7.3.3 Salient Spectral Geometric Features

If M is a domain in the Euclidean plane, the Laplace-Beltrami operator reduces to the well-known expression

Thus, we are interested in the Helmholtz equation

where λ is a real scalar. This equation is important because the family of eigenvalues 0 ≤ λ0 ≤ λ1 ≤ · · · ≤ +∞ is the shape spectrum which is isometric invariant1 as it only depends on the gradient and the divergence, dependent only on the Riemannian structure of the manifold. Another important property is that the LaplaceBeltrami operator is Hermitian, so the eigenvectors vi and vj corresponding to different eigenvalues λi and λj are orthogonal:

vi · vj = vi vj = 0, i = j (7.17)

M

In addition, the i-th eigenvector defines a coefficient over the function f as fol-

Given a 3D mesh, we can calculate the discrete Laplace-Beltrami operator at each vertex with the cotangent scheme proposed by Meyer et al. [77]:

where Ai is the Voronoi region area around pi , αij and βij are the angles opposite to the arc pi pj and N1(pi ) is the set of pi ’s adjacent vertices. See Fig. 7.5 for details.

The computation of Ai , the Voronoi region around a vertex pi , must be performed as follows. We accumulate a region for each adjacent face T to pi , if T is non-obtuse we add the Voronoi region for that face by using Eq. (7.21). If T is obtuse, we add area(T)/2 if the angle of T at pi is obtuse, or area(T)/4 otherwise.

1The shape spectrum of two shapes with a non-rigid transformation is approximately the same.

The shape spectrum is the set of eigenvalues {λ0, λ1, λ2, . . . , λn−1}. If the shape is closed, λ0 = 0.

7.3.3.1 Feature Points Detection

Recalling the problem of geometric reconstruction, we have the vertices set P = [X Y Z] where X, Y and Z are column vectors containing the values of their coordinates. In addition, we define a coefficient matrix C = PT SV where S is the diagonal matrix as stated before and V is a N × N matrix with each column being an eigenvector. The eigenvectors are ordered according to their associated eigenvalues, from low to high. Thus, we can obtain the vertices set with:

We introduce the matrix notation A1−p,1−q as the submatrix with rows from 1 to p and columns from 1 to q. Using this notation, we can express the reconstruction problem using the first k eigenvectors:

P(k) = V1−n,1−k CT− − (7.29)

1 3,1 k

As can be noted, P(k) has the same size as P, but the vertex position is dependent on k. We can use P(k) with the original connections for evaluating the information gained with different k values (see Fig. 7.6). The authors defined a geometric energy at a vertex i corresponding to the kth eigenvector as:

The authors evaluated E in order to detect interest points. A vertex pi is selected if E(i, k) > E(j, h) for all vertex pj adjacent to pi in the original mesh and for all

h [k − 1, k, k + 1]. Furthermore, each selected vertex has an associated scale fac-

√

tor sf = 1/ λk . An important fact to be considered here is that the same vertex can be selected with different associated scale factors. The authors proposed to evaluate only the first 100 eigenvectors because they claimed that more eigenvectors do not contribute significant spatial and spectral energy. The first eigenvectors are associated to the coarse structure of the mesh and later eigenvectors are associated with the detailed geometry. Thus the last eigenvectors could be associated with noise. In this respect, the number of vertices could give us an insight concerning how many eigenvectors might be used, because the maximum number of eigenvectors equals the number of vertices.

Fig. 7.6 Effect of reconstructing P(k) using k = 5, 20, 50, 100, 300, and 1000, respectively over a shape with 8000+ vertices