A Manifold of Planar Triangular Meshes with Complete Riemannian Metric

TL;DR

This paper introduces a smooth manifold structure for planar triangular meshes and proposes a geodesically complete Riemannian metric that preserves mesh quality during shape analysis tasks.

Contribution

It defines a new Riemannian metric on the manifold of triangular meshes that prevents mesh degradation and maintains connectivity during geodesic computations.

Findings

The metric preserves mesh connectivity and aspect ratios.

Numerical experiments confirm mesh quality is maintained.

The approach enables stable shape interpolation and registration.

Abstract

Shape spaces are fundamental in a variety of applications including image registration, morphing, matching, interpolation, and shape optimization. In this work, we consider two-dimensional shapes represented by triangular meshes of a given connectivity. We show that the collection of admissible configurations representable by such meshes form a smooth manifold. For this manifold of planar triangular meshes we propose a geodesically complete Riemannian metric. It is a distinguishing feature of this metric that it preserves the mesh connectivity and prevents the mesh from degrading along geodesic curves. We detail a symplectic numerical integrator for the geodesic equation in its Hamiltonian formulation. Numerical experiments show that the proposed metric keeps the cell aspect ratios bounded away from zero and thus avoids mesh degradation along arbitrarily long geodesic curves.