Integer Coordinates for Intrinsic Geometry Processing

TL;DR

This paper introduces a robust, efficient integer coordinate-based representation for intrinsic triangulations, enabling high-quality geometry processing on low-quality meshes with strong guarantees and broad applicability.

Contribution

It extends normal coordinates and roundabouts to support various mesh operations, providing a general, robust data structure for intrinsic triangulations with provable guarantees.

Findings

Successfully generated high-quality intrinsic Delaunay refinements on the Thingi10k dataset.

Supports a wide range of mesh processing operations with robustness guarantees.

Enables existing geometry algorithms to work directly on low-quality meshes.

Abstract

In this work, we present a general, efficient, and provably robust representation for intrinsic triangulations. These triangulations have emerged as a powerful tool for robust geometry processing of surface meshes, taking a low-quality mesh and retriangulating it with high-quality intrinsic triangles. However, existing representations either support only edge flips, or do not offer a robust procedure to recover the common subdivision, that is, how the intrinsic triangulation sits along the original surface. To build a general-purpose robust structure, we extend the framework of normal coordinates, which have been deeply studied in topology, as well as the more recent idea of roundabouts from geometry processing, to support a variety of mesh processing operations like vertex insertions, edge splits, etc. The basic idea is to store an integer per mesh edge counting the number of times a curve crosses that edge. We show that this paradigm offers a highly effective representation for intrinsic triangulations with strong robustness guarantees. The resulting data structure is general and efficient, while offering a guarantee of always encoding a valid subdivision. Among other things, this allows us to generate a high-quality intrinsic Delaunay refinement of all manifold meshes in the challenging Thingi10k dataset for the first time. This enables a broad class of existing surface geometry algorithms to be applied out-of-the-box to low-quality triangulations.