On the optimality of spectral compression of mesh data

Reviewer: Bruce E. Litow

Ben-Chen and Gotsman present strong evidence that in one and two dimensions, optimal compression of mesh geometries is achieved by using the eigenvectors of the mesh Laplacian, ordered from least to largest corresponding eigenvalue (the Laplacian is a symmetric real matrix, so its eigenvalues are all real). The evidence rests on a theorem that up to a scalar multiplier, dependent only on the mesh boundary conditions, the covariance matrix of a random mesh equals the mesh Laplacian. In one dimension, the uniform independent distribution for mesh geometries can be assumed, but in two dimensions a somewhat more complicated distribution is required. There are three issues with two dimensions. First, the proof is an asymptotic result that introduces the requirement that the mesh average valence goes to infinity as the mesh size goes to infinity. This brings up the second issue, which is rate of convergence to the limit distribution as a function of valence. This matter is left open, although some experimental evidence suggests that already good agreement is attained for a valence of six. Finally, there is the issue of the realism of the mesh geometry distribution needed for the proof. The authors provide reasonable experimental evidence that it can represent a wide range of interesting two-dimensional mesh geometries. The extension of this result to three dimensions is mainly an open problem. It is worth mentioning that the authors look at, but do not resolve, the rate of decrease of the eigenvalues of the Laplacian. Of course, in any single application, this could be computed, but it would be interesting to have more general, useful estimates since this goes directly to the efficiency of compression. A sensitivity analysis, that is, what happens to the mesh when only a few majorant eigenvalues are used, is not carried out. Online Computing Reviews Service