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<div class="title">imageProcessing/at-u2-v0 </div> </div>
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<div class="textblock"><p>Computes a piecewise smooth approximation of a grey-level or color image, by optimizing the Ambrosio-Tortorelli functional (with u a 2-form and v a 0-form).</p>
<dl class="section user"><dt>Author(s) of this documentation:\n Marion Foare, Jacques-Olivier Lachaud</dt><dd></dd></dl>
<p><b>Usage:</b> at-u2-v0 -i [input.pgm]</p>
<p>(for grey-level image restoration)</p>
<p><b>Usage:</b> at-u2-v0 -i [input.ppm]</p>
<p>(for color image restoration)</p>
<p>The Ambrosio-Tortorelli functional is a classical relaxation of the Mumford-Shah functional.</p>
<p>Given an input grayscale image, defined in an open bounded domain \( \Omega \), we represent its gray levels by a function \( g \in L^{\infty}(\Omega) \). In the Ambrosio-Tortorelli functional [1], one wants to find a function \( u \in SBV(\Omega) \) which is a smooth approximation of the input image \( g \). The Ambrosio-Tortorelli functional [1] is defined by </p><p class="formulaDsp">
\[ \displaystyle AT_{\varepsilon}(u,v) = \int_\Omega \alpha |u-g|^2 + v^2 |\nabla u|^2 + \lambda \varepsilon |\nabla v|^2 + \frac{\lambda}{4 \varepsilon} |1-v|^2 dx, \]
</p>
<p> for functions \( u,v \in W^{1,2}(\Omega)\) with \( 0 \leq v \leq 1 \).</p>
<p>In AT functional, function \( v \) is a smooth approximation of the set of discontinuities, and takes value close to 0 in this set, while being close to 1 outside discontinuities. A remarkable property of this functional is that it \( \Gamma \)-converges to (a relaxation of) MS functional as \( \varepsilon \) tends to 0 (see [1]). The intuition is that a large \( \varepsilon \) induces a solution with a fuzzy set of discontinuities, which is then progressively narrowed to the crisp 1-dimensional set of discontinuites as \( \varepsilon \) goes to 0.</p>
<p>We discretize AT with discrete calculus and we set \( u \) and \( g \) to live on the faces and \( v \) to live on the vertices and edges. Pixels are faces, so functions \( u \) and \( g \) are 2-forms since they represent the gray levels of each pixel. On the contrary, we set \( v \) in-between cells of non null measure, so in this case on vertices as a 0-form, and on edges by averaging with \( \mathbf{M} \). We call this formulation AT20. The DEC reformulation is straightforward, except for the second term, where we use matrix \( \mathbf{M} \) to transport the 0-form \( v \) onto edges :</p>
<p class="formulaDsp">
\[ \displaystyle AT20(u,v) = \Sigma_{i=1}^n \alpha \langle u_i - g_i , u_i - g_i \rangle_2 + \langle \mathbf{M} v , \bar{\mathbf{\star}} \bar{\mathbf{d_0}} \mathbf{\star} u_i \rangle_1 ^2 \\ + \lambda \varepsilon \langle \mathbf{d_0} v , \mathbf{d_0} v \rangle_1 + \frac{\lambda}{4\varepsilon} \langle 1 - v , 1 - v \rangle_0. \]
</p>
<p>For more details, see <a class="el" href="moduleAT.html">Image restoration and inpainting with Ambrosio-Tortorelli functional</a></p>
<p><b>Allowed</b> <b>options</b> <b>are:</b> </p>
<div class="fragment"><div class="line">Positionals:</div>
<div class="line"> 1 TEXT:FILE REQUIRED the input image PPM filename.</div>
<div class="line"> </div>
<div class="line">Options:</div>
<div class="line"> -h,--help Print <span class="keyword">this</span> help message and exit</div>
<div class="line"> -i,--input TEXT:FILE REQUIRED the input image PPM filename.</div>
<div class="line"> -m,--inpainting-mask TEXT the input inpainting mask filename.</div>
<div class="line"> -o,--output TEXT=AT the output image basename.</div>
<div class="line"> -l,--lambda FLOAT the parameter lambda.</div>
<div class="line"> -M,--metric-average use metric average to smooth L1-metric.</div>
<div class="line"> -1,--lambda-1 FLOAT=0.3125 the initial parameter lambda (l1).</div>
<div class="line"> -2,--lambda-2 FLOAT=0.0005 the <span class="keyword">final</span> parameter lambda (l2).</div>
<div class="line"> -q,--lambda-ratio FLOAT=1.41421 the division ratio <span class="keywordflow">for</span> lambda from l1 to l2.</div>
<div class="line"> -a,--alpha FLOAT=1 the parameter alpha.</div>
<div class="line"> -e,--epsilon the initial and <span class="keyword">final</span> parameter epsilon of AT functional at the same time.</div>
<div class="line"> --epsilon-1 FLOAT=2 the initial parameter epsilon.</div>
<div class="line"> --epsilon-2 FLOAT=0.25 the <span class="keyword">final</span> parameter epsilon.</div>
<div class="line"> --epsilon-r FLOAT=2 sets the ratio between two consecutive epsilon values of AT functional.</div>
<div class="line"> -n,--nbiter INT=10 the maximum number of iterations.</div>
<div class="line"> --image-snr TEXT the input image without deterioration <span class="keywordflow">if</span> you wish to compute the SNR.</div>
<div class="line"> -p,--pixel-size INT=1 the pixel size <span class="keywordflow">for</span> outputing images (useful when one wants to see the discontinuities v on top of u).</div>
<div class="line"> -c,--color-v TEXT=0xff0000 the color chosen <span class="keywordflow">for</span> displaying the singularities v (e.g. red is 0xff0000).</div>
<div class="line"> -v,--verbose INT=0 the verbose level (0: silent, 1: less silent, etc).</div>
</div><!-- fragment --><p><b>example:</b> </p>
<div class="fragment"><div class="line">./imageProcessing/at-u2-v0 -i ../imageProcessing/Images/degrade-b04.pgm --image-snr ../imageProcessing/Images/degrade.pgm -a 0.05 --epsilon-1 4 --epsilon-2 0.25 -l 0.0075 -p 2 -c 0xff0000 -o degrade</div>
</div><!-- fragment --><center> <table class="doxtable">
<tr>
<td>Input image <em>g</em> </td><td>Reconstructed image <em>u</em> </td><td>Perfect image </td></tr>
<tr>
<td><div class="image">
<img src="degrade-b04.png" alt=""/>
<div class="caption">
Input image (noise = 0.4)</div></div>
</td><td><div class="image">
<img src="degrade-a0.05000-l0.0075000-u2.png" alt=""/>
<div class="caption">
AT20 alpha=0.05 lambda=0.0075 </div></div>
</td><td><div class="image">
<img src="degrade.png" alt=""/>
<div class="caption">
Perfect image</div></div>
</td></tr>
<tr>
<td>SNR of <em>g</em> = 21.9183 </td><td>SNR of <em>u</em> = 34.3655 </td><td>Perfect image </td></tr>
</table>
</center><dl class="section note"><dt>Note</dt><dd>Other restoration examples, parameter analysis, and image inpainting examples may be found in <a class="el" href="moduleAT.html">Image restoration and inpainting with Ambrosio-Tortorelli functional</a>.</dd></dl>
<p>[1] Luigi Ambrosio, and Vincenzo Maria Tortorelli. "Approximation of functional depending on jumps by elliptic functional via \(\Gamma\)‐convergence." Communications on Pure and Applied Mathematics 43.8 (1990): 999-1036.</p>
<p>[2] Marion Foare, Jacques-Olivier Lachaud, and Hugues Talbot. "Image restoration
and segmentation using the Ambrosio-Tortorelli functional and discrete calculus." In Proceedings of the IAPR International Conference on Pattern Recognition (ICPR2016), Cancun, Mexico, 2016.</p>
<p>[3] Marion Foare, Jacques-Olivier Lachaud, and Hugues Talbot. "Numerical implementation of the Ambrosio-Tortorelli functional using discrete calculus and application to image restoration and inpainting." In Proceedings of 1st Workshop on Reproducible Research In Pattern Recognition (RRPR 2016), Springer LNCS. To appear.</p>
<p>[4] Matteo Focardi. "On the variational approximation of free-
discontinuity problems in the vectorial case." Mathematical Models and Methods in Applied Sciences 11.04 (2001): 663-684. </p>
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