Finsler-Laplace-Beltrami Operators with Application to Shape Analysis

TL;DR

This work extends intrinsic shape analysis beyond Riemannian geometry by introducing Finsler manifolds and a theoretically grounded Finsler-Laplace-Beltrami operator (FLBO). Building on Randers metrics, the authors derive a Finsler heat equation and its heat kernel, linking diffusion in anisotropic, direction-dependent spaces to a robust operator for geometry processing. They propose a practical discretization that yields an ALBO-like operator with a diffusion tensor $D_{\mathcal{F}_x^*}$ and a corresponding anisotropic convolution framework, enabling efficient spectral filtering across multiple orientations. Experiments on standard shape datasets (FAUST, SCAPE, SHREC) demonstrate FLBO’s effectiveness for full and partial shape matching, with competitive performance to state-of-the-art methods and accessible open-source code, highlighting the potential of Finsler geometry in computer vision and geometric DL.

Abstract

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.

Figures (9)

• Figure 1: Simplified overview of the derivation of the Finsler-Laplace-Beltrami operator (FLBO), built from a Finsler metric. It generalizes the traditional heuristic anisotropic Laplace-Beltrami operators (ALBO) boscaini2016li2020 (top). The FLBO can directly replace the ALBOs in surface processing tasks like shape correspondence, notably by constructing shape-dependent anisotropic convolutions in their spectral domain.
• Figure 2: Due to non-uniform wind currents, the geodesic curve from $A$ to $B$ differs from the one from $B$ to $A$. While not possible in Riemannian geometry, such asymmetries are characteristic of Finsler manifolds.
• Figure 3: Notations for unit vectors and angles on the triangular mesh. The figure mimics the illustration in boscaini2016.
• Figure 4: Visual shape correspondence results on the FAUST Remeshed (row 1), SCAPE remeshed (row 2), and SHREC'16 Partial Cuts (columns 1 and 2 in rows 3 to 5), and SHREC'16 Partial Holes (columns 1 and 3 in rows 3 to 5) datasets. The source shape is on the left, on which we perform dense correspondence estimation on the shapes to the right. See the supplementary material for more visual results on other shapes.
• Figure 5: Finsler-based anisotropic kernels $g_{\alpha\theta,x}$ centered at the same point $x$ with different rotation angles $\theta$ equal to $0$ (left) and $\tfrac{3\pi}{8}$ (right). Here the filter is chosen to be the Chebychev polynomial $\hat{g}(\lambda) = T_5(\lambda)$.

Theorems & Definitions (8)

• Definition 1: Finsler metric
• Definition 2: Randers metric
• Definition 3: Dual Finsler metric
• Lemma 1: Dual Randers metric
• Definition 4: Finsler heat equation
• Proposition 1: Simplified Randers heat equation - distance trick
• proof
• Proposition 2