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<div class="section" id="spherical-polar-fourier-imaging">
<h1>Spherical Polar Fourier Imaging<a class="headerlink" href="#spherical-polar-fourier-imaging" title="Permalink to this headline">¶</a></h1>
<div class="contents local topic" id="table-of-contents">
<p class="topic-title first">Table of Contents</p>
<ul class="simple">
<li><a class="reference internal" href="#theory" id="id19">Theory</a></li>
<li><a class="reference internal" href="#synthetic-data-experiment" id="id20">Synthetic data Experiment</a><ul>
<li><a class="reference internal" href="#dwi-data-generation" id="id21">DWI data generation</a></li>
<li><a class="reference internal" href="#dl-spfi-reconstruction" id="id22">DL-SPFI reconstruction</a></li>
<li><a class="reference internal" href="#l1-spfi-reconstruction" id="id23">L1-SPFI reconstruction</a></li>
<li><a class="reference internal" href="#l2-spfi-reconstruction" id="id24">L2-SPFI reconstruction</a></li>
<li><a class="reference internal" href="#analytical-reconstruction-of-eap-and-odf" id="id25">Analytical reconstruction of EAP and ODF</a></li>
<li><a class="reference internal" href="#analytical-reconstruction-of-scalar-maps" id="id26">Analytical reconstruction of scalar maps</a></li>
</ul>
</li>
<li><a class="reference internal" href="#real-data-experiment-for-human-connectome-project" id="id27">Real data Experiment for Human Connectome Project</a><ul>
<li><a class="reference internal" href="#hcp-data" id="id28">HCP data</a></li>
<li><a class="reference internal" href="#preprocess-for-dmritool" id="id29">Preprocess for DMRITool</a></li>
<li><a class="reference internal" href="#dl-spfi-reconstruction-of-spf-coefficients" id="id30">DL-SPFI Reconstruction of SPF coefficients</a></li>
<li><a class="reference internal" href="#reconstruction-of-eaps" id="id31">Reconstruction of EAPs</a></li>
<li><a class="reference internal" href="#reconstruction-of-scalar-maps" id="id32">Reconstruction of scalar maps</a></li>
</ul>
</li>
</ul>
</div>
<div class="section" id="theory">
<h2><a class="toc-backref" href="#id19">Theory</a><a class="headerlink" href="#theory" title="Permalink to this headline">¶</a></h2>
<p>Spherical Polar Fourier Imaging (SPFI) is a reconstruction method to estimate continuous diffusion signal,
Ensemble Average Propagator (EAP), diffusion Orientation Distribution Function (dODF),
some meaningful quantities (RTO, MSD, PFA) from samples of DWI signals.</p>
<p>SPFI represents the signal using 3D SPF basis <a class="reference internal" href="#assemlal2009" id="id1">[Assemlal2009]</a>:</p>
<div class="math">
<p><img src="_images/math/00bc375063eaf68526455e644a4761f5e05e88c6.png" alt="E(q\uu|\zeta) = \sum_{n=0}^N\sum_{l=0}^L\sum_{m=-l}^l a_{nlm} G_n(q|\zeta) Y_l^m(\uu)
G_n(q|\zeta)=\left[ \frac{2n!}{\zeta ^{3/2}\Gamma(n+3/2)}\right ]^{1/2}\exp\left(-\frac{q^2}{2\zeta}\right) L_n^{1/2}(\frac{q^2}{\zeta})"/></p>
</div><p>where <img class="math" src="_images/math/ff35339b080003073c00951d178b021f1976c6df.png" alt="\q=q\uu, \uu\in \mathbb{S}^2"/>, <img class="math" src="_images/math/ed5732c0a59dba6fb368c3830d2193896fdc906d.png" alt="\zeta"/> is the scale parameter and <img class="math" src="_images/math/8e918e81df364447e8c1b2607f68a7f01eabea70.png" alt="Y_l^m(\uu)"/> is the real spherical harmonic basis.</p>
<p>The reconstruction in SPFI has two steps:</p>
<ol class="arabic simple">
<li>Estimate the coefficients of SPF basis from measurements of diffusion signals using compressed sensing, dictionary learning, etc.</li>
<li>Analytically reconstruct EAP, dODF, scalar maps from the SPF coefficients <a class="reference internal" href="#cheng2010a" id="id2">[Cheng2010a]</a> <a class="reference internal" href="#cheng2010b" id="id3">[Cheng2010b]</a>.</li>
</ol>
<p>SPFI can be seen as a generalization of Q-Ball Imaging <a class="reference internal" href="#tuch2004" id="id4">[Tuch2004]</a>. Compared to QBI which works only for single shell data, SPFI works for arbitrarily sampled data.
Compared to Diffusion spectrum imaging (DSI), SPFI requires relatively less number of samples and lower b values.
SPFI uses analytical transforms, avoiding numerical Fourier transform and numerical integral in DSI.</p>
<p>A typical sampling scheme for SPFI has 2 or 3 b-values, maximal b value more than 3000 <img class="math" src="_images/math/b4135658c5b2507746b5e254a8217089d59841ec.png" alt="s/mm^2"/>, and each shell has more than 30 samples.
For example, b values are 1500 and 3000, 40 samples per shell.
The samples in different shell are better to be staggered.
See <a class="reference internal" href="tutorial_qspacesampling.html#uniformsampling"><span class="std std-ref">the tutorial on uniform sampling scheme</span></a>.</p>
</div>
<div class="section" id="synthetic-data-experiment">
<h2><a class="toc-backref" href="#id20">Synthetic data Experiment</a><a class="headerlink" href="#synthetic-data-experiment" title="Permalink to this headline">¶</a></h2>
<div class="section" id="dwi-data-generation">
<h3><a class="toc-backref" href="#id21">DWI data generation</a><a class="headerlink" href="#dwi-data-generation" title="Permalink to this headline">¶</a></h3>
<p>Generate DWI data in 3 shells.
See <a class="reference internal" href="tutorial_dwisimulation.html#dwisimulator"><span class="std std-ref">the tutorial on DWI data simulation</span></a>.</p>
<div class="highlight-shell"><div class="highlight"><pre><span></span><span class="nb">export</span> <span class="nv">DMRITOOL_EXAMPLE_DIR</span><span class="o">=</span><span class="si">${</span><span class="nv">DMRITOOL_SOURCE_DIR</span><span class="si">}</span>/Examples
<span class="nv">b</span><span class="o">=</span><span class="m">1000</span>,2000,3000
DWISimulator <span class="si">${</span><span class="nv">DMRITOOL_EXAMPLE_DIR</span><span class="si">}</span>/dwi_circle_crossing.txt --outdwi dwi.nii.gz --outodf odfTrue.nii.gz --outeap eapTrue_r0.015.nii.gz --qorientations <span class="si">${</span><span class="nv">DMRITOOL_EXAMPLE_DIR</span><span class="si">}</span>/Elec060.txt --bvalues <span class="si">${</span><span class="nv">b</span><span class="si">}</span> --rorientations <span class="si">${</span><span class="nv">DMRITOOL_EXAMPLE_DIR</span><span class="si">}</span>/directions_t4.txt --rvalues <span class="m">0</span>.015 --noisesigma <span class="m">0</span>.0 --outb0 dwi_diagonal_b0.nii.gz --outputdwitype EACHSHELL
</pre></div>
</div>
</div>
<div class="section" id="dl-spfi-reconstruction">
<h3><a class="toc-backref" href="#id22">DL-SPFI reconstruction</a><a class="headerlink" href="#dl-spfi-reconstruction" title="Permalink to this headline">¶</a></h3>
<p id="dl-spfi">Reconstruction of SPF coefficients using DL-SPFI (with the default scale).</p>
<div class="highlight-shell"><div class="highlight"><pre><span></span>SphericalPolarFourierImaging dwi.txt --sh <span class="m">8</span> --ra <span class="m">4</span> --signal signalSPF.nii.gz --radius <span class="m">0</span>.015 --estimation L1_DL --lambdaL1 1e-7
</pre></div>
</div>
<ul class="simple">
<li>The above commend is to perform DL-SPFI using SH rank 8 and radial rank 4, regularization parameter lambda 1e-7.</li>
<li>It uses default scale <img class="math" src="_images/math/eec95afa1b5cea0380951aebf491d1de97e09083.png" alt="\zeta = 1.0 / (8 \pi^2 \tau D_0)"/>, default mean diffusivity <img class="math" src="_images/math/4a61646080ed70d14ba0aa09cbaedd256d9d22d7.png" alt="D_0=0.7\times 10^{-3} mm^2/s"/> for all voxels. You can set default mean diffusivity in <code class="docutils literal"><span class="pre">--md0</span></code></li>
<li>You can try different regularization <code class="docutils literal"><span class="pre">lamdaL1</span></code> around <img class="math" src="_images/math/5c3cd4c7e476e953d161b895a5053e30bb403db1.png" alt="1e-6"/> in range <img class="math" src="_images/math/aef6ed8bfb27d9db23ad2aa651f338ec6ad68a88.png" alt="[1e-8, 1e-5]"/></li>
<li>For real data or general synthetic data with unknown mean diffusivity,
It is better to estimate mean diffusivity and adaptively set scale for each voxel.</li>
</ul>
<p>We use a <strong>generalized high order tensor (GHOT)</strong> model to estimate the mean diffusivity <a class="reference internal" href="#chengthesis2012" id="id5">[ChengThesis2012]</a>.
Then use DL-SPFI with adaptive scale for adaptive dictionary <a class="reference internal" href="#cheng2013" id="id6">[Cheng2013]</a>.</p>
<div class="highlight-shell"><div class="highlight"><pre><span></span>MeanDiffusivityEstimator dwi.txt D_sh4_ra1.nii.gz --sh <span class="m">4</span> --ra <span class="m">1</span>
SphericalPolarFourierImaging dwi.txt --sh <span class="m">8</span> --ra <span class="m">4</span> --signal signalSPF.nii.gz --radius <span class="m">0</span>.015 --estimation L1_DL --lambdaL1 1e-7 --mdImage D_sh4_ra1.nii.gz
</pre></div>
</div>
<div class="admonition note">
<p class="first admonition-title">Note</p>
<p class="last">You <strong>cannot</strong> use different ranks from SH rank 8 and radial rank 4 for <strong>DL-SPFI</strong>, because the used dictionary was learned using rank <code class="docutils literal"><span class="pre">(8,4)</span></code>.</p>
</div>
</div>
<div class="section" id="l1-spfi-reconstruction">
<h3><a class="toc-backref" href="#id23">L1-SPFI reconstruction</a><a class="headerlink" href="#l1-spfi-reconstruction" title="Permalink to this headline">¶</a></h3>
<p>Without using learned dictionary, you can try L1-SPFI which uses least squares with L1 norm regularization based on compressed sensing <a class="reference internal" href="#cheng2011" id="id7">[Cheng2011]</a>.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">SphericalPolarFourierImaging</span> <span class="n">dwi</span><span class="o">.</span><span class="n">txt</span> <span class="o">--</span><span class="n">sh</span> <span class="mi">8</span> <span class="o">--</span><span class="n">ra</span> <span class="mi">4</span> <span class="o">--</span><span class="n">lambdaSH</span> <span class="mf">1e-9</span> <span class="o">--</span><span class="n">lambdaRA</span> <span class="mf">1e-9</span> <span class="o">--</span><span class="n">signal</span> <span class="n">signalSPF</span><span class="o">.</span><span class="n">nii</span><span class="o">.</span><span class="n">gz</span> <span class="o">--</span><span class="n">radius</span> <span class="mf">0.015</span> <span class="o">--</span><span class="n">estimation</span> <span class="n">L1_2</span> <span class="o">--</span><span class="n">mdImage</span> <span class="n">D_sh4_ra1</span><span class="o">.</span><span class="n">nii</span><span class="o">.</span><span class="n">gz</span>
</pre></div>
</div>
<ul class="simple">
<li>Without <code class="docutils literal"><span class="pre">--mdImage</span> <span class="pre">D_sh4_ra1.nii.gz</span></code>, it uses default scale for all voxels.</li>
<li>You can try different ranks, and different regularizations <code class="docutils literal"><span class="pre">lambdaSH</span></code> and <code class="docutils literal"><span class="pre">lambdaRA</span></code> around <img class="math" src="_images/math/286bf95983d478c294fad1ed6670ae4cc91f84dd.png" alt="1e-8"/>.</li>
</ul>
</div>
<div class="section" id="l2-spfi-reconstruction">
<h3><a class="toc-backref" href="#id24">L2-SPFI reconstruction</a><a class="headerlink" href="#l2-spfi-reconstruction" title="Permalink to this headline">¶</a></h3>
<p>Besides DL-SPFI and L1-SPFI, you can also try L2-SPFI which uses least squares with L2 norm regularization <a class="reference internal" href="#cheng2010a" id="id8">[Cheng2010a]</a>.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">SphericalPolarFourierImaging</span> <span class="n">dwi</span><span class="o">.</span><span class="n">txt</span> <span class="o">--</span><span class="n">sh</span> <span class="mi">6</span> <span class="o">--</span><span class="n">ra</span> <span class="mi">2</span> <span class="o">--</span><span class="n">lambdaSH</span> <span class="mf">1e-9</span> <span class="o">--</span><span class="n">lambdaRA</span> <span class="mf">1e-9</span> <span class="o">--</span><span class="n">signal</span> <span class="n">signalSPF</span><span class="o">.</span><span class="n">nii</span><span class="o">.</span><span class="n">gz</span> <span class="o">--</span><span class="n">radius</span> <span class="mf">0.015</span> <span class="o">--</span><span class="n">estimation</span> <span class="n">LS</span> <span class="o">--</span><span class="n">mdImage</span> <span class="n">D_sh4_ra1</span><span class="o">.</span><span class="n">nii</span><span class="o">.</span><span class="n">gz</span>
</pre></div>
</div>
<ul class="simple">
<li>Without <code class="docutils literal"><span class="pre">--mdImage</span> <span class="pre">D_sh4_ra1.nii.gz</span></code>, it uses the default scale for all voxels based on the default mean diffusivity.</li>
<li>You can try different ranks, and different regularizations <code class="docutils literal"><span class="pre">lambdaSH</span></code> and <code class="docutils literal"><span class="pre">lambdaRA</span></code> around <img class="math" src="_images/math/286bf95983d478c294fad1ed6670ae4cc91f84dd.png" alt="1e-8"/>.</li>
<li>You may need to use lower ranks in L2-SPFI than L1-SPFI and DL-SPFI.</li>
</ul>
<div class="admonition note">
<p class="first admonition-title">Note</p>
<p class="last">We encourage you to use DL-SPFI with adaptive scales other than L2-SPFI or L1-SPFI.</p>
</div>
</div>
<div class="section" id="analytical-reconstruction-of-eap-and-odf">
<h3><a class="toc-backref" href="#id25">Analytical reconstruction of EAP and ODF</a><a class="headerlink" href="#analytical-reconstruction-of-eap-and-odf" title="Permalink to this headline">¶</a></h3>
<p>SPFI has analytical relationship between SPF coefficients and EAP profiles, ODFs.
See <a class="reference internal" href="#cheng2010a" id="id9">[Cheng2010a]</a> and <a class="reference internal" href="#cheng2010b" id="id10">[Cheng2010b]</a>.
You can use different methods (L2-SPFI, L1-SPFI or DL-SPFI) to reconstruct SPF coefficients,
then you always can efficiently obtain the EAP profiles and ODFs from the SPF coefficients.
But please set the scale in SPF basis correctly based the mean diffusivity used in SPF coefficients reconstruction, also set the ranks correctly.</p>
<ul class="simple">
<li>Analytically obtain EAP profiles and ODFs (using default scale based on default mean diffusivity).</li>
</ul>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">SPFToProfile</span> <span class="n">signalSPF</span><span class="o">.</span><span class="n">nii</span><span class="o">.</span><span class="n">gz</span> <span class="n">eap_r0</span><span class="o">.</span><span class="mf">015.</span><span class="n">nii</span><span class="o">.</span><span class="n">gz</span> <span class="o">--</span><span class="n">sh</span> <span class="mi">8</span> <span class="o">--</span><span class="n">ra</span> <span class="mi">4</span> <span class="o">--</span><span class="n">radius</span> <span class="mf">0.015</span> <span class="o">--</span><span class="n">fourier</span>
<span class="n">SPFToODF</span> <span class="n">signalSPF</span><span class="o">.</span><span class="n">nii</span><span class="o">.</span><span class="n">gz</span> <span class="n">odf</span><span class="o">.</span><span class="n">nii</span><span class="o">.</span><span class="n">gz</span> <span class="o">--</span><span class="n">sh</span> <span class="mi">8</span> <span class="o">--</span><span class="n">ra</span> <span class="mi">4</span>
</pre></div>
</div>
<ul class="simple">
<li>Analytically obtain EAP profiles and ODFs (using adaptive scale based on adaptive mean diffusivity).</li>
</ul>
<div class="highlight-shell"><div class="highlight"><pre><span></span>SPFToProfile signalSPF.nii.gz eap_r0.015.nii.gz --sh <span class="m">8</span> --ra <span class="m">4</span> --radius <span class="m">0</span>.015 --fourier --mdImage D_sh4_ra1.nii.gz
SPFToODF signalSPF.nii.gz odf.nii.gz --sh <span class="m">8</span> --ra <span class="m">4</span> --mdImage D_sh4_ra1.nii.gz
</pre></div>
</div>
<ul class="simple">
<li>Visualization of EAP profiles and ODFs.</li>
</ul>
<div class="highlight-shell"><div class="highlight"><pre><span></span>MeshFromSHCoefficients eap_r0.015.nii.gz -o eap_r0.015_vis.vtk --tessorder <span class="m">4</span> --scale 8e-6
VTKPolyData.py --vtk eap_r0.015_vis.vtk --png synthetic_eap_r0.015.png --zoom <span class="m">1</span>.3
MeshFromSHCoefficients odf.nii.gz -o odf_vis.vtk --tessorder <span class="m">4</span> --scale <span class="m">1</span>.5
VTKPolyData.py --vtk odf_vis.vtk --png synthetic_odf.png --zoom <span class="m">1</span>.3
</pre></div>
</div>
<div class="figure align-center" id="id16">
<a class="reference internal image-reference" href="_images/synthetic_eap_r0.015.png"><img alt="synthetic_eap_r0.015.png" src="_images/synthetic_eap_r0.015.png" style="width: 480.0px; height: 480.0px;" /></a>
<p class="caption"><span class="caption-text"><strong>eap_r0.015</strong></span></p>
</div>
<div class="figure align-center" id="id17">
<a class="reference internal image-reference" href="_images/synthetic_odf.png"><img alt="synthetic_odf.png" src="_images/synthetic_odf.png" style="width: 480.0px; height: 480.0px;" /></a>
<p class="caption"><span class="caption-text"><strong>odf</strong></span></p>
</div>
</div>
<div class="section" id="analytical-reconstruction-of-scalar-maps">
<h3><a class="toc-backref" href="#id26">Analytical reconstruction of scalar maps</a><a class="headerlink" href="#analytical-reconstruction-of-scalar-maps" title="Permalink to this headline">¶</a></h3>
<p>We can analytically generate some scalar maps using the estimated SPF coefficients.
See <a class="reference internal" href="#wu2007" id="id11">[Wu2007]</a>, <a class="reference internal" href="#cheng2010a" id="id12">[Cheng2010a]</a>, <a class="reference internal" href="#chengthesis2012" id="id13">[ChengThesis2012]</a>.</p>
<ul class="simple">
<li><strong>Return-To-Origin probability (RTO)</strong> is the EAP with <img class="math" src="_images/math/25f2430ba1c87922798f129f9cdeee9a2ebc04eb.png" alt="r=0"/>, i.e. <img class="math" src="_images/math/d2451d1ddc162f8887e2108f2ee862d749eb4e58.png" alt="P(0)"/>.</li>
</ul>
<div class="highlight-shell"><div class="highlight"><pre><span></span>SPFToScalarMap signalSPF.nii.gz rto.nii.gz --mapType RTO --sh <span class="m">8</span> --ra <span class="m">4</span> --mdImage D_sh4_ra1.nii.gz
</pre></div>
</div>
<ul class="simple">
<li><strong>Mean Squared Displacement (MSD)</strong> is the variance of the probability, i.e. <img class="math" src="_images/math/b1d7828d3748ea3fe8f03f9fe113bf57a0433ac1.png" alt="\int_{\mathbb{R}^3} P(R) R^TR d R"/>.</li>
</ul>
<div class="highlight-shell"><div class="highlight"><pre><span></span>SPFToScalarMap signalSPF.nii.gz msd.nii.gz --mapType MSD --sh <span class="m">8</span> --ra <span class="m">4</span> --mdImage D_sh4_ra1.nii.gz
</pre></div>
</div>
<ul class="simple">
<li><strong>Propagator Fractional Anisotropy (PFA)</strong> is a generalization of FA for non-Gaussian EAP.
It is defined as the normalized L2 distance between the EAP to its nearest EAP. See <a class="reference internal" href="#chengthesis2012" id="id14">[ChengThesis2012]</a>.</li>
</ul>
<div class="highlight-shell"><div class="highlight"><pre><span></span>SPFToScalarMap signalSPF.nii.gz pfa.nii.gz --mapType PFA --sh <span class="m">8</span> --ra <span class="m">4</span> --mdImage D_sh4_ra1.nii.gz
</pre></div>
</div>
<ul class="simple">
<li>From the above estimated EAP profiles and ODFs, you can also generate <strong>generalized FA (GFA)</strong>. See <a class="reference internal" href="#tuch2004" id="id15">[Tuch2004]</a>.</li>
</ul>
<div class="highlight-shell"><div class="highlight"><pre><span></span>SHCoefficientsToGFA eap_r0.015.nii.gz eap_r0.015_gfa.nii.gz
SHCoefficientsToGFA odf.nii.gz odf_gfa.nii.gz
</pre></div>
</div>
<ul class="simple">
<li>We can visualize eap profile using scalar maps as background.
<code class="docutils literal"><span class="pre">--image-range</span></code> is to control the contrast in visualization.
If it is not given, it maps the minimal value to black and the maximal value to white.</li>
</ul>
<div class="highlight-shell"><div class="highlight"><pre><span></span>VTKPolyData.py --vtk eap_r0.015_vis.vtk --image eap_r0.015_gfa.nii.gz --image-range <span class="m">0</span>,1 --png synthetic_eap_r0.015_withgfa.png --zoom <span class="m">1</span>.3
</pre></div>
</div>
<div class="figure align-center" id="id18">
<a class="reference internal image-reference" href="_images/synthetic_eap_r0.015_withgfa.png"><img alt="synthetic_eap_r0.015_withgfa.png" src="_images/synthetic_eap_r0.015_withgfa.png" style="width: 480.0px; height: 480.0px;" /></a>
<p class="caption"><span class="caption-text"><strong>eap_r0.015</strong></span></p>
</div>
</div>
</div>
<div class="section" id="real-data-experiment-for-human-connectome-project">
<h2><a class="toc-backref" href="#id27">Real data Experiment for Human Connectome Project</a><a class="headerlink" href="#real-data-experiment-for-human-connectome-project" title="Permalink to this headline">¶</a></h2>
<div class="section" id="hcp-data">
<h3><a class="toc-backref" href="#id28">HCP data</a><a class="headerlink" href="#hcp-data" title="Permalink to this headline">¶</a></h3>
<p>We downloaded a preprocessed subject data (ID: 100307) from <a class="reference external" href="http://www.humanconnectome.org/documentation/Q3/">HCP data</a>.
The data has three shells with b values of 1000, 2000, 3000 <img class="math" src="_images/math/b4135658c5b2507746b5e254a8217089d59841ec.png" alt="s/mm^2"/>, 90 samples per shell.
It also contains 18 volumes with <code class="docutils literal"><span class="pre">b=0</span></code>, thus totally it has 288 volumes.
Considering the data size is big <code class="docutils literal"><span class="pre">(145,</span> <span class="pre">174,</span> <span class="pre">145,</span> <span class="pre">288)</span></code>, we cropped only one slice for this tutorial.
You can download it from <a class="reference external" href="https://www.dropbox.com/s/ct8hsbn8wfp1diw/HCP_100307_c88.zip?dl=0">this link</a>.</p>
</div>
<div class="section" id="preprocess-for-dmritool">
<h3><a class="toc-backref" href="#id29">Preprocess for DMRITool</a><a class="headerlink" href="#preprocess-for-dmritool" title="Permalink to this headline">¶</a></h3>
<p>The data has already preprocessed.
Here we just use <a class="reference internal" href="commands/DWIPreprocess.html"><span class="doc">DWIPreprocess</span></a> to normalize the DWI data using the baseline image.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">echo</span> <span class="n">b_raw</span><span class="o">.</span><span class="n">txt</span> <span class="n">grad_raw</span><span class="o">.</span><span class="n">txt</span> <span class="n">dwi_c88</span><span class="o">.</span><span class="n">nii</span><span class="o">.</span><span class="n">gz</span> <span class="o">></span> <span class="n">data_c88_raw</span><span class="o">.</span><span class="n">txt</span>
<span class="n">DWIPreprocess</span> <span class="n">data_c88_raw</span><span class="o">.</span><span class="n">txt</span> <span class="n">data_c88_normalize</span><span class="o">.</span><span class="n">txt</span> <span class="o">--</span><span class="n">oEachShell</span> <span class="o">--</span><span class="n">bThreshold</span> <span class="mi">15</span> <span class="o">--</span><span class="n">mask</span> <span class="n">mask_c88</span><span class="o">.</span><span class="n">nii</span><span class="o">.</span><span class="n">gz</span> <span class="o">--</span><span class="n">odwi</span> <span class="n">dwi_c88_normalize</span><span class="o">.</span><span class="n">nii</span><span class="o">.</span><span class="n">gz</span> <span class="o">--</span><span class="n">ograd</span> <span class="n">grad_normalize</span><span class="o">.</span><span class="n">txt</span> <span class="o">--</span><span class="n">ob0Image</span> <span class="n">dwi_c88_b0</span><span class="o">.</span><span class="n">nii</span><span class="o">.</span><span class="n">gz</span>
</pre></div>
</div>
<ul class="simple">
<li>The b values in the HCP data all larger than 0. There is no exact <code class="docutils literal"><span class="pre">b=0</span></code>. Thus we consider b values smaller than the <code class="docutils literal"><span class="pre">bThreshold</span></code> as <code class="docutils literal"><span class="pre">b=0</span></code>.</li>
<li>The b values are not exactly equal 1000, 2000, or 3000. <a class="reference internal" href="commands/DWIPreprocess.html"><span class="doc">DWIPreprocess</span></a> can group b values and replace b values using the mean of the group.</li>
<li>It outputs <code class="docutils literal"><span class="pre">data_c88_normalize.txt</span></code> which contains the b values, gradients and the normalized DWI data.</li>
</ul>
</div>
<div class="section" id="dl-spfi-reconstruction-of-spf-coefficients">
<h3><a class="toc-backref" href="#id30">DL-SPFI Reconstruction of SPF coefficients</a><a class="headerlink" href="#dl-spfi-reconstruction-of-spf-coefficients" title="Permalink to this headline">¶</a></h3>
<p>Estimate the mean diffusivity first.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">MeanDiffusivityEstimator</span> <span class="n">data_c88_normalize</span><span class="o">.</span><span class="n">txt</span> <span class="n">D_sh4_ra1</span><span class="o">.</span><span class="n">nii</span><span class="o">.</span><span class="n">gz</span> <span class="o">--</span><span class="n">sh</span> <span class="mi">4</span> <span class="o">--</span><span class="n">ra</span> <span class="mi">1</span>
</pre></div>
</div>
<p>Perform DL-SPFI to estimate the SPF coefficients.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">SphericalPolarFourierImaging</span> <span class="n">data_c88_normalize</span><span class="o">.</span><span class="n">txt</span> <span class="o">--</span><span class="n">estimation</span> <span class="n">L1_DL</span> <span class="o">--</span><span class="n">sh</span> <span class="mi">8</span> <span class="o">--</span><span class="n">ra</span> <span class="mi">4</span> <span class="o">--</span><span class="n">lambdaL1</span> <span class="mf">1e-6</span> <span class="o">--</span><span class="n">signal</span> <span class="n">signalSPF</span><span class="o">.</span><span class="n">nii</span><span class="o">.</span><span class="n">gz</span> <span class="o">--</span><span class="n">solver</span> <span class="n">SPAMS</span> <span class="o">--</span><span class="n">mask</span> <span class="n">mask_c88</span><span class="o">.</span><span class="n">nii</span><span class="o">.</span><span class="n">gz</span> <span class="o">--</span><span class="n">mdImage</span> <span class="n">D_sh4_ra1</span><span class="o">.</span><span class="n">nii</span><span class="o">.</span><span class="n">gz</span>
</pre></div>
</div>
</div>
<div class="section" id="reconstruction-of-eaps">
<h3><a class="toc-backref" href="#id31">Reconstruction of EAPs</a><a class="headerlink" href="#reconstruction-of-eaps" title="Permalink to this headline">¶</a></h3>
<p>Obtain EAP profiles from the SPF coefficients, and obtain GFA of EAP profiles.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">SPFToProfile</span> <span class="n">signalSPF</span><span class="o">.</span><span class="n">nii</span><span class="o">.</span><span class="n">gz</span> <span class="n">eap_r0</span><span class="o">.</span><span class="mf">015.</span><span class="n">nii</span><span class="o">.</span><span class="n">gz</span> <span class="o">--</span><span class="n">radius</span> <span class="mf">0.015</span> <span class="o">--</span><span class="n">ra</span> <span class="mi">4</span> <span class="o">--</span><span class="n">sh</span> <span class="mi">8</span> <span class="o">--</span><span class="n">mdImage</span> <span class="n">D_sh4_ra1</span><span class="o">.</span><span class="n">nii</span><span class="o">.</span><span class="n">gz</span> <span class="o">--</span><span class="n">fourier</span>
<span class="n">SHCoefficientsToGFA</span> <span class="n">eap_r0</span><span class="o">.</span><span class="mf">015.</span><span class="n">nii</span><span class="o">.</span><span class="n">gz</span> <span class="n">eap_r0</span><span class="o">.</span><span class="mi">015</span><span class="n">_gfa</span><span class="o">.</span><span class="n">nii</span><span class="o">.</span><span class="n">gz</span>
</pre></div>
</div>
<p>Generate a coarse mesh (<code class="docutils literal"><span class="pre">--tessorder</span> <span class="pre">3</span></code>) for visualization the EAP profiles in a whole slice.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">MeshFromSHCoefficients</span> <span class="n">eap_r0</span><span class="o">.</span><span class="mf">015.</span><span class="n">nii</span><span class="o">.</span><span class="n">gz</span> <span class="o">-</span><span class="n">o</span> <span class="n">eap_r0</span><span class="o">.</span><span class="mi">015</span><span class="n">_visall</span><span class="o">.</span><span class="n">vtk</span> <span class="o">--</span><span class="n">tessorder</span> <span class="mi">3</span> <span class="o">--</span><span class="n">scale</span> <span class="mf">1e-5</span>
</pre></div>
</div>
<p>Generate a fine mesh (<code class="docutils literal"><span class="pre">--tessorder</span> <span class="pre">4</span></code>) for visualization the EAP profiles in a ROI.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">MeshFromSHCoefficients</span> <span class="n">eap_r0</span><span class="o">.</span><span class="mf">015.</span><span class="n">nii</span><span class="o">.</span><span class="n">gz</span> <span class="o">-</span><span class="n">o</span> <span class="n">eap_r0</span><span class="o">.</span><span class="mi">015</span><span class="n">_vis</span><span class="o">.</span><span class="n">vtk</span> <span class="o">--</span><span class="n">tessorder</span> <span class="mi">4</span> <span class="o">--</span><span class="n">scale</span> <span class="mf">1e-5</span> <span class="o">--</span><span class="n">box</span> <span class="mi">80</span><span class="p">,</span><span class="mi">100</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">70</span><span class="p">,</span><span class="mi">90</span>
</pre></div>
</div>
<p>You can use <a class="reference internal" href="commands/VTKPolyData.py.html"><span class="doc">VTKPolyData.py</span></a> or <a class="reference external" href="http://www.paraview.org">paraview</a> to visualize the vtk files.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">VTKPolyData</span><span class="o">.</span><span class="n">py</span> <span class="o">--</span><span class="n">vtk</span> <span class="n">eap_r0</span><span class="o">.</span><span class="mi">015</span><span class="n">_visall</span><span class="o">.</span><span class="n">vtk</span>
<span class="n">VTKPolyData</span><span class="o">.</span><span class="n">py</span> <span class="o">--</span><span class="n">vtk</span> <span class="n">eap_r0</span><span class="o">.</span><span class="mi">015</span><span class="n">_vis</span><span class="o">.</span><span class="n">vtk</span>
</pre></div>
</div>
<p>You can also put scalar map like GFA map in the background.
See the following results obtained by <a class="reference external" href="http://www.paraview.org">paraview</a>.</p>
<table border="1" class="docutils">
<colgroup>
<col width="51%" />
<col width="49%" />
</colgroup>
<tbody valign="top">
<tr class="row-odd"><td><a class="reference internal" href="_images/HCP_DL-SPFI_1e-6_eap_r0.015_whole.png"><img alt="HCP_eap_r0.015_whole" src="_images/HCP_DL-SPFI_1e-6_eap_r0.015_whole.png" style="width: 518.4px; height: 433.8px;" /></a></td>
<td><a class="reference internal" href="_images/HCP_DL-SPFI_1e-6_eap_r0.015_ROI.png"><img alt="HCP_eap_r0.015_ROI" src="_images/HCP_DL-SPFI_1e-6_eap_r0.015_ROI.png" style="width: 400.4px; height: 405.3px;" /></a></td>
</tr>
<tr class="row-even"><td>eap_r0.015_visall.vtk</td>
<td>eap_r0.015_vis.vtk</td>
</tr>
</tbody>
</table>
</div>
<div class="section" id="reconstruction-of-scalar-maps">
<h3><a class="toc-backref" href="#id32">Reconstruction of scalar maps</a><a class="headerlink" href="#reconstruction-of-scalar-maps" title="Permalink to this headline">¶</a></h3>
<p>Analytical reconstruction of RTO, MSD and PFA maps.</p>
<div class="highlight-shell"><div class="highlight"><pre><span></span>SPFToScalarMap signalSPF.nii.gz rto.nii.gz --mapType RTO --sh <span class="m">8</span> --ra <span class="m">4</span> --mdImage D_sh4_ra1.nii.gz
SPFToScalarMap signalSPF.nii.gz msd.nii.gz --mapType MSD --sh <span class="m">8</span> --ra <span class="m">4</span> --mdImage D_sh4_ra1.nii.gz
SPFToScalarMap signalSPF.nii.gz pfa.nii.gz --mapType PFA --sh <span class="m">8</span> --ra <span class="m">4</span> --mdImage D_sh4_ra1.nii.gz
</pre></div>
</div>
<p>You can use many tools to visualize the 3D volume, e.g. <a class="reference internal" href="commands/VTKPolyData.py.html"><span class="doc">VTKPolyData.py</span></a>.
Here are pictures for the scalar maps I got by <code class="docutils literal"><span class="pre">fslview</span></code> in <a class="reference external" href="http://fsl.fmrib.ox.ac.uk/fsl/fslwiki">FSL</a>.</p>
<table border="1" class="docutils">
<colgroup>
<col width="33%" />
<col width="33%" />
<col width="33%" />
</colgroup>
<tbody valign="top">
<tr class="row-odd"><td><a class="reference internal" href="_images/HCP_DL-SPFI_1e-6_RTO.png"><img alt="HCP_RTO" src="_images/HCP_DL-SPFI_1e-6_RTO.png" style="width: 282.1px; height: 240.8px;" /></a></td>
<td><a class="reference internal" href="_images/HCP_DL-SPFI_1e-6_MSD.png"><img alt="HCP_MSD" src="_images/HCP_DL-SPFI_1e-6_MSD.png" style="width: 282.1px; height: 240.8px;" /></a></td>
<td><a class="reference internal" href="_images/HCP_DL-SPFI_1e-6_PFA.png"><img alt="HCP_PFA" src="_images/HCP_DL-SPFI_1e-6_PFA.png" style="width: 282.1px; height: 240.8px;" /></a></td>
</tr>
<tr class="row-even"><td>RTO map</td>
<td>MSD map</td>
<td>PFA map</td>
</tr>
</tbody>
</table>
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<tr><td class="label"><a class="fn-backref" href="#id1">[Assemlal2009]</a></td><td>Haz-Edine Assemlal, David Tschumperle, Luc Brun,
<a class="reference external" href="http://dx.doi.org/10.1016/j.media.2009.06.004">Efficient and robust computation of PDF features from diffusion MR signal</a>, Medical Image Analysis, vol 13, p. 715-729, 2009</td></tr>
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<tr><td class="label">[Tuch2004]</td><td><em>(<a class="fn-backref" href="#id4">1</a>, <a class="fn-backref" href="#id15">2</a>)</em> David S. Tuch, <a class="reference external" href="http://onlinelibrary.wiley.com/doi/10.1002/mrm.20279/full">Q-Ball Imaging</a>, Magnetic Resonance in Medicine, vol 52, p. 1358-1372, 2004.</td></tr>
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<tr><td class="label"><a class="fn-backref" href="#id11">[Wu2007]</a></td><td>Yu-Chien Wu, Andrew L. Alexanderb, <a class="reference external" href="http://www.ncbi.nlm.nih.gov/pubmed/17481920">Hybrid diffusion imaging</a>, NeuroImage, vol 36, p. 617-629, 2007.</td></tr>
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<tr><td class="label">[Cheng2010a]</td><td><em>(<a class="fn-backref" href="#id2">1</a>, <a class="fn-backref" href="#id8">2</a>, <a class="fn-backref" href="#id9">3</a>, <a class="fn-backref" href="#id12">4</a>)</em> Jian Cheng, Aurobrata Ghosh, Rachid Deriche, Tianzi Jiang,
<a class="reference external" href="https://hal.inria.fr/inria-00496932/document">Model-free and Analytical EAP Reconstruction via Spherical Polar Fourier Diffusion MRI</a>,
13th International Conference on Medical Image Computing and Computer Assisted Intervention (MICCAI‘10), Beijing, September 20-24, 2010</td></tr>
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<table class="docutils citation" frame="void" id="cheng2010b" rules="none">
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<tr><td class="label">[Cheng2010b]</td><td><em>(<a class="fn-backref" href="#id3">1</a>, <a class="fn-backref" href="#id10">2</a>)</em> Jian Cheng, Aurobrata Ghosh, Rachid Deriche, Tianzi Jiang,
<a class="reference external" href="https://hal.archives-ouvertes.fr/docs/00/49/69/29/PDF/ODF_MICCAI2010.pdf">Model-free, regularized, fast, and robust analytical orientation distribution function estimation</a>,
13th International Conference on Medical Image Computing and Computer Assisted Intervention (MICCAI‘10), Beijing, September 20-24, 2010</td></tr>
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<tr><td class="label"><a class="fn-backref" href="#id7">[Cheng2011]</a></td><td>Jian Cheng, Sylvain Merlet, Emmanuel Caruyer, Aurobrata Ghosh, Rachid Deriche, Tianzi Jiang,
<a class="reference external" href="http://hal.cirad.fr/inria-00615434/document">Compressive Sensing Ensemble Average Propagator Estimation via L1 Spherical Polar Fourier Imaging</a>,
MICCAI Workshop on Computational Diffusion MRI (CDMRI‘11), Toronto, Canada, September, 2011</td></tr>
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<tr><td class="label">[ChengThesis2012]</td><td><em>(<a class="fn-backref" href="#id5">1</a>, <a class="fn-backref" href="#id13">2</a>, <a class="fn-backref" href="#id14">3</a>)</em> Jian Cheng, Estimation and Processing of Ensemble Average Propagator and Its Features in Diffusion MRI, Universite Nice Sophia Antipolis, 2012</td></tr>
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<table class="docutils citation" frame="void" id="cheng2013" rules="none">
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<tr><td class="label"><a class="fn-backref" href="#id6">[Cheng2013]</a></td><td>Jian Cheng, Tianzi Jiang, Rachid Deriche, Dinggang Shen, Pew-Thian Yap,
<a class="reference external" href="https://hal.archives-ouvertes.fr/file/index/docid/824507/filename/DL_SPFI_MICCAI.pdf">Regularized Spherical Polar Fourier Diffusion MRI with Optimal Dictionary Learning</a>,
16th International Conference on Medical Image Computing and Computer Assisted Intervention (MICCAI‘13), Nagoya, September 22-26, 2013</td></tr>
</tbody>
</table>
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