Flexible Mesh Segmentation via Reeb Graph Representation of Geometrical and Topological Features

TL;DR

This work addresses robust mesh segmentation by integrating geometric and topological cues through a Reeb-graph representation. It introduces a three-phase pipeline: (i) Reeb graph construction via an enhanced topological skeleton, (ii) topological simplification using a novel cancellation algorithm that handles both simple and degenerate critical points while preserving vertex-graph correspondence, and (iii) region growing guided by either a standard or Reeb-constrained cost to ensure region continuity. Key contributions include a persistence- and area-based criterion for simplification, a self-balancing BST-driven optimization for cancellations with $O(e \log e)$ complexity, and a region-growing scheme yielding contiguous segments in $O(n \log n)$. The method is validated on Shape Diameter Function and Shape Index applications, demonstrating flexibility across part-based and chart-based segmentation, while also highlighting limitations in saddle-dominated transitions and the need for parameter-tuning. Overall, the framework offers a scalable, geometry-topology integrated approach for advanced geometric analysis of meshes with potential for multi-scale and multivariate extensions.

Abstract

This paper presents a new mesh segmentation method that integrates geometrical and topological features through a flexible Reeb graph representation. The algorithm consists of three phases: construction of the Reeb graph using the improved topological skeleton approach, topological simplification of the graph by cancelling critical points while preserving essential features, and generation of contiguous segments via an adaptive region-growth process that takes geometric and topological criteria into account. Operating with a computational complexity of O(n log(n)) for a mesh of n vertices, the method demonstrates both efficiency and scalability. An evaluation through case studies, including part-based decomposition with Shape Diameter Function and terrain analysis with Shape Index, validates the effectiveness of the method in completely different applications. The results establish this approach as a robust framework for advanced geometric analysis of meshes, connecting the geometric and topological features of shapes.

Figures (10)

• Figure 1: Classification of vertex configurations in discrete Morse theory.
• Figure 2: Level-set configurations: regular vertex (a) and Morse saddle vertex (b) with their isolines.
• Figure 3: Level-sets and Reeb graphs of a torus, comparing Morse (a) and non-Morse (b) height functions.
• Figure 4: Strip of triangles surrounding the level-set (purple) of a regular point (yellow). Discrete contours at the top (red) and bottom (blue) represent this level-set.
• Figure 5: Critical node cancellation operations illustrated through Reeb graphs (top) and continuous function representations (bottom). Node colors indicate scalar values (blue: minimum to red: maximum). Critical nodes shown as triangles (extrema) or circles (saddle nodes). Purple highlights mark cancellation regions and affected edges.