Differentiable Geodesic Distance for Intrinsic Minimization on Triangle Meshes

TL;DR

The paper tackles the challenge of optimizing distance-based objectives on triangle meshes by making geodesic distance differentiable. It introduces a differentiable geodesic distance via a variational shortest-path formulation and implicit differentiation, yielding closed-form first and second derivatives suitable for Newton-type solvers. The framework supports elastic geodesic networks, geodesic membranes, two-way coupling with hosts, differentiable geodesic Voronoi diagrams, and Karcher means, demonstrating substantial convergence and performance gains over first-order methods. This intrinsic minimization approach enables robust, second-order optimization in complex geometry processing tasks, with broad implications for geometric modeling and simulation. The method preserves intrinsic geometry, avoids path-tracking complexities, and leverages exact geodesics to achieve accurate, fast convergence across diverse topologies and applications.

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.

Figures (18)

• Figure 1: Embedded two-way coupling. We simulate a geodesic elastic network embedded in the surface of a deformable bunny, modeled with solid finite elements. Our analytical geodesic distance derivatives allow us to use Newton's method for simulating the deformations induced by tightening the network.
• Figure 2: Behavior of geodesic paths passing over different types of vertices. Fixing one endpoint of a geodesic, we translate the other such that the path moves across the center vertex. Whereas the geodesic passes through the center vertex in the planar and hyperbolic case, it jumps over the spherical vertex to avoid the local distance maximum.
• Figure 3: Mollification for edge intersection points. We smoothly blend linear edge intersection trajectories $\mathbf{x}(t)$ with cubic functions (b) to achieve $C^2$-continuity when geodesics pass through mesh vertices (a).
• Figure 4: Elastic geodesic spring networks. We initialize the nodes (shown in red) in close proximity to either the vertices of the hosting mesh or its edge midpoints. Our method converges robustly in both scenarios.
• Figure 5: Qualitative comparisons with the Vector Heat Method sharp2019vector for computing Karcher Means on a selection of meshes. The yellow curves show the optimization trajectories toward the Karcher mean of a given set of points (shown in red). The initial guesses (chosen randomly) are shown in white and the Karcher mean in blue. The color gradient indicates steps toward the solution. As can be seen from these examples, our second-order solver allows for larger steps toward the minima (col 1-3) and significantly fewer iterations to convergence (col 4-5).