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.

翻译：拉普拉斯-贝尔特拉米算子（LBO）源于对配备黎曼度量的流形的研究。它常被称为几何处理领域的瑞士军刀，能够捕捉内在形状信息，并衍生出热扩散、测地距离以及众多形状描述子。该算子在几何深度学习中也发挥着核心作用。本文探索了作为黎曼流形推广的芬斯勒流形。我们重新审视了芬斯勒热方程，推导出芬斯勒热核与芬斯勒-拉普拉斯-贝尔特拉米算子（FLBO）：一种新颖且具有理论依据的各向异性拉普拉斯-贝尔特拉米算子（ALBO）。实验评估表明，所提出的FLBO在空间滤波和形状对应估计中，是传统黎曼LBO及ALBO的优质替代方案。我们期望所提出的芬斯勒热核与FLBO能激发计算机视觉领域对芬斯勒几何的进一步探索。