GeometryProcessing/Spring2009/Schedule/Spectral Processing3

• John Meier

I feel like I'm getting a better grasp of the concepts spectral processing is based on and what can be accomplished using it. This is the high-level understanding I have so far (comments/corrections welcome):

Method

Build a (roughly) symmetric matrix encoding a useful pairwise relationship between vertices in the mesh (e.g., Discrete approx. of the Laplacian or Laplace-Beltrami operators). Since the resulting matrix is similar to a symmetric matrix, it has real eigenvalues that we can compute. Since the eigenvectors form an n-dimensional orthogonal basis, we can project each mesh vertex into the resulting space and operate on the mesh "signal".

Applications

Smoothing - Taubin proposed a method for effectively low-pass filtering the mesh in the lower dimensions of the eigenspace that does not even require explicit computation of the eigen information.

Segmentation - Liu et. al. propose a method for performing mesh segmentation by examining the projection of the mesh in lower-dimension subspaces of the eigen space of two Laplacian operators.

I'm still a little unclear on what the distinction is between eigenvectors and eigenfunctions (interpolations of the respective eigenvectors?), and how the latter are used to produce the mesh colorings we saw at the end of lecture last week.