Seamless Parametrization in Penner Coordinates

TL;DR

This paper presents a simple, efficient algorithm for seamless surface parametrization with prescribed holonomy, enabling better control in quad layouts and texture mapping, demonstrated on large mesh datasets.

Contribution

It introduces a novel algorithm for seamless parametrization with prescribed holonomy signatures, combining simplicity and efficiency, and provides empirical evidence of its effectiveness.

Findings

Performs well on large datasets with over 16,000 meshes

Converges in an average of 9 iterations

Shows promising empirical results despite lacking formal guarantees

Abstract

We introduce a conceptually simple and efficient algorithm for seamless parametrization, a key element in constructing quad layouts and texture charts on surfaces. More specifically, we consider the construction of parametrizations with prescribed holonomy signatures i.e., a set of angles at singularities, and rotations along homology loops, preserving which is essential for constructing parametrizations following an input field, as well as for user control of the parametrization structure. Our algorithm performs exceptionally well on a large dataset based on Thingi10k [Zhou and Jacobson 2016], (16156 meshes) as well as on a challenging smaller dataset of [Myles et al. 2014], converging, on average, in 9 iterations. Although the algorithm lacks a formal mathematical guarantee, presented empirical evidence and the connections between convex optimization and closely related algorithms, suggest that a similar formulation can be found for this algorithm in the future.