Mesh Deformation

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This tour explores deformation of 2D mesh using Laplacian interpolation. The dense deformation field is obtained from a sparse set of displaced anchor point by computing harmonic interpolation.

In [2]:

Mesh Creation

We create a simple mesh with fine scale details.

We generate point on a square.

In [3]:
p = 150;
[Y,X] = meshgrid(linspace(-1,1,p),linspace(-1,1,p));
vertex0 = [X(:)'; Y(:)'; zeros(1,p^2)];
n = size(vertex0,2);

We generate a triangulation of a square.

In [4]:
I = reshape(1:p^2,p,p);
a = I(1:p-1,1:p-1); b = I(2:p,1:p-1); c = I(1:p-1,2:p);
d = I(2:p,1:p-1); e = I(2:p,2:p); f = I(1:p-1,2:p);
faces = cat(1, [a(:) b(:) c(:)], [d(:) e(:) f(:)])';

Width and height of the bumps.

In [5]:
sigma = .03;
h = .35;
q = 8;

Elevate the surface using bumps.

In [6]:
t = linspace(-1,1,q+2); t([1 length(t)]) = [];
vertex = vertex0;
for i=1:q
    for j=1:q
        d = (X(:)'-t(i)).^2 + (Y(:)'-t(j)).^2;
        vertex(3,:) = vertex(3,:) + h * exp( -d/(2*sigma^2)  );

Display the surface.

In [7]:

Compute its geometric (cotan) Laplacian

In [8]:
W = sparse(n,n);
for i=1:3
   i1 = mod(i-1,3)+1;
   i2 = mod(i  ,3)+1;
   i3 = mod(i+1,3)+1;
   pp = vertex(:,faces(i2,:)) - vertex(:,faces(i1,:));
   qq = vertex(:,faces(i3,:)) - vertex(:,faces(i1,:));
   % normalize the vectors   
   pp = pp ./ repmat( sqrt(sum(pp.^2,1)), [3 1] );
   qq = qq ./ repmat( sqrt(sum(qq.^2,1)), [3 1] );
   % compute angles
   ang = acos(sum(pp.*qq,1));
   u = cot(ang);
   u = clamp(u, 0.01,100);
   W = W + sparse(faces(i2,:),faces(i3,:),u,n,n);
   W = W + sparse(faces(i3,:),faces(i2,:),u,n,n);

Compute the symmetric Laplacian matrix.

In [9]:
d = full( sum(W,1) );
D = spdiags(d(:), 0, n,n);
L = D - W;

Boundary Modification

We modify the domain by modifying its boundary.

Select boundary indexes.

In [10]:
I = find( abs(X(:))==1 | abs(Y(:))==1 );

Compute the deformation field (zeros outsize the handle, proportional to the normal otherwise).

In [11]:
Delta0 = zeros(3,n);
d = ( vertex(1,I) + vertex(2,I) ) / 2;
Delta0(3,I) = sign(d) .* abs(d).^3;

Modify the Laplacian to take into account the fixed handles.

In [12]:
L1 = L;
L1(I,:) = 0;
L1(I + (I-1)*n) = 1;

Compute the full deformation by solving for Laplacian=0 on each coordinate.

In [13]:
Delta = ( L1 \ Delta0' )';

Compute the deformed mesh.

In [14]:
vertex1 = vertex+Delta;

Display it.

In [15]:

Exercise 1

Perform a more complicated deformation of the boundary. eform isplay it.

In [16]:
In [17]:
%% Insert your code here.

Exercise 2

Move both the inside and the boundary.

aplacian eform isplay it.

In [18]:
In [19]:
%% Insert your code here.

Exercise 3

Apply the mesh deformation method to a real mesh, with both large scale and fine scale details.

In [20]:
In [21]:
%% Insert your code here.

Non-linear Deformation

Linear methods give poor results for large deformation.

It is possible to obtain better result by applying the linear deformation only to a low pass version of the mesh (coarse scale modifications). The remaining details are then added in the direction of the normal, in a local frame that is rotated to match the deformation of the coarse surface.

Exercise 4

Apply the deformation to the coarse mesh |vertex0| to obtain |vertex1|. Important: you need to compute and use the cotan Laplacian of the coarse mesh, not of the original mesh! ompute laplacian

aplacian eform isplay it.

In [22]:
In [23]:
%% Insert your code here.

Compute the residual vector contribution along the normal (which is vertical).

In [24]:
normal = compute_normal(vertex0,faces);
d = repmat( sum(normal .* (vertex-vertex0)), [3 1]);

Exercise 5

Add the normal contribution |d.*normal| to |vertex1|, but after replacing the normal of |vertex0| by the normal of |vertex1|. isplay it.

In [25]: