Differentiable Geodesic Distance for Intrinsic Minimization on Triangle Meshes

TL;DR

This paper introduces a novel differentiable geodesic distance framework on triangle meshes, enabling efficient intrinsic minimization for complex geometry processing tasks using second-order methods.

Contribution

It presents a variational formulation for intrinsic distances that allows computation of derivatives, facilitating Newton-type minimization on discrete surfaces.

Findings

Second-order methods outperform first-order in geodesic minimization.

Framework successfully applied to geodesic networks and shape analysis.

Demonstrates efficiency and versatility on complex surfaces.

Abstract

Computing intrinsic distances on discrete surfaces is at the heart of many minimization problems in geometry processing and beyond. Solving these problems is extremely challenging as it demands the computation of on-surface distances along with their derivatives. We present a novel approach for intrinsic minimization of distance-based objectives defined on triangle meshes. Using a variational formulation of shortest-path geodesics, we compute first and second-order distance derivatives based on the implicit function theorem, thus opening the door to efficient Newton-type minimization solvers. We demonstrate our differentiable geodesic distance framework on a wide range of examples, including geodesic networks and membranes on surfaces of arbitrary genus, two-way coupling between hosting surface and embedded system, differentiable geodesic Voronoi diagrams, and efficient computation of Karcher means on complex shapes. Our analysis shows that second-order descent methods based on our differentiable geodesics outperform existing first-order and quasi-Newton methods by large margins.