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This tour studies the computation of the medial axis using the Fast Marching.
addpath('toolbox_signal')
addpath('toolbox_general')
addpath('toolbox_graph')
addpath('solutions/shapes_6_medialaxis')
The Voronoi diagram is the segmentation of the image given by the region of influence of the set of starting points.
Load a distance map.
n = 200;
W = load_image('mountain', n);
W = rescale(W,.25,1);
Select seed points.
pstart = [[20;20] [120;100] [180;30] [60;160]];
nbound = size(pstart,2);
Display the map and the points.
ms = 20;
clf; hold on;
imageplot(W);
h = plot(pstart(2,:), pstart(1,:), '.r'); set(h, 'MarkerSize', ms);
Compute the geodesic distant to the whole set of points.
[D,S,Q] = perform_fast_marching(W, pstart);
Display the geodesic distance.
clf; hold on;
imageplot(convert_distance_color(D, W));
h = plot(pstart(2,:), pstart(1,:), '.r'); set(h, 'MarkerSize', ms);
Display the Voronoi Segmentation.
clf; hold on;
imageplot(Q);
h = plot(pstart(2,:), pstart(1,:), '.r'); set(h, 'MarkerSize', ms);
colormap jet(256);
The medial axis is difficult to extract from the singularity of the distance map. It is much more robust to extract it from the discontinuities in the Voronoi index map |Q|.
Compute the derivative, the gradient.
G = grad(Q);
Take it modulo |nbound|.
G(G<-nbound/2) = G(G<-nbound/2) + nbound;
G(G>nbound/2) = G(G>nbound/2) - nbound;
Compute the norm of the gadient.
G = sqrt(sum(G.^2,3));
Compute the medial axis by thresholding the gradient magnitude.
B = 1 - (G>.1);
Display.
clf; hold on;
imageplot(B);
h = plot(pstart(2,:), pstart(1,:), '.r'); set(h, 'MarkerSize', ms);
The sekeleton, also called Medial Axis, is the set of points where the geodesic distance is singular.
A binary shape is represented as a binary image.
n = 200;
name = 'chicken';
M = load_image(name,n);
M = perform_blurring(M,5);
M = double( rescale( M )>.5 );
if M(1)==1
M = 1-M;
end
Compute its boundary, that is going to be the set of starting points.
pstart = compute_shape_boundary(M);
nbound = size(pstart,2);
Display the metric.
lw = 2;
clf; hold on;
imageplot(-M);
h = plot(pstart(2,:), pstart(1,:), 'r'); set(h, 'LineWidth', lw); axis ij;
Parameters for the Fast Marching: constant speed |W|, but retricted using |L| to the inside of the shape.
W = ones(n);
L = zeros(n)-Inf; L(M==1) = +Inf;
Compute the fast marching, from the boundary points.
options.constraint_map = L;
[D,S,Q] = perform_fast_marching(W, pstart, options);
D(M==0) = Inf;
Display the distance function to the boundary.
clf;
hold on;
display_shape_function(D);
h = plot(pstart(2,:), pstart(1,:), 'r'); set(h, 'LineWidth', lw); axis ij;
Display the index of the closest boundary point.
clf;
hold on;
display_shape_function(Q);
h = plot(pstart(2,:), pstart(1,:), 'r'); set(h, 'LineWidth', lw); axis ij;
Exercise 1
Compute the norm of the gradient |G| modulo |nbound|. Be careful to remove the boundary of the shape from this indicator. Display the thresholded gradient map. radient ompute the norm of the gadient. emove the boundary to the skeletton.
exo1()
%% Insert your code here.
Exercise 2
Display the Skeleton obtained for different threshold values.
exo2()
%% Insert your code here.