📄 tp1.html
字号:
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"><html xmlns="http://www.w3.org/TR/REC-html40"><head><meta http-equiv=Content-Type content="text/html; charset=iso-8859-1"><link rel=File-List href="index_files/filelist.xml"><title>Lecture 1 - Active Contours and Level Sets</title><link href="styles.css" rel="stylesheet" type="text/css"></head><body bgcolor="#FFFFFF" lang=FR><div > <h1>Lecture 1 - Active Contours and Level Sets</h1></div><blockquote> <blockquote> <blockquote> <blockquote> <blockquote> <p align="justify"><strong>Abstract : </strong>The goals of this lecture is to use the level set framework in order to do curve evolution. The mean curvature motion is the basic tool, and it can be extended into edge-based (geodesic active contours) and region-based (Chan-Vese) snakes. </p> </blockquote> </blockquote> </blockquote> </blockquote></blockquote><h2> Setting up Matlab. </h2><ul> <li>First download the Matlab toolbox <a href="matlab/toolbox_fast_marching.zip"><font face="Courier New, Courier, mono">toolbox_fast_marching.zip</font></a>. Unzip it into your working directory. You should have a directory toolbox_fast_marching/ in your path. </li> <li>The first thing to do is to install this toolbox in your path.</li> <blockquote> <p><font face="Courier New, Courier, mono">path(path, 'toolbox_fast_marching/');<br> path(path, 'toolbox_fast_marching/data/');<br> path(path, 'toolbox_fast_marching/toolbox/');</font></p> </blockquote> <li>Recompile the mex file for your machine (this can produce some warning). If it does not work, either use the already compiled mex file (they should be available in <font face="Courier New, Courier, mono">toolbox_fast_marching/</font> for MacOs and Unix) or try to set up matlab with a C compiler (e.g. gcc) using '<font face="Courier New, Courier, mono">mex -setup</font>'. </li> <blockquote> <p><font face="Courier New, Courier, mono">cd toolbox_fast_marching<br> compile_mex;<br> cd ..</font></p> </blockquote> </ul> <h2> Managing level set function. </h2> <ul> <li>In order to perform curve evolution, we will deal with a distance function stored in a 2D image D. The curve will be embedded in the level set D=0. This curve will be evolved by modifying the image D. A curve evolution ODE can be replaced by an PDE on D. This allows to deal with topological changes when a curve split or two curves merge.</li> <blockquote> <p><font face="Courier New, Courier, mono">n = 200; % size of the image<br> % load a distance function<br> D0 = compute_levelset_shape('circlerect2', n);<br> % type 'help compute_levelset_shape' to see other <br> % basic curve you can load.<br> <br> % display the curve<br> clf; hold on;<br> imagesc(D); axis image; axis off; axis([1 n 1 n]);<br> [c,h] = contour(D,[0 0], 'r');<br> set(h, 'LineWidth', 2);<br> hold off;<br> colormap gray(256);<br> <br> % do the union of two curves<br> options.center = [0.15 0.15]*n;<br> options.radius = 0.1*n;<br> D1 = compute_levelset_shape('circle', n,options);<br> imagesc(min(D0,D1)<0);</font></p> </blockquote> <li>During the curve evolution, the image D might become far from being a distance function. In order to stabilize the algorithm, one needs to re-compute this distance function.</li> <blockquote> <p><font face="Courier New, Courier, mono">% here we simulate a modification of the distance function<br> [Y,X] = meshgrid(1:n,1:n);<br> D = (D0.^3) .* (X+n/3); <br> D1 = perform_redistancing(D);<br> % display both the original and the new, <br> % redistanced, curve (should be very close)<br> ...</font></p> </blockquote></ul><h2> Mean Curvature Motion. </h2><ul> <li>In order to compute differential quantities (tangent, normal, curvature, etc) on the curve, you can compute derivatives of the image D. </li> <blockquote> <p><font face="Courier New, Courier, mono">% the gradient<br> g0 = divgrad(D);<br> % display the gradient (as arrow field with 'quiver', ...)<br> ...<br> % the normalized gradient<br> d = max(eps, sqrt(sum(g0.^2,3)) );<br> g = g0 ./ repmat( d, [1 1 2] );<br> % display<br> ...<br> % the curvature<br> K = d .* divgrad( g );<br> % display<br> ...</font></p> </blockquote> <li>The mean curvature motion of the level sets of some image u is driven be the following equation.<br> <font face="Courier New, Courier, mono"> </font><font face="Courier New, Courier, mono"> </font><font face="Courier New, Courier, mono"> </font><img src="images/tp1b/eq-mean-curvature.png" width="250"><br> Implement this evolution explicitly in time using finite differences.</li></ul><ul> <blockquote> <p><font face="Courier New, Courier, mono">Tmax = 1000; % maximum time of evolution<br> dt = 0.4; % time step (should be small)<br> niter = round(Tmax/dt); % number of iterations<br> D = D0; % initialization</font><font face="Courier New, Courier, mono"><br> for i=1:niter<br> % compute the right hand size of the PDE<br> ...<br> % update the distance field<br> D = ...;<br> % redistance the function from time to time<br> if mod(i,30)==0<br> D = perform_redistancing(D);<br> end <br> % display from time to time<br> if mod(i,30)=1<br> % display here<br> ...<br> end <br> end</font></p> </blockquote> <table width="1100" border="0" cellspacing="0" cellpadding="0" align="center"> <tr align="center"> <td> <img src="images/tp1b/mean-01.png" width="200"/> <img src="images/tp1b/mean-02.png" width="200"/> <img src="images/tp1b/mean-03.png" width="200"/> <img src="images/tp1b/mean-04.png" width="200"/> <img src="images/tp1b/mean-05.png" width="200"/> </td> </tr> <tr> <td align="center">Curve evolution under the mean curvature motion (the background is the distance function D).</td> </tr> </table></ul><h2> Edge-based Segmentation with Geodesic Active Contour (snakes + level set). </h2><ul> <li>Given a background image M to segment, one needs to compute an edge-stopping function E. It should be small in area of high gradient, and high in area of large gradient.</li> <blockquote> <p><font face="Courier New, Courier, mono">% load an image<br> name = 'brain';<br> M = rescale( sum( load_image(name, n), 3) );<br> % display it<br> ...<br> % compute a smoothed gradient<br> sigma = 4; % blurring size<br> G = divgrad( perform_blurring(M,sigma) );<br> % compute the norm of the gradient<br> d = ...<br> % compute the edge-stopping function<br> E = ...<br> % rescale it so that it is in realistic ranges<br> E = rescale(E,0.3,1);</font></p> </blockquote> <li>The geodesic active contour evolution is a mean curvature motion modulated by the edge-stopping function:<br> <font face="Courier New, Courier, mono"> </font><font face="Courier New, Courier, mono"> </font><font face="Courier New, Courier, mono"> </font><img src="images/tp1b/eq-snake.png" width="350"> <br> Implement this evolution explicitly in time using finite differences.</li> <blockquote> <p><font face="Courier New, Courier, mono">...</font> </p> </blockquote> <table width="1100" border="0" cellspacing="0" cellpadding="0" align="center"> <tr align="center"> <td> <img src="images/tp1b/snake-01.png" width="200"/> <img src="images/tp1b/snake-02.png" width="200"/> <img src="images/tp1b/snake-03.png" width="200"/> <img src="images/tp1b/snake-04.png" width="200"/> <img src="images/tp1b/snake-05.png" width="200"/> </td> </tr> <tr> <td align="center">Segmentation with geodesic active contours.</td> </tr> </table></p> </ul><h2> Region-based Segmentation with Chan-Vese (Mumord-Shah + level sets). </h2><ul> <li>The geodesic active contour uses an edge-based energy. It has lots of local minima and is very sensitive to initialization. In order to circumvent these drawbacks, one can use a region based energy like the Mumford-Shah functional. Re-casted into the level set framework, it reads:<br> <font face="Courier New, Courier, mono"> </font><font face="Courier New, Courier, mono"> </font><font face="Courier New, Courier, mono"> </font><img src="images/tp1b/eq-chan-vese-def.png" width="550"> <br> The corresponding gradient descent is the Chan-Vese active contour method:<br> <font face="Courier New, Courier, mono"> </font><font face="Courier New, Courier, mono"> </font><font face="Courier New, Courier, mono"> </font><img src="images/tp1b/eq-chan-vese.png" width="520"> <br> Implement this evolution explicitly in time using finite differences, when c1 and c2 are known in advanced.</li> <blockquote> <p><font face="Courier New, Courier, mono">% initialize with a complex distance function <br> D0 = compute_levelset_shape('small-disks', n); <br> % set parameters<br> lambda = 0.8;<br> c1 = ...; % black<br> c2 = ...; % gray<br> ...</font> <table width="1100" border="0" cellspacing="0" cellpadding="0" align="center"> <tr align="center"> <td> <img src="images/tp1b/chan-vese-01.png" width="200"> <img src="images/tp1b/chan-vese-02.png" width="200"> <img src="images/tp1b/chan-vese-03.png" width="200"> <img src="images/tp1b/chan-vese-04.png" width="200"> <img src="images/tp1b/chan-vese-05.png" width="200"> </td> </tr> <tr> <td align="center">Segmentation with Chan-Vese active contour without edges. </td> </tr> </table> </blockquote> <li>In the case that one does not know in advance the constants c0 and c1, how to update them automatically during the evolution ? Implement this method.</li> <blockquote><br/> Copyright © 2006 <a href="http://www.cmap.polytechnique.fr/%7Epeyre/">Gabriel Peyré</a> </blockquote></ul></body></html>⌨️ 快捷键说明
复制代码
Ctrl + C
搜索代码
Ctrl + F
全屏模式
F11
切换主题
Ctrl + Shift + D
显示快捷键
?
增大字号
Ctrl + =
减小字号
Ctrl + -