Metric Optimization in Penner Coordinates

TL;DR

This paper explores how Penner coordinates can be utilized to efficiently solve a broad class of metric optimization problems in geometry processing, ensuring solution existence and robustness.

Contribution

It introduces a novel approach to applying Penner coordinates for metric optimization, extending their use beyond discrete conformal maps to general metric problems.

Findings

Penner coordinates enable solving diverse metric optimization problems.

The approach guarantees solution existence and robustness.

Applications include optimization and interpolation of mesh metrics.

Abstract

Many parametrization and mapping-related problems in geometry processing can be viewed as metric optimization problems, i.e., computing a metric minimizing a functional and satisfying a set of constraints, such as flatness. Penner coordinates are global coordinates on the space of metrics on meshes with a fixed vertex set and topology, but varying connectivity, making it homeomorphic to the Euclidean space of dimension equal to the number of edges in the mesh, without any additional constraints imposed. These coordinates play an important role in the theory of discrete conformal maps, enabling recent development of highly robust algorithms with convergence and solution existence guarantees for computing such maps. We demonstrate how Penner coordinates can be used to solve a general class of optimization problems involving metrics, including optimization and interpolation, while retaining the key solution existence guarantees available for discrete conformal maps.