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Wrapping Cycles in Delaunay Complexes: Bridging Persistent Homology and Discrete Morse Theory

Ulrich Bauer, Fabian Roll

TL;DR

This work builds a rigorous bridge between persistent homology and discrete Morse theory in the context of Delaunay/Wrapping constructions. By embedding persistence reductions into an algebraic Morse framework and introducing reduction gradients on a carefully chosen basis, the authors prove that lexicographically minimal cycles are always supported on the Wrap complex for the corresponding radius, effectively unifying Morse-theoretic and homology-based shape reconstruction approaches. The key contributions include a detailed formalization of the algebraic flow, its stabilization, and the equivalence between lexicographic minimality and flow invariance, plus a general result linking descending complexes to both reduction matrices and discrete gradients. These results illuminate how to combine Wrap and lexicographically minimal cycles to produce robust, watertight reconstructions from point clouds, with implications for persistent homology computations and practical shape reconstruction algorithms.

Abstract

We study the connection between discrete Morse theory and persistent homology in the context of shape reconstruction methods. Specifically, we consider the construction of Wrap complexes, introduced by Edelsbrunner as a subcomplex of the Delaunay complex, and the construction of lexicographic optimal homologous cycles, also considered by Cohen-Steiner, Lieutier, and Vuillamy in a similar setting. We show that for any cycle in a Delaunay complex for a given radius parameter, the lexicographically optimal homologous cycle is supported on the Wrap complex for the same parameter, thereby establishing a close connection between the two methods. We obtain this result by establishing a fundamental connection between reduction of cycles in the computation of persistent homology and gradient flows in the algebraic generalization of discrete Morse theory.

Wrapping Cycles in Delaunay Complexes: Bridging Persistent Homology and Discrete Morse Theory

TL;DR

This work builds a rigorous bridge between persistent homology and discrete Morse theory in the context of Delaunay/Wrapping constructions. By embedding persistence reductions into an algebraic Morse framework and introducing reduction gradients on a carefully chosen basis, the authors prove that lexicographically minimal cycles are always supported on the Wrap complex for the corresponding radius, effectively unifying Morse-theoretic and homology-based shape reconstruction approaches. The key contributions include a detailed formalization of the algebraic flow, its stabilization, and the equivalence between lexicographic minimality and flow invariance, plus a general result linking descending complexes to both reduction matrices and discrete gradients. These results illuminate how to combine Wrap and lexicographically minimal cycles to produce robust, watertight reconstructions from point clouds, with implications for persistent homology computations and practical shape reconstruction algorithms.

Abstract

We study the connection between discrete Morse theory and persistent homology in the context of shape reconstruction methods. Specifically, we consider the construction of Wrap complexes, introduced by Edelsbrunner as a subcomplex of the Delaunay complex, and the construction of lexicographic optimal homologous cycles, also considered by Cohen-Steiner, Lieutier, and Vuillamy in a similar setting. We show that for any cycle in a Delaunay complex for a given radius parameter, the lexicographically optimal homologous cycle is supported on the Wrap complex for the same parameter, thereby establishing a close connection between the two methods. We obtain this result by establishing a fundamental connection between reduction of cycles in the computation of persistent homology and gradient flows in the algebraic generalization of discrete Morse theory.
Paper Structure (17 sections, 29 theorems, 16 equations, 9 figures, 4 algorithms)

This paper contains 17 sections, 29 theorems, 16 equations, 9 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Persistent homology and apparent pairs
  4. Lexicographic optimality
  5. Discrete Morse theory and apparent pairs
  6. Descending complexes and gradient refinements
  7. Algebraic Morse theory and persistence
  8. Gradient pairs from persistence pairs
  9. The flow of an algebraic gradient
  10. Relating the algebraic flow and lexicographically minimal cycles
  11. Relating algebraic reduction gradients and discrete gradients
  12. Algebraic Morse theory and persistence
  13. Gradient pairs from persistence pairs
  14. The flow of an algebraic gradient
  15. Relating the algebraic flow and lexicographically minimal cycles
  16. ...and 2 more sections

Key Result

Theorem 1

Let $X\subset \mathbb{R}^d$ be a finite subset in general position, let $r \in \mathbb R$, and let $h \in H_*(\operatorname{Del}_{r}(X))$ be a homology class of the Delaunay complex $\operatorname{Del}_{r}(X)$. Then the lexicographically minimal cycle of $h$, with respect to the Delaunay-lexicograph

Figures (9)

  • Figure 1: Left: Delaunay triangulation of a point cloud, with critical simplices highlighted. Middle: Wrap complex for a small radius parameter. Right: lexicographically minimal cycle for the most persistent feature (black contour), shown together with its bounding chain (shaded blue).
  • Figure 2: The lexicographically minimal cycle corresponding to the most persistent feature of the Delaunay filtration for 3D scan point clouds scans_stanford yields an accurate reconstruction of the surface.
  • Figure 3: Left: Generalized discrete gradient (blue) with corresponding descending complex (green). Right: lexicographic gradient refinement (blue) with corresponding descending complex (green).
  • Figure 4: Discrete gradient (blue) with corresponding descending complex (green). Left: Cycle $c$ (red). Right: Stabilized cycle $\Phi(c)=\Phi^\infty(c)$ (red), supported on the descending complex (green).
  • Figure 5: Discrete gradient (blue) together with the unique critical simplex (orange).
  • ...and 4 more figures

Theorems & Definitions (45)

  • Theorem 1
  • Corollary 2
  • Definition 1
  • Lemma 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Example 6
  • Proposition 7
  • Definition 8
  • ...and 35 more