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<div class="section" id="an-1d-example">
<h1>An 1D example<a class="headerlink" href="#an-1d-example" title="Permalink to this headline">ΒΆ</a></h1>
<p><strong>Import pynufft module</strong></p>
<p>In python environment, import pynufft module:</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>
</pre></div>
</div>
<p>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><strong>Planning</strong></p>
<p>The <span class="math">\(M\)</span> locations of non-uniform samples (<span class="math">\(om\)</span>) must be provided:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</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">1512</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="c1"># om is an M x 1 ndarray: locations of M points. *om* is normalized between [-pi, pi]</span>
<span class="c1"># Here M = 1512</span>
</pre></div>
</div>
<p>Also, 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="n">Kd</span> <span class="o">=</span> <span class="p">(</span><span class="mi">512</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>
</pre></div>
</div>
<p>Now provide 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">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="n">time_data</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="mi">256</span><span class="p">,</span> <span class="p">)</span>
<span class="n">time_data</span><span class="p">[</span><span class="mi">96</span><span class="p">:</span><span class="mi">128</span><span class="o">+</span><span class="mi">32</span><span class="p">]</span> <span class="o">=</span> <span class="mf">1.0</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">time_data</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mi">1</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">show</span><span class="p">()</span>
</pre></div>
</div>
<p>This generates a time series <a class="reference internal" href="#box-function"><span class="std std-numref">Fig. 2</span></a>.</p>
<div class="figure" id="id1">
<span id="box-function"></span><a class="reference internal image-reference" href="../_images/box_function.png"><img alt="../_images/box_function.png" src="../_images/box_function.png" style="width: 75%;" /></a>
<p class="caption"><span class="caption-number">Fig. 2 </span><span class="caption-text">A box function time series</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">nufft_freq_data</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">time_data</span><span class="p">)</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="n">nufft_freq_data</span><span class="o">.</span><span class="n">real</span><span class="p">,</span><span class="s1">'.'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">'real'</span><span class="p">)</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="n">nufft_freq_data</span><span class="o">.</span><span class="n">imag</span><span class="p">,</span><span class="s1">'r.'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">'imag'</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">legend</span><span class="p">()</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 non-Cartesian spectrum <a class="reference internal" href="#non-cartesian-spectrum"><span class="std std-numref">Fig. 3</span></a>.</p>
<div class="figure" id="id2">
<span id="non-cartesian-spectrum"></span><a class="reference internal image-reference" href="../_images/non_Cartesian_spectrum.png"><img alt="../_images/non_Cartesian_spectrum.png" src="../_images/non_Cartesian_spectrum.png" style="width: 75%;" /></a>
<p class="caption"><span class="caption-number">Fig. 3 </span><span class="caption-text">Non-Cartesian spectrum of box function in <a class="reference internal" href="#box-function"><span class="std std-numref">Fig. 2</span></a>. Note the non-uniform density.</span></p>
</div>
<p><strong>Signal restoration through "solve()"</strong></p>
<p>The signal can be solved by the solve() method</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">restore_time</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">nufft_freq_data</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">30</span><span class="p">)</span>
<span class="n">restore_time1</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">nufft_freq_data</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">30</span><span class="p">,</span><span class="n">rho</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">restore_time2</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">nufft_freq_data</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">30</span><span class="p">,</span><span class="n">rho</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
</pre></div>
</div>
<p>Now display the restored signals:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">im1</span><span class="p">,</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">numpy</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">time_data</span><span class="p">),</span><span class="s1">'r'</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">im2</span><span class="p">,</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">numpy</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">restore_time1</span><span class="p">),</span><span class="s1">'b:'</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">im3</span><span class="p">,</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">numpy</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">restore_time2</span><span class="p">),</span><span class="s1">'k--'</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">im4</span><span class="p">,</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">numpy</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">restore_time</span><span class="p">),</span><span class="s1">'r:'</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">'conjugate_gradient_method'</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="n">im1</span><span class="p">,</span> <span class="n">im2</span><span class="p">,</span> <span class="n">im3</span><span class="p">,</span><span class="n">im4</span><span class="p">])</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="solve"></span><a class="reference internal image-reference" href="../_images/script_1D_solve.png"><img alt="../_images/script_1D_solve.png" src="../_images/script_1D_solve.png" style="width: 75%;" /></a>
<p class="caption"><span class="caption-number">Fig. 4 </span><span class="caption-text">Signals restored by "solve()". L1TVOLS and L1TVOLS are more close to <a class="reference internal" href="#box-function"><span class="std std-numref">Fig. 2</span></a> as cg suffers from distortions.</span></p>
</div>
<p>The complete code is:</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">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">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="p">,</span> <span class="n">NUFFT_hsa</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">1512</span><span class="p">,</span><span class="mi">1</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="c1"># time grid, tuple</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="c1"># frequency grid, tuple</span>
<span class="n">Jd</span> <span class="o">=</span> <span class="p">(</span><span class="mi">7</span><span class="p">,)</span> <span class="c1"># interpolator </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">time_data</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="mi">256</span><span class="p">,</span> <span class="p">)</span>
<span class="n">time_data</span><span class="p">[</span><span class="mi">64</span><span class="p">:</span><span class="mi">192</span><span class="p">]</span> <span class="o">=</span> <span class="mf">1.0</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">time_data</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mi">1</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">show</span><span class="p">()</span>
<span class="n">nufft_freq_data</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">time_data</span><span class="p">)</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="n">nufft_freq_data</span><span class="o">.</span><span class="n">real</span><span class="p">,</span><span class="s1">'.'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">'real'</span><span class="p">)</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="n">nufft_freq_data</span><span class="o">.</span><span class="n">imag</span><span class="p">,</span><span class="s1">'r.'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">'imag'</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">legend</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">restore_time</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">nufft_freq_data</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">30</span><span class="p">)</span>
<span class="n">restore_time1</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">nufft_freq_data</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">30</span><span class="p">,</span><span class="n">rho</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">restore_time2</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">nufft_freq_data</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">30</span><span class="p">,</span><span class="n">rho</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">im1</span><span class="p">,</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">numpy</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">time_data</span><span class="p">),</span><span class="s1">'r'</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">im2</span><span class="p">,</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">numpy</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">restore_time1</span><span class="p">),</span><span class="s1">'b:'</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">im3</span><span class="p">,</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">numpy</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">restore_time2</span><span class="p">),</span><span class="s1">'k--'</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">im4</span><span class="p">,</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">numpy</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">restore_time</span><span class="p">),</span><span class="s1">'r:'</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">'conjugate_gradient_method'</span><span class="p">)</span>
<span class="n">pyplot</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="n">im1</span><span class="p">,</span> <span class="n">im2</span><span class="p">,</span> <span class="n">im3</span><span class="p">,</span><span class="n">im4</span><span class="p">])</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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