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This tour explores geodesic computations on 3D meshes.
from __future__ import division
import nt_toolbox as nt
from nt_solutions import fastmarching_4_mesh as solutions
%matplotlib inline
%load_ext autoreload
%autoreload 2
Using the fast marching on a triangulated surface, one can compute the distance from a set of input points. This function also returns the segmentation of the surface into geodesic Voronoi cells.
Load a 3D mesh.
name = 'elephant-50kv'
[vertex, faces] = read_mesh(name)
nvert = size(vertex, 2)
Starting points for the distance computation.
nstart = 15
pstarts = floor(rand(nstart, 1)*nvert) + 1
options.start_points = pstarts
No end point for the propagation.
clear options
options.end_points = []
Use a uniform, constant, metric for the propagation.
options.W = ones(nvert, 1)
Compute the distance using Fast Marching.
options.nb_iter_max = Inf
[D, S, Q] = perform_fast_marching_mesh(vertex, faces, pstarts, options)
Display the distance on the 3D mesh.
plot_fast_marching_mesh(vertex, faces, D, [], options)
Extract precisely the voronoi regions, and display it.
[Qexact, DQ, voronoi_edges] = compute_voronoi_mesh(vertex, faces, pstarts, options)
options.voronoi_edges = voronoi_edges
plot_fast_marching_mesh(vertex, faces, D, [], options)
Exercise 1
Using |options.nb_iter_max|, display the progression of the propagation.
solutions.exo1()
## Insert your code here.
Geodesic path are extracted using gradient descent of the distance map.
Select random endding points, from which the geodesic curves start.
nend = 40
pend = floor(rand(nend, 1)*nvert) + 1
Compute the vertices 1-ring.
vring = compute_vertex_ring(faces)
Exercise 2
For each point |pend(k)|, compute a discrete geodesic path |path| such that |path(1)=pend(k)| and |D(path(i+1))<D(path(i))| with |[path(i), path(i+1)]| being an edge of the mesh. This means that |path(i+1)| is an element of |vring{path(i)}|. Display the paths on the mesh.
solutions.exo2()
## Insert your code here.
In order to extract a smooth path, one needs to use a gradient descent.
options.method = 'continuous'
paths = compute_geodesic_mesh(D, vertex, faces, pend, options)
Display the smooth paths.
plot_fast_marching_mesh(vertex, faces, Q, paths, options)
In order to extract salient features of a surface, one can define a speed function that depends on some curvature measure of the surface.
Load a mesh with sharp features.
clear options
name = 'fandisk'
[vertex, faces] = read_mesh(name)
options.name = name
nvert = size(vertex, 2)
Display it.
plot_mesh(vertex, faces, options)
Compute the curvature.
options.verb = 0
[Umin, Umax, Cmin, Cmax] = compute_curvature(vertex, faces, options)
Compute some absolute measure of curvature.
C = abs(Cmin) + abs(Cmax)
C = min(C, .1)
Display the curvature on the surface
options.face_vertex_color = rescale(C)
plot_mesh(vertex, faces, options)
colormap jet(256)
Compute a metric that depends on the curvature. Should be small in area that the geodesic should follow.
epsilon = .5
W = rescale(-min(C, 0.1), .1, 1)
Display the metric on the surface
options.face_vertex_color = rescale(W)
plot_mesh(vertex, faces, options)
colormap jet(256)
Starting points.
pstarts = [2564; 16103; 15840]
options.start_points = pstarts
Compute the distance using Fast Marching.
options.W = W
options.nb_iter_max = Inf
[D, S, Q] = perform_fast_marching_mesh(vertex, faces, pstarts, options)
Display the distance on the 3D mesh.
options.colorfx = 'equalize'
plot_fast_marching_mesh(vertex, faces, D, [], options)
Exercise 3
Using |options.nb_iter_max|, display the progression of the propagation for constant |W|.
solutions.exo3()
## Insert your code here.
Exercise 4
Using |options.nb_iter_max|, display the progression of the propagation for a curvature based |W|.
solutions.exo4()
## Insert your code here.
Exercise 5
Extract geodesics. ompute distances ompute paths isplay
solutions.exo5()
## Insert your code here.
One can take into account a texture to design the speed function.
clear options
options.base_mesh = 'ico'
options.relaxation = 1
options.keep_subdivision = 0
[vertex, faces] = compute_semiregular_sphere(7, options)
nvert = size(vertex, 2)
Load a function on the mesh.
name = 'earth'
f = load_spherical_function(name, vertex, options)
options.name = name
Starting points.
pstarts = [2844; 5777]
options.start_points = pstarts
Display the function.
plot_fast_marching_mesh(vertex, faces, f, [], options)
colormap gray(256)
Load and display the gradient magnitude of the function.
g = load_spherical_function('earth-grad', vertex, options)
Display it.
plot_fast_marching_mesh(vertex, faces, g, [], options)
colormap gray(256)
Design a metric.
W = rescale(-min(g, 10), 0.01, 1)
Display it.
plot_fast_marching_mesh(vertex, faces, W, [], options)
colormap gray(256)
Exercise 6
Using |options.nb_iter_max|, display the progression of the propagation for a curvature based |W|.
solutions.exo6()
## Insert your code here.
Exercise 7
Extract geodesics. ompute distances ompute paths isplay
solutions.exo7()
## Insert your code here.