Seamless Parametrization in Penner Coordinates

TL;DR

A conceptually simple and efficient algorithm for seamless parametrization, a key element in constructing quad layouts and texture charts on surfaces, which performs exceptionally well on a large dataset based on Thingi10k as well as on a challenging smaller dataset, converging, on average, in 9 iterations.

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.

Figures (14)

• Figure 1: Holonomy angle notation. The signed loop holonomy angles $d_m^j \alpha_m^j$ measure the rotation between dual edges.
• Figure 2: Intrinsic flip. Week's flip algorithm uses Ptolemy formula for $\ell(e')$ which coincides with the Euclidean length if the angles opposite to $e'$ sum up to $\pi$.
• Figure 3: Penner cell decomposition of cone metrics with 3 vertices and 3 edges (Figure from capouellez2023metric)
• Figure 4: Dual loop update for a flip.
• Figure 5: Top: Distributions of mesh face counts, cross-field cone counts, and surface genus for the dataset of Myles:2014. Bottom: Distributions of iteration counts, average linear solve times, and RMSRE errors for our method. Outliers are aggregated in the rightmost bin.

Theorems & Definitions (1)

• definition 1