Test_GF_with_correct_gradient.html
Test_GF_with_correct_gradient.html
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<!DOCTYPE html> <html> <head><meta charset="utf-8" /> <title>Test_GF_with_correct_gradient</title> <script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script> <style type="text/css"> /*! * * Twitter Bootstrap * *//*! normalize.css v3.0.2 | MIT License | git.io/normalize */html{font-family:sans-serif;-ms-text-size-adjust:100%;-webkit-text-size-adjust:100%;font-size:10px;-webkit-tap-highlight-color:transparent}article,aside,details,figcaption,figure,footer,header,hgroup,main,menu,nav,section,summary{display:block}audio,canvas,progress,video{display:inline-block;vertical-align:baseline}audio:not([controls]){display:none;height:0}[hidden],template{display:none}a{background-color:transparent}a:active,a:hover{outline:0}abbr[title]{border-bottom:1px dotted}b,optgroup,strong{font-weight:700}dfn{font-style:italic}h1{font-size:2em;margin:.67em 0}mark{background:#ff0;color:#000}small{font-size:80%}sub,sup{font-size:75%;line-height:0;position:relative;vertical-align:baseline}sup{top:-.5em}sub{bottom:-.25em}img{border:0;vertical-align:middle}svg:not(:root){overflow:hidden}hr{-moz-box-sizing:content-box;box-sizing:content-box;height:0}pre,textarea{overflow:auto}code,kbd,pre,samp{font-size:1em}button,input,optgroup,select,textarea{color:inherit;font:inherit;margin:0}button{overflow:visible}button,select{text-transform:none}button,html input[type=button],input[type=reset],input[type=submit]{-webkit-appearance:button;cursor:pointer}button[disabled],html input[disabled]{cursor:default}button::-moz-focus-inner,input::-moz-focus-inner{border:0;padding:0}input{line-height:normal}input[type=checkbox],input[type=radio]{box-sizing:border-box;padding:0}input[type=number]::-webkit-inner-spin-button,input[type=number]::-webkit-outer-spin-button{height:auto}input[type=search]::-webkit-search-cancel-button,input[type=search]::-webkit-search-decoration{-webkit-appearance:none}table{border-collapse:collapse;border-spacing:0}td,th{padding:0}/*! 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avoid; } div.output { display: block; page-break-inside: avoid; } } </style> <!-- Custom stylesheet, it must be in the same directory as the html file --> <link rel="stylesheet" href="custom.css"> <!-- Loading mathjax macro --> <!-- Load mathjax --> <script src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS_HTML"></script> <!-- MathJax configuration --> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ tex2jax: { inlineMath: [ ['$','$'], ["\\(","\\)"] ], displayMath: [ ['$$','$$'], ["\\[","\\]"] ], processEscapes: true, processEnvironments: true }, // Center justify equations in code and markdown cells. Elsewhere // we use CSS to left justify single line equations in code cells. displayAlign: 'center', "HTML-CSS": { styles: {'.MathJax_Display': {"margin": 0}}, linebreaks: { automatic: true } } }); </script> <!-- End of mathjax configuration --></head> <body> <div tabindex="-1" id="notebook" class="border-box-sizing"> <div class="container" id="notebook-container"> <div class="cell border-box-sizing text_cell rendered"> <div class="prompt input_prompt"> </div> <div class="inner_cell"> <div class="text_cell_render border-box-sizing rendered_html"> <h1 id="Gradient-inversion-with-the-mirror-flipped-gradient-operator">Gradient inversion with the mirror-flipped gradient operator<a class="anchor-link" href="#Gradient-inversion-with-the-mirror-flipped-gradient-operator">¶</a></h1> </div> </div> </div> <div class="cell border-box-sizing code_cell rendered"> <div class="input"> <div class="prompt input_prompt">In [1]:</div> <div class="inner_cell"> <div class="input_area"> <div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span> <span class="kn">import</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">p</span> <span class="kn">import</span> <span class="nn">math</span> <span class="o">%</span><span class="k">matplotlib</span> inline <span class="kn">import</span> <span class="nn">scipy.ndimage.filters</span> <span class="k">as</span> <span class="nn">filters</span> <span class="kn">from</span> <span class="nn">scipy</span> <span class="k">import</span> <span class="n">misc</span> <span class="kn">from</span> <span class="nn">numpy</span> <span class="k">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sin</span> <span class="n">matplotlib</span><span class="o">.</span><span class="n">rcParams</span><span class="p">[</span><span class="s1">'figure.figsize'</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="mf">20.0</span><span class="p">,</span> <span class="mf">10.0</span><span class="p">)</span> </pre></div> </div> </div> </div> </div> <div class="cell border-box-sizing text_cell rendered"> <div class="prompt input_prompt"> </div> <div class="inner_cell"> <div class="text_cell_render border-box-sizing rendered_html"> <p>We will test our routine on a test image (a Matlab example image). I chose this image intentionally to be non-square, so that I make sure to get the sequence of dimensions right everywhere.</p> </div> </div> </div> <div class="cell border-box-sizing code_cell rendered"> <div class="input"> <div class="prompt input_prompt">In [2]:</div> <div class="inner_cell"> <div class="input_area"> <div class=" highlight hl-ipython3"><pre><span></span><span class="n">img</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">misc</span><span class="o">.</span><span class="n">imread</span><span class="p">(</span><span class="s1">'cell.png'</span><span class="p">),</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">float</span><span class="p">)</span> <span class="c1"># cell is one of the Matlab demo images</span> <span class="c1"># img = np.sum(misc.imread('cameraman.png'),2).astype(float) # another Matlab demo image</span> <span class="c1"># img = np.sum(misc.imread('SheppLogan_Phantom_128.png'),2).astype(float) # cell is one of the Matlab demo images</span> <span class="n">img</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="n">img</span><span class="o">.</span><span class="n">dtype</span> <span class="n">p</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">img</span><span class="p">,</span><span class="n">cmap</span><span class="o">=</span><span class="n">p</span><span class="o">.</span><span class="n">get_cmap</span><span class="p">(</span><span class="s1">'gray'</span><span class="p">))</span> <span class="c1">#define the pixel size</span> <span class="p">[</span><span class="n">Nyimg</span><span class="p">,</span><span class="n">Nximg</span><span class="p">]</span> <span class="o">=</span> <span class="n">img</span><span class="o">.</span><span class="n">shape</span> <span class="nb">print</span><span class="p">(</span><span class="s2">"Size of img: "</span><span class="p">,</span><span class="n">Nximg</span><span class="p">,</span><span class="s2">"x"</span><span class="p">,</span><span class="n">Nyimg</span><span class="p">,</span><span class="s2">"pixels"</span><span class="p">)</span> </pre></div> </div> </div> </div> <div class="output_wrapper"> <div class="output"> <div class="output_area"><div class="prompt"></div> <div class="output_subarea output_stream output_stdout output_text"> <pre>Size of img: 191 x 159 pixels </pre> </div> </div> <div class="output_area"><div class="prompt"></div> <div class="output_png output_subarea "> <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAsMAAAJNCAYAAADUNnjqAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz AAALEgAACxIB0t1+/AAAIABJREFUeJzsvTuMdFuW57VORGZGZOZXt7odCqm6u+6VkHCx2sGgS4wx 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</div> </div> </div> </div> <div class="cell border-box-sizing text_cell rendered"> <div class="prompt input_prompt"> </div> <div class="inner_cell"> <div class="text_cell_render border-box-sizing rendered_html"> <p>Next, lets define a mirror padded version of our input image. It will be this image that we are recovering. We do the mirror padding, in order to ensure periodic boundary conditions. This allows us to use FFTs to invert the gradient.</p> </div> </div> </div> <div class="cell border-box-sizing code_cell rendered"> <div class="input"> <div class="prompt input_prompt">In [3]:</div> <div class="inner_cell"> <div class="input_area"> <div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#define the pixel size</span> <span class="n">imgPad</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">lib</span><span class="o">.</span><span class="n">pad</span><span class="p">(</span><span class="n">img</span><span class="p">,</span> <span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="n">Nyimg</span><span class="p">),(</span><span class="mi">0</span><span class="p">,</span><span class="n">Nximg</span><span class="p">)),</span> <span class="s1">'symmetric'</span><span class="p">)</span> <span class="p">[</span><span class="n">Ny</span><span class="p">,</span><span class="n">Nx</span><span class="p">]</span> <span class="o">=</span> <span class="n">imgPad</span><span class="o">.</span><span class="n">shape</span> <span class="nb">print</span><span class="p">(</span><span class="s2">"Size of imgPad: "</span><span class="p">,</span><span class="n">Nx</span><span class="p">,</span><span class="s2">"x"</span><span class="p">,</span><span class="n">Ny</span><span class="p">,</span><span class="s2">"pixels"</span><span class="p">)</span> <span class="n">p</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">imgPad</span><span class="p">,</span><span class="n">cmap</span><span class="o">=</span><span class="n">p</span><span class="o">.</span><span class="n">get_cmap</span><span class="p">(</span><span class="s1">'gray'</span><span class="p">)),</span> <span class="n">p</span><span class="o">.</span><span class="n">colorbar</span><span class="p">()</span> </pre></div> </div> </div> </div> <div class="output_wrapper"> <div class="output"> <div class="output_area"><div class="prompt"></div> <div class="output_subarea output_stream output_stdout output_text"> <pre>Size of imgPad: 382 x 318 pixels </pre> </div> </div> <div class="output_area"><div class="prompt output_prompt">Out[3]:</div> <div class="output_text output_subarea output_execute_result"> <pre>(<matplotlib.image.AxesImage at 0x84d20b8>, <matplotlib.colorbar.Colorbar at 0x873bd30>)</pre> </div> </div> <div class="output_area"><div class="prompt"></div> <div class="output_png output_subarea "> <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAy4AAAJOCAYAAABRMFNFAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz AAALEgAACxIB0t1+/AAAIABJREFUeJzsvUuMa+mW5/XftsMOR9iOx3lE5r1J3byle1UFVcOqLoSu RD1Aggk9QagfI0BigEAlRogR6lGrhBBihhAwQ2qgJRAjKLV0dVWFSnRRaiRUTatLXeJ2dWWeZ7xs hyPCDptB5O/zf6/49rYjTpw8JzO9pNCJE/be+3usbz3+67GLxWKhDW1oQxva0IY2tKENbWhDG/qY qfGhB7ChDW1oQxva0IY2tKENbWhDq2jjuGxoQxva0IY2tKENbWhDG/roaeO4bGhDG9rQhja0oQ1t 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nJKur+ePSPr81b8/L+mvvNIRvcFUVdU/k7QdPq7j5w9L+uWqqs6qqnpf0p/oUsY7qqEa/kqXchzp R9TxtzVVVfVhVVV/dPXvA0lf1aXB0cnvHVANf9+9+rqT3zugqqqOrv45o8vXNlTq5PfOqIa/Uie/ Hb0BdF+Oy7uS/j/7/69roPg7uhlVkn4rpfSllNJ/evXZo6qqnkiXh62kjXsb3dtBGzX8jPL8gTp5 vin99ZTSH6WU/keDgnT8vSGllP6cpO+R9Luq1wcdf29Ixt/fu/qok987oJRSL6X0LyR9KOm3qqr6 kjr5vTOq4a/UyW9HbwB1xflvD/3Fqqq+T9JflvSfp5T+DQ2iKFCX2r1b6vh5t/TfS/pXqqr6Hl0e qP/NPY/njaYrGNOvSfrpq8xApw/ukAr87eT3jqiqqouqqr5Xl5nCH0gp/QV18ntnVODvd6mT347e ELovx+UDSR+3///Y1Wcd3ZCqqvrm1X+fSfpHukzlPkkpPZIyLvvp/Y3wraA6fn4g6dvtd50834Cq qnpmuOn/QQM4QsffMSmlNKlLo/p/rqrq168+7uT3jqjE305+756qqtqT9NuSPqtOfu+cnL+d/Hb0 ptB9OS5fkvSplNInUkrTkn5c0m/c01jeeEopzV1F/5RSmpf0lyR9WZc8/Ymrn/1VSb9evEFHdZQ0 jPmt4+dvSPrxlNJ0SumTkj4l6fdf1SDfYBri75UxAv2Hkv6vq393/B2f/idJX6mq6r+zzzr5vTu6 xt9Ofu+GUkrrwJRSSrOSfkiXdUSd/N4B1fD3jzv57ehNocn7eGhVVecppb8u6Qu6dJ7+blVVX72P sbwl9EjSP0wpVbpc01+qquoLKaU/kPSrKaWflPQ1ST92n4N8kyil9Pcl/duS1lJKfybpZyX9vKT/ JfKzqqqvpJR+VdJXJJ1K+qmu40oz1fD3B1NK3yPpQtL7kv4zqePvuJRS+ouS/hNJX77CsVeS/itJ v6CCPuj4Ox418Pc/7uT3TugdSZ+/6j7ak/QrVVX9Zkrpd9XJ711QHX//Xie/Hb0JdC/tkDvqqKOO Ouqoo4466qijjsahrji/o4466qijjjrqqKOOOnrtqXNcOuqoo4466qijjjrqqKPXnjrHpaOOOuqo o4466qijjjp67alzXDrqqKOOOuqoo4466qij1546x6WjjjrqqKOOOuqoo446eu2pc1w66qijjjrq qKOOOuqoo9eeOselo4466qijjjrqqKOOOnrtqXNcOuqoo4466qijjjrqqKPXnv5/BJsko7sx7L4A AAAASUVORK5CYII= " > </div> </div> </div> </div> </div> <div class="cell border-box-sizing code_cell rendered"> <div class="input"> <div class="prompt input_prompt">In [4]:</div> <div class="inner_cell"> <div class="input_area"> <div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># define the real space grid</span> <span class="n">grid2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mgrid</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="n">Ny</span><span class="p">:</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">:</span><span class="n">Nx</span><span class="p">:</span><span class="mi">1</span><span class="p">]</span> <span class="n">x</span> <span class="o">=</span> <span class="n">grid2</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="n">y</span> <span class="o">=</span> <span class="n">grid2</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="n">dx</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">x</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="n">dy</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="o">-</span><span class="n">y</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="nb">print</span><span class="p">(</span><span class="s2">"Pixel size: "</span><span class="p">,</span><span class="n">dx</span><span class="p">,</span><span class="n">dy</span><span class="p">)</span> </pre></div> </div> </div> </div> <div class="output_wrapper"> <div class="output"> <div class="output_area"><div class="prompt"></div> <div class="output_subarea output_stream output_stdout output_text"> <pre>Pixel size: 1 1 </pre> </div> </div> </div> </div> </div> <div class="cell border-box-sizing text_cell rendered"> <div class="prompt input_prompt"> </div> <div class="inner_cell"> <div class="text_cell_render border-box-sizing rendered_html"> <p>We will do the gradient computation in reciprocal space, but in the forward direction this can certainly also be done more quickly in real space. But in order to make sure that the forward and inverse calculation correspond to one another, we need to ensure that we define the gradient filters as follows: $$\frac{\partial f}{\partial x} = \frac{f(x+dx,y)-f(f-dx,y)}{2 dx} = IFT\left[\tilde{f}\left( k_x,k_y\right) \cdot i \sin(k_x) \right]$$ $$\frac{\partial f}{\partial y} = \frac{f(x,y+dy)-f(f,y-dy)}{2 dy} = IFT\left[\tilde{f}\left( k_x,k_y\right) \cdot i \sin(k_y) \right]$$ where $\tilde{f}\left( k_x,k_y\right)$ is the Fourier transform of $f(x,y)$ so that $$f(x,y) = \sum_{x=1}^{N_x} \sum_{y=1}^{Ny} \tilde{f}\left( k_x,k_y\right) \exp\left[ik_x x+ik_y y \right] $$</p> <p>Using these definitions we define the following differentiating operator $$ D[f] = \left(\frac{\partial}{\partial x} + i\frac{\partial}{\partial y} \right)f(x,y) = IFT\left[\tilde{f}\left( k_x,k_y\right) \cdot \left(i \sin(k_x) -\sin(k_y) \right) \right] = g(x,y)$$ The integrating inverse operator of $D[f]$ is then $$ D^{-1}[g] = \left(\frac{\partial}{\partial x} + i\frac{\partial}{\partial y} \right)^{-1}g(x,y) = \frac{1}{N_x N_y}IFT\left[\frac{\tilde{g}\left( k_x,k_y\right)}{i \sin(k_x) -\sin(k_y)} \right]$$ where again $\tilde{f}\left( k_x,k_y\right)$ is the Fourier transform of $g(x,y)$</p> </div> </div> </div> <div class="cell border-box-sizing code_cell rendered"> <div class="input"> <div class="prompt input_prompt">In [ ]:</div> <div class="inner_cell"> <div class="input_area"> <div class=" highlight hl-ipython3"><pre><span></span> </pre></div> </div> </div> </div> </div> <div class="cell border-box-sizing code_cell rendered"> <div class="input"> <div class="prompt input_prompt">In [5]:</div> <div class="inner_cell"> <div class="input_area"> <div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># construct reciprocal space</span> <span class="n">kx1</span> <span class="o">=</span> <span class="n">np</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="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">fftfreq</span><span class="p">(</span><span class="n">Nx</span><span class="p">,</span><span class="n">dx</span><span class="p">))</span> <span class="n">ky1</span> <span class="o">=</span> <span class="n">np</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="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">fftfreq</span><span class="p">(</span><span class="n">Ny</span><span class="p">,</span><span class="n">dy</span><span class="p">))</span> <span class="n">kx</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">tile</span><span class="p">(</span><span class="n">kx1</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">Nx</span><span class="p">),(</span><span class="n">Ny</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span> <span class="n">ky</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">tile</span><span class="p">(</span><span class="n">ky1</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">Ny</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="n">Nx</span><span class="p">))</span> <span class="c1"># compute the sin(kx,ky) because this is the reciprocal space filter that corresponds to </span> <span class="c1"># the finite difference derivative operator df/dx = [f(x+dx)-f(x-dx)]/2dx</span> <span class="n">kxs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">kx</span><span class="p">)</span> <span class="n">kys</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">ky</span><span class="p">)</span> <span class="c1"># finally we need to define k=isin(kx)-sin(ky) and ik=1/k</span> <span class="n">k</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="n">j</span><span class="o">*</span><span class="n">kxs</span><span class="o">-</span><span class="n">kys</span><span class="p">)</span> <span class="n">kmod</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">kx</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">ky</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="n">ksmod</span> <span class="o">=</span> <span class="n">kxs</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">kys</span><span class="o">**</span><span class="mi">2</span> <span class="c1"># ik = np.where(ksmod > 1e-8,(-1j*kxs-kys)/ksmod,1)</span> <span class="n">ik</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">ksmod</span> <span class="o">></span> <span class="mi">1</span><span class="n">e</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">1</span><span class="o">/</span><span class="n">k</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span> <span class="c1"># we also want to apply some slight low-pass filtering in the inversion step, </span> <span class="c1"># in order to not enhance noise </span> <span class="n">kmax</span> <span class="o">=</span> <span class="mf">0.5</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">amax</span><span class="p">(</span><span class="n">kmod</span><span class="p">)</span> <span class="c1"># ik = np.where(kmod>kmax, ik*0.5*(1+np.cos(pi*(kmod-kmax)/(np.amax(kmod)-kmax))), ik) </span> <span class="c1"># Now, let's define the functions D and D^{-1}:</span> <span class="c1"># Since we have padded these images, we can safely wrap around the edges.</span> <span class="k">def</span> <span class="nf">D</span><span class="p">(</span><span class="n">f</span><span class="p">):</span> <span class="n">fmean</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="n">Df</span> <span class="o">=</span> <span class="mf">0.5</span><span class="o">*</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span><span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">))</span><span class="o">+</span><span class="mi">1</span><span class="n">j</span><span class="o">*</span><span class="mf">0.5</span><span class="o">*</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span><span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">))</span> <span class="c1"># Just to convince ourselves that above math is correct, we can, alternatively</span> <span class="c1"># also do this calculation using Fourier transforms (will be a bit slower though):</span> <span class="c1"># D = np.fft.ifft2(np.fft.fft2(f)*k)</span> <span class="k">return</span> <span class="n">Df</span><span class="p">,</span> <span class="n">fmean</span> <span class="k">def</span> <span class="nf">iD</span><span class="p">(</span><span class="n">g</span><span class="p">,</span><span class="n">gmean</span><span class="p">):</span> <span class="n">iDg</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">fft2</span><span class="p">(</span><span class="n">g</span><span class="p">)</span><span class="o">*</span><span class="n">ik</span> <span class="n">iDg</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="o">=</span> <span class="n">gmean</span><span class="o">*</span><span class="n">Nx</span><span class="o">*</span><span class="n">Ny</span> <span class="n">iDg</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">ifft2</span><span class="p">(</span><span class="n">iDg</span><span class="p">))</span> <span class="k">return</span> <span class="n">iDg</span> </pre></div> </div> </div> </div> <div class="output_wrapper"> <div class="output"> <div class="output_area"><div class="prompt"></div> <div class="output_subarea output_stream output_stderr output_text"> <pre>C:\Program Files\Anaconda3\lib\site-packages\ipykernel\__main__.py:15: RuntimeWarning: divide by zero encountered in true_divide C:\Program Files\Anaconda3\lib\site-packages\ipykernel\__main__.py:15: RuntimeWarning: invalid value encountered in true_divide </pre> </div> </div> </div> </div> </div> <div class="cell border-box-sizing text_cell rendered"> <div class="prompt input_prompt"> </div> <div class="inner_cell"> <div class="text_cell_render border-box-sizing rendered_html"> <p>The above operator can now also be used to compute the second order derivatives!</p> </div> </div> </div> <div class="cell border-box-sizing code_cell rendered"> <div class="input"> <div class="prompt input_prompt">In [6]:</div> <div class="inner_cell"> <div class="input_area"> <div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># Test the above routine:</span> <span class="p">[</span><span class="n">imgD</span><span class="p">,</span><span class="n">imgMean</span><span class="p">]</span> <span class="o">=</span> <span class="n">D</span><span class="p">(</span><span class="n">imgPad</span><span class="p">)</span> <span class="nb">print</span><span class="p">(</span><span class="s2">"Mean of image: "</span><span class="p">,</span><span class="n">imgMean</span><span class="p">)</span> <span class="n">p</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">p</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">imgD</span><span class="p">),</span><span class="n">cmap</span><span class="o">=</span><span class="n">p</span><span class="o">.</span><span class="n">get_cmap</span><span class="p">(</span><span class="s1">'gray'</span><span class="p">)),</span> <span class="n">p</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(),</span> <span class="n">p</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"df/dx"</span><span class="p">)</span> <span class="n">p</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">p</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">imag</span><span class="p">(</span><span class="n">imgD</span><span class="p">),</span><span class="n">cmap</span><span class="o">=</span><span class="n">p</span><span class="o">.</span><span class="n">get_cmap</span><span class="p">(</span><span class="s1">'gray'</span><span class="p">)),</span> <span class="n">p</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(),</span> <span class="n">p</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"df/dy"</span><span class="p">)</span> <span class="nb">print</span><span class="p">(</span><span class="s2">"Fraction of nonzero elements in dfdx: "</span><span class="p">,</span><span class="n">np</span><span class="o">.</span><span class="n">count_nonzero</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">imgD</span><span class="p">))</span><span class="o">/</span><span class="p">(</span><span class="n">Nx</span><span class="o">*</span><span class="n">Ny</span><span class="p">))</span> <span class="nb">print</span><span class="p">(</span><span class="s2">"Fraction of nonzero elements in dfdy: "</span><span class="p">,</span><span class="n">np</span><span class="o">.</span><span class="n">count_nonzero</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">imag</span><span class="p">(</span><span class="n">imgD</span><span class="p">))</span><span class="o">/</span><span class="p">(</span><span class="n">Nx</span><span class="o">*</span><span class="n">Ny</span><span class="p">))</span> <span class="c1"># Test the inversion !!! </span> <span class="n">D2</span> <span class="o">=</span> <span class="n">iD</span><span class="p">(</span><span class="n">imgD</span><span class="p">,</span><span class="n">imgMean</span><span class="p">)</span> <span class="n">p</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">p</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">D2</span><span class="p">,</span><span class="n">cmap</span><span class="o">=</span><span class="n">p</span><span class="o">.</span><span class="n">get_cmap</span><span class="p">(</span><span class="s1">'gray'</span><span class="p">)),</span> <span class="n">p</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(),</span> <span class="n">p</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"recovered from gradient"</span><span class="p">)</span> <span class="n">p</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">p</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">imgPad</span><span class="p">,</span><span class="n">cmap</span><span class="o">=</span><span class="n">p</span><span class="o">.</span><span class="n">get_cmap</span><span class="p">(</span><span class="s1">'gray'</span><span class="p">)),</span> <span class="n">p</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(),</span> <span class="n">p</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"original"</span><span class="p">)</span> </pre></div> </div> </div> </div> <div class="output_wrapper"> <div class="output"> <div class="output_area"><div class="prompt"></div> <div class="output_subarea output_stream output_stdout output_text"> <pre>Mean of image: 608.110639139 Fraction of nonzero elements in dfdx: 0.22460403701142612 Fraction of nonzero elements in dfdy: 0.23148605485857288 </pre> </div> </div> <div class="output_area"><div class="prompt output_prompt">Out[6]:</div> <div class="output_text output_subarea output_execute_result"> <pre>(<matplotlib.image.AxesImage at 0x8290898>, <matplotlib.colorbar.Colorbar at 0x835a8d0>, <matplotlib.text.Text at 0x82daf60>)</pre> </div> </div> <div class="output_area"><div class="prompt"></div> <div class="output_png output_subarea "> <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAA/wAAAJZCAYAAAAUBz4SAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz AAALEgAACxIB0t1+/AAAIABJREFUeJzsvWmMXWd6Jvacu+/7fmu5xSKLJVIiWy2r1XJ3JFtwd2ba W2zDQTyJjYxh/xhnBoNgMsh4DDjAJB23DSQxxkF+ZWJ4Bm4bcXsmaC8w2ous3uwWtZCiuBSLtd59 3/fl5EfV8/JUiVqsZklq+XsAgnXvPefes3zn+97leZ9X03UdCgoKCgoKCgoKCgoKCgoKHy+YPuwD UFBQUFBQUFBQUFBQUFBQePRQDr+CgoKCgoKCgoKCgoKCwscQyuFXUFBQUFBQUFBQUFBQUPgYQjn8 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