Title: Sparse Approximation of 3D Meshes using the Spectral Geometry of the Hamiltonian Operator,
Abstract: The discrete Laplace operator is ubiquitous in spectral shape analysis, since
its eigenfunctions are provably optimal in representing smooth functions
defined on the surface of the shape. Indeed, subspaces defined by its
eigenfunctions have been utilized for shape compression, treating the
coordinates as smooth functions defined on the given surface. However, surfaces
of shapes in nature often contain geometric structures for which the general
smoothness assumption may fail to hold. At the other end, some explicit mesh
compression algorithms utilize the order by which vertices that represent the
surface are traversed, a property which has been ignored in spectral
approaches. Here, we incorporate the order of vertices into an operator that
defines a novel spectral domain. We propose a method for representing 3D meshes
using the spectral geometry of the Hamiltonian operator, integrated within a
sparse approximation framework. We adapt the concept of a potential function
from quantum physics and incorporate vertex ordering information into the
potential, yielding a novel data-dependent operator. The potential function
modifies the spectral geometry of the Laplacian to focus on regions with finer
details of the given surface. By sparsely encoding the geometry of the shape
using the proposed data-dependent basis, we improve compression performance
compared to previous results that use the standard Laplacian basis and spectral
graph wavelets.