Phase space measurement with forward modeling

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<div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p><b>Phase space measurement with forward modeling</b> is one approach to address the scattering issue in biomedical imaging. <a href="/wiki/Scattering" title="Scattering">Scattering</a> is one of the biggest problems in biomedical imaging, given that scattered light is eventually defocused, thus resulting in diffused images.<sup id="cite_ref-scattering_1-0" class="reference"><a href="#cite_note-scattering-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Instead of removing the scattered light, this approach uses the information of scattered light to reconstruct the original light signals. This approach requires the phase space data of light in imaging system and a forward model to describe scattering events in a turbid medium. Phase space of light can be obtained by using <a href="/wiki/Digital_micromirror_device" title="Digital micromirror device">digital micromirror device</a> (DMD)<sup id="cite_ref-liu_2-0" class="reference"><a href="#cite_note-liu-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> or <a href="/wiki/Light_field_microscopy" title="Light field microscopy">light field microscopy</a>.<sup id="cite_ref-pegard_3-0" class="reference"><a href="#cite_note-pegard-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Phase space measurement with forward modeling can be used in neuroscience to record neuronal activity in the brain. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Concepts">Concepts</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phase_space_measurement_with_forward_modeling&action=edit&section=1" title="Edit section: Concepts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Phase space of light is used to delineate the space and spatial frequency of light.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> As light propagates or scatters it will change its phase space as well. For example, as the position of light changes while staying in the same angle, simple propagation of light will shear the phase space of light. For scattering, since it diverges the light angle, the phase will be broadened after scattering. Therefore, scattering, and propagation of light can be modeled by the Wigner function which can generally describe light in wave optics.<sup id="cite_ref-liu_2-1" class="reference"><a href="#cite_note-liu-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> With a forward model to describe the propagation and scattering event in a scattering tissue, such as brain, a light field of a surface from point sources in a tissue can be estimated. To find the location of point sources of a target in a scattering medium, first, a light field of whole targets should be measured. Then simulated intensity plane is made by a phase space with all possible coordinates that may account for measured phase space. By applying optimization process with the <a href="/wiki/Non-negative_least_squares" title="Non-negative least squares">non-negative least squares</a> and a sparsity constraint, a sparse vector set that would correspond to the locations of targets of interest would be obtained by getting rid of non-possible options.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (May 2020)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="An_example_of_using_a_forward_model_for_scattering_events_in_a_turbid_medium">An example of using a forward model for scattering events in a turbid medium</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phase_space_measurement_with_forward_modeling&action=edit&section=2" title="Edit section: An example of using a forward model for scattering events in a turbid medium"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Wigner_quasiprobability_distribution" title="Wigner quasiprobability distribution">Wigner quasiprobability distribution</a> can be used for a forward model <sup id="cite_ref-liu_2-2" class="reference"><a href="#cite_note-liu-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W(r,u)=\iint \limits _{D}<{\tilde {f}}^{*}(u+u'/2){\tilde {f}}(u-u'/2)>e^{i2\pi u'r}d^{2}u'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </munder> <mo><</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>u</mi> <mo>+</mo> <msup> <mi>u</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo>−</mo> <msup> <mi>u</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>></mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <msup> <mi>u</mi> <mo>′</mo> </msup> <mi>r</mi> </mrow> </msup> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>u</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W(r,u)=\iint \limits _{D}<{\tilde {f}}^{*}(u+u'/2){\tilde {f}}(u-u'/2)>e^{i2\pi u'r}d^{2}u'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e643101cec7f711c4eb79f6b055df522397dcb1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:55.609ex; height:7.343ex;" alt="{\displaystyle W(r,u)=\iint \limits _{D}<{\tilde {f}}^{*}(u+u'/2){\tilde {f}}(u-u'/2)>e^{i2\pi u'r}d^{2}u'}"></span> (1) </p><p>Eventually, scattering and propagation of light can be described as <sup id="cite_ref-liu_2-3" class="reference"><a href="#cite_note-liu-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W(r,u)={\frac {-Nr^{2}}{2\pi \lambda ^{2}\sigma ^{2}(Zd-Zs)^{2}}}e^{{\frac {Nr^{2}}{2\lambda ^{2}\sigma ^{2}(Zd-Zs)^{2}}}(r-rs+\lambda (Zd-{\frac {Zd-Zs}{Nr}})u)^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>N</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>π</mi> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>Z</mi> <mi>d</mi> <mo>−</mo> <mi>Z</mi> <mi>s</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>N</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>Z</mi> <mi>d</mi> <mo>−</mo> <mi>Z</mi> <mi>s</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>−</mo> <mi>r</mi> <mi>s</mi> <mo>+</mo> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>Z</mi> <mi>d</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mi>d</mi> <mo>−</mo> <mi>Z</mi> <mi>s</mi> </mrow> <mrow> <mi>N</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> <mi>u</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W(r,u)={\frac {-Nr^{2}}{2\pi \lambda ^{2}\sigma ^{2}(Zd-Zs)^{2}}}e^{{\frac {Nr^{2}}{2\lambda ^{2}\sigma ^{2}(Zd-Zs)^{2}}}(r-rs+\lambda (Zd-{\frac {Zd-Zs}{Nr}})u)^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba8c89f2f021946b51b17261737b317d807277b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:60.898ex; height:7.176ex;" alt="{\displaystyle W(r,u)={\frac {-Nr^{2}}{2\pi \lambda ^{2}\sigma ^{2}(Zd-Zs)^{2}}}e^{{\frac {Nr^{2}}{2\lambda ^{2}\sigma ^{2}(Zd-Zs)^{2}}}(r-rs+\lambda (Zd-{\frac {Zd-Zs}{Nr}})u)^{2}}}"></span> (2) </p><p>The weight sum of decomposed contribution is <sup id="cite_ref-liu_2-4" class="reference"><a href="#cite_note-liu-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {I}}(r,u)=\sum _{rs,Zs}C(rs,Zs)W(r,u;rs,Zs)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>I</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>s</mi> <mo>,</mo> <mi>Z</mi> <mi>s</mi> </mrow> </munder> <mi>C</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mi>s</mi> <mo>,</mo> <mi>Z</mi> <mi>s</mi> <mo stretchy="false">)</mo> <mi>W</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>u</mi> <mo>;</mo> <mi>r</mi> <mi>s</mi> <mo>,</mo> <mi>Z</mi> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {I}}(r,u)=\sum _{rs,Zs}C(rs,Zs)W(r,u;rs,Zs)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f0b257bfdedcdd8c66d3a540c895007272a33c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:38.312ex; height:5.843ex;" alt="{\displaystyle {\hat {I}}(r,u)=\sum _{rs,Zs}C(rs,Zs)W(r,u;rs,Zs)}"></span> (3) <br /> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(rs,Zs)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mi>s</mi> <mo>,</mo> <mi>Z</mi> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(rs,Zs)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4285cc7925bef57a86d68ad4fe758011bb3a4ded" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.52ex; height:2.843ex;" alt="{\displaystyle C(rs,Zs)}"></span> is an coefficient that represents the intensity of light from a point source at the location <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (rs,Zs)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>r</mi> <mi>s</mi> <mo>,</mo> <mi>Z</mi> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (rs,Zs)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/751282925680776523fb45d21265d0bf66164fd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.753ex; height:2.843ex;" alt="{\displaystyle (rs,Zs)}"></span> </p><p>To obtain a sparse vector set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(rs,Zs)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mi>s</mi> <mo>,</mo> <mi>Z</mi> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(rs,Zs)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4285cc7925bef57a86d68ad4fe758011bb3a4ded" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.52ex; height:2.843ex;" alt="{\displaystyle C(rs,Zs)}"></span>, solve the <a href="/wiki/Lasso_(statistics)" title="Lasso (statistics)">lasso</a> problem <sup id="cite_ref-liu_2-5" class="reference"><a href="#cite_note-liu-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \min _{c\geqslant 0}\sum _{r,u}\left\vert I(r,u)-{\hat {I}}(r,u)\right\vert ^{2}+\mu \sum _{rs,Zs}\left\vert c(rs,Zs)\right\vert }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo>⩾</mo> <mn>0</mn> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>,</mo> <mi>u</mi> </mrow> </munder> <msup> <mrow> <mo>|</mo> <mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>I</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>μ</mi> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>s</mi> <mo>,</mo> <mi>Z</mi> <mi>s</mi> </mrow> </munder> <mrow> <mo>|</mo> <mrow> <mi>c</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mi>s</mi> <mo>,</mo> <mi>Z</mi> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \min _{c\geqslant 0}\sum _{r,u}\left\vert I(r,u)-{\hat {I}}(r,u)\right\vert ^{2}+\mu \sum _{rs,Zs}\left\vert c(rs,Zs)\right\vert }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e10b897dbe32682444ddc366a45ea82c80a4bc27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:45.328ex; height:6.343ex;" alt="{\displaystyle \min _{c\geqslant 0}\sum _{r,u}\left\vert I(r,u)-{\hat {I}}(r,u)\right\vert ^{2}+\mu \sum _{rs,Zs}\left\vert c(rs,Zs)\right\vert }"></span> (4) <br /> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(r,u)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(r,u)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bd6515d736c3f48b9894909636a8719c14ac125" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.393ex; height:2.843ex;" alt="{\displaystyle I(r,u)}"></span> is the actual measured phase space and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> is an arbitrary coefficient that favors sparsity. </p> <div class="mw-heading mw-heading2"><h2 id="Application">Application</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phase_space_measurement_with_forward_modeling&action=edit&section=3" title="Edit section: Application"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Phase space measurement with forward modeling can be used in neuroscience to record neuronal activity in the brain. Researchers have been widely using two-photon scanning microscopy to visualize neurons and their activity by imaging fluorescence emitted from calcium indicators expressed in neurons.<sup id="cite_ref-pegard_3-1" class="reference"><a href="#cite_note-pegard-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> However, <a href="/wiki/Two-photon_excitation_microscopy" title="Two-photon excitation microscopy">two-photon excitation microscopy</a> is slow, because it has to scan all the pixels one by one in the target of interest. One advantage of using Phase space measurement with forward modeling is fast, which largely depends on the speed of camera being used.<sup id="cite_ref-pegard_3-2" class="reference"><a href="#cite_note-pegard-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> A light field camera can capture an image with all the pixels in one frame at a time to speed up the frame rate of their system. This feature can facilitate voltage imaging in the brain to record action potentials.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (May 2020)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phase_space_measurement_with_forward_modeling&action=edit&section=4" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-scattering-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-scattering_1-0">^</a></b></span> <span class="reference-text">Elizabeth M. C. Hillman et al., (2007) <a rel="nofollow" class="external text" href="https://www.spiedigitallibrary.org/journals/Journal-of-Biomedical-Optics/volume-12/issue-05/051402/Optical-brain-imaging-in-vivo--techniques-and-applications-from/10.1117/1.2789693.full">"Optical brain imaging in vivo: techniques and applications from animal to man"</a></span> </li> <li id="cite_note-liu-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-liu_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-liu_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-liu_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-liu_2-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-liu_2-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-liu_2-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text">Liu et al., 2015 <a rel="nofollow" class="external text" href="https://www.osapublishing.org/oe/abstract.cfm?uri=oe-23-11-14461">"3D imaging in volumetric scattering media using phase-space measurements"</a></span> </li> <li id="cite_note-pegard-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-pegard_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-pegard_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-pegard_3-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Pegard et al., 2016 <a rel="nofollow" class="external text" href="https://www.osapublishing.org/optica/abstract.cfm?uri=optica-3-5-517">"Compressive light-field microscopy for 3D neural activity recording"</a></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><i>M.J.Bastiaans et al., 2009 </i><a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/abs/pii/0030401878900809">"The Wigner distribution function applied to optical signals and systems"</a></span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Phase_space_measurement_with_forward_modeling&oldid=1114472312">https://en.wikipedia.org/w/index.php?title=Phase_space_measurement_with_forward_modeling&oldid=1114472312</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Medical_imaging" title="Category:Medical imaging">Medical imaging</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_May_2020" title="Category:Articles with unsourced statements from May 2020">Articles with unsourced statements from May 2020</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 6 October 2022, at 17:32<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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Phase space measurement with forward modeling - Wikipedia