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.

通过研究配备了黎曼度量的流形，拉普拉斯 - 贝尔特拉米 (Laplace-Beltrami) 算子产生。本文探索了 Finsler 流形作为黎曼流形的一种推广，并提出了 Finsler 热核和 Finsler-Laplace-Beltrami 算子 (FLBO)，为空间滤波和形状对应估计提供了一种有效的替代方法。希望这个提议能够激发计算机视觉领域对 Finsler 几何的进一步探索。