Mesh Parameterization

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This tour explores 2-D parameterization of 3D surfaces using linear methods.

A review paper for mesh parameterization can be found in:

M.S. Floater and K. Hormann, Surface Parameterization: a Tutorial and Survey in Advances in multiresolution for geometric modelling, p. 157-186, 2005.

See also:

K. Hormann, K. Polthier and A. Sheffer Mesh parameterization: theory and practice, Siggraph Asia Course Notes

In [2]:

Conformal Laplacian

The conformal Laplacian uses the cotan weights to obtain an accurate discretization of the Laplace Beltrami Laplacian.

They where first introduces as a linear finite element approximation of the Laplace-Beltrami operator in:

U. Pinkall and K. Polthier, Computing discrete minimal surfaces and their conjugates Experimental Mathematics, 2(1):15-36, 1993.

First load a mesh. The faces are stored in a matrix $F = (f_j)_{j=1}^m \in \RR^{3 \times m}$ of $m$ faces $f_j \in \{1,\ldots,n\}^3$. The position of the vertices are stored in a matrix $X = (x_i)_{i=1}^n \in \RR^{3 \times n}$ of $n$ triplets of points $x_k \in \RR^3$

In [3]:
name = 'nefertiti';
[X,F] = read_mesh(name);
n = size(X,2);
clear options; = name;

In order to perform mesh parameterization, it is important that this mesh has the topology of a disk, i.e. it should have a single B.

First we compute the boundary $B = (i_1,\ldots,i_p)$ of the mesh. By definition, for the edges $(i_k,i_{k+1})$, there is a single adjacent face $ (i_k,i_{k+1},\ell) $.

In [4]:
options.verb = 0;
B = compute_boundary(F, options);

Length of the boundary.

In [5]:
p = length(B);

Display the boundary.

In [6]:
clf; hold on;
hh = plot3( X(1,B),X(2,B),X(3,B), 'r' );

The conformal Laplacian weight matrix $W \in \RR^{n \times n}$ is defined as $$ W_{i,j} = \choice{ \text{cotan}(\al_{i,j}) + \text{cotan}(\be_{i,j}) \qifq i \sim j \\ \quad 0 \quad \text{otherwise}. } $$ Here, $i \times j$ means that there exists two faces $ (i,j,k) $ and $ (i,j,\ell) $ in the mesh (note that for B faces, one has $k=\ell$).

The angles are the angles centered as $k$ and $\ell$, i.e. $$ \al_{i,j} = \widehat{x_i x_k x_j } \qandq \be_{i,j} = \widehat{x_i x_\ell x_j }. $$

Compute the conformal 'cotan' weights. Note that each angle $\alpha$ in the mesh contributes with $1/\text{tan}(\alpha)$ to the weight of the opposite edge. We compute $\alpha$ as $$ \alpha = \text{acos}\pa{ \frac{\dotp{u}{v}}{\norm{u}\norm{v}} } $$ where $u \in \RR^3, v \in \RR^3$ are the edges of the adjacent vertices that defines $\al$.

In [7]:
W = make_sparse(n,n);
for i=1:3
   i2 = mod(i  ,3)+1;
   i3 = mod(i+1,3)+1;
   u = X(:,F(i2,:)) - X(:,F(i,:));
   v = X(:,F(i3,:)) - X(:,F(i,:));
   % normalize the vectors   
   u = u ./ repmat( sqrt(sum(u.^2,1)), [3 1] );
   v = v ./ repmat( sqrt(sum(v.^2,1)), [3 1] );
   % compute angles
   alpha = 1 ./ tan( acos(sum(u.*v,1)) );
   alpha = max(alpha, 1e-2); % avoid degeneracy
   W = W + make_sparse(F(i2,:),F(i3,:), alpha, n, n );
   W = W + make_sparse(F(i3,:),F(i2,:), alpha, n, n );

Compute the symmetric Laplacian matrix $$ L = D-W \qwhereq D = \diag_i\pa{\sum_j W_{i,j}}. $$

In [8]:
d = full( sum(W,1) );
D = spdiags(d(:), 0, n,n);
L = D - W;

Fixed Boundary Harmonic Parameterization

The problem of mesh parameterization corresponds to finding 2-D locations $(y_i = (y_i^1,y_i^2) \in \RR^2$ for each original vertex, where $ Y = (y_i)_{i=1}^n \in \RR^{2 \times n} $ denotes the flattened positions.

The goal is for this parameterization to be valid, i.e. the 2-D mesh obtained by replacing $X$ by $Y$ but keeping the same face connectivity $F$ should not contained flipped faces (all face should have the same orientation in the plane).

We consider here a linear methods, that finds the parameterization, that impose that the coordinates are harmonic inside the domain, and have fixed position on the boundary (Dirichlet conditions) $$ \forall s=1,2, \quad \forall i \notin B, \quad (L y^s)_i = 0, \qandq \forall j \in B, \quad y^s_j = z_j^s. $$

In order for this method to define a valid parameterization, it is necessary that the fixed position $ z_j = (z^1_j,z^2_j) \in \RR^2 $ are consecutive points along a convex polygon.

Compute the fixed positions $Z=(z_j)_j$ for the vertices on $B$. Here we use a circle.

In [9]:
p = length(B);
t = linspace(0,2*pi(),p+1); t(p+1) = [];
Z = [cos(t); sin(t)];

Computing the parameterization requires to solve two independent linear system $$ \forall s=1,2, \quad L_1 y^s = r^s $$ where $L_1$ is a modified Laplacian, the is obtained from $L$ by $$ \choice{ \forall i \notin B, \quad (L_0)_{i,j} = L_{i,j} \\ \forall i \in B, \quad (L_0)_{i,i}=1, \\ \forall i \in B, \forall j \neq i, \quad (L_0)_{i,i}=0, } $$ i.e. replacing each row indexed by $B$ by a 1 on the diagonal.

In [10]:
L1 = L;
L1(B,:) = 0;
for i=1:length(B)
    L1(B(i),B(i)) = 1;

Set up the right hand size $R$ with the fixed position.

In [11]:
R = zeros(2, n);
R(:,B) = Z;

Solve the two linear systems. For Scilab user: this might take some time ...

In [12]:
if using_matlab()
    Y = (L1 \ R')'; 
    options.maxit = 300;
    Y(1,:) = perform_cg(L1,R(1,:)',options)';
    Y(2,:) = perform_cg(L1,R(2,:)',options)';

Display the parameterization.

In [13]:
clf; = 0;

Mesh Parameterization on a Square

One can perform a fixed B parameterization on a square. This is useful to compute a geometry image (a color image storring the position of the vertices).

Exercise 1

Compute the fixed positions $Z$ of the points indexed by $B$ that are along a square. Warning: $p$ is not divisible by 4.

In [14]:
In [15]:
%% Insert your code here.

Exercise 2

Compute the parameterization $Y$ on a square.

In [16]: