Sparse Approximation of 3D Meshes using the Spectral Geometry of the Hamiltonian Operator

TL;DR

This paper introduces a novel spectral method for 3D mesh compression that incorporates vertex order and local geometric details via a Hamiltonian operator with a data-dependent potential, outperforming traditional spectral approaches.

Contribution

It proposes a new spectral domain based on a Hamiltonian operator with a potential function that encodes vertex order, enhancing mesh compression accuracy.

Findings

Improved mesh compression performance over standard Laplacian methods.

Effective encoding of local geometric details using the potential function.

Enhanced spectral representation tailored to surface features.

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.