Test_GF_with_correct_gradient.html
Test_GF_with_correct_gradient.html
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<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>
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<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>
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<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>
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<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>
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<pre>Size of img: 191 x 159 pixels
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<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>
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<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>
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<pre>Size of imgPad: 382 x 318 pixels
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<pre>(<matplotlib.image.AxesImage at 0x84d20b8>,
<matplotlib.colorbar.Colorbar at 0x873bd30>)</pre>
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<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>
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<pre>Pixel size: 1 1
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<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>
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<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>
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<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
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<p>The above operator can now also be used to compute the second order derivatives!</p>
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<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>
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<div class="output">
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<pre>Mean of image: 608.110639139
Fraction of nonzero elements in dfdx: 0.22460403701142612
Fraction of nonzero elements in dfdy: 0.23148605485857288
</pre>
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</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>
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