Segmenting a Surface Mesh into Pants Using Morse Theory

TL;DR

This paper introduces two Morse theory-based algorithms for decomposing surface meshes into pants, demonstrating improved efficiency and robustness over existing methods, applicable to surfaces with or without boundaries.

Contribution

The paper presents novel Morse theory algorithms for pants decomposition of surface meshes, with one leveraging handle identification and the other using Reeb graphs, both outperforming prior methods.

Findings

Both algorithms run faster than existing methods.

The Reeb graph based algorithm is the most time-efficient.

Algorithms are robust against noise in surface data.

Abstract

A pair of pants is a genus zero orientable surface with three boundary components. A pants decomposition of a surface is a finite collection of unordered pairwise disjoint simple closed curves embedded in the surface that decompose the surface into pants. In this paper we present two Morse theory based algorithms for pants decomposition of a surface mesh. Both algorithms operates on a choice of an appropriate Morse function on the surface. The first algorithm uses this Morse function to identify handles that are glued systematically to obtain a pant decomposition. The second algorithm uses the Reeb graph of the Morse function to obtain a pant decomposition. Both algorithms work for surfaces with or without boundaries. Our preliminary implementation of the two algorithms shows that both algorithms run in much less time than an existing state-of-the-art method, and the Reeb graph based algorithm achieves the best time efficiency. Finally, we demonstrate the robustness of our algorithms against noise.