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<div class="section" id="a-2d-example">
<h1>A 2D example<a class="headerlink" href="#a-2d-example" title="Permalink to this headline">ΒΆ</a></h1>
<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>Loading the X-Y locations("om")</strong></p>
<p>It requires the x-y coordinates of <span class="math">\(M\)</span> points to plan NufftObj.</p>
<p>A 2D trajectory from my PROPELLER MRI research is provided in pynufft package.:</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">om</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">'om2D.npz'</span><span class="p">)[</span><span class="s1">'arr_0'</span><span class="p">]</span>
</pre></div>
</div>
<p>The <span class="math">\(M\)</span> locations of non-uniform samples (<span class="math">\(om\)</span>) is an M x 2 numpy.ndarray</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">om</span><span class="p">)</span>
<span class="p">[[</span><span class="o">-</span><span class="mf">3.12932086</span> <span class="mf">0.28225246</span><span class="p">]</span>
<span class="p">[</span><span class="o">-</span><span class="mf">3.1047771</span> <span class="mf">0.28225246</span><span class="p">]</span>
<span class="p">[</span><span class="o">-</span><span class="mf">3.08023357</span> <span class="mf">0.28225246</span><span class="p">]</span>
<span class="o">....</span>
<span class="p">[</span><span class="o">-</span><span class="mf">2.99815702</span> <span class="mf">0.76063216</span><span class="p">]</span>
<span class="p">[</span><span class="o">-</span><span class="mf">3.02239823</span> <span class="mf">0.76447165</span><span class="p">]</span>
<span class="p">[</span><span class="o">-</span><span class="mf">3.04663992</span> <span class="mf">0.76831114</span><span class="p">]]</span>
</pre></div>
</div>
<p>You can see the 2D <span class="math">\(M\)</span> locations by plotting <span class="math">\(x\)</span> versus <span class="math">\(y\)</span>:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">om</span><span class="p">[::</span><span class="mi">10</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span><span class="n">om</span><span class="p">[::</span><span class="mi">10</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span><span class="s1">'o'</span><span class="p">)</span>
<span class="n">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'non-uniform coordinates'</span><span class="p">)</span>
<span class="n">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">'axis 0'</span><span class="p">)</span>
<span class="n">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">'axis 1'</span><span class="p">)</span>
<span class="n">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
<p>As can be seen in <a class="reference internal" href="#propeller-trajectory"><span class="std std-numref">Fig. 5</span></a>:</p>
<div class="figure" id="id1">
<span id="propeller-trajectory"></span><a class="reference internal image-reference" href="../_images/propeller_trajectory.png"><img alt="../_images/propeller_trajectory.png" src="../_images/propeller_trajectory.png" style="width: 50%;" /></a>
<p class="caption"><span class="caption-number">Fig. 5 </span><span class="caption-text">The 2D PROPELLER trajectory of M points.</span></p>
</div>
<p><strong>Planning</strong>
Create a pynufft object NufftObj:</p>
<div class="highlight-default"><div class="highlight"><pre><span></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">256</span><span class="p">,</span> <span class="mi">256</span><span class="p">)</span> <span class="c1"># image size</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'setting image dimension Nd...'</span><span class="p">,</span> <span class="n">Nd</span><span class="p">)</span>
<span class="n">Kd</span> <span class="o">=</span> <span class="p">(</span><span class="mi">512</span><span class="p">,</span> <span class="mi">512</span><span class="p">)</span> <span class="c1"># k-space size</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'setting spectrum dimension Kd...'</span><span class="p">,</span> <span class="n">Kd</span><span class="p">)</span>
<span class="n">Jd</span> <span class="o">=</span> <span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span> <span class="c1"># interpolation size</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'setting interpolation size Jd...'</span><span class="p">,</span> <span class="n">Jd</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="n">image</span> <span class="o">=</span> <span class="n">scipy</span><span class="o">.</span><span class="n">misc</span><span class="o">.</span><span class="n">ascent</span><span class="p">()</span>
<span class="n">image</span> <span class="o">=</span> <span class="n">scipy</span><span class="o">.</span><span class="n">misc</span><span class="o">.</span><span class="n">imresize</span><span class="p">(</span><span class="n">image</span><span class="p">,</span> <span class="p">(</span><span class="mi">256</span><span class="p">,</span><span class="mi">256</span><span class="p">))</span>
<span class="n">image</span><span class="o">=</span><span class="n">image</span><span class="o">/</span><span class="n">numpy</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">image</span><span class="p">[</span><span class="o">...</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'loading image...'</span><span class="p">)</span>
<span class="n">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">image</span><span class="o">.</span><span class="n">real</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">matplotlib</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">gray</span><span class="p">)</span>
<span class="n">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">show</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. 6</span></a>.</p>
<div class="figure" id="id2">
<span id="d-example-image"></span><a class="reference internal image-reference" href="../_images/2d_example_image.png"><img alt="../_images/2d_example_image.png" src="../_images/2d_example_image.png" style="width: 75%;" /></a>
<p class="caption"><span class="caption-number">Fig. 6 </span><span class="caption-text">The 2D image from scipy.misc.ascent()</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">image0</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">y</span><span class="p">,</span> <span class="n">solver</span><span class="o">=</span><span class="s1">'cg'</span><span class="p">,</span><span class="n">maxiter</span><span class="o">=</span><span class="mi">50</span><span class="p">)</span>
<span class="n">image3</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">y</span><span class="p">,</span> <span class="n">solver</span><span class="o">=</span><span class="s1">'L1TVOLS'</span><span class="p">,</span><span class="n">maxiter</span><span class="o">=</span><span class="mi">50</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">image4</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">y</span><span class="p">,</span> <span class="n">solver</span><span class="o">=</span><span class="s1">'L1TVLAD'</span><span class="p">,</span><span class="n">maxiter</span><span class="o">=</span><span class="mi">50</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">image2</span> <span class="o">=</span> <span class="n">NufftObj</span><span class="o">.</span><span class="n">adjoint</span><span class="p">(</span><span class="n">y</span> <span class="p">)</span> <span class="c1"># adjoint</span>
<span class="n">matplotlib</span><span class="o">.</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">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'Restored image (cg)'</span><span class="p">)</span>
<span class="n">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">image0</span><span class="o">.</span><span class="n">real</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">matplotlib</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">gray</span><span class="p">,</span> <span class="n">norm</span><span class="o">=</span><span class="n">matplotlib</span><span class="o">.</span><span class="n">colors</span><span class="o">.</span><span class="n">Normalize</span><span class="p">(</span><span class="n">vmin</span><span class="o">=</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="mi">1</span><span class="p">))</span>
<span class="n">matplotlib</span><span class="o">.</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">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">image2</span><span class="o">.</span><span class="n">real</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">matplotlib</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">gray</span><span class="p">,</span> <span class="n">norm</span><span class="o">=</span><span class="n">matplotlib</span><span class="o">.</span><span class="n">colors</span><span class="o">.</span><span class="n">Normalize</span><span class="p">(</span><span class="n">vmin</span><span class="o">=</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="mi">5</span><span class="p">))</span>
<span class="n">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'Adjoint transform'</span><span class="p">)</span>
<span class="n">matplotlib</span><span class="o">.</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">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'L1TV OLS'</span><span class="p">)</span>
<span class="n">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">image3</span><span class="o">.</span><span class="n">real</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">matplotlib</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">gray</span><span class="p">,</span> <span class="n">norm</span><span class="o">=</span><span class="n">matplotlib</span><span class="o">.</span><span class="n">colors</span><span class="o">.</span><span class="n">Normalize</span><span class="p">(</span><span class="n">vmin</span><span class="o">=</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="mi">1</span><span class="p">))</span>
<span class="n">matplotlib</span><span class="o">.</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">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'L1TV LAD'</span><span class="p">)</span>
<span class="n">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">image4</span><span class="o">.</span><span class="n">real</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">matplotlib</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">gray</span><span class="p">,</span> <span class="n">norm</span><span class="o">=</span><span class="n">matplotlib</span><span class="o">.</span><span class="n">colors</span><span class="o">.</span><span class="n">Normalize</span><span class="p">(</span><span class="n">vmin</span><span class="o">=</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="mi">1</span><span class="p">))</span>
<span class="n">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
<div class="figure" id="id3">
<span id="d-restore"></span><a class="reference internal image-reference" href="../_images/2D_restoration.png"><img alt="../_images/2D_restoration.png" src="../_images/2D_restoration.png" style="width: 100%;" /></a>
<p class="caption"><span class="caption-number">Fig. 7 </span><span class="caption-text">Image restoration through solve() 'cg', 'L1TVOLS', 'L1TVLAD' and adjoint().</span></p>
</div>
<p>The spectrum of the restored image:</p>
<div class="figure" id="id4">
<span id="d-spectrum"></span><a class="reference internal image-reference" href="../_images/2D_spectrum.png"><img alt="../_images/2D_spectrum.png" src="../_images/2D_spectrum.png" style="width: 75%;" /></a>
<p class="caption"><span class="caption-number">Fig. 8 </span><span class="caption-text">The spectrum of the restored image solved by cg.</span></p>
</div>
<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>
<span class="c1"># load k-space points</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">om</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">'om2D.npz'</span><span class="p">)[</span><span class="s1">'arr_0'</span><span class="p">]</span>
<span class="nb">print</span><span class="p">(</span><span class="n">om</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'setting non-uniform coordinates...'</span><span class="p">)</span>
<span class="n">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">om</span><span class="p">[::</span><span class="mi">10</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span><span class="n">om</span><span class="p">[::</span><span class="mi">10</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span><span class="s1">'o'</span><span class="p">)</span>
<span class="n">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'non-uniform coordinates'</span><span class="p">)</span>
<span class="n">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">'axis 0'</span><span class="p">)</span>
<span class="n">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">'axis 1'</span><span class="p">)</span>
<span class="n">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="n">NufftObj</span> <span class="o">=</span> <span class="n">NUFFT_cpu</span><span class="p">()</span>
<span class="n">Nd</span> <span class="o">=</span> <span class="p">(</span><span class="mi">256</span><span class="p">,</span> <span class="mi">256</span><span class="p">)</span> <span class="c1"># image size</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'setting image dimension Nd...'</span><span class="p">,</span> <span class="n">Nd</span><span class="p">)</span>
<span class="n">Kd</span> <span class="o">=</span> <span class="p">(</span><span class="mi">512</span><span class="p">,</span> <span class="mi">512</span><span class="p">)</span> <span class="c1"># k-space size</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'setting spectrum dimension Kd...'</span><span class="p">,</span> <span class="n">Kd</span><span class="p">)</span>
<span class="n">Jd</span> <span class="o">=</span> <span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span> <span class="c1"># interpolation size</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'setting interpolation size Jd...'</span><span class="p">,</span> <span class="n">Jd</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">image</span> <span class="o">=</span> <span class="n">scipy</span><span class="o">.</span><span class="n">misc</span><span class="o">.</span><span class="n">ascent</span><span class="p">()</span>
<span class="n">image</span> <span class="o">=</span> <span class="n">scipy</span><span class="o">.</span><span class="n">misc</span><span class="o">.</span><span class="n">imresize</span><span class="p">(</span><span class="n">image</span><span class="p">,</span> <span class="p">(</span><span class="mi">256</span><span class="p">,</span><span class="mi">256</span><span class="p">))</span>
<span class="n">image</span><span class="o">=</span><span class="n">image</span><span class="o">/</span><span class="n">numpy</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">image</span><span class="p">[</span><span class="o">...</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'loading image...'</span><span class="p">)</span>
<span class="n">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">image</span><span class="o">.</span><span class="n">real</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">matplotlib</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">gray</span><span class="p">)</span>
<span class="n">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">show</span><span class="p">()</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>
<span class="nb">print</span><span class="p">(</span><span class="s1">'setting non-uniform data'</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'y is an (M,) list'</span><span class="p">,</span><span class="nb">type</span><span class="p">(</span><span class="n">y</span><span class="p">),</span> <span class="n">y</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="n">matplotlib</span><span class="o">.</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">image0</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">y</span><span class="p">,</span> <span class="n">solver</span><span class="o">=</span><span class="s1">'cg'</span><span class="p">,</span><span class="n">maxiter</span><span class="o">=</span><span class="mi">50</span><span class="p">)</span>
<span class="n">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'Restored image (cg)'</span><span class="p">)</span>
<span class="n">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">image0</span><span class="o">.</span><span class="n">real</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">matplotlib</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">gray</span><span class="p">,</span> <span class="n">norm</span><span class="o">=</span><span class="n">matplotlib</span><span class="o">.</span><span class="n">colors</span><span class="o">.</span><span class="n">Normalize</span><span class="p">(</span><span class="n">vmin</span><span class="o">=</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="mi">1</span><span class="p">))</span>
<span class="n">matplotlib</span><span class="o">.</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">image2</span> <span class="o">=</span> <span class="n">NufftObj</span><span class="o">.</span><span class="n">adjoint</span><span class="p">(</span><span class="n">y</span> <span class="p">)</span>
<span class="n">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">image2</span><span class="o">.</span><span class="n">real</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">matplotlib</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">gray</span><span class="p">,</span> <span class="n">norm</span><span class="o">=</span><span class="n">matplotlib</span><span class="o">.</span><span class="n">colors</span><span class="o">.</span><span class="n">Normalize</span><span class="p">(</span><span class="n">vmin</span><span class="o">=</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="mi">5</span><span class="p">))</span>
<span class="n">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'Adjoint transform'</span><span class="p">)</span>
<span class="n">matplotlib</span><span class="o">.</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">image3</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">y</span><span class="p">,</span> <span class="n">solver</span><span class="o">=</span><span class="s1">'L1TVOLS'</span><span class="p">,</span><span class="n">maxiter</span><span class="o">=</span><span class="mi">50</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">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'L1TV OLS'</span><span class="p">)</span>
<span class="n">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">image3</span><span class="o">.</span><span class="n">real</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">matplotlib</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">gray</span><span class="p">,</span> <span class="n">norm</span><span class="o">=</span><span class="n">matplotlib</span><span class="o">.</span><span class="n">colors</span><span class="o">.</span><span class="n">Normalize</span><span class="p">(</span><span class="n">vmin</span><span class="o">=</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="mi">1</span><span class="p">))</span>
<span class="n">matplotlib</span><span class="o">.</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">image4</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">y</span><span class="p">,</span> <span class="n">solver</span><span class="o">=</span><span class="s1">'L1TVLAD'</span><span class="p">,</span><span class="n">maxiter</span><span class="o">=</span><span class="mi">50</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">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'L1TV LAD'</span><span class="p">)</span>
<span class="n">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">image4</span><span class="o">.</span><span class="n">real</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">matplotlib</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">gray</span><span class="p">,</span> <span class="n">norm</span><span class="o">=</span><span class="n">matplotlib</span><span class="o">.</span><span class="n">colors</span><span class="o">.</span><span class="n">Normalize</span><span class="p">(</span><span class="n">vmin</span><span class="o">=</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="mi">1</span><span class="p">))</span>
<span class="n">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="n">shifted_kspectrum</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">fftshift</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">fftn</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">fftshift</span><span class="p">(</span><span class="n">image0</span><span class="p">)))</span>
<span class="c1"># print('getting the k-space spectrum, shape =',shifted_kspectrum.shape)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'Showing the shifted k-space spectrum'</span><span class="p">)</span>
<span class="n">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span> <span class="n">shifted_kspectrum</span><span class="o">.</span><span class="n">real</span><span class="p">,</span> <span class="n">cmap</span> <span class="o">=</span> <span class="n">matplotlib</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">gray</span><span class="p">,</span> <span class="n">norm</span><span class="o">=</span><span class="n">matplotlib</span><span class="o">.</span><span class="n">colors</span><span class="o">.</span><span class="n">Normalize</span><span class="p">(</span><span class="n">vmin</span><span class="o">=-</span><span class="mi">100</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="mi">100</span><span class="p">))</span>
<span class="n">matplotlib</span><span class="o">.</span><span class="n">pyplot</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'shifted k-space spectrum'</span><span class="p">)</span>
<span class="n">matplotlib</span><span class="o">.</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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