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<!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 2 - Front Propagation in 2D and 3D</title><link href="styles.css" rel="stylesheet" type="text/css"></head><body bgcolor="#FFFFFF" lang=FR><div > <h1>Lecture 2 - Front propagation<br> in 2D and 3D</h1></div><blockquote> <blockquote> <blockquote> <blockquote> <blockquote> <p align="justify"><strong>Abstract : </strong>The goals of this lecture is to manipulate the fast marching algorithm in 2D and 3D. Application to shortest path extraction (e.g. road tracking and tubular structure extraction in medical images) and Voronoi cell segmentation is presented.</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> Front propagation in 2D. </h2> <ul> <li>We can now load an image and build a speed propagation function <font face="Courier New, Courier, mono">W</font>. Note that area with <font face="Courier New, Courier, mono">W</font> values near 0 (black pixels) are those in which the front propagate slowly, so that <font face="Courier New, Courier, mono">W </font>is the inverse of the potential that weight the metric.</li> <blockquote> <p><font face="Courier New, Courier, mono">n = 128;<br> name = 'cavern'; % other possibilities are 'mountain' and 'road2'<br> W = load_image(name, n);<br> W = rescale(W, 0.01, 1); % set up a reasonable range for the potential<br> % display the weighting function</font><font face="Courier New, Courier, mono"><br> clf; imagesc(W); colormap gray(256);</font> </p> </blockquote> <li>After asking to the user the starting and ending points, we proceed to the front propagation. Note that the propagation terminates when the front reaches the ending point.</li> <blockquote> <p><font face="Courier New, Courier, mono">[start_points,end_points] = pick_start_end_point(W);<br> options.end_points = end_points;<br> [D,S] = perform_fast_marching_2d(W, start_points, options);<br> % display the distance function<br> clf; imagesc(D);<br> D(I==Inf) = 0; % remove Inf values that make contour crash<br> figure; contour(D,50); </font></p> </blockquote> <li>The shortest path is extracted by performing a gradient descent of D, starting from the ending point.</li> <blockquote> <p><font face="Courier New, Courier, mono">p = extract_path_2d(D,end_points, options);</font> <font face="Courier New, Courier, mono"><br> % display the path<br> clf; plot_fast_marching_2d(W,S,p,start_points,end_points);</font></p> </blockquote> <li>Now that you know the basic commands, you can try other speed functions loaded from other files, and you can try more complicated potential than just the value of the image. You can try a potential based on the gradient of the image computed using <font face="Courier New, Courier, mono">grad(W)</font>.</li></ul><ul> <table width="1200" border="0" cellspacing="0" cellpadding="0" align="center"> <tr align="center"> <td> <img src="images/tp1/cavern-dist-function.jpg" width="300"/> <img src="images/tp1/cavern-dist-function-contour.jpg" width="300"/> <img src="images/tp1/cavern-shortest-path-2d.jpg" width="300"/><br/> <img src="images/tp1/mountain-dist-function.jpg" width="300"/> <img src="images/tp1/mountain-dist-func-cont.jpg" width="300"/> <img src="images/tp1/mountain-shortest-path-2d.jpg" width="300"/><br/> <img src="images/tp1/road2-dist-function.jpg" width="300"/> <img src="images/tp1/road2-dist-function-contour.jpg" width="300"/> <img src="images/tp1/road2-shortest-path-2d.jpg" width="300"/> </td> </tr> <tr> <td align="center">Left : distance function for 3 different W maps. <br> Center : the corresponding level set maps. <br> Right : shortest path together with the explored area (in red).</td> </tr> </table></ul><h2> 3D Volumetric Shortest Paths. </h2><ul> <li> <div align="justify">Firs you need to load a 3D array of data <font face="Courier New, Courier, mono">M</font>. Such a volumetric dataset is more difficult to visualize than a standard 2D image. You can render slices like <font face="Courier New, Courier, mono">M(:,:,i)</font> or use a volumetric renderer such a <font face="Courier New, Courier, mono">vol3d</font>. In order to do so, you need to set up a correct alpha mapping to make transparent some parts of the volume. If you rotate the data, then you need to re-render the volume using <font face="Courier New, Courier, mono">vol3d(h)</font>. You can download the 3D data set <a href="matlab/brain1-crop-256.mat">here</a>.</div> </li> <blockquote> <p><font face="Courier New, Courier, mono">% load the whole volume<br> load brain1-crop-256.mat<br> % crop to retain only the central part<br> n = 100;<br> M = rescale( crop(M,n) ); <br> % display some horizontal slices<br> imageplot(M(:,:,50));<br> % same thing in the other directions<br> ...</font></p> </blockquote> <table width="1000" border="0" cellspacing="0" cellpadding="0" align="center"> <tr align="center"> <td> <img src="images/tp1/brain-img-1.png" width="200"/> <img src="images/tp1/brain-img-2.png" width="200"/> <img src="images/tp1/brain-img-3.png" width="200"/> <img src="images/tp1/brain-img-4.png" width="200"/> <br/> <img src="images/tp1/brain-img-5.png" width="200"/> <img src="images/tp1/brain-img-6.png" width="200"/> <img src="images/tp1/brain-img-7.png" width="200"/> <img src="images/tp1/brain-img-8.png" width="200"/> </td> </tr> <tr> <td align="center">Cross sections of the data-set.</td> </tr> </table> <br> <li> <div align="justify">Another, more efficient way, to render volumetric data is to display a semi-transparent volumetric image. This can be achieved using the function <font face="Courier New, Courier, mono">vol3d</font> than render semi-transparent slices of the data orthogonal to the viewing direction. In order to do so, you need to set up a correct alpha mapping to make transparent some parts of the volume. If you rotate the data, then you need to re-render the volume using <font face="Courier New, Courier, mono">vol3d(h)</font>.</div> </li> <blockquote> <p><font face="Courier New, Courier, mono">clf;<br> h = vol3d('cdata',M,'texture','2D');<br> view(3); axis off; <br> % set up a colormap<br> colormap bone(256);<br> % set up an alpha map<br> options.center = ...; % here a value in [0,1]<br> options.sigma = .08; % control the width of the non-transparent region<br> a = compute_alpha_map('gaussian', options); % you can plot(a) to see the alphamap <br> colormap bone(256);<br> % refresh the rendering<br> vol3d(h); <br> % try with other alphamapping and colormapping<br> ... </font></p> </blockquote> <table width="1000" border="0" cellspacing="0" cellpadding="0" align="center"> <tr align="center"> <td> <img src="images/tp1/brain-vol3d-1.png" width="300"/> <img src="images/tp1/brain-vol3d-2.png" width="300"/> <img src="images/tp1/brain-vol3d-3.png" width="300"/><br/> <img src="images/tp1/brain-vol3d-4.png" width="300"/> <img src="images/tp1/brain-vol3d-5.png" width="300"/> <img src="images/tp1/brain-vol3d-6.png" width="300"/> </td> </tr> <tr> <td align="center">Volumetric rendering with different alpha-mapping. Each time the<font face="Courier New, Courier, mono"> options.center</font> value is increased.</td> </tr> </table> <br> <li>Geodesic distances can be computed on a 3D volume using the Fast Marching. The important point here is to define the correct potential field W that should be large in the region where you want the front to move fast. It means that geodesic will follow these regions. In order to do so, we will ask the user to click on a starting point in a given horizontal slice <font face="Courier New, Courier, mono">W(:,:,delta)</font>. The potential is then computed in order to be large for value of <font face="Courier New, Courier, mono">M</font> that are close to the value at this starting point.</li> <blockquote> <p><font face="Courier New, Courier, mono">% ask to the user for some input point<br> </font><font face="Courier New, Courier, mono">delta = 5;<br> clf; imageplot(M(:,:,delta));<br> title('Pick starting point');<br> start_point = round( ginput(1) );<br> start_point = [start_point(2); start_point(1); delta];<br> % compute a potential that is high only very close <br> % to the value of M at the selected point<br> W = ...;<br> W = rescale(W,.001,1);<br> % perform the front propagation</font><font face="Courier New, Courier, mono"><br> options.nb_iter_max = Inf;<br> [D,S] = perform_fast_marching_3d(W, start_point, options);<br> % display the results using vol3d<br> ... </font> </p> </blockquote> <table width="1000" border="0" cellspacing="0" cellpadding="0" align="center">⌨️ 快捷键说明
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