Finsler-Laplace-Beltrami Operators with Application to Shape Analysis
Simon Weber Thomas Dagès Maolin Gao Daniel Cremers
The Laplace-Beltrami operator (LBO) emerges from studying manifolds equipped
with a Riemannian metric. It is often called the Swiss army knife of geometry
processing as it allows to capture intrinsic shape information and gives rise
to heat diffusion, geodesic distances, and a multitude of shape descriptors. It
also plays a central role in geometric deep learning. In this work, we explore
Finsler manifolds as a generalization of Riemannian manifolds. We revisit the
Finsler heat equation and derive a Finsler heat kernel and a
Finsler-Laplace-Beltrami Operator (FLBO): a novel theoretically justified
anisotropic Laplace-Beltrami operator (ALBO). In experimental evaluations we
demonstrate that the proposed FLBO is a valuable alternative to the traditional
Riemannian-based LBO and ALBOs for spatial filtering and shape correspondence
estimation. We hope that the proposed Finsler heat kernel and the FLBO will
inspire further exploration of Finsler geometry in the computer vision
community.