Laplace-Beltrami Eigenfunctions Towards an Algorithm That "Understands" Geometry

TLDR

Geometry processing seeks to derive higher‑level representations from raw data, such as parameterizations that enable information attachment and conversion between forms, and more broadly aims to construct structured function bases on surfaces. The paper investigates hierarchical function bases defined by Laplace‑Beltrami eigenfunctions. The authors model the eigenfunctions as vibration‑mode analogues, provide an intuitive explanation, then detail how to approximate them numerically and demonstrate applications in geometry processing. On a sphere the basis reduces to spherical harmonics, while on general shapes it yields a geometry‑ and topology‑adapted basis, and the authors demonstrate practical computation and applications in geometry processing.

Abstract

One of the challenges in geometry processing is to automatically reconstruct a higher-level representation from raw geometric data. For instance, computing a parameterization of an object helps attaching information to it and converting between various representations. More generally, this family of problems may be thought of in terms of constructing structured function bases attached to surfaces. In this paper, we study a specific type of hierarchical function bases, defined by the eigenfunctions of the Laplace-Beltrami operator. When applied to a sphere, this function basis corresponds to the classical spherical harmonics. On more general objects, this defines a function basis well adapted to the geometry and the topology of the object. Based on physical analogies (vibration modes), we first give an intuitive view before explaining the underlying theory. We then explain in practice how to compute an approximation of the eigenfunctions of a differential operator, and show possible applications in geometry processing