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In the Search for Good Neck Cuts

Sam Ruggerio, Sariel Har-Peled

TL;DR

The paper tackles the problem of extracting neck-like features on genus-zero surfaces by formalizing bottlenecks as high-tightness cuts within an isoperimetric framework. It introduces a practical, shortest-path–based algorithm that avoids heavy preprocessing and achieves sub-quadratic runtimes in real meshes, underpinned by a polynomial-time approximation for the collar under well-behavedness assumptions. Key contributions include a formal tightness/alpha-bottleneck notion, a scalable pipeline for salient-point detection, and a skeleton-driven search for high-quality neck-curves, with empirical validation on standard 3D mesh datasets. The approach yields a simple, implementable method that produces locally near-optimal neck cuts, offering improvements over topological and algebraic methods while remaining practical for robotics, mesh segmentation, and related applications.

Abstract

We study the problem of finding neck-like features on a surface. Applications for such cuts include robotics, mesh segmentation, and algorithmic applications. We provide a new definition for a surface bottleneck -- informally, it is the shortest cycle relative to the size of the areas it separates. Inspired by the isoperimetric inequality, we formally define such optimal cuts, study their properties, and present several algorithms inspired by these ideas that work surprisingly well in practice. For examples of our algorithms, see https://neckcut.space.

In the Search for Good Neck Cuts

TL;DR

The paper tackles the problem of extracting neck-like features on genus-zero surfaces by formalizing bottlenecks as high-tightness cuts within an isoperimetric framework. It introduces a practical, shortest-path–based algorithm that avoids heavy preprocessing and achieves sub-quadratic runtimes in real meshes, underpinned by a polynomial-time approximation for the collar under well-behavedness assumptions. Key contributions include a formal tightness/alpha-bottleneck notion, a scalable pipeline for salient-point detection, and a skeleton-driven search for high-quality neck-curves, with empirical validation on standard 3D mesh datasets. The approach yields a simple, implementable method that produces locally near-optimal neck cuts, offering improvements over topological and algebraic methods while remaining practical for robotics, mesh segmentation, and related applications.

Abstract

We study the problem of finding neck-like features on a surface. Applications for such cuts include robotics, mesh segmentation, and algorithmic applications. We provide a new definition for a surface bottleneck -- informally, it is the shortest cycle relative to the size of the areas it separates. Inspired by the isoperimetric inequality, we formally define such optimal cuts, study their properties, and present several algorithms inspired by these ideas that work surprisingly well in practice. For examples of our algorithms, see https://neckcut.space.
Paper Structure (21 sections, 4 theorems, 14 equations, 10 figures, 2 tables)

This paper contains 21 sections, 4 theorems, 14 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background and prior work
  3. Our approach
  4. Isoperimetry, bottleneck cuts, and salient points
  5. Isoperimetry and bottlenecks
  6. Well behaved surfaces
  7. Salient points to a bottleneck
  8. Approximating the optimal collar
  9. Identifying the neck where the collar lies
  10. The algorithm for computing the collar
  11. A practical algorithm
  12. Computing salient points
  13. Step I: Salient point via approximate diameter.
  14. Stage II: Collecting candidates for salient points.
  15. Stage III: Filtering the candidates.
  16. ...and 6 more sections

Key Result

Lemma 2.10

For $s = \mathsf{s}(\mathcalb{o}, \mathcalb{b})$, we have that $\mathsf{d}_{\mathcal{M}\xspace}\mleft(s,\mathcalb{o}\mright) \geq \left\| \mathcalb{o} \right\|$, if $\alpha \geq 4 \tau$.

Figures (10)

  • Figure 2.1: Middle, Right: For some optimal bottleneck $\xi$, we consider the geodesic $\sigma$. Left: If the geodesic were to cross $\xi$ at $x$, it would be a contradiction, as shortcutting along $\xi$ would be shorter.
  • Figure 3.1: Left: A long neck with a stable collar. Right: Every geodesic cycle from the beak to the left foot.
  • Figure 3.2:
  • Figure 3.3:
  • Figure 4.1: The salient points of a human mesh, with no filtering (left), $r=10$ (middle), and $r=20$ (right). The bottleneck curves are shown in the right mesh.
  • ...and 5 more figures

Theorems & Definitions (17)

  • Definition 2.1
  • Example 2.2
  • Definition 2.4
  • Definition 2.5
  • Example 2.6
  • Definition 2.7
  • Definition 2.8
  • Remark 2.9
  • Lemma 2.10: salient points are far
  • Definition 3.1
  • ...and 7 more