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<div class="section" id="multi-dimensional-nufft">
<h1>Multi-dimensional NUFFT<a class="headerlink" href="#multi-dimensional-nufft" title="Permalink to this headline">ΒΆ</a></h1>
<p>Multi-dimensional transforms are supported in PyNUFFT.</p>
<p><strong>Import pynufft module</strong></p>
<p>In python environment, import pynufft module and other packages:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span>
<span class="kn">import</span> <span class="nn">scipy.misc</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span>
<span class="kn">from</span> <span class="nn">pynufft.pynufft</span> <span class="k">import</span> <span class="n">NUFFT_cpu</span>
</pre></div>
</div>
<p><strong>Planning</strong>
Create a pynufft object NufftObj:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">pynufft.pynufft</span> <span class="k">import</span> <span class="n">NUFFT_cpu</span><span class="p">,</span> <span class="n">NUFFT_hsa</span>
<span class="n">NufftObj</span> <span class="o">=</span> <span class="n">NUFFT_cpu</span><span class="p">()</span>
</pre></div>
</div>
<p>Provided <span class="math">\(om\)</span>, the size of time series (<span class="math">\(Nd\)</span>), oversampled grid (<span class="math">\(Kd\)</span>), and interpolatro size (<span class="math">\(Jd\)</span>) are:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">Nd</span> <span class="o">=</span> <span class="p">(</span><span class="mi">64</span><span class="p">,</span><span class="mi">64</span><span class="p">,</span><span class="mi">64</span><span class="p">)</span> <span class="c1"># time grid, tuple</span>
<span class="n">Kd</span> <span class="o">=</span> <span class="p">(</span><span class="mi">64</span><span class="p">,</span><span class="mi">64</span><span class="p">,</span><span class="mi">64</span><span class="p">)</span> <span class="c1"># frequency grid, tuple</span>
<span class="n">Jd</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span> <span class="c1"># interpolator</span>
<span class="c1"># om= numpy.load(DATA_PATH+'om3D.npz')['arr_0']</span>
<span class="n">om</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">15120</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">om</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
</pre></div>
</div>
<p>Now we can plan NufftObj with these parameters:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">NufftObj</span><span class="o">.</span><span class="n">plan</span><span class="p">(</span><span class="n">om</span><span class="p">,</span> <span class="n">Nd</span><span class="p">,</span> <span class="n">Kd</span><span class="p">,</span> <span class="n">Jd</span><span class="p">)</span>
</pre></div>
</div>
<p><strong>Forward transform</strong></p>
<p>Now NufftObj has been prepared and is ready for computations. Let continue with an example.:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">pkg_resources</span>
<span class="n">DATA_PATH</span> <span class="o">=</span> <span class="n">pkg_resources</span><span class="o">.</span><span class="n">resource_filename</span><span class="p">(</span><span class="s1">'pynufft'</span><span class="p">,</span> <span class="s1">'./src/data/'</span><span class="p">)</span>
<span class="n">image</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="n">DATA_PATH</span> <span class="o">+</span><span class="s1">'phantom_3D_128_128_128.npz'</span><span class="p">)[</span><span class="s1">'arr_0'</span><span class="p">][</span><span class="mi">0</span><span class="p">::</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">::</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">::</span><span class="mi">2</span><span class="p">]</span>
<span class="nb">print</span><span class="p">(</span><span class="n">special_license</span><span class="p">)</span>
</pre></div>
</div>
<p>This display the image <a class="reference internal" href="#d-example-image"><span class="std std-numref">Fig. 9</span></a>.</p>
<div class="figure" id="id1">
<span id="d-example-image"></span><a class="reference internal image-reference" href="../_images/3D_phantom.png"><img alt="../_images/3D_phantom.png" src="../_images/3D_phantom.png" style="width: 75%;" /></a>
<p class="caption"><span class="caption-number">Fig. 9 </span><span class="caption-text">The 3D phantom</span></p>
</div>
<p>NufftObj transform the time_data to non-Cartesian locations:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">y</span> <span class="o">=</span> <span class="n">NufftObj</span><span class="o">.</span><span class="n">forward</span><span class="p">(</span><span class="n">image</span><span class="p">)</span>
</pre></div>
</div>
<p><strong>Image restoration with solve()</strong>:</p>
<p>The image can be restored from non-Cartesian samples y:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">restore_image</span> <span class="o">=</span> <span class="n">NufftObj</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">kspace</span><span class="p">,</span><span class="s1">'cg'</span><span class="p">,</span> <span class="n">maxiter</span><span class="o">=</span><span class="mi">500</span><span class="p">)</span>
<span class="n">restore_image1</span> <span class="o">=</span> <span class="n">NufftObj</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">kspace</span><span class="p">,</span><span class="s1">'L1TVLAD'</span><span class="p">,</span> <span class="n">maxiter</span><span class="o">=</span><span class="mi">500</span><span class="p">,</span><span class="n">rho</span><span class="o">=</span><span class="mf">0.1</span><span class="p">)</span>
<span class="c1">#</span>
<span class="n">restore_image2</span> <span class="o">=</span> <span class="n">NufftObj</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">kspace</span><span class="p">,</span><span class="s1">'L1TVOLS'</span><span class="p">,</span> <span class="n">maxiter</span><span class="o">=</span><span class="mi">500</span><span class="p">,</span><span class="n">rho</span><span class="o">=</span><span class="mf">0.1</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">image</span><span class="p">[:,:,</span><span class="mi">32</span><span class="p">]),</span> <span class="n">label</span><span class="o">=</span><span class="s1">'original signal'</span><span class="p">,</span><span class="n">cmap</span><span class="o">=</span><span class="n">gray</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'original'</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">restore_image1</span><span class="p">[:,:,</span><span class="mi">32</span><span class="p">]),</span> <span class="n">label</span><span class="o">=</span><span class="s1">'L1TVLAD'</span><span class="p">,</span><span class="n">cmap</span><span class="o">=</span><span class="n">gray</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'L1TVLAD'</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">restore_image2</span><span class="p">[:,:,</span><span class="mi">32</span><span class="p">]),</span> <span class="n">label</span><span class="o">=</span><span class="s1">'L1TVOLS'</span><span class="p">,</span><span class="n">cmap</span><span class="o">=</span><span class="n">gray</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'L1TVOLS'</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">4</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">restore_image</span><span class="p">[:,:,</span><span class="mi">32</span><span class="p">]),</span> <span class="n">label</span><span class="o">=</span><span class="s1">'CG'</span><span class="p">,</span><span class="n">cmap</span><span class="o">=</span><span class="n">gray</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'CG'</span><span class="p">)</span>
</pre></div>
</div>
<div class="figure" id="id2">
<span id="d-restore"></span><a class="reference internal image-reference" href="../_images/3D_restore.png"><img alt="../_images/3D_restore.png" src="../_images/3D_restore.png" style="width: 100%;" /></a>
<p class="caption"><span class="caption-number">Fig. 10 </span><span class="caption-text">Image restoration using'cg', 'L1TVOLS', 'L1TVLAD'.</span></p>
</div>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">special_license</span><span class="o">=</span><span class="s1">'''</span>
<span class="s1">The license of the 3D Shepp-Logan phantom:</span>
<span class="s1">Copyright (c) 2006, Matthias Schabel </span>
<span class="s1">All rights reserved.</span>
<span class="s1">Redistribution and use in source and binary forms, with or without </span>
<span class="s1">modification, are permitted provided that the following conditions are </span>
<span class="s1">met:</span>
<span class="s1">* Redistributions of source code must retain the above copyright </span>
<span class="s1">notice, this list of conditions and the following disclaimer. </span>
<span class="s1">* Redistributions in binary form must reproduce the above copyright </span>
<span class="s1">notice, this list of conditions and the following disclaimer in </span>
<span class="s1">the documentation and/or other materials provided with the distribution </span>
<span class="s1">* Neither the name of the University of Utah Department of Radiology nor the names </span>
<span class="s1">of its contributors may be used to endorse or promote products derived </span>
<span class="s1">from this software without specific prior written permission.</span>
<span class="s1">THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" </span>
<span class="s1">AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE </span>
<span class="s1">IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE </span>
<span class="s1">ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE </span>
<span class="s1">LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR </span>
<span class="s1">CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF </span>
<span class="s1">SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS </span>
<span class="s1">INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN </span>
<span class="s1">CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) </span>
<span class="s1">ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE </span>
<span class="s1">POSSIBILITY OF SUCH DAMAGE.</span>
<span class="s1">'''</span>
<span class="kn">import</span> <span class="nn">numpy</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">pyplot</span>
<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="k">import</span> <span class="n">cm</span>
<span class="n">gray</span> <span class="o">=</span> <span class="n">cm</span><span class="o">.</span><span class="n">gray</span>
<span class="kn">import</span> <span class="nn">pkg_resources</span>
<span class="n">DATA_PATH</span> <span class="o">=</span> <span class="n">pkg_resources</span><span class="o">.</span><span class="n">resource_filename</span><span class="p">(</span><span class="s1">'pynufft'</span><span class="p">,</span> <span class="s1">'./src/data/'</span><span class="p">)</span>
<span class="n">image</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="n">DATA_PATH</span> <span class="o">+</span><span class="s1">'phantom_3D_128_128_128.npz'</span><span class="p">)[</span><span class="s1">'arr_0'</span><span class="p">][</span><span class="mi">0</span><span class="p">::</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">::</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">::</span><span class="mi">2</span><span class="p">]</span>
<span class="nb">print</span><span class="p">(</span><span class="n">special_license</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">image</span><span class="p">[:,:,</span><span class="mi">32</span><span class="p">]),</span> <span class="n">label</span><span class="o">=</span><span class="s1">'original signal'</span><span class="p">,</span><span class="n">cmap</span><span class="o">=</span><span class="n">gray</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="n">Nd</span> <span class="o">=</span> <span class="p">(</span><span class="mi">64</span><span class="p">,</span><span class="mi">64</span><span class="p">,</span><span class="mi">64</span><span class="p">)</span> <span class="c1"># time grid, tuple</span>
<span class="n">Kd</span> <span class="o">=</span> <span class="p">(</span><span class="mi">64</span><span class="p">,</span><span class="mi">64</span><span class="p">,</span><span class="mi">64</span><span class="p">)</span> <span class="c1"># frequency grid, tuple</span>
<span class="n">Jd</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span> <span class="c1"># interpolator </span>
<span class="c1"># om= numpy.load(DATA_PATH+'om3D.npz')['arr_0']</span>
<span class="n">om</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">15120</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">om</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="kn">from</span> <span class="nn">pynufft.pynufft</span> <span class="k">import</span> <span class="n">NUFFT_cpu</span><span class="p">,</span> <span class="n">NUFFT_hsa</span>
<span class="n">NufftObj</span> <span class="o">=</span> <span class="n">NUFFT_cpu</span><span class="p">()</span>
<span class="n">NufftObj</span><span class="o">.</span><span class="n">plan</span><span class="p">(</span><span class="n">om</span><span class="p">,</span> <span class="n">Nd</span><span class="p">,</span> <span class="n">Kd</span><span class="p">,</span> <span class="n">Jd</span><span class="p">)</span>
<span class="n">kspace</span> <span class="o">=</span><span class="n">NufftObj</span><span class="o">.</span><span class="n">forward</span><span class="p">(</span><span class="n">image</span><span class="p">)</span>
<span class="n">restore_image</span> <span class="o">=</span> <span class="n">NufftObj</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">kspace</span><span class="p">,</span><span class="s1">'cg'</span><span class="p">,</span> <span class="n">maxiter</span><span class="o">=</span><span class="mi">500</span><span class="p">)</span>
<span class="n">restore_image1</span> <span class="o">=</span> <span class="n">NufftObj</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">kspace</span><span class="p">,</span><span class="s1">'L1TVLAD'</span><span class="p">,</span> <span class="n">maxiter</span><span class="o">=</span><span class="mi">500</span><span class="p">,</span><span class="n">rho</span><span class="o">=</span><span class="mf">0.1</span><span class="p">)</span>
<span class="c1"># </span>
<span class="n">restore_image2</span> <span class="o">=</span> <span class="n">NufftObj</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">kspace</span><span class="p">,</span><span class="s1">'L1TVOLS'</span><span class="p">,</span> <span class="n">maxiter</span><span class="o">=</span><span class="mi">500</span><span class="p">,</span><span class="n">rho</span><span class="o">=</span><span class="mf">0.1</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">image</span><span class="p">[:,:,</span><span class="mi">32</span><span class="p">]),</span> <span class="n">label</span><span class="o">=</span><span class="s1">'original signal'</span><span class="p">,</span><span class="n">cmap</span><span class="o">=</span><span class="n">gray</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'original'</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">restore_image1</span><span class="p">[:,:,</span><span class="mi">32</span><span class="p">]),</span> <span class="n">label</span><span class="o">=</span><span class="s1">'L1TVLAD'</span><span class="p">,</span><span class="n">cmap</span><span class="o">=</span><span class="n">gray</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'L1TVLAD'</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">restore_image2</span><span class="p">[:,:,</span><span class="mi">32</span><span class="p">]),</span> <span class="n">label</span><span class="o">=</span><span class="s1">'L1TVOLS'</span><span class="p">,</span><span class="n">cmap</span><span class="o">=</span><span class="n">gray</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'L1TVOLS'</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">4</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">restore_image</span><span class="p">[:,:,</span><span class="mi">32</span><span class="p">]),</span> <span class="n">label</span><span class="o">=</span><span class="s1">'CG'</span><span class="p">,</span><span class="n">cmap</span><span class="o">=</span><span class="n">gray</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'CG'</span><span class="p">)</span>
<span class="c1"># pyplot.legend([im1, im im4])</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
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