Training Dataset</span></th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.13.13.14.1.2"><em class="ltx_emph ltx_font_italic" id="S4.T1.13.13.14.1.2.1">wiindoor</em></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.13.13.14.1.3"><em class="ltx_emph ltx_font_italic" id="S4.T1.13.13.14.1.3.1">etoile center</em></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.13.13.14.1.4"><em class="ltx_emph ltx_font_italic" id="S4.T1.13.13.14.1.4.1">etoile</em></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.13.13.14.1.5"><em class="ltx_emph ltx_font_italic" id="S4.T1.13.13.14.1.5.1">munich</em></td> </tr> <tr class="ltx_tr" id="S4.T1.13.13.15.2"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.13.13.15.2.1"><span class="ltx_text ltx_font_bold" id="S4.T1.13.13.15.2.1.1">Scale</span></th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.13.13.15.2.2">indoor room</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.13.13.15.2.3">isolated building</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.13.13.15.2.4">city blocks</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.13.13.15.2.5">urban city</td> </tr> <tr class="ltx_tr" id="S4.T1.4.4.4"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.4.4.4.5"><span class="ltx_text ltx_font_bold" id="S4.T1.4.4.4.5.1">Covered area</span></th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.1.1.1.1"><math alttext="10\times 10\;\rm{m}^{2}" class="ltx_Math" display="inline" id="S4.T1.1.1.1.1.m1.1"><semantics id="S4.T1.1.1.1.1.m1.1a"><mrow id="S4.T1.1.1.1.1.m1.1.1" xref="S4.T1.1.1.1.1.m1.1.1.cmml"><mrow id="S4.T1.1.1.1.1.m1.1.1.2" xref="S4.T1.1.1.1.1.m1.1.1.2.cmml"><mn id="S4.T1.1.1.1.1.m1.1.1.2.2" xref="S4.T1.1.1.1.1.m1.1.1.2.2.cmml">10</mn><mo id="S4.T1.1.1.1.1.m1.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S4.T1.1.1.1.1.m1.1.1.2.1.cmml">×</mo><mn id="S4.T1.1.1.1.1.m1.1.1.2.3" xref="S4.T1.1.1.1.1.m1.1.1.2.3.cmml">10</mn></mrow><mo id="S4.T1.1.1.1.1.m1.1.1.1" lspace="0.280em" xref="S4.T1.1.1.1.1.m1.1.1.1.cmml">⁢</mo><msup id="S4.T1.1.1.1.1.m1.1.1.3" xref="S4.T1.1.1.1.1.m1.1.1.3.cmml"><mi id="S4.T1.1.1.1.1.m1.1.1.3.2" mathvariant="normal" xref="S4.T1.1.1.1.1.m1.1.1.3.2.cmml">m</mi><mn id="S4.T1.1.1.1.1.m1.1.1.3.3" xref="S4.T1.1.1.1.1.m1.1.1.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.T1.1.1.1.1.m1.1b"><apply id="S4.T1.1.1.1.1.m1.1.1.cmml" xref="S4.T1.1.1.1.1.m1.1.1"><times id="S4.T1.1.1.1.1.m1.1.1.1.cmml" xref="S4.T1.1.1.1.1.m1.1.1.1"></times><apply id="S4.T1.1.1.1.1.m1.1.1.2.cmml" xref="S4.T1.1.1.1.1.m1.1.1.2"><times id="S4.T1.1.1.1.1.m1.1.1.2.1.cmml" xref="S4.T1.1.1.1.1.m1.1.1.2.1"></times><cn id="S4.T1.1.1.1.1.m1.1.1.2.2.cmml" type="integer" xref="S4.T1.1.1.1.1.m1.1.1.2.2">10</cn><cn id="S4.T1.1.1.1.1.m1.1.1.2.3.cmml" type="integer" xref="S4.T1.1.1.1.1.m1.1.1.2.3">10</cn></apply><apply id="S4.T1.1.1.1.1.m1.1.1.3.cmml" xref="S4.T1.1.1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.T1.1.1.1.1.m1.1.1.3.1.cmml" xref="S4.T1.1.1.1.1.m1.1.1.3">superscript</csymbol><ci id="S4.T1.1.1.1.1.m1.1.1.3.2.cmml" xref="S4.T1.1.1.1.1.m1.1.1.3.2">m</ci><cn id="S4.T1.1.1.1.1.m1.1.1.3.3.cmml" type="integer" xref="S4.T1.1.1.1.1.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.1.1.1.1.m1.1c">10\times 10\;\rm{m}^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.T1.1.1.1.1.m1.1d">10 × 10 roman_m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.2.2.2.2"><math alttext="150\times 160\;\rm{m}^{2}" class="ltx_Math" display="inline" id="S4.T1.2.2.2.2.m1.1"><semantics id="S4.T1.2.2.2.2.m1.1a"><mrow id="S4.T1.2.2.2.2.m1.1.1" xref="S4.T1.2.2.2.2.m1.1.1.cmml"><mrow id="S4.T1.2.2.2.2.m1.1.1.2" xref="S4.T1.2.2.2.2.m1.1.1.2.cmml"><mn id="S4.T1.2.2.2.2.m1.1.1.2.2" xref="S4.T1.2.2.2.2.m1.1.1.2.2.cmml">150</mn><mo id="S4.T1.2.2.2.2.m1.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S4.T1.2.2.2.2.m1.1.1.2.1.cmml">×</mo><mn id="S4.T1.2.2.2.2.m1.1.1.2.3" xref="S4.T1.2.2.2.2.m1.1.1.2.3.cmml">160</mn></mrow><mo id="S4.T1.2.2.2.2.m1.1.1.1" lspace="0.280em" xref="S4.T1.2.2.2.2.m1.1.1.1.cmml">⁢</mo><msup id="S4.T1.2.2.2.2.m1.1.1.3" xref="S4.T1.2.2.2.2.m1.1.1.3.cmml"><mi id="S4.T1.2.2.2.2.m1.1.1.3.2" mathvariant="normal" xref="S4.T1.2.2.2.2.m1.1.1.3.2.cmml">m</mi><mn id="S4.T1.2.2.2.2.m1.1.1.3.3" xref="S4.T1.2.2.2.2.m1.1.1.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.T1.2.2.2.2.m1.1b"><apply id="S4.T1.2.2.2.2.m1.1.1.cmml" xref="S4.T1.2.2.2.2.m1.1.1"><times id="S4.T1.2.2.2.2.m1.1.1.1.cmml" xref="S4.T1.2.2.2.2.m1.1.1.1"></times><apply id="S4.T1.2.2.2.2.m1.1.1.2.cmml" xref="S4.T1.2.2.2.2.m1.1.1.2"><times id="S4.T1.2.2.2.2.m1.1.1.2.1.cmml" xref="S4.T1.2.2.2.2.m1.1.1.2.1"></times><cn id="S4.T1.2.2.2.2.m1.1.1.2.2.cmml" type="integer" xref="S4.T1.2.2.2.2.m1.1.1.2.2">150</cn><cn id="S4.T1.2.2.2.2.m1.1.1.2.3.cmml" type="integer" xref="S4.T1.2.2.2.2.m1.1.1.2.3">160</cn></apply><apply id="S4.T1.2.2.2.2.m1.1.1.3.cmml" xref="S4.T1.2.2.2.2.m1.1.1.3"><csymbol cd="ambiguous" id="S4.T1.2.2.2.2.m1.1.1.3.1.cmml" xref="S4.T1.2.2.2.2.m1.1.1.3">superscript</csymbol><ci id="S4.T1.2.2.2.2.m1.1.1.3.2.cmml" xref="S4.T1.2.2.2.2.m1.1.1.3.2">m</ci><cn id="S4.T1.2.2.2.2.m1.1.1.3.3.cmml" type="integer" xref="S4.T1.2.2.2.2.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.2.2.2.2.m1.1c">150\times 160\;\rm{m}^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.T1.2.2.2.2.m1.1d">150 × 160 roman_m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.3.3.3.3"><math alttext="853\times 676\;\rm{m}^{2}" class="ltx_Math" display="inline" id="S4.T1.3.3.3.3.m1.1"><semantics id="S4.T1.3.3.3.3.m1.1a"><mrow id="S4.T1.3.3.3.3.m1.1.1" xref="S4.T1.3.3.3.3.m1.1.1.cmml"><mrow id="S4.T1.3.3.3.3.m1.1.1.2" xref="S4.T1.3.3.3.3.m1.1.1.2.cmml"><mn id="S4.T1.3.3.3.3.m1.1.1.2.2" xref="S4.T1.3.3.3.3.m1.1.1.2.2.cmml">853</mn><mo id="S4.T1.3.3.3.3.m1.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S4.T1.3.3.3.3.m1.1.1.2.1.cmml">×</mo><mn id="S4.T1.3.3.3.3.m1.1.1.2.3" xref="S4.T1.3.3.3.3.m1.1.1.2.3.cmml">676</mn></mrow><mo id="S4.T1.3.3.3.3.m1.1.1.1" lspace="0.280em" xref="S4.T1.3.3.3.3.m1.1.1.1.cmml">⁢</mo><msup id="S4.T1.3.3.3.3.m1.1.1.3" xref="S4.T1.3.3.3.3.m1.1.1.3.cmml"><mi id="S4.T1.3.3.3.3.m1.1.1.3.2" mathvariant="normal" xref="S4.T1.3.3.3.3.m1.1.1.3.2.cmml">m</mi><mn id="S4.T1.3.3.3.3.m1.1.1.3.3" xref="S4.T1.3.3.3.3.m1.1.1.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.T1.3.3.3.3.m1.1b"><apply id="S4.T1.3.3.3.3.m1.1.1.cmml" xref="S4.T1.3.3.3.3.m1.1.1"><times id="S4.T1.3.3.3.3.m1.1.1.1.cmml" xref="S4.T1.3.3.3.3.m1.1.1.1"></times><apply id="S4.T1.3.3.3.3.m1.1.1.2.cmml" xref="S4.T1.3.3.3.3.m1.1.1.2"><times id="S4.T1.3.3.3.3.m1.1.1.2.1.cmml" xref="S4.T1.3.3.3.3.m1.1.1.2.1"></times><cn id="S4.T1.3.3.3.3.m1.1.1.2.2.cmml" type="integer" xref="S4.T1.3.3.3.3.m1.1.1.2.2">853</cn><cn id="S4.T1.3.3.3.3.m1.1.1.2.3.cmml" type="integer" xref="S4.T1.3.3.3.3.m1.1.1.2.3">676</cn></apply><apply id="S4.T1.3.3.3.3.m1.1.1.3.cmml" xref="S4.T1.3.3.3.3.m1.1.1.3"><csymbol cd="ambiguous" id="S4.T1.3.3.3.3.m1.1.1.3.1.cmml" xref="S4.T1.3.3.3.3.m1.1.1.3">superscript</csymbol><ci id="S4.T1.3.3.3.3.m1.1.1.3.2.cmml" xref="S4.T1.3.3.3.3.m1.1.1.3.2">m</ci><cn id="S4.T1.3.3.3.3.m1.1.1.3.3.cmml" type="integer" xref="S4.T1.3.3.3.3.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.3.3.3.3.m1.1c">853\times 676\;\rm{m}^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.T1.3.3.3.3.m1.1d">853 × 676 roman_m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.4.4.4.4"><math alttext="1475\times 1205\;\rm{m}^{2}" class="ltx_Math" display="inline" id="S4.T1.4.4.4.4.m1.1"><semantics id="S4.T1.4.4.4.4.m1.1a"><mrow id="S4.T1.4.4.4.4.m1.1.1" xref="S4.T1.4.4.4.4.m1.1.1.cmml"><mrow id="S4.T1.4.4.4.4.m1.1.1.2" xref="S4.T1.4.4.4.4.m1.1.1.2.cmml"><mn id="S4.T1.4.4.4.4.m1.1.1.2.2" xref="S4.T1.4.4.4.4.m1.1.1.2.2.cmml">1475</mn><mo id="S4.T1.4.4.4.4.m1.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S4.T1.4.4.4.4.m1.1.1.2.1.cmml">×</mo><mn id="S4.T1.4.4.4.4.m1.1.1.2.3" xref="S4.T1.4.4.4.4.m1.1.1.2.3.cmml">1205</mn></mrow><mo id="S4.T1.4.4.4.4.m1.1.1.1" lspace="0.280em" xref="S4.T1.4.4.4.4.m1.1.1.1.cmml">⁢</mo><msup id="S4.T1.4.4.4.4.m1.1.1.3" xref="S4.T1.4.4.4.4.m1.1.1.3.cmml"><mi id="S4.T1.4.4.4.4.m1.1.1.3.2" mathvariant="normal" xref="S4.T1.4.4.4.4.m1.1.1.3.2.cmml">m</mi><mn id="S4.T1.4.4.4.4.m1.1.1.3.3" xref="S4.T1.4.4.4.4.m1.1.1.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.T1.4.4.4.4.m1.1b"><apply id="S4.T1.4.4.4.4.m1.1.1.cmml" xref="S4.T1.4.4.4.4.m1.1.1"><times id="S4.T1.4.4.4.4.m1.1.1.1.cmml" xref="S4.T1.4.4.4.4.m1.1.1.1"></times><apply id="S4.T1.4.4.4.4.m1.1.1.2.cmml" xref="S4.T1.4.4.4.4.m1.1.1.2"><times id="S4.T1.4.4.4.4.m1.1.1.2.1.cmml" xref="S4.T1.4.4.4.4.m1.1.1.2.1"></times><cn id="S4.T1.4.4.4.4.m1.1.1.2.2.cmml" type="integer" xref="S4.T1.4.4.4.4.m1.1.1.2.2">1475</cn><cn id="S4.T1.4.4.4.4.m1.1.1.2.3.cmml" type="integer" xref="S4.T1.4.4.4.4.m1.1.1.2.3">1205</cn></apply><apply id="S4.T1.4.4.4.4.m1.1.1.3.cmml" xref="S4.T1.4.4.4.4.m1.1.1.3"><csymbol cd="ambiguous" id="S4.T1.4.4.4.4.m1.1.1.3.1.cmml" xref="S4.T1.4.4.4.4.m1.1.1.3">superscript</csymbol><ci id="S4.T1.4.4.4.4.m1.1.1.3.2.cmml" xref="S4.T1.4.4.4.4.m1.1.1.3.2">m</ci><cn id="S4.T1.4.4.4.4.m1.1.1.3.3.cmml" type="integer" xref="S4.T1.4.4.4.4.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.4.4.4.4.m1.1c">1475\times 1205\;\rm{m}^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.T1.4.4.4.4.m1.1d">1475 × 1205 roman_m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S4.T1.8.8.8"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.8.8.8.5"><span class="ltx_text ltx_font_bold" id="S4.T1.8.8.8.5.1">Transmitters</span></th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.5.5.5.1"><math alttext="175" class="ltx_Math" display="inline" id="S4.T1.5.5.5.1.m1.1"><semantics id="S4.T1.5.5.5.1.m1.1a"><mn id="S4.T1.5.5.5.1.m1.1.1" xref="S4.T1.5.5.5.1.m1.1.1.cmml">175</mn><annotation-xml encoding="MathML-Content" id="S4.T1.5.5.5.1.m1.1b"><cn id="S4.T1.5.5.5.1.m1.1.1.cmml" type="integer" xref="S4.T1.5.5.5.1.m1.1.1">175</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.5.5.5.1.m1.1c">175</annotation><annotation encoding="application/x-llamapun" id="S4.T1.5.5.5.1.m1.1d">175</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.6.6.6.2"><math alttext="375" class="ltx_Math" display="inline" id="S4.T1.6.6.6.2.m1.1"><semantics id="S4.T1.6.6.6.2.m1.1a"><mn id="S4.T1.6.6.6.2.m1.1.1" xref="S4.T1.6.6.6.2.m1.1.1.cmml">375</mn><annotation-xml encoding="MathML-Content" id="S4.T1.6.6.6.2.m1.1b"><cn id="S4.T1.6.6.6.2.m1.1.1.cmml" type="integer" xref="S4.T1.6.6.6.2.m1.1.1">375</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.6.6.6.2.m1.1c">375</annotation><annotation encoding="application/x-llamapun" id="S4.T1.6.6.6.2.m1.1d">375</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.7.7.7.3"><math alttext="175" class="ltx_Math" display="inline" id="S4.T1.7.7.7.3.m1.1"><semantics id="S4.T1.7.7.7.3.m1.1a"><mn id="S4.T1.7.7.7.3.m1.1.1" xref="S4.T1.7.7.7.3.m1.1.1.cmml">175</mn><annotation-xml encoding="MathML-Content" id="S4.T1.7.7.7.3.m1.1b"><cn id="S4.T1.7.7.7.3.m1.1.1.cmml" type="integer" xref="S4.T1.7.7.7.3.m1.1.1">175</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.7.7.7.3.m1.1c">175</annotation><annotation encoding="application/x-llamapun" id="S4.T1.7.7.7.3.m1.1d">175</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.8.8.8.4"><math alttext="175" class="ltx_Math" display="inline" id="S4.T1.8.8.8.4.m1.1"><semantics id="S4.T1.8.8.8.4.m1.1a"><mn id="S4.T1.8.8.8.4.m1.1.1" xref="S4.T1.8.8.8.4.m1.1.1.cmml">175</mn><annotation-xml encoding="MathML-Content" id="S4.T1.8.8.8.4.m1.1b"><cn id="S4.T1.8.8.8.4.m1.1.1.cmml" type="integer" xref="S4.T1.8.8.8.4.m1.1.1">175</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.8.8.8.4.m1.1c">175</annotation><annotation encoding="application/x-llamapun" id="S4.T1.8.8.8.4.m1.1d">175</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S4.T1.12.12.12"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.12.12.12.5"><span class="ltx_text ltx_font_bold" id="S4.T1.12.12.12.5.1">Receivers</span></th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.9.9.9.1"><math alttext="100\times 100\times 2" class="ltx_Math" display="inline" id="S4.T1.9.9.9.1.m1.1"><semantics id="S4.T1.9.9.9.1.m1.1a"><mrow id="S4.T1.9.9.9.1.m1.1.1" xref="S4.T1.9.9.9.1.m1.1.1.cmml"><mn id="S4.T1.9.9.9.1.m1.1.1.2" xref="S4.T1.9.9.9.1.m1.1.1.2.cmml">100</mn><mo id="S4.T1.9.9.9.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.T1.9.9.9.1.m1.1.1.1.cmml">×</mo><mn id="S4.T1.9.9.9.1.m1.1.1.3" xref="S4.T1.9.9.9.1.m1.1.1.3.cmml">100</mn><mo id="S4.T1.9.9.9.1.m1.1.1.1a" lspace="0.222em" rspace="0.222em" xref="S4.T1.9.9.9.1.m1.1.1.1.cmml">×</mo><mn id="S4.T1.9.9.9.1.m1.1.1.4" xref="S4.T1.9.9.9.1.m1.1.1.4.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T1.9.9.9.1.m1.1b"><apply id="S4.T1.9.9.9.1.m1.1.1.cmml" xref="S4.T1.9.9.9.1.m1.1.1"><times id="S4.T1.9.9.9.1.m1.1.1.1.cmml" xref="S4.T1.9.9.9.1.m1.1.1.1"></times><cn id="S4.T1.9.9.9.1.m1.1.1.2.cmml" type="integer" xref="S4.T1.9.9.9.1.m1.1.1.2">100</cn><cn id="S4.T1.9.9.9.1.m1.1.1.3.cmml" type="integer" xref="S4.T1.9.9.9.1.m1.1.1.3">100</cn><cn id="S4.T1.9.9.9.1.m1.1.1.4.cmml" type="integer" xref="S4.T1.9.9.9.1.m1.1.1.4">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.9.9.9.1.m1.1c">100\times 100\times 2</annotation><annotation encoding="application/x-llamapun" id="S4.T1.9.9.9.1.m1.1d">100 × 100 × 2</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.10.10.10.2"><math alttext="30\times 32\times 2" class="ltx_Math" display="inline" id="S4.T1.10.10.10.2.m1.1"><semantics id="S4.T1.10.10.10.2.m1.1a"><mrow id="S4.T1.10.10.10.2.m1.1.1" xref="S4.T1.10.10.10.2.m1.1.1.cmml"><mn id="S4.T1.10.10.10.2.m1.1.1.2" xref="S4.T1.10.10.10.2.m1.1.1.2.cmml">30</mn><mo id="S4.T1.10.10.10.2.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.T1.10.10.10.2.m1.1.1.1.cmml">×</mo><mn id="S4.T1.10.10.10.2.m1.1.1.3" xref="S4.T1.10.10.10.2.m1.1.1.3.cmml">32</mn><mo id="S4.T1.10.10.10.2.m1.1.1.1a" lspace="0.222em" rspace="0.222em" xref="S4.T1.10.10.10.2.m1.1.1.1.cmml">×</mo><mn id="S4.T1.10.10.10.2.m1.1.1.4" xref="S4.T1.10.10.10.2.m1.1.1.4.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T1.10.10.10.2.m1.1b"><apply id="S4.T1.10.10.10.2.m1.1.1.cmml" xref="S4.T1.10.10.10.2.m1.1.1"><times id="S4.T1.10.10.10.2.m1.1.1.1.cmml" xref="S4.T1.10.10.10.2.m1.1.1.1"></times><cn id="S4.T1.10.10.10.2.m1.1.1.2.cmml" type="integer" xref="S4.T1.10.10.10.2.m1.1.1.2">30</cn><cn id="S4.T1.10.10.10.2.m1.1.1.3.cmml" type="integer" xref="S4.T1.10.10.10.2.m1.1.1.3">32</cn><cn id="S4.T1.10.10.10.2.m1.1.1.4.cmml" type="integer" xref="S4.T1.10.10.10.2.m1.1.1.4">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.10.10.10.2.m1.1c">30\times 32\times 2</annotation><annotation encoding="application/x-llamapun" id="S4.T1.10.10.10.2.m1.1d">30 × 32 × 2</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.11.11.11.3"><math alttext="86\times 68\times 2" class="ltx_Math" display="inline" id="S4.T1.11.11.11.3.m1.1"><semantics id="S4.T1.11.11.11.3.m1.1a"><mrow id="S4.T1.11.11.11.3.m1.1.1" xref="S4.T1.11.11.11.3.m1.1.1.cmml"><mn id="S4.T1.11.11.11.3.m1.1.1.2" xref="S4.T1.11.11.11.3.m1.1.1.2.cmml">86</mn><mo id="S4.T1.11.11.11.3.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.T1.11.11.11.3.m1.1.1.1.cmml">×</mo><mn id="S4.T1.11.11.11.3.m1.1.1.3" xref="S4.T1.11.11.11.3.m1.1.1.3.cmml">68</mn><mo id="S4.T1.11.11.11.3.m1.1.1.1a" lspace="0.222em" rspace="0.222em" xref="S4.T1.11.11.11.3.m1.1.1.1.cmml">×</mo><mn id="S4.T1.11.11.11.3.m1.1.1.4" xref="S4.T1.11.11.11.3.m1.1.1.4.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T1.11.11.11.3.m1.1b"><apply id="S4.T1.11.11.11.3.m1.1.1.cmml" xref="S4.T1.11.11.11.3.m1.1.1"><times id="S4.T1.11.11.11.3.m1.1.1.1.cmml" xref="S4.T1.11.11.11.3.m1.1.1.1"></times><cn id="S4.T1.11.11.11.3.m1.1.1.2.cmml" type="integer" xref="S4.T1.11.11.11.3.m1.1.1.2">86</cn><cn id="S4.T1.11.11.11.3.m1.1.1.3.cmml" type="integer" xref="S4.T1.11.11.11.3.m1.1.1.3">68</cn><cn id="S4.T1.11.11.11.3.m1.1.1.4.cmml" type="integer" xref="S4.T1.11.11.11.3.m1.1.1.4">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.11.11.11.3.m1.1c">86\times 68\times 2</annotation><annotation encoding="application/x-llamapun" id="S4.T1.11.11.11.3.m1.1d">86 × 68 × 2</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.12.12.12.4"><math alttext="148\times 121\times 2" class="ltx_Math" display="inline" id="S4.T1.12.12.12.4.m1.1"><semantics id="S4.T1.12.12.12.4.m1.1a"><mrow id="S4.T1.12.12.12.4.m1.1.1" xref="S4.T1.12.12.12.4.m1.1.1.cmml"><mn id="S4.T1.12.12.12.4.m1.1.1.2" xref="S4.T1.12.12.12.4.m1.1.1.2.cmml">148</mn><mo id="S4.T1.12.12.12.4.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.T1.12.12.12.4.m1.1.1.1.cmml">×</mo><mn id="S4.T1.12.12.12.4.m1.1.1.3" xref="S4.T1.12.12.12.4.m1.1.1.3.cmml">121</mn><mo id="S4.T1.12.12.12.4.m1.1.1.1a" lspace="0.222em" rspace="0.222em" xref="S4.T1.12.12.12.4.m1.1.1.1.cmml">×</mo><mn id="S4.T1.12.12.12.4.m1.1.1.4" xref="S4.T1.12.12.12.4.m1.1.1.4.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T1.12.12.12.4.m1.1b"><apply id="S4.T1.12.12.12.4.m1.1.1.cmml" xref="S4.T1.12.12.12.4.m1.1.1"><times id="S4.T1.12.12.12.4.m1.1.1.1.cmml" xref="S4.T1.12.12.12.4.m1.1.1.1"></times><cn id="S4.T1.12.12.12.4.m1.1.1.2.cmml" type="integer" xref="S4.T1.12.12.12.4.m1.1.1.2">148</cn><cn id="S4.T1.12.12.12.4.m1.1.1.3.cmml" type="integer" xref="S4.T1.12.12.12.4.m1.1.1.3">121</cn><cn id="S4.T1.12.12.12.4.m1.1.1.4.cmml" type="integer" xref="S4.T1.12.12.12.4.m1.1.1.4">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.12.12.12.4.m1.1c">148\times 121\times 2</annotation><annotation encoding="application/x-llamapun" id="S4.T1.12.12.12.4.m1.1d">148 × 121 × 2</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S4.T1.13.13.13"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_b ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.13.13.13.2"><span class="ltx_text ltx_font_bold" id="S4.T1.13.13.13.2.1">Antenna patterns</span></th> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" colspan="4" id="S4.T1.13.13.13.1"><math alttext="4" class="ltx_Math" display="inline" id="S4.T1.13.13.13.1.m1.1"><semantics id="S4.T1.13.13.13.1.m1.1a"><mn id="S4.T1.13.13.13.1.m1.1.1" xref="S4.T1.13.13.13.1.m1.1.1.cmml">4</mn><annotation-xml encoding="MathML-Content" id="S4.T1.13.13.13.1.m1.1b"><cn id="S4.T1.13.13.13.1.m1.1.1.cmml" type="integer" xref="S4.T1.13.13.13.1.m1.1.1">4</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.13.13.13.1.m1.1c">4</annotation><annotation encoding="application/x-llamapun" id="S4.T1.13.13.13.1.m1.1d">4</annotation></semantics></math></td> </tr> </tbody> </table> </figure> </section> <section class="ltx_subsubsection" id="S4.SS1.SSS2"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection"><span class="ltx_text" id="S4.SS1.SSS2.4.1.1">IV-A</span>2 </span>Training setups</h4> <div class="ltx_para" id="S4.SS1.SSS2.p1"> <p class="ltx_p" id="S4.SS1.SSS2.p1.6">In our experiments, <math alttext="n" class="ltx_Math" display="inline" id="S4.SS1.SSS2.p1.1.m1.1"><semantics id="S4.SS1.SSS2.p1.1.m1.1a"><mi id="S4.SS1.SSS2.p1.1.m1.1.1" xref="S4.SS1.SSS2.p1.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.p1.1.m1.1b"><ci id="S4.SS1.SSS2.p1.1.m1.1.1.cmml" xref="S4.SS1.SSS2.p1.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.p1.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.p1.1.m1.1d">italic_n</annotation></semantics></math> rays are initially launched from receivers to find <math alttext="n" class="ltx_Math" display="inline" id="S4.SS1.SSS2.p1.2.m2.1"><semantics id="S4.SS1.SSS2.p1.2.m2.1a"><mi id="S4.SS1.SSS2.p1.2.m2.1.1" xref="S4.SS1.SSS2.p1.2.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.p1.2.m2.1b"><ci id="S4.SS1.SSS2.p1.2.m2.1.1.cmml" xref="S4.SS1.SSS2.p1.2.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.p1.2.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.p1.2.m2.1d">italic_n</annotation></semantics></math> nearest light probes. Each light probe is attached to <math alttext="K" class="ltx_Math" display="inline" id="S4.SS1.SSS2.p1.3.m3.1"><semantics id="S4.SS1.SSS2.p1.3.m3.1a"><mi id="S4.SS1.SSS2.p1.3.m3.1.1" xref="S4.SS1.SSS2.p1.3.m3.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.p1.3.m3.1b"><ci id="S4.SS1.SSS2.p1.3.m3.1.1.cmml" xref="S4.SS1.SSS2.p1.3.m3.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.p1.3.m3.1c">K</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.p1.3.m3.1d">italic_K</annotation></semantics></math> different point clouds. The hyperparameters <math alttext="n" class="ltx_Math" display="inline" id="S4.SS1.SSS2.p1.4.m4.1"><semantics id="S4.SS1.SSS2.p1.4.m4.1a"><mi id="S4.SS1.SSS2.p1.4.m4.1.1" xref="S4.SS1.SSS2.p1.4.m4.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.p1.4.m4.1b"><ci id="S4.SS1.SSS2.p1.4.m4.1.1.cmml" xref="S4.SS1.SSS2.p1.4.m4.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.p1.4.m4.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.p1.4.m4.1d">italic_n</annotation></semantics></math> and <math alttext="K" class="ltx_Math" display="inline" id="S4.SS1.SSS2.p1.5.m5.1"><semantics id="S4.SS1.SSS2.p1.5.m5.1a"><mi id="S4.SS1.SSS2.p1.5.m5.1.1" xref="S4.SS1.SSS2.p1.5.m5.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.p1.5.m5.1b"><ci id="S4.SS1.SSS2.p1.5.m5.1.1.cmml" xref="S4.SS1.SSS2.p1.5.m5.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.p1.5.m5.1c">K</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.p1.5.m5.1d">italic_K</annotation></semantics></math> are selected as <math alttext="8" class="ltx_Math" display="inline" id="S4.SS1.SSS2.p1.6.m6.1"><semantics id="S4.SS1.SSS2.p1.6.m6.1a"><mn id="S4.SS1.SSS2.p1.6.m6.1.1" xref="S4.SS1.SSS2.p1.6.m6.1.1.cmml">8</mn><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.p1.6.m6.1b"><cn id="S4.SS1.SSS2.p1.6.m6.1.1.cmml" type="integer" xref="S4.SS1.SSS2.p1.6.m6.1.1">8</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.p1.6.m6.1c">8</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.p1.6.m6.1d">8</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS1.SSS2.p2"> <p class="ltx_p" id="S4.SS1.SSS2.p2.4">Our model is trained with a batch size of <math alttext="1000" class="ltx_Math" display="inline" id="S4.SS1.SSS2.p2.1.m1.1"><semantics id="S4.SS1.SSS2.p2.1.m1.1a"><mn id="S4.SS1.SSS2.p2.1.m1.1.1" xref="S4.SS1.SSS2.p2.1.m1.1.1.cmml">1000</mn><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.p2.1.m1.1b"><cn id="S4.SS1.SSS2.p2.1.m1.1.1.cmml" type="integer" xref="S4.SS1.SSS2.p2.1.m1.1.1">1000</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.p2.1.m1.1c">1000</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.p2.1.m1.1d">1000</annotation></semantics></math> and a learning rate of <math alttext="0.0001" class="ltx_Math" display="inline" id="S4.SS1.SSS2.p2.2.m2.1"><semantics id="S4.SS1.SSS2.p2.2.m2.1a"><mn id="S4.SS1.SSS2.p2.2.m2.1.1" xref="S4.SS1.SSS2.p2.2.m2.1.1.cmml">0.0001</mn><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.p2.2.m2.1b"><cn id="S4.SS1.SSS2.p2.2.m2.1.1.cmml" type="float" xref="S4.SS1.SSS2.p2.2.m2.1.1">0.0001</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.p2.2.m2.1c">0.0001</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.p2.2.m2.1d">0.0001</annotation></semantics></math>. We train the model for <math alttext="500" class="ltx_Math" display="inline" id="S4.SS1.SSS2.p2.3.m3.1"><semantics id="S4.SS1.SSS2.p2.3.m3.1a"><mn id="S4.SS1.SSS2.p2.3.m3.1.1" xref="S4.SS1.SSS2.p2.3.m3.1.1.cmml">500</mn><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.p2.3.m3.1b"><cn id="S4.SS1.SSS2.p2.3.m3.1.1.cmml" type="integer" xref="S4.SS1.SSS2.p2.3.m3.1.1">500</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.p2.3.m3.1c">500</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.p2.3.m3.1d">500</annotation></semantics></math> epochs, which typically takes between <math alttext="1.5\sim 10" class="ltx_Math" display="inline" id="S4.SS1.SSS2.p2.4.m4.1"><semantics id="S4.SS1.SSS2.p2.4.m4.1a"><mrow id="S4.SS1.SSS2.p2.4.m4.1.1" xref="S4.SS1.SSS2.p2.4.m4.1.1.cmml"><mn id="S4.SS1.SSS2.p2.4.m4.1.1.2" xref="S4.SS1.SSS2.p2.4.m4.1.1.2.cmml">1.5</mn><mo id="S4.SS1.SSS2.p2.4.m4.1.1.1" xref="S4.SS1.SSS2.p2.4.m4.1.1.1.cmml">∼</mo><mn id="S4.SS1.SSS2.p2.4.m4.1.1.3" xref="S4.SS1.SSS2.p2.4.m4.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.p2.4.m4.1b"><apply id="S4.SS1.SSS2.p2.4.m4.1.1.cmml" xref="S4.SS1.SSS2.p2.4.m4.1.1"><csymbol cd="latexml" id="S4.SS1.SSS2.p2.4.m4.1.1.1.cmml" xref="S4.SS1.SSS2.p2.4.m4.1.1.1">similar-to</csymbol><cn id="S4.SS1.SSS2.p2.4.m4.1.1.2.cmml" type="float" xref="S4.SS1.SSS2.p2.4.m4.1.1.2">1.5</cn><cn id="S4.SS1.SSS2.p2.4.m4.1.1.3.cmml" type="integer" xref="S4.SS1.SSS2.p2.4.m4.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.p2.4.m4.1c">1.5\sim 10</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.p2.4.m4.1d">1.5 ∼ 10</annotation></semantics></math> hours in a GPU environment using an NVIDIA GeForce RTX 3090 Ti. We utilize the Adam optimizer <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib67" title="">67</a>]</cite> and the mean square error (MSE) loss function for received power optimization.</p> </div> </section> <section class="ltx_subsubsection" id="S4.SS1.SSS3"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection"><span class="ltx_text" id="S4.SS1.SSS3.4.1.1">IV-A</span>3 </span>Evaluation Metric</h4> <div class="ltx_para" id="S4.SS1.SSS3.p1"> <p class="ltx_p" id="S4.SS1.SSS3.p1.1">The evaluation metric serves as a quantitative measure to assess the performance of the proposed method in predicting radio path loss maps. It quantifies the accuracy of the predictions by comparing them to ground truth (GT) data or measurements. The specific evaluation metrics that will be used in our numerical experiments include Mean Square Error (MSE), and Peak Signal-to-Noise Ratio (PSNR).</p> </div> <div class="ltx_para" id="S4.SS1.SSS3.p2"> <table class="ltx_equationgroup ltx_eqn_table" id="S4.E9"> <tbody> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S4.E9X"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\centering\begin{cases}\rm{MSE}=\sum_{i=0}^{N}(o_{i}-o_{gt})^{2}% \\ \rm{PSNR}=20\rm{log}_{10}({\rm{max}{(o_{i})}}/{\sqrt{MSE}})\\ \end{cases}" class="ltx_Math" display="inline" id="S4.E9X.2.1.1.m1.1"><semantics id="S4.E9X.2.1.1.m1.1a"><mrow id="S4.E9.m1.2.2.2.2.2.2a" xref="S4.E9X.2.1.1.m1.1.1.1.cmml"><mo id="S4.E9.m1.2.2.2.2.2.2a.3" xref="S4.E9X.2.1.1.m1.1.1.1.1.cmml">{</mo><mtable columnspacing="5pt" id="S4.E9.m1.2.2.2.2.2.2.2a" rowspacing="0pt" xref="S4.E9X.2.1.1.m1.1.1.1.cmml"><mtr id="S4.E9.m1.2.2.2.2.2.2.2aa" 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roman_N end_POSTSUPERSCRIPT ( roman_o start_POSTSUBSCRIPT roman_i end_POSTSUBSCRIPT - roman_o start_POSTSUBSCRIPT roman_gt end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL roman_PSNR = 20 roman_l roman_o roman_g start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ( roman_max ( roman_o start_POSTSUBSCRIPT roman_i end_POSTSUBSCRIPT ) / square-root start_ARG roman_MSE end_ARG ) end_CELL start_CELL end_CELL end_ROW</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"><span class="ltx_tag ltx_tag_equationgroup ltx_align_right">(9)</span></td> </tr> </tbody> </table> </div> </section> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S4.SS2.4.1.1">IV-B</span> </span><span class="ltx_text ltx_font_italic" id="S4.SS2.5.2" style="color:#000000;">Validation and Verification</span> </h3> <section class="ltx_subsubsection" id="S4.SS2.SSS1"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection"><span class="ltx_text" id="S4.SS2.SSS1.4.1.1">IV-B</span>1 </span>Assessment in Learning EM Propagation Physics</h4> <div class="ltx_para" id="S4.SS2.SSS1.p1"> <p class="ltx_p" id="S4.SS2.SSS1.p1.1">In this subsection, we evaluate our framework’s ability to model EM propagation physics. Our objective is to determine how effectively the neural network learns and understands key EM principles such as reflection, transmission, and diffraction, particularly in the context of various building structures. To assess this, we train the neural network using scenes with a single isolated building at the center. This setup allows us to isolate and analyze the model’s performance in understanding EM propagation in the presence of architectural elements. We provide two separate datasets for training: one including diffraction effects and the other without. The validation results, shown in Fig. <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:fig:_Evaluation_EM_physics_understanding</span>, demonstrate the model’s proficiency in accurately capturing essential features of EM propagation physics. Moreover, enhancements in prediction accuracy can be achieved through the refinement of the training dataset.</p> </div> </section> <section class="ltx_subsubsection" id="S4.SS2.SSS2"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection"><span class="ltx_text" id="S4.SS2.SSS2.4.1.1">IV-B</span>2 </span>Comparison to Other Neural Surrogates</h4> <div class="ltx_para" id="S4.SS2.SSS2.p1"> <p class="ltx_p" id="S4.SS2.SSS2.p1.1">While our primary focus is on 3D end-to-end channel power prediction, the scarcity of open-source 3D-based neural surrogates led us to evaluate our model against a 2D-based neural surrogate PMNet <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib20" title="">20</a>]</cite> and a standard Multi-Layer Perceptron (MLP) model. The MLP network is designed with four hidden layers of sizes 64, 64, 32, and 64, using leaky ReLU as the activation function.</p> </div> <div class="ltx_para" id="S4.SS2.SSS2.p2"> <p class="ltx_p" id="S4.SS2.SSS2.p2.2">Figure <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:fig:_Comparison_with_others</span> and Table <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S4.T2" title="TABLE II ‣ IV-B2 Comparison to Other Neural Surrogates ‣ IV-B Validation and Verification ‣ IV Numerical Experiments ‣ RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_tag">II</span></a> present the visualization and quantitative comparison. The results indicate that our prediction closely matches the ground truth (ray-tracing simulator) and outperforms other neural surrogates. Notably, even evaluating the 2D path loss map (at a certain height), our 3D RayProNet shows a significant advantage over the 2D pipeline (PMNet). In our numerical experiments, the MSE score of PMNet is substantially lower than their 2D validations (approximately <math alttext="\sim 10^{-2}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p2.1.m1.1"><semantics id="S4.SS2.SSS2.p2.1.m1.1a"><mrow id="S4.SS2.SSS2.p2.1.m1.1.1" xref="S4.SS2.SSS2.p2.1.m1.1.1.cmml"><mi id="S4.SS2.SSS2.p2.1.m1.1.1.2" xref="S4.SS2.SSS2.p2.1.m1.1.1.2.cmml"></mi><mo id="S4.SS2.SSS2.p2.1.m1.1.1.1" xref="S4.SS2.SSS2.p2.1.m1.1.1.1.cmml">∼</mo><msup id="S4.SS2.SSS2.p2.1.m1.1.1.3" xref="S4.SS2.SSS2.p2.1.m1.1.1.3.cmml"><mn id="S4.SS2.SSS2.p2.1.m1.1.1.3.2" xref="S4.SS2.SSS2.p2.1.m1.1.1.3.2.cmml">10</mn><mrow id="S4.SS2.SSS2.p2.1.m1.1.1.3.3" xref="S4.SS2.SSS2.p2.1.m1.1.1.3.3.cmml"><mo id="S4.SS2.SSS2.p2.1.m1.1.1.3.3a" xref="S4.SS2.SSS2.p2.1.m1.1.1.3.3.cmml">−</mo><mn id="S4.SS2.SSS2.p2.1.m1.1.1.3.3.2" xref="S4.SS2.SSS2.p2.1.m1.1.1.3.3.2.cmml">2</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p2.1.m1.1b"><apply id="S4.SS2.SSS2.p2.1.m1.1.1.cmml" xref="S4.SS2.SSS2.p2.1.m1.1.1"><csymbol cd="latexml" id="S4.SS2.SSS2.p2.1.m1.1.1.1.cmml" xref="S4.SS2.SSS2.p2.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S4.SS2.SSS2.p2.1.m1.1.1.2.cmml" xref="S4.SS2.SSS2.p2.1.m1.1.1.2">absent</csymbol><apply id="S4.SS2.SSS2.p2.1.m1.1.1.3.cmml" xref="S4.SS2.SSS2.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS2.p2.1.m1.1.1.3.1.cmml" xref="S4.SS2.SSS2.p2.1.m1.1.1.3">superscript</csymbol><cn id="S4.SS2.SSS2.p2.1.m1.1.1.3.2.cmml" type="integer" xref="S4.SS2.SSS2.p2.1.m1.1.1.3.2">10</cn><apply id="S4.SS2.SSS2.p2.1.m1.1.1.3.3.cmml" xref="S4.SS2.SSS2.p2.1.m1.1.1.3.3"><minus id="S4.SS2.SSS2.p2.1.m1.1.1.3.3.1.cmml" xref="S4.SS2.SSS2.p2.1.m1.1.1.3.3"></minus><cn id="S4.SS2.SSS2.p2.1.m1.1.1.3.3.2.cmml" type="integer" xref="S4.SS2.SSS2.p2.1.m1.1.1.3.3.2">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p2.1.m1.1c">\sim 10^{-2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p2.1.m1.1d">∼ 10 start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT</annotation></semantics></math> in our isolated building environment and approximately <math alttext="\sim 10^{-4}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p2.2.m2.1"><semantics id="S4.SS2.SSS2.p2.2.m2.1a"><mrow id="S4.SS2.SSS2.p2.2.m2.1.1" xref="S4.SS2.SSS2.p2.2.m2.1.1.cmml"><mi id="S4.SS2.SSS2.p2.2.m2.1.1.2" xref="S4.SS2.SSS2.p2.2.m2.1.1.2.cmml"></mi><mo id="S4.SS2.SSS2.p2.2.m2.1.1.1" xref="S4.SS2.SSS2.p2.2.m2.1.1.1.cmml">∼</mo><msup id="S4.SS2.SSS2.p2.2.m2.1.1.3" xref="S4.SS2.SSS2.p2.2.m2.1.1.3.cmml"><mn id="S4.SS2.SSS2.p2.2.m2.1.1.3.2" xref="S4.SS2.SSS2.p2.2.m2.1.1.3.2.cmml">10</mn><mrow id="S4.SS2.SSS2.p2.2.m2.1.1.3.3" xref="S4.SS2.SSS2.p2.2.m2.1.1.3.3.cmml"><mo id="S4.SS2.SSS2.p2.2.m2.1.1.3.3a" xref="S4.SS2.SSS2.p2.2.m2.1.1.3.3.cmml">−</mo><mn id="S4.SS2.SSS2.p2.2.m2.1.1.3.3.2" xref="S4.SS2.SSS2.p2.2.m2.1.1.3.3.2.cmml">4</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p2.2.m2.1b"><apply id="S4.SS2.SSS2.p2.2.m2.1.1.cmml" xref="S4.SS2.SSS2.p2.2.m2.1.1"><csymbol cd="latexml" id="S4.SS2.SSS2.p2.2.m2.1.1.1.cmml" xref="S4.SS2.SSS2.p2.2.m2.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S4.SS2.SSS2.p2.2.m2.1.1.2.cmml" xref="S4.SS2.SSS2.p2.2.m2.1.1.2">absent</csymbol><apply id="S4.SS2.SSS2.p2.2.m2.1.1.3.cmml" xref="S4.SS2.SSS2.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS2.p2.2.m2.1.1.3.1.cmml" xref="S4.SS2.SSS2.p2.2.m2.1.1.3">superscript</csymbol><cn id="S4.SS2.SSS2.p2.2.m2.1.1.3.2.cmml" type="integer" xref="S4.SS2.SSS2.p2.2.m2.1.1.3.2">10</cn><apply id="S4.SS2.SSS2.p2.2.m2.1.1.3.3.cmml" xref="S4.SS2.SSS2.p2.2.m2.1.1.3.3"><minus id="S4.SS2.SSS2.p2.2.m2.1.1.3.3.1.cmml" xref="S4.SS2.SSS2.p2.2.m2.1.1.3.3"></minus><cn id="S4.SS2.SSS2.p2.2.m2.1.1.3.3.2.cmml" type="integer" xref="S4.SS2.SSS2.p2.2.m2.1.1.3.3.2">4</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p2.2.m2.1c">\sim 10^{-4}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p2.2.m2.1d">∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math> in their USC campus setting). The main reason for this difference is the size of the training dataset. The dataset in this experiment uses only 100 transmitters, whereas PMNet validates its pipeline with 19,016 configurations on the USC campus. Typically, a larger dataset size leads to better performance.</p> </div> <figure class="ltx_figure" id="S4.F14.sf1"> </figure> <figure class="ltx_table" id="S4.T2"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S4.T2.12.1.1" style="font-size:90%;">TABLE II</span>: </span><span class="ltx_text ltx_font_bold" id="S4.T2.13.2" style="font-size:90%;">Comparison with other neural surrogates:<span class="ltx_text ltx_font_medium" id="S4.T2.13.2.1"> MSE loss and PSNR score comparison of power between ray-tracing results (ground truth) and various neural predictions (Ours, MLP, PMNet) in our isolated building environment. The upward arrow indicates better performance with larger values, while the downward arrow denotes better performance with smaller values. The best scores and lowest errors are highlighted in </span>bold<span class="ltx_text ltx_font_medium" id="S4.T2.13.2.2"> font.</span></span></figcaption> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S4.T2.8.8"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S4.T2.8.8.9.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_l ltx_border_r ltx_border_t" id="S4.T2.8.8.9.1.1">-</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S4.T2.8.8.9.1.2">Ours</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S4.T2.8.8.9.1.3">MLP</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S4.T2.8.8.9.1.4">PMNet</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T2.4.4.4"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t" id="S4.T2.1.1.1.1">MSE <math alttext="\downarrow" class="ltx_Math" 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xref="S4.T2.2.2.2.2.m1.1.1.1.cmml">×</mo><msup id="S4.T2.2.2.2.2.m1.1.1.3" xref="S4.T2.2.2.2.2.m1.1.1.3.cmml"><mn id="S4.T2.2.2.2.2.m1.1.1.3.2" xref="S4.T2.2.2.2.2.m1.1.1.3.2.cmml">𝟏𝟎</mn><mrow id="S4.T2.2.2.2.2.m1.1.1.3.3" xref="S4.T2.2.2.2.2.m1.1.1.3.3.cmml"><mo class="ltx_mathvariant_bold" id="S4.T2.2.2.2.2.m1.1.1.3.3a" mathvariant="bold" xref="S4.T2.2.2.2.2.m1.1.1.3.3.cmml">−</mo><mn id="S4.T2.2.2.2.2.m1.1.1.3.3.2" xref="S4.T2.2.2.2.2.m1.1.1.3.3.2.cmml">𝟒</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.2.2.2.2.m1.1b"><apply id="S4.T2.2.2.2.2.m1.1.1.cmml" xref="S4.T2.2.2.2.2.m1.1.1"><times id="S4.T2.2.2.2.2.m1.1.1.1.cmml" xref="S4.T2.2.2.2.2.m1.1.1.1"></times><cn id="S4.T2.2.2.2.2.m1.1.1.2.cmml" type="integer" xref="S4.T2.2.2.2.2.m1.1.1.2">3</cn><apply id="S4.T2.2.2.2.2.m1.1.1.3.cmml" xref="S4.T2.2.2.2.2.m1.1.1.3"><csymbol cd="ambiguous" id="S4.T2.2.2.2.2.m1.1.1.3.1.cmml" xref="S4.T2.2.2.2.2.m1.1.1.3">superscript</csymbol><cn 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encoding="MathML-Content" id="S4.T2.4.4.4.4.m1.1b"><cn id="S4.T2.4.4.4.4.m1.1.1.cmml" type="float" xref="S4.T2.4.4.4.4.m1.1.1">0.039</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.4.4.4.4.m1.1c">0.039</annotation><annotation encoding="application/x-llamapun" id="S4.T2.4.4.4.4.m1.1d">0.039</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S4.T2.8.8.8"> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_l ltx_border_r ltx_border_t" id="S4.T2.5.5.5.1">PSNR <math alttext="\uparrow" class="ltx_Math" display="inline" id="S4.T2.5.5.5.1.m1.1"><semantics id="S4.T2.5.5.5.1.m1.1a"><mo id="S4.T2.5.5.5.1.m1.1.1" stretchy="false" xref="S4.T2.5.5.5.1.m1.1.1.cmml">↑</mo><annotation-xml encoding="MathML-Content" id="S4.T2.5.5.5.1.m1.1b"><ci id="S4.T2.5.5.5.1.m1.1.1.cmml" xref="S4.T2.5.5.5.1.m1.1.1">↑</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.5.5.5.1.m1.1c">\uparrow</annotation><annotation encoding="application/x-llamapun" id="S4.T2.5.5.5.1.m1.1d">↑</annotation></semantics></math> </td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T2.6.6.6.2"><math alttext="\boldsymbol{35.24}" class="ltx_Math" display="inline" id="S4.T2.6.6.6.2.m1.1"><semantics id="S4.T2.6.6.6.2.m1.1a"><mn class="ltx_mathvariant_bold" id="S4.T2.6.6.6.2.m1.1.1" mathvariant="bold" xref="S4.T2.6.6.6.2.m1.1.1.cmml">35.24</mn><annotation-xml encoding="MathML-Content" id="S4.T2.6.6.6.2.m1.1b"><cn id="S4.T2.6.6.6.2.m1.1.1.cmml" type="float" xref="S4.T2.6.6.6.2.m1.1.1">35.24</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.6.6.6.2.m1.1c">\boldsymbol{35.24}</annotation><annotation encoding="application/x-llamapun" id="S4.T2.6.6.6.2.m1.1d">bold_35.24</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T2.7.7.7.3"><math alttext="23.90" class="ltx_Math" display="inline" id="S4.T2.7.7.7.3.m1.1"><semantics id="S4.T2.7.7.7.3.m1.1a"><mn id="S4.T2.7.7.7.3.m1.1.1" xref="S4.T2.7.7.7.3.m1.1.1.cmml">23.90</mn><annotation-xml encoding="MathML-Content" id="S4.T2.7.7.7.3.m1.1b"><cn id="S4.T2.7.7.7.3.m1.1.1.cmml" type="float" xref="S4.T2.7.7.7.3.m1.1.1">23.90</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.7.7.7.3.m1.1c">23.90</annotation><annotation encoding="application/x-llamapun" id="S4.T2.7.7.7.3.m1.1d">23.90</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T2.8.8.8.4"><math alttext="14.27" class="ltx_Math" display="inline" id="S4.T2.8.8.8.4.m1.1"><semantics id="S4.T2.8.8.8.4.m1.1a"><mn id="S4.T2.8.8.8.4.m1.1.1" xref="S4.T2.8.8.8.4.m1.1.1.cmml">14.27</mn><annotation-xml encoding="MathML-Content" id="S4.T2.8.8.8.4.m1.1b"><cn id="S4.T2.8.8.8.4.m1.1.1.cmml" type="float" xref="S4.T2.8.8.8.4.m1.1.1">14.27</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.8.8.8.4.m1.1c">14.27</annotation><annotation encoding="application/x-llamapun" id="S4.T2.8.8.8.4.m1.1d">14.27</annotation></semantics></math></td> </tr> </tbody> </table> </figure> </section> <section class="ltx_subsubsection" id="S4.SS2.SSS3"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection"><span class="ltx_text" id="S4.SS2.SSS3.4.1.1">IV-B</span>3 </span><span class="ltx_text" id="S4.SS2.SSS3.5.2">Verification through Ablation Experiment</span> </h4> <div class="ltx_para" id="S4.SS2.SSS3.p1"> <p class="ltx_p" id="S4.SS2.SSS3.p1.2"><span class="ltx_text" id="S4.SS2.SSS3.p1.2.2">One of the key ingredients in RayProNet is the introduction of light probes. Hence, we will evaluate the impact of this module by performing an ablation experiment. If we remove the light probe module, receivers will directly shoot rays to find the <math alttext="K" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p1.1.1.m1.1"><semantics id="S4.SS2.SSS3.p1.1.1.m1.1a"><mi id="S4.SS2.SSS3.p1.1.1.m1.1.1" xref="S4.SS2.SSS3.p1.1.1.m1.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p1.1.1.m1.1b"><ci id="S4.SS2.SSS3.p1.1.1.m1.1.1.cmml" xref="S4.SS2.SSS3.p1.1.1.m1.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p1.1.1.m1.1c">K</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p1.1.1.m1.1d">italic_K</annotation></semantics></math> closest point clouds, rather than the <math alttext="n" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p1.2.2.m2.1"><semantics id="S4.SS2.SSS3.p1.2.2.m2.1a"><mi id="S4.SS2.SSS3.p1.2.2.m2.1.1" xref="S4.SS2.SSS3.p1.2.2.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p1.2.2.m2.1b"><ci id="S4.SS2.SSS3.p1.2.2.m2.1.1.cmml" xref="S4.SS2.SSS3.p1.2.2.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p1.2.2.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p1.2.2.m2.1d">italic_n</annotation></semantics></math> closest light probes. For this ablation experiment, we use a similar ray sampling strategy to <em class="ltx_emph ltx_font_italic" id="S4.SS2.SSS3.p1.2.2.1">NPLF</em> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib23" title="">23</a>]</cite>. Both models are trained for 12 hours.</span></p> </div> <div class="ltx_para" id="S4.SS2.SSS3.p2"> <p class="ltx_p" id="S4.SS2.SSS3.p2.1">The results of our ablation experiment are shown in Figure <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:fig:_Ablation_results</span> and Table <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S4.T3" title="TABLE III ‣ IV-B3 Verification through Ablation Experiment ‣ IV-B Validation and Verification ‣ IV Numerical Experiments ‣ RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_tag">III</span></a>. These results align with our expectation that in both large outdoor scenes and small indoor scenes, it is common for a ray beam to be shot from an antenna but not reach any buildings (point clouds in our pipeline) nearby, causing the ray to be wasted as a default latent feature. The data in Figure <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:fig:_Ablation_results</span> and Table <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S4.T3" title="TABLE III ‣ IV-B3 Verification through Ablation Experiment ‣ IV-B Validation and Verification ‣ IV Numerical Experiments ‣ RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_tag">III</span></a> support our analysis that power prediction is significantly limited when light probes are removed. Since light probes cover all areas in space, each antenna can always find a nearby light probe and extract propagation features from it. Hence, our proposed approach consistently aligns well with ray-tracing ground truth, even in large outdoor scenes.</p> </div> <figure class="ltx_figure" id="S4.F20.sf1"> </figure> <figure class="ltx_table" id="S4.T3"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S4.T3.18.1.1" style="font-size:90%;">TABLE III</span>: </span><span class="ltx_text ltx_font_bold" id="S4.T3.19.2" style="font-size:90%;">Ablation experiment:<span class="ltx_text ltx_font_medium" id="S4.T3.19.2.1"> This table presents the MSE loss and PSNR score of power for both our dataset (<em class="ltx_emph ltx_font_italic" id="S4.T3.19.2.1.1">etoile</em>) and <em class="ltx_emph ltx_font_italic" id="S4.T3.19.2.1.2">WINERT</em>’s dataset (<em class="ltx_emph ltx_font_italic" id="S4.T3.19.2.1.3">wiindoor</em>).</span></span></figcaption> <div class="ltx_block ltx_minipage ltx_align_center ltx_align_middle" id="S4.T3.12.12" style="width:208.1pt;"> <table class="ltx_tabular ltx_minipage ltx_align_top" id="S4.T3.6.6.6.6" style="width:208.1pt;"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S4.T3.6.6.6.6.7.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T3.6.6.6.6.7.1.1"><em class="ltx_emph ltx_font_italic" id="S4.T3.6.6.6.6.7.1.1.1">etoile</em></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S4.T3.6.6.6.6.7.1.2">Ours</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S4.T3.6.6.6.6.7.1.3">Ablation</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T3.3.3.3.3.3"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T3.1.1.1.1.1.1">MSE <math alttext="\downarrow" class="ltx_Math" display="inline" id="S4.T3.1.1.1.1.1.1.m1.1"><semantics id="S4.T3.1.1.1.1.1.1.m1.1a"><mo id="S4.T3.1.1.1.1.1.1.m1.1.1" stretchy="false" xref="S4.T3.1.1.1.1.1.1.m1.1.1.cmml">↓</mo><annotation-xml encoding="MathML-Content" id="S4.T3.1.1.1.1.1.1.m1.1b"><ci id="S4.T3.1.1.1.1.1.1.m1.1.1.cmml" xref="S4.T3.1.1.1.1.1.1.m1.1.1">↓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.1.1.1.1.1.1.m1.1c">\downarrow</annotation><annotation encoding="application/x-llamapun" id="S4.T3.1.1.1.1.1.1.m1.1d">↓</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.2.2.2.2.2.2"><math alttext="\boldsymbol{0.0011}" class="ltx_Math" display="inline" id="S4.T3.2.2.2.2.2.2.m1.1"><semantics id="S4.T3.2.2.2.2.2.2.m1.1a"><mn class="ltx_mathvariant_bold" id="S4.T3.2.2.2.2.2.2.m1.1.1" mathvariant="bold" xref="S4.T3.2.2.2.2.2.2.m1.1.1.cmml">0.0011</mn><annotation-xml encoding="MathML-Content" id="S4.T3.2.2.2.2.2.2.m1.1b"><cn id="S4.T3.2.2.2.2.2.2.m1.1.1.cmml" type="float" xref="S4.T3.2.2.2.2.2.2.m1.1.1">0.0011</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.2.2.2.2.2.2.m1.1c">\boldsymbol{0.0011}</annotation><annotation encoding="application/x-llamapun" id="S4.T3.2.2.2.2.2.2.m1.1d">bold_0.0011</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.3.3.3.3.3.3"><math alttext="0.005" class="ltx_Math" display="inline" id="S4.T3.3.3.3.3.3.3.m1.1"><semantics id="S4.T3.3.3.3.3.3.3.m1.1a"><mn id="S4.T3.3.3.3.3.3.3.m1.1.1" xref="S4.T3.3.3.3.3.3.3.m1.1.1.cmml">0.005</mn><annotation-xml encoding="MathML-Content" id="S4.T3.3.3.3.3.3.3.m1.1b"><cn id="S4.T3.3.3.3.3.3.3.m1.1.1.cmml" type="float" xref="S4.T3.3.3.3.3.3.3.m1.1.1">0.005</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.3.3.3.3.3.3.m1.1c">0.005</annotation><annotation encoding="application/x-llamapun" id="S4.T3.3.3.3.3.3.3.m1.1d">0.005</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S4.T3.6.6.6.6.6"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_b ltx_border_l ltx_border_r ltx_border_t" id="S4.T3.4.4.4.4.4.1">PSNR <math alttext="\uparrow" class="ltx_Math" display="inline" id="S4.T3.4.4.4.4.4.1.m1.1"><semantics id="S4.T3.4.4.4.4.4.1.m1.1a"><mo id="S4.T3.4.4.4.4.4.1.m1.1.1" stretchy="false" xref="S4.T3.4.4.4.4.4.1.m1.1.1.cmml">↑</mo><annotation-xml encoding="MathML-Content" id="S4.T3.4.4.4.4.4.1.m1.1b"><ci id="S4.T3.4.4.4.4.4.1.m1.1.1.cmml" xref="S4.T3.4.4.4.4.4.1.m1.1.1">↑</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.4.4.4.4.4.1.m1.1c">\uparrow</annotation><annotation encoding="application/x-llamapun" id="S4.T3.4.4.4.4.4.1.m1.1d">↑</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T3.5.5.5.5.5.2"><math alttext="\boldsymbol{29.38}" class="ltx_Math" display="inline" id="S4.T3.5.5.5.5.5.2.m1.1"><semantics id="S4.T3.5.5.5.5.5.2.m1.1a"><mn class="ltx_mathvariant_bold" id="S4.T3.5.5.5.5.5.2.m1.1.1" mathvariant="bold" xref="S4.T3.5.5.5.5.5.2.m1.1.1.cmml">29.38</mn><annotation-xml encoding="MathML-Content" id="S4.T3.5.5.5.5.5.2.m1.1b"><cn id="S4.T3.5.5.5.5.5.2.m1.1.1.cmml" type="float" xref="S4.T3.5.5.5.5.5.2.m1.1.1">29.38</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.5.5.5.5.5.2.m1.1c">\boldsymbol{29.38}</annotation><annotation encoding="application/x-llamapun" id="S4.T3.5.5.5.5.5.2.m1.1d">bold_29.38</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T3.6.6.6.6.6.3"><math alttext="23.07" class="ltx_Math" display="inline" id="S4.T3.6.6.6.6.6.3.m1.1"><semantics id="S4.T3.6.6.6.6.6.3.m1.1a"><mn id="S4.T3.6.6.6.6.6.3.m1.1.1" xref="S4.T3.6.6.6.6.6.3.m1.1.1.cmml">23.07</mn><annotation-xml encoding="MathML-Content" id="S4.T3.6.6.6.6.6.3.m1.1b"><cn id="S4.T3.6.6.6.6.6.3.m1.1.1.cmml" type="float" xref="S4.T3.6.6.6.6.6.3.m1.1.1">23.07</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.6.6.6.6.6.3.m1.1c">23.07</annotation><annotation encoding="application/x-llamapun" id="S4.T3.6.6.6.6.6.3.m1.1d">23.07</annotation></semantics></math></td> </tr> </tbody> </table> <table class="ltx_tabular ltx_minipage ltx_align_top" id="S4.T3.12.12.12.6" style="width:208.1pt;"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S4.T3.12.12.12.6.7.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T3.12.12.12.6.7.1.1"><em class="ltx_emph ltx_font_italic" id="S4.T3.12.12.12.6.7.1.1.1">wiindoor</em></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S4.T3.12.12.12.6.7.1.2">Ours</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S4.T3.12.12.12.6.7.1.3">Ablation</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T3.9.9.9.3.3"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T3.7.7.7.1.1.1">MSE <math alttext="\downarrow" class="ltx_Math" display="inline" id="S4.T3.7.7.7.1.1.1.m1.1"><semantics id="S4.T3.7.7.7.1.1.1.m1.1a"><mo id="S4.T3.7.7.7.1.1.1.m1.1.1" stretchy="false" xref="S4.T3.7.7.7.1.1.1.m1.1.1.cmml">↓</mo><annotation-xml encoding="MathML-Content" id="S4.T3.7.7.7.1.1.1.m1.1b"><ci id="S4.T3.7.7.7.1.1.1.m1.1.1.cmml" xref="S4.T3.7.7.7.1.1.1.m1.1.1">↓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.7.7.7.1.1.1.m1.1c">\downarrow</annotation><annotation encoding="application/x-llamapun" id="S4.T3.7.7.7.1.1.1.m1.1d">↓</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.8.8.8.2.2.2"><math alttext="\boldsymbol{0.0017}" class="ltx_Math" display="inline" id="S4.T3.8.8.8.2.2.2.m1.1"><semantics id="S4.T3.8.8.8.2.2.2.m1.1a"><mn class="ltx_mathvariant_bold" id="S4.T3.8.8.8.2.2.2.m1.1.1" mathvariant="bold" xref="S4.T3.8.8.8.2.2.2.m1.1.1.cmml">0.0017</mn><annotation-xml encoding="MathML-Content" id="S4.T3.8.8.8.2.2.2.m1.1b"><cn id="S4.T3.8.8.8.2.2.2.m1.1.1.cmml" type="float" xref="S4.T3.8.8.8.2.2.2.m1.1.1">0.0017</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.8.8.8.2.2.2.m1.1c">\boldsymbol{0.0017}</annotation><annotation encoding="application/x-llamapun" id="S4.T3.8.8.8.2.2.2.m1.1d">bold_0.0017</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.9.9.9.3.3.3"><math alttext="0.022" class="ltx_Math" display="inline" id="S4.T3.9.9.9.3.3.3.m1.1"><semantics id="S4.T3.9.9.9.3.3.3.m1.1a"><mn id="S4.T3.9.9.9.3.3.3.m1.1.1" xref="S4.T3.9.9.9.3.3.3.m1.1.1.cmml">0.022</mn><annotation-xml encoding="MathML-Content" id="S4.T3.9.9.9.3.3.3.m1.1b"><cn id="S4.T3.9.9.9.3.3.3.m1.1.1.cmml" type="float" xref="S4.T3.9.9.9.3.3.3.m1.1.1">0.022</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.9.9.9.3.3.3.m1.1c">0.022</annotation><annotation encoding="application/x-llamapun" id="S4.T3.9.9.9.3.3.3.m1.1d">0.022</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S4.T3.12.12.12.6.6"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_b ltx_border_l ltx_border_r ltx_border_t" id="S4.T3.10.10.10.4.4.1">PSNR <math alttext="\uparrow" class="ltx_Math" display="inline" id="S4.T3.10.10.10.4.4.1.m1.1"><semantics id="S4.T3.10.10.10.4.4.1.m1.1a"><mo id="S4.T3.10.10.10.4.4.1.m1.1.1" stretchy="false" xref="S4.T3.10.10.10.4.4.1.m1.1.1.cmml">↑</mo><annotation-xml encoding="MathML-Content" id="S4.T3.10.10.10.4.4.1.m1.1b"><ci id="S4.T3.10.10.10.4.4.1.m1.1.1.cmml" xref="S4.T3.10.10.10.4.4.1.m1.1.1">↑</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.10.10.10.4.4.1.m1.1c">\uparrow</annotation><annotation encoding="application/x-llamapun" id="S4.T3.10.10.10.4.4.1.m1.1d">↑</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T3.11.11.11.5.5.2"><math alttext="\boldsymbol{27.69}" class="ltx_Math" display="inline" id="S4.T3.11.11.11.5.5.2.m1.1"><semantics id="S4.T3.11.11.11.5.5.2.m1.1a"><mn class="ltx_mathvariant_bold" id="S4.T3.11.11.11.5.5.2.m1.1.1" mathvariant="bold" xref="S4.T3.11.11.11.5.5.2.m1.1.1.cmml">27.69</mn><annotation-xml encoding="MathML-Content" id="S4.T3.11.11.11.5.5.2.m1.1b"><cn id="S4.T3.11.11.11.5.5.2.m1.1.1.cmml" type="float" xref="S4.T3.11.11.11.5.5.2.m1.1.1">27.69</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.11.11.11.5.5.2.m1.1c">\boldsymbol{27.69}</annotation><annotation encoding="application/x-llamapun" id="S4.T3.11.11.11.5.5.2.m1.1d">bold_27.69</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T3.12.12.12.6.6.3"><math alttext="16.60" class="ltx_Math" display="inline" id="S4.T3.12.12.12.6.6.3.m1.1"><semantics id="S4.T3.12.12.12.6.6.3.m1.1a"><mn id="S4.T3.12.12.12.6.6.3.m1.1.1" xref="S4.T3.12.12.12.6.6.3.m1.1.1.cmml">16.60</mn><annotation-xml encoding="MathML-Content" id="S4.T3.12.12.12.6.6.3.m1.1b"><cn id="S4.T3.12.12.12.6.6.3.m1.1.1.cmml" type="float" xref="S4.T3.12.12.12.6.6.3.m1.1.1">16.60</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.12.12.12.6.6.3.m1.1c">16.60</annotation><annotation encoding="application/x-llamapun" id="S4.T3.12.12.12.6.6.3.m1.1d">16.60</annotation></semantics></math></td> </tr> </tbody> </table> </div> </figure> </section> </section> <section class="ltx_subsection" id="S4.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S4.SS3.4.1.1">IV-C</span> </span><span class="ltx_text ltx_font_italic" id="S4.SS3.5.2" style="color:#000000;">Evaluation in Large-scale, 3D Wireless Scenes</span> </h3> <section class="ltx_subsubsection" id="S4.SS3.SSS1"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection"><span class="ltx_text" id="S4.SS3.SSS1.4.1.1">IV-C</span>1 </span>Large-scale Environment</h4> <div class="ltx_para" id="S4.SS3.SSS1.p1"> <p class="ltx_p" id="S4.SS3.SSS1.p1.1"><span class="ltx_text" id="S4.SS3.SSS1.p1.1.1">This subsection aims to validate the scalability of the proposed RayProNet in predicting EM propagation across different scene scales, ranging from small indoor rooms to expansive urban cities. By evaluating our model on three distinct scene scales, as depicted in Fig. <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:fig:_various_scales</span>, we demonstrate its versatility and robustness. The results show a high degree of consistency with ray-tracing simulation results, confirming the accuracy of our model in diverse settings.</span></p> </div> <div class="ltx_para" id="S4.SS3.SSS1.p2"> <p class="ltx_p" id="S4.SS3.SSS1.p2.1">In particular, the experiment involving a small-scale indoor room showcases our model’s capability to accurately capture complex ray trajectories. Despite the inherent complexity of the ray paths, the model effectively recognizes the intricate propagation patterns. This validation underscores our methodology’s ability to handle a wide range of scenarios, making it suitable for applications in both indoor and outdoor wireless environments.</p> </div> <div class="ltx_para" id="S4.SS3.SSS1.p3"> <p class="ltx_p" id="S4.SS3.SSS1.p3.5">We also provide a time performance evaluation comparing our model to traditional ray-tracing (Table <a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#S4.T4" title="TABLE IV ‣ IV-C1 Large-scale Environment ‣ IV-C Evaluation in Large-scale, 3D Wireless Scenes ‣ IV Numerical Experiments ‣ RayProNet: A Neural Point Field Framework for Radio Propagation Modeling in 3D Environments"><span class="ltx_text ltx_ref_tag">IV</span></a>). The validation dataset consists of <math alttext="25" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p3.1.m1.1"><semantics id="S4.SS3.SSS1.p3.1.m1.1a"><mn id="S4.SS3.SSS1.p3.1.m1.1.1" xref="S4.SS3.SSS1.p3.1.m1.1.1.cmml">25</mn><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p3.1.m1.1b"><cn id="S4.SS3.SSS1.p3.1.m1.1.1.cmml" type="integer" xref="S4.SS3.SSS1.p3.1.m1.1.1">25</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p3.1.m1.1c">25</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p3.1.m1.1d">25</annotation></semantics></math> transmitters, <math alttext="4" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p3.2.m2.1"><semantics id="S4.SS3.SSS1.p3.2.m2.1a"><mn id="S4.SS3.SSS1.p3.2.m2.1.1" xref="S4.SS3.SSS1.p3.2.m2.1.1.cmml">4</mn><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p3.2.m2.1b"><cn id="S4.SS3.SSS1.p3.2.m2.1.1.cmml" type="integer" xref="S4.SS3.SSS1.p3.2.m2.1.1">4</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p3.2.m2.1c">4</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p3.2.m2.1d">4</annotation></semantics></math> antenna patterns, and <math alttext="148\times 121" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p3.3.m3.1"><semantics id="S4.SS3.SSS1.p3.3.m3.1a"><mrow id="S4.SS3.SSS1.p3.3.m3.1.1" xref="S4.SS3.SSS1.p3.3.m3.1.1.cmml"><mn id="S4.SS3.SSS1.p3.3.m3.1.1.2" xref="S4.SS3.SSS1.p3.3.m3.1.1.2.cmml">148</mn><mo id="S4.SS3.SSS1.p3.3.m3.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.SS3.SSS1.p3.3.m3.1.1.1.cmml">×</mo><mn id="S4.SS3.SSS1.p3.3.m3.1.1.3" xref="S4.SS3.SSS1.p3.3.m3.1.1.3.cmml">121</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p3.3.m3.1b"><apply id="S4.SS3.SSS1.p3.3.m3.1.1.cmml" xref="S4.SS3.SSS1.p3.3.m3.1.1"><times id="S4.SS3.SSS1.p3.3.m3.1.1.1.cmml" xref="S4.SS3.SSS1.p3.3.m3.1.1.1"></times><cn id="S4.SS3.SSS1.p3.3.m3.1.1.2.cmml" type="integer" xref="S4.SS3.SSS1.p3.3.m3.1.1.2">148</cn><cn id="S4.SS3.SSS1.p3.3.m3.1.1.3.cmml" type="integer" xref="S4.SS3.SSS1.p3.3.m3.1.1.3">121</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p3.3.m3.1c">148\times 121</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p3.3.m3.1d">148 × 121</annotation></semantics></math> receivers (<math alttext="100\times 100" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p3.4.m4.1"><semantics id="S4.SS3.SSS1.p3.4.m4.1a"><mrow id="S4.SS3.SSS1.p3.4.m4.1.1" xref="S4.SS3.SSS1.p3.4.m4.1.1.cmml"><mn id="S4.SS3.SSS1.p3.4.m4.1.1.2" xref="S4.SS3.SSS1.p3.4.m4.1.1.2.cmml">100</mn><mo id="S4.SS3.SSS1.p3.4.m4.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.SS3.SSS1.p3.4.m4.1.1.1.cmml">×</mo><mn id="S4.SS3.SSS1.p3.4.m4.1.1.3" xref="S4.SS3.SSS1.p3.4.m4.1.1.3.cmml">100</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p3.4.m4.1b"><apply id="S4.SS3.SSS1.p3.4.m4.1.1.cmml" xref="S4.SS3.SSS1.p3.4.m4.1.1"><times id="S4.SS3.SSS1.p3.4.m4.1.1.1.cmml" xref="S4.SS3.SSS1.p3.4.m4.1.1.1"></times><cn id="S4.SS3.SSS1.p3.4.m4.1.1.2.cmml" type="integer" xref="S4.SS3.SSS1.p3.4.m4.1.1.2">100</cn><cn id="S4.SS3.SSS1.p3.4.m4.1.1.3.cmml" type="integer" xref="S4.SS3.SSS1.p3.4.m4.1.1.3">100</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p3.4.m4.1c">100\times 100</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p3.4.m4.1d">100 × 100</annotation></semantics></math> in a small-scale indoor room scene). This setup results in 100 different configurations. The results show our methodology is at least 80 times faster than traditional ray-tracing, with an average time consumption of at most <math alttext="3.2" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p3.5.m5.1"><semantics id="S4.SS3.SSS1.p3.5.m5.1a"><mn id="S4.SS3.SSS1.p3.5.m5.1.1" xref="S4.SS3.SSS1.p3.5.m5.1.1.cmml">3.2</mn><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p3.5.m5.1b"><cn id="S4.SS3.SSS1.p3.5.m5.1.1.cmml" type="float" xref="S4.SS3.SSS1.p3.5.m5.1.1">3.2</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p3.5.m5.1c">3.2</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p3.5.m5.1d">3.2</annotation></semantics></math> seconds per configuration. <span class="ltx_text" id="S4.SS3.SSS1.p3.5.1"></span></p> </div> <figure class="ltx_figure" id="S4.F26.sf1"> </figure> <figure class="ltx_table" id="S4.T4"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S4.T4.15.5.1" style="font-size:90%;">TABLE IV</span>: </span><span class="ltx_text ltx_font_bold" id="S4.T4.8.8.4" style="font-size:90%;">Runtime comparison between our model and ray-tracing:<span class="ltx_text ltx_font_medium" id="S4.T4.8.8.4.4"> In this table, we present a comparison of runtime performance between our model and ray-tracing with a validation set consisting of <math alttext="25" class="ltx_Math" display="inline" id="S4.T4.5.5.1.1.m1.1"><semantics id="S4.T4.5.5.1.1.m1.1b"><mn id="S4.T4.5.5.1.1.m1.1.1" xref="S4.T4.5.5.1.1.m1.1.1.cmml">25</mn><annotation-xml encoding="MathML-Content" id="S4.T4.5.5.1.1.m1.1c"><cn id="S4.T4.5.5.1.1.m1.1.1.cmml" type="integer" xref="S4.T4.5.5.1.1.m1.1.1">25</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.5.5.1.1.m1.1d">25</annotation><annotation encoding="application/x-llamapun" id="S4.T4.5.5.1.1.m1.1e">25</annotation></semantics></math> transmitters, <math alttext="4" class="ltx_Math" display="inline" id="S4.T4.6.6.2.2.m2.1"><semantics id="S4.T4.6.6.2.2.m2.1b"><mn id="S4.T4.6.6.2.2.m2.1.1" xref="S4.T4.6.6.2.2.m2.1.1.cmml">4</mn><annotation-xml encoding="MathML-Content" id="S4.T4.6.6.2.2.m2.1c"><cn id="S4.T4.6.6.2.2.m2.1.1.cmml" type="integer" xref="S4.T4.6.6.2.2.m2.1.1">4</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.6.6.2.2.m2.1d">4</annotation><annotation encoding="application/x-llamapun" id="S4.T4.6.6.2.2.m2.1e">4</annotation></semantics></math> antenna patterns, and <math alttext="148\times 121" class="ltx_Math" display="inline" id="S4.T4.7.7.3.3.m3.1"><semantics id="S4.T4.7.7.3.3.m3.1b"><mrow id="S4.T4.7.7.3.3.m3.1.1" xref="S4.T4.7.7.3.3.m3.1.1.cmml"><mn id="S4.T4.7.7.3.3.m3.1.1.2" xref="S4.T4.7.7.3.3.m3.1.1.2.cmml">148</mn><mo id="S4.T4.7.7.3.3.m3.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.T4.7.7.3.3.m3.1.1.1.cmml">×</mo><mn id="S4.T4.7.7.3.3.m3.1.1.3" xref="S4.T4.7.7.3.3.m3.1.1.3.cmml">121</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T4.7.7.3.3.m3.1c"><apply id="S4.T4.7.7.3.3.m3.1.1.cmml" xref="S4.T4.7.7.3.3.m3.1.1"><times id="S4.T4.7.7.3.3.m3.1.1.1.cmml" xref="S4.T4.7.7.3.3.m3.1.1.1"></times><cn id="S4.T4.7.7.3.3.m3.1.1.2.cmml" type="integer" xref="S4.T4.7.7.3.3.m3.1.1.2">148</cn><cn id="S4.T4.7.7.3.3.m3.1.1.3.cmml" type="integer" xref="S4.T4.7.7.3.3.m3.1.1.3">121</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.7.7.3.3.m3.1d">148\times 121</annotation><annotation encoding="application/x-llamapun" id="S4.T4.7.7.3.3.m3.1e">148 × 121</annotation></semantics></math> receivers (<math alttext="100\times 100" class="ltx_Math" display="inline" id="S4.T4.8.8.4.4.m4.1"><semantics id="S4.T4.8.8.4.4.m4.1b"><mrow id="S4.T4.8.8.4.4.m4.1.1" xref="S4.T4.8.8.4.4.m4.1.1.cmml"><mn id="S4.T4.8.8.4.4.m4.1.1.2" xref="S4.T4.8.8.4.4.m4.1.1.2.cmml">100</mn><mo id="S4.T4.8.8.4.4.m4.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.T4.8.8.4.4.m4.1.1.1.cmml">×</mo><mn id="S4.T4.8.8.4.4.m4.1.1.3" xref="S4.T4.8.8.4.4.m4.1.1.3.cmml">100</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T4.8.8.4.4.m4.1c"><apply id="S4.T4.8.8.4.4.m4.1.1.cmml" xref="S4.T4.8.8.4.4.m4.1.1"><times id="S4.T4.8.8.4.4.m4.1.1.1.cmml" xref="S4.T4.8.8.4.4.m4.1.1.1"></times><cn id="S4.T4.8.8.4.4.m4.1.1.2.cmml" type="integer" xref="S4.T4.8.8.4.4.m4.1.1.2">100</cn><cn id="S4.T4.8.8.4.4.m4.1.1.3.cmml" type="integer" xref="S4.T4.8.8.4.4.m4.1.1.3">100</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.8.8.4.4.m4.1d">100\times 100</annotation><annotation encoding="application/x-llamapun" id="S4.T4.8.8.4.4.m4.1e">100 × 100</annotation></semantics></math> in small-scale indoor room scene).</span></span></figcaption> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S4.T4.12.12"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S4.T4.12.12.5.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T4.12.12.5.1.1">Dataset</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S4.T4.12.12.5.1.2">urban city</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S4.T4.12.12.5.1.3">indoor room</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T4.10.10.2"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T4.10.10.2.3">Runtime (ours)</th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.9.9.1.1"><math alttext="\boldsymbol{32.86s}" class="ltx_Math" display="inline" id="S4.T4.9.9.1.1.m1.1"><semantics id="S4.T4.9.9.1.1.m1.1a"><mrow id="S4.T4.9.9.1.1.m1.1.1" xref="S4.T4.9.9.1.1.m1.1.1.cmml"><mn class="ltx_mathvariant_bold" id="S4.T4.9.9.1.1.m1.1.1.2" mathvariant="bold" xref="S4.T4.9.9.1.1.m1.1.1.2.cmml">32.86</mn><mo id="S4.T4.9.9.1.1.m1.1.1.1" xref="S4.T4.9.9.1.1.m1.1.1.1.cmml">⁢</mo><mi id="S4.T4.9.9.1.1.m1.1.1.3" xref="S4.T4.9.9.1.1.m1.1.1.3.cmml">𝒔</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.T4.9.9.1.1.m1.1b"><apply id="S4.T4.9.9.1.1.m1.1.1.cmml" xref="S4.T4.9.9.1.1.m1.1.1"><times id="S4.T4.9.9.1.1.m1.1.1.1.cmml" xref="S4.T4.9.9.1.1.m1.1.1.1"></times><cn id="S4.T4.9.9.1.1.m1.1.1.2.cmml" type="float" xref="S4.T4.9.9.1.1.m1.1.1.2">32.86</cn><ci id="S4.T4.9.9.1.1.m1.1.1.3.cmml" xref="S4.T4.9.9.1.1.m1.1.1.3">𝒔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.9.9.1.1.m1.1c">\boldsymbol{32.86s}</annotation><annotation encoding="application/x-llamapun" id="S4.T4.9.9.1.1.m1.1d">bold_32.86 bold_italic_s</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.10.10.2.2"><math alttext="\boldsymbol{13.17s}" class="ltx_Math" display="inline" id="S4.T4.10.10.2.2.m1.1"><semantics id="S4.T4.10.10.2.2.m1.1a"><mrow id="S4.T4.10.10.2.2.m1.1.1" xref="S4.T4.10.10.2.2.m1.1.1.cmml"><mn class="ltx_mathvariant_bold" id="S4.T4.10.10.2.2.m1.1.1.2" mathvariant="bold" xref="S4.T4.10.10.2.2.m1.1.1.2.cmml">13.17</mn><mo id="S4.T4.10.10.2.2.m1.1.1.1" xref="S4.T4.10.10.2.2.m1.1.1.1.cmml">⁢</mo><mi id="S4.T4.10.10.2.2.m1.1.1.3" xref="S4.T4.10.10.2.2.m1.1.1.3.cmml">𝒔</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.T4.10.10.2.2.m1.1b"><apply id="S4.T4.10.10.2.2.m1.1.1.cmml" xref="S4.T4.10.10.2.2.m1.1.1"><times id="S4.T4.10.10.2.2.m1.1.1.1.cmml" xref="S4.T4.10.10.2.2.m1.1.1.1"></times><cn id="S4.T4.10.10.2.2.m1.1.1.2.cmml" type="float" xref="S4.T4.10.10.2.2.m1.1.1.2">13.17</cn><ci id="S4.T4.10.10.2.2.m1.1.1.3.cmml" xref="S4.T4.10.10.2.2.m1.1.1.3">𝒔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.10.10.2.2.m1.1c">\boldsymbol{13.17s}</annotation><annotation encoding="application/x-llamapun" id="S4.T4.10.10.2.2.m1.1d">bold_13.17 bold_italic_s</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S4.T4.12.12.4"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_b ltx_border_l ltx_border_r ltx_border_t" id="S4.T4.12.12.4.3">Runtime (ray-tracing)</th> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T4.11.11.3.1"><math alttext="2642s" class="ltx_Math" display="inline" id="S4.T4.11.11.3.1.m1.1"><semantics id="S4.T4.11.11.3.1.m1.1a"><mrow id="S4.T4.11.11.3.1.m1.1.1" xref="S4.T4.11.11.3.1.m1.1.1.cmml"><mn id="S4.T4.11.11.3.1.m1.1.1.2" xref="S4.T4.11.11.3.1.m1.1.1.2.cmml">2642</mn><mo id="S4.T4.11.11.3.1.m1.1.1.1" xref="S4.T4.11.11.3.1.m1.1.1.1.cmml">⁢</mo><mi id="S4.T4.11.11.3.1.m1.1.1.3" xref="S4.T4.11.11.3.1.m1.1.1.3.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.T4.11.11.3.1.m1.1b"><apply id="S4.T4.11.11.3.1.m1.1.1.cmml" xref="S4.T4.11.11.3.1.m1.1.1"><times id="S4.T4.11.11.3.1.m1.1.1.1.cmml" xref="S4.T4.11.11.3.1.m1.1.1.1"></times><cn id="S4.T4.11.11.3.1.m1.1.1.2.cmml" type="integer" xref="S4.T4.11.11.3.1.m1.1.1.2">2642</cn><ci id="S4.T4.11.11.3.1.m1.1.1.3.cmml" xref="S4.T4.11.11.3.1.m1.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.11.11.3.1.m1.1c">2642s</annotation><annotation encoding="application/x-llamapun" id="S4.T4.11.11.3.1.m1.1d">2642 italic_s</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T4.12.12.4.2"><math alttext="2771s" class="ltx_Math" display="inline" id="S4.T4.12.12.4.2.m1.1"><semantics id="S4.T4.12.12.4.2.m1.1a"><mrow id="S4.T4.12.12.4.2.m1.1.1" xref="S4.T4.12.12.4.2.m1.1.1.cmml"><mn id="S4.T4.12.12.4.2.m1.1.1.2" xref="S4.T4.12.12.4.2.m1.1.1.2.cmml">2771</mn><mo id="S4.T4.12.12.4.2.m1.1.1.1" xref="S4.T4.12.12.4.2.m1.1.1.1.cmml">⁢</mo><mi id="S4.T4.12.12.4.2.m1.1.1.3" xref="S4.T4.12.12.4.2.m1.1.1.3.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.T4.12.12.4.2.m1.1b"><apply id="S4.T4.12.12.4.2.m1.1.1.cmml" xref="S4.T4.12.12.4.2.m1.1.1"><times id="S4.T4.12.12.4.2.m1.1.1.1.cmml" xref="S4.T4.12.12.4.2.m1.1.1.1"></times><cn id="S4.T4.12.12.4.2.m1.1.1.2.cmml" type="integer" xref="S4.T4.12.12.4.2.m1.1.1.2">2771</cn><ci id="S4.T4.12.12.4.2.m1.1.1.3.cmml" xref="S4.T4.12.12.4.2.m1.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.12.12.4.2.m1.1c">2771s</annotation><annotation encoding="application/x-llamapun" id="S4.T4.12.12.4.2.m1.1d">2771 italic_s</annotation></semantics></math></td> </tr> </tbody> </table> </figure> </section> <section class="ltx_subsubsection" id="S4.SS3.SSS2"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection"><span class="ltx_text" id="S4.SS3.SSS2.4.1.1">IV-C</span>2 </span>Antenna Radiation Pattern</h4> <div class="ltx_para" id="S4.SS3.SSS2.p1"> <p class="ltx_p" id="S4.SS3.SSS2.p1.1"><span class="ltx_text" id="S4.SS3.SSS2.p1.1.1">Furthermore, our RayProNet is capable of accommodating various types of trained antenna radiation patterns as input. This versatility allows the model to adapt to different antenna configurations, enhancing its applicability in wireless planning scenarios. The evaluation of these different antenna radiation patterns, as shown in Fig. <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:fig:_Evaluation_Antenna_radiation_pattern</span>, reveals a substantial agreement between our predictions and the ray tracing results. Such capability is crucial for applications requiring detailed antenna placement, highlighting the practical utility and versatility of our proposed methodology in diverse wireless environments. <span class="ltx_text" id="S4.SS3.SSS2.p1.1.1.1"></span></span></p> </div> <figure class="ltx_figure" id="S4.F30.sf1"> </figure> </section> <section class="ltx_subsubsection" id="S4.SS3.SSS3"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection"><span class="ltx_text" id="S4.SS3.SSS3.4.1.1">IV-C</span>3 </span>Quantitative Measurements</h4> <div class="ltx_para" id="S4.SS3.SSS3.p1"> <p class="ltx_p" id="S4.SS3.SSS3.p1.10"><span class="ltx_text" id="S4.SS3.SSS3.p1.10.10">So far, our results are primarily displayed in the format of 2D coverage maps for visualization. However, it is important to emphasize that our approach is essentially an end-to-end pipeline capable of predicting received signal strength at designated locations. To rigorously evaluate our model’s performance, we selected five distinct receiver locations on the map: (-167.5 <span class="ltx_text ltx_markedasmath" id="S4.SS3.SSS3.p1.10.10.1">m</span>, 22.5 <span class="ltx_text ltx_markedasmath" id="S4.SS3.SSS3.p1.10.10.2">m</span>), (-162.5 <span class="ltx_text ltx_markedasmath" id="S4.SS3.SSS3.p1.10.10.3">m</span>, 52.5 <span class="ltx_text ltx_markedasmath" id="S4.SS3.SSS3.p1.10.10.4">m</span>), (-157.5 <span class="ltx_text ltx_markedasmath" id="S4.SS3.SSS3.p1.10.10.5">m</span>, 62.5 <span class="ltx_text ltx_markedasmath" id="S4.SS3.SSS3.p1.10.10.6">m</span>), (-147.5 <span class="ltx_text ltx_markedasmath" id="S4.SS3.SSS3.p1.10.10.7">m</span>, 72.5 <span class="ltx_text ltx_markedasmath" id="S4.SS3.SSS3.p1.10.10.8">m</span>), and (-137.5 <span class="ltx_text ltx_markedasmath" id="S4.SS3.SSS3.p1.10.10.9">m</span>, 97.5 <span class="ltx_text ltx_markedasmath" id="S4.SS3.SSS3.p1.10.10.10">m</span>).</span></p> </div> <div class="ltx_para" id="S4.SS3.SSS3.p2"> <p class="ltx_p" id="S4.SS3.SSS3.p2.3">For each of these horizontal locations, we assessed the model’s predictions at three different heights: 7.5 <span class="ltx_text ltx_markedasmath" id="S4.SS3.SSS3.p2.3.1">m</span>, 10.5 <span class="ltx_text ltx_markedasmath" id="S4.SS3.SSS3.p2.3.2">m</span>, and 13.5 <span class="ltx_text ltx_markedasmath" id="S4.SS3.SSS3.p2.3.3">m</span>, resulting in a total of 15 evaluation points. This comprehensive selection allows us to test the model’s accuracy and reliability across various spatial configurations. The precise locations of these points, along with the corresponding results, are illustrated in Figure <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:fig:_quantitative_measurements</span>. This detailed analysis demonstrates our model’s robustness and flexibility in accurately predicting power propagation in 3D environments. <span class="ltx_text" id="S4.SS3.SSS3.p2.3.4"></span></p> </div> <figure class="ltx_figure" id="S4.F30.sf2"> </figure> </section> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">V </span><span class="ltx_text ltx_font_smallcaps" id="S5.1.1">Conclusion</span> </h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.1"><span class="ltx_text" id="S5.p1.1.1">To the best of our knowledge, this work represents the first effort in 3D neural wireless channel modeling capable of handling large-scale input scenes. Most prior works have focused on 2D image tasks that do not explicitly require explicit geometry representation. A recent work in <em class="ltx_emph ltx_font_italic" id="S5.p1.1.1.1">Winert</em> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.16907v1#bib.bib22" title="">22</a>]</cite> was primarily designed for small indoor scenes, as its pipeline necessitates mapping the intersection between a ray and a specific mesh triangle into a one-hot vector - an approach that is impractical for large scenes due to its excessive memory requirements.<span class="ltx_text" id="S5.p1.1.1.2"></span></span></p> </div> <div class="ltx_para" id="S5.p2"> <p class="ltx_p" id="S5.p2.1"><span class="ltx_text" id="S5.p2.1.1">Our proposed method offers a significant advancement in rapid wireless channel modeling for extensive 3D scenes, achieving speeds <math alttext="80\sim 200" class="ltx_Math" display="inline" id="S5.p2.1.1.m1.1"><semantics id="S5.p2.1.1.m1.1a"><mrow id="S5.p2.1.1.m1.1.1" xref="S5.p2.1.1.m1.1.1.cmml"><mn id="S5.p2.1.1.m1.1.1.2" xref="S5.p2.1.1.m1.1.1.2.cmml">80</mn><mo id="S5.p2.1.1.m1.1.1.1" xref="S5.p2.1.1.m1.1.1.1.cmml">∼</mo><mn id="S5.p2.1.1.m1.1.1.3" xref="S5.p2.1.1.m1.1.1.3.cmml">200</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.1.1.m1.1b"><apply id="S5.p2.1.1.m1.1.1.cmml" xref="S5.p2.1.1.m1.1.1"><csymbol cd="latexml" id="S5.p2.1.1.m1.1.1.1.cmml" xref="S5.p2.1.1.m1.1.1.1">similar-to</csymbol><cn id="S5.p2.1.1.m1.1.1.2.cmml" type="integer" xref="S5.p2.1.1.m1.1.1.2">80</cn><cn id="S5.p2.1.1.m1.1.1.3.cmml" type="integer" xref="S5.p2.1.1.m1.1.1.3">200</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.1.1.m1.1c">80\sim 200</annotation><annotation encoding="application/x-llamapun" id="S5.p2.1.1.m1.1d">80 ∼ 200</annotation></semantics></math> times faster than GPU-accelerated ray tracing methods. This efficiency is particularly beneficial in scenarios where transmitter and receiver locations frequently change, such as in wireless deployment and planning.<span class="ltx_text" id="S5.p2.1.1.1"></span></span></p> </div> <div class="ltx_para" id="S5.p3"> <p class="ltx_p" id="S5.p3.1"><span class="ltx_text" id="S5.p3.1.1">Our framework does have a notable limitation: geometry and occlusion information are embedded within the neural networks. Consequently, any changes to the scene geometry necessitate re-training the pipeline. Future research will be focused on developing a more flexible framework capable of adapting to geometry changes without the need for re-training, enhancing its applicability and efficiency.<span class="ltx_text" id="S5.p3.1.1.1"></span></span></p> </div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> <span class="ltx_tag ltx_tag_bibitem">[1]</span> <span class="ltx_bibblock"> H. L. Bertoni, <em class="ltx_emph ltx_font_italic" id="bib.bib1.1.1">Radio Propagation for Modern Wireless Systems</em>. Pearson Education, 2009. [Online]. Available: <span class="ltx_ref ltx_nolink ltx_url ltx_font_typewriter ltx_ref_self">https://books.google.com/books?id=YF-s90or91sC</span> </span> </li> <li class="ltx_bibitem" id="bib.bib2"> <span class="ltx_tag ltx_tag_bibitem">[2]</span> <span class="ltx_bibblock"> B. Mondal, T. A. Thomas, E. Visotsky, F. W. Vook, A. 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