A Splitting Scheme for Flip-Free Distortion Energies

TL;DR

This paper presents a novel, robust optimization algorithm based on operator splitting and ADMM for minimizing flip-free distortion energies in geometric parametrization and deformation tasks, ensuring robustness and efficiency.

Contribution

The paper introduces a new ADMM-based splitting scheme tailored for non-convex, non-smooth distortion energies, with proven convergence and parallelizable local steps.

Findings

Efficient global step involving a single matrix multiplication

Closed-form local steps per element, highly parallelizable

Robustness to flipped elements during optimization

Abstract

We introduce a robust optimization method for flip-free distortion energies used, for example, in parametrization, deformation, and volume correspondence. This method can minimize a variety of distortion energies, such as the symmetric Dirichlet energy and our new symmetric gradient energy. We identify and exploit the special structure of distortion energies to employ an operator splitting technique, leading us to propose a novel Alternating Direction Method of Multipliers (ADMM) algorithm to deal with the non-convex, non-smooth nature of distortion energies. The scheme results in an efficient method where the global step involves a single matrix multiplication and the local steps are closed-form per-triangle/per-tetrahedron expressions that are highly parallelizable. The resulting general-purpose optimization algorithm exhibits robustness to flipped triangles and tetrahedra in initial data as well as during the optimization. We establish the convergence of our proposed algorithm under certain conditions and demonstrate applications to parametrization, deformation, and volume correspondence.