Finsler-Laplace-Beltrami Operators with Application to Shape Analysis

Simon Weber, Thomas Dagès, Maolin Gao, Daniel Cremers; Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2024, pp. 3131-3140

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.