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This tour explores geodesic remeshing of surfaces.
This method is introduced in
Geodesic Remeshing Using Front Propagation Gabriel Peyr and Laurent Cohen, International Journal on Computer Vision, Vol. 69(1), p.145-156, Aug. 2006.
addpath('toolbox_signal')
addpath('toolbox_general')
addpath('toolbox_graph')
addpath('solutions/fastmarching_6_sampling_surf')
An uniform sampling of points on a surface is obtained using a greedy farthest point sampling.
Load a 3D mesh.
clear options;
name = 'bunny';
[vertex,faces] = read_mesh(name);
n = size(vertex,2);
options.name = name;
Display it.
clf;
plot_mesh(vertex,faces, options);
Pick a first point.
landmarks = [100];
Compute the geodesic distance to this point.
[D,Z,Q] = perform_fast_marching_mesh(vertex, faces, landmarks);
Display the geodesic distance to the point.
clf; hold on;
options.face_vertex_color = mod( 20*D/max(D),1 );
plot_mesh(vertex,faces, options);
colormap jet(256);
h = plot3(vertex(1,landmarks), vertex(2,landmarks), vertex(3,landmarks), 'r.');
set(h, 'MarkerSize', 20);
Select as the next sampling point the farthest point.
[tmp,landmarks(end+1)] = max(D);
Update the distance map using a local propagation.
options.constraint_map = D;
[D1,Z,Q] = perform_fast_marching_mesh(vertex, faces, landmarks,options);
D = min(D,D1);
Display the update distance map.
clf; hold on;
options.face_vertex_color = mod( 20*D/max(D),1 );
plot_mesh(vertex,faces, options);
colormap jet(256);
h = plot3(vertex(1,landmarks), vertex(2,landmarks), vertex(3,landmarks), 'r.');
set(h, 'MarkerSize', 20);
Exercise 1
Perform the farthest point sampling of |m=500| points. nitialize
exo1()
%% Insert your code here.
An intrinsic triangulation of the point is obtained using the geodesic Delaunay triangulation.
Compute the voronoi map |Q| of the segmentation.
[D,Z,Q] = perform_fast_marching_mesh(vertex, faces, landmarks);
Display the update distance map.
[B,I,J] = unique(Q);
v = randperm(m)'; J = v(J);
clf; hold on;
options.face_vertex_color = J;
plot_mesh(vertex,faces, options);
colormap jet(256);
h = plot3(vertex(1,landmarks), vertex(2,landmarks), vertex(3,landmarks), 'k.');
set(h, 'MarkerSize', 15);
Count the number |d(i)| of different voronoi indexes for each face |i|.
V = Q(faces); V = sort(V,1);
V = unique(V', 'rows')';
d = 1 + (V(1,:)~=V(2,:)) + (V(2,:)~=V(3,:));
Select the faces with 3 different indexe, they corresponds to geodesic Delaunay faces.
I = find(d==3); I = sort(I);
Build the Delaunay faces set.
z = zeros(n,1);
z(landmarks) = (1:m)';
facesV = z(V(:,I));
Position of the vertices of the subsampled mesh.
vertexV = vertex(:,landmarks);
Re-orient the faces so that they point outward of the mesh.
options.method = 'slow';
options.verb = 0;
facesV = perform_faces_reorientation(vertexV,facesV, options);
Display the sub-sampled mesh.
clf;
options.face_vertex_color = [];
plot_mesh(vertexV,facesV, options);
shading faceted;
It is possible to seed more point on a given part of the mesh.
Create a density function by designing an isotropic metric. Here we use a metric that is slower in the left part.
W = ones(n,1);
W(vertex(1,:)<median(vertex(1,:))) = .4;
options.W = W;
Display the speed function.
clf;
hold on;
options.face_vertex_color = W;
plot_mesh(vertex,faces, options);
colormap jet(256);
Perform front propagation using this speed function.
landmarks = [5000];
options.constraint_map = [];
[D,Z,Q] = perform_fast_marching_mesh(vertex, faces, landmarks, options);
Display the distance map.
clf;
hold on;
options.face_vertex_color = mod( 20*D/max(D),1 );
plot_mesh(vertex,faces, options);
colormap jet(256);
h = plot3(vertex(1,landmarks), vertex(2,landmarks), vertex(3,landmarks), 'r.');
set(h, 'MarkerSize', 20);
Exercise 2
Perform a spacially adative remeshing. nitialize
exo2()
%% Insert your code here.
A better remeshing quality is obtained by sampling more densly sharp features. This is achieved using a spatially varying metric, so that the front propagate slowly near regions of high curvature.
Compute the curvature of the mesh.
[Umin,Umax,Cmin,Cmax,Cmean,Cgauss,Normal] = compute_curvature(vertex,faces,options);
Compute the total curvature.
C = abs(Cmin)+abs(Cmax);
Display it.
clf;
hold on;
options.face_vertex_color = min(C,.1);
plot_mesh(vertex,faces, options);
colormap jet(256);
Exercise 3
Design a metric |W| so that the sampling is densed in area where |C| is large.
isplay
exo3()
%% Insert your code here.
Exercise 4
Use such a metric to perform feature sensitive remeshing. Tune the metric to reduce as much as possible the Hausdorff approximation error.
exo4()
%% Insert your code here.