$\newcommand{\dotp}[2]{\langle #1, #2 \rangle}$ $\newcommand{\enscond}[2]{\lbrace #1, #2 \rbrace}$ $\newcommand{\pd}[2]{ \frac{ \partial #1}{\partial #2} }$ $\newcommand{\umin}[1]{\underset{#1}{\min}\;}$ $\newcommand{\umax}[1]{\underset{#1}{\max}\;}$ $\newcommand{\umin}[1]{\underset{#1}{\min}\;}$ $\newcommand{\uargmin}[1]{\underset{#1}{argmin}\;}$ $\newcommand{\norm}[1]{\|#1\|}$ $\newcommand{\abs}[1]{\left|#1\right|}$ $\newcommand{\choice}[1]{ \left\{ \begin{array}{l} #1 \end{array} \right. }$ $\newcommand{\pa}[1]{\left(#1\right)}$ $\newcommand{\diag}[1]{{diag}\left( #1 \right)}$ $\newcommand{\qandq}{\quad\text{and}\quad}$ $\newcommand{\qwhereq}{\quad\text{where}\quad}$ $\newcommand{\qifq}{ \quad \text{if} \quad }$ $\newcommand{\qarrq}{ \quad \Longrightarrow \quad }$ $\newcommand{\ZZ}{\mathbb{Z}}$ $\newcommand{\CC}{\mathbb{C}}$ $\newcommand{\RR}{\mathbb{R}}$ $\newcommand{\EE}{\mathbb{E}}$ $\newcommand{\Zz}{\mathcal{Z}}$ $\newcommand{\Ww}{\mathcal{W}}$ $\newcommand{\Vv}{\mathcal{V}}$ $\newcommand{\Nn}{\mathcal{N}}$ $\newcommand{\NN}{\mathcal{N}}$ $\newcommand{\Hh}{\mathcal{H}}$ $\newcommand{\Bb}{\mathcal{B}}$ $\newcommand{\Ee}{\mathcal{E}}$ $\newcommand{\Cc}{\mathcal{C}}$ $\newcommand{\Gg}{\mathcal{G}}$ $\newcommand{\Ss}{\mathcal{S}}$ $\newcommand{\Pp}{\mathcal{P}}$ $\newcommand{\Ff}{\mathcal{F}}$ $\newcommand{\Xx}{\mathcal{X}}$ $\newcommand{\Mm}{\mathcal{M}}$ $\newcommand{\Ii}{\mathcal{I}}$ $\newcommand{\Dd}{\mathcal{D}}$ $\newcommand{\Ll}{\mathcal{L}}$ $\newcommand{\Tt}{\mathcal{T}}$ $\newcommand{\si}{\sigma}$ $\newcommand{\al}{\alpha}$ $\newcommand{\la}{\lambda}$ $\newcommand{\ga}{\gamma}$ $\newcommand{\Ga}{\Gamma}$ $\newcommand{\La}{\Lambda}$ $\newcommand{\si}{\sigma}$ $\newcommand{\Si}{\Sigma}$ $\newcommand{\be}{\beta}$ $\newcommand{\de}{\delta}$ $\newcommand{\De}{\Delta}$ $\newcommand{\phi}{\varphi}$ $\newcommand{\th}{\theta}$ $\newcommand{\om}{\omega}$ $\newcommand{\Om}{\Omega}$
This tour explores geodesic computations on 3D meshes.
warning off
addpath('toolbox_signal')
addpath('toolbox_general')
addpath('toolbox_graph')
addpath('toolbox_wavelet_meshes')
addpath('solutions/fastmarching_4_mesh')
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.
clf;
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.
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.
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.
clf;
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);
clf;
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);
clf;
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';
clf;
plot_fast_marching_mesh(vertex,faces, D, [], options);
Exercise 3
Using |options.nb_iter_max|, display the progression of the propagation for constant |W|.
exo3()
%% Insert your code here.
Exercise 4
Using |options.nb_iter_max|, display the progression of the propagation for a curvature based |W|.
exo4()
%% Insert your code here.
Exercise 5
Extract geodesics. ompute distances ompute paths isplay
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.
clf;
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.
clf;
plot_fast_marching_mesh(vertex,faces, g, [], options);
colormap gray(256);
Design a metric.
W = rescale(-min(g,10),0.01,1);
Display it.
clf;
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|.
exo6()
%% Insert your code here.
Exercise 7
Extract geodesics. ompute distances ompute paths isplay
exo7()
%% Insert your code here.