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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.
from __future__ import division
import nt_toolbox as nt
from nt_solutions import fastmarching_6_sampling_surf as solutions
%matplotlib inline
%load_ext autoreload
%autoreload 2
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.
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
solutions.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.
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.
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.
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
solutions.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.
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
solutions.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.
solutions.exo4()
## Insert your code here.