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MATTopo: Topology-preserving Medial Axis Transform with Restricted Power Diagram

Ningna Wang, Hui Huang, Shibo Song, Bin Wang, Wenping Wang, Xiaohu Guo

TL;DR

This approach is the first to adaptively and directly revise the medial mesh without globally modifying the dependent structure, such as voxel size or sampling density, while preserving its topology and medial features.

Abstract

We present a novel topology-preserving 3D medial axis computation framework based on volumetric restricted power diagram (RPD), while preserving the medial features and geometric convergence simultaneously, for both 3D CAD and organic shapes. The volumetric RPD discretizes the input 3D volume into sub-regions given a set of medial spheres. With this intermediate structure, we convert the homotopy equivalency between the generated medial mesh and the input 3D shape into a localized contractibility checking for each restricted element (power cell, power face, power edge), by checking their connected components and Euler characteristics. We further propose a fractional Euler characteristic algorithm for efficient GPU-based computation of Euler characteristic for each restricted element on the fly while computing the volumetric RPD. Compared with existing voxel-based or point-cloud-based methods, our approach is the first to adaptively and directly revise the medial mesh without globally modifying the dependent structure, such as voxel size or sampling density, while preserving its topology and medial features. In comparison with the feature preservation method MATFP, our method provides geometrically comparable results with fewer spheres and more robustly captures the topology of the input 3D shape.

MATTopo: Topology-preserving Medial Axis Transform with Restricted Power Diagram

TL;DR

This approach is the first to adaptively and directly revise the medial mesh without globally modifying the dependent structure, such as voxel size or sampling density, while preserving its topology and medial features.

Abstract

We present a novel topology-preserving 3D medial axis computation framework based on volumetric restricted power diagram (RPD), while preserving the medial features and geometric convergence simultaneously, for both 3D CAD and organic shapes. The volumetric RPD discretizes the input 3D volume into sub-regions given a set of medial spheres. With this intermediate structure, we convert the homotopy equivalency between the generated medial mesh and the input 3D shape into a localized contractibility checking for each restricted element (power cell, power face, power edge), by checking their connected components and Euler characteristics. We further propose a fractional Euler characteristic algorithm for efficient GPU-based computation of Euler characteristic for each restricted element on the fly while computing the volumetric RPD. Compared with existing voxel-based or point-cloud-based methods, our approach is the first to adaptively and directly revise the medial mesh without globally modifying the dependent structure, such as voxel size or sampling density, while preserving its topology and medial features. In comparison with the feature preservation method MATFP, our method provides geometrically comparable results with fewer spheres and more robustly captures the topology of the input 3D shape.
Paper Structure (39 sections, 1 theorem, 7 equations, 22 figures, 3 tables)

This paper contains 39 sections, 1 theorem, 7 equations, 22 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Medial Axis Computation
  4. Algebraic methods
  5. Voxel-based methods
  6. Point-cloud-based methods
  7. Homotopy Equivalence
  8. Medial Feature Preservation
  9. Thinness
  10. Centeredness
  11. Reconstructibility
  12. Scalability
  13. Restricted Voronoi and Power Diagram in Volumes
  14. Preliminaries
  15. Medial Axis and Medial Mesh
  16. ...and 24 more sections

Key Result

Theorem 1

If for each finite subset $J\subseteq I$, the set $\bigcap_{j\in J} U_j$ is either empty or contractible, then $\mathcal{N}(\mathcal{U})$ is homotopy-equivalent to $X=\bigcup \mathcal{U}$, that is, $\mathcal{U}$ is a good cover leray1950anneau.

Figures (22)

  • Figure 1: We propose a novel volumetric RPD-based framework for computing the medial axis while preserving topology, medial features, and geometry. (a) Input tetrahedral mesh with pre-detected surface sharp features; (b) the RPD; (c) the generated medial mesh; (d) the generated external (in black) and internal (in red) features; (e) a zoomed-in view of the generated medial mesh.
  • Figure 2: Illustration of the importance of the medial axis's homotopy equivalence property. (a) Two medial meshes generated by MATFP 2022MATFP and our method. Both initial medial meshes contain approximately $5k$ medial spheres. The Euler characteristic of MATFP's mesh is $4$, while our result is $1$ (the ground truth is $1$). The connected component (CC) results are $3$ and $1$, respectively (the ground truth is $1$). (b) Two simplified medial meshes generated using the simplification algorithm Q-MAT li2015qmat with the target number of medial spheres set to $170$. Both the initial and simplified medial meshes from MATFP 2022MATFP exhibit 'broken' legs, whereas our method preserves the structure.
  • Figure 3: (a) The medial axis $\mathcal{M}$ of a shape $\mathcal{S}$ in $\mathbb{R}^2$. (b) The medial cone as a linear interpolation of two medial spheres $\mathbf{m}_i$ and $\mathbf{m}_j$. (c) The medial slab as a linear interpolation of three spheres $\mathbf{m}_i$, $\mathbf{m}_j$, and $\mathbf{m}_k$.
  • Figure 4: The duality between the volumetric RPD (a) of three medial spheres and the generated medial mesh (b)
  • Figure 5: The duality between the medial mesh (top) and volumetric RPD (bottom) for a 3D torus shape, where the homotopy equivalence does not hold for case (a) and case (b). An additional medial sphere $\mathbf{m}_3$ needs to be inserted in order to maintain the homotopy equivalence, as shown in (c). The dual edges are depicted as black dotted lines.
  • ...and 17 more figures

Theorems & Definitions (2)

  • Definition 1: Nerve
  • Theorem 1: Nerve Theorem