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The Mid-sphere Cousin of the Medial Axis Transform

Herbert Edelsbrunner, Elizabeth Stephenson, Martin Hafskjold Thoresen

TL;DR

This work introduces the mid-sphere axis transform as a topological generalization of the medial axis for smoothly embedded surfaces in ${\mathbb{R}}^3$, capturing ridge-like features through a global, discrete-algebraic criterion. It centers on Faustian interchanges of saddles in the squared-distance function $f_x(y)=\|y-x\|^2$, using persistent homology to pair critical points and identify mid-sphere axis events. A vineyard-based algorithm computes staircase-like approximations of the axis on triangulated surfaces, with pruning strategies to mitigate noise and artifacts, and the authors provide code and demonstrations on canonical shapes. The approach generalizes to higher dimensions and opens avenues for scale-aware and Voronoi-based refinements, broadening the toolkit for robust, feature-preserving surface analysis.

Abstract

The medial axis of a smoothly embedded surface in $\mathbb{R}^3$ consists of all points for which the Euclidean distance function on the surface has at least two minima. We generalize this notion to the mid-sphere axis, which consists of all points for which the Euclidean distance function has two interchanging saddles that swap their partners in the pairing by persistent homology. It offers a discrete-algebraic multi-scale approach to computing ridge-like structures on the surface. As a proof of concept, an algorithm that computes stair-case approximations of the mid-sphere axis is provided.

The Mid-sphere Cousin of the Medial Axis Transform

TL;DR

This work introduces the mid-sphere axis transform as a topological generalization of the medial axis for smoothly embedded surfaces in , capturing ridge-like features through a global, discrete-algebraic criterion. It centers on Faustian interchanges of saddles in the squared-distance function , using persistent homology to pair critical points and identify mid-sphere axis events. A vineyard-based algorithm computes staircase-like approximations of the axis on triangulated surfaces, with pruning strategies to mitigate noise and artifacts, and the authors provide code and demonstrations on canonical shapes. The approach generalizes to higher dimensions and opens avenues for scale-aware and Voronoi-based refinements, broadening the toolkit for robust, feature-preserving surface analysis.

Abstract

The medial axis of a smoothly embedded surface in consists of all points for which the Euclidean distance function on the surface has at least two minima. We generalize this notion to the mid-sphere axis, which consists of all points for which the Euclidean distance function has two interchanging saddles that swap their partners in the pairing by persistent homology. It offers a discrete-algebraic multi-scale approach to computing ridge-like structures on the surface. As a proof of concept, an algorithm that computes stair-case approximations of the mid-sphere axis is provided.

Paper Structure

This paper contains 7 sections, 1 theorem, 1 equation, 6 figures.

Table of Contents

  1. Introduction
  2. The Mid-sphere Axis Transform
  3. Morse Theory of Squared Distance Function
  4. Faustian Interchanges
  5. Pairing of Critical Points
  6. Computation and Examples
  7. Discussion

Key Result

theorem 1

A point $x \in {\mathbb R}{\hbox{${\mathbb R}$}}^3$ with radius $r > 0$ belongs to the mid-sphere axis transform of $F$ iff $f_x$ has a Faustian interchange of two saddles at $r^2$.

Figures (6)

  • Figure 1: A mid-sphere touching an ellipsoid in two antipodal points. The vertical plane that passes through these two points intersects the ellipsoid in a curve that lies inside the sphere, and the horizontal plane that passes through the same two points intersects the sphere in a great-circle filled by a disk that both lie inside the ellipsoid.
  • Figure 2: Two locally cylindrical pieces of a blue surface with elliptic cross-sections as shown. The smaller cylinder runs inside the wider cylinder, and both share an axis normal to the drawing plane. The two yellow spheres touch the two cylinders in two points each, and because these spheres are concentric but have different radii, their center belongs to two different red sheets of the mid-sphere axis, which intersect in a line that runs along the common central axis of the two cylinders.
  • Figure 3: The six non-empty sublevel sets at non-critical values interleaved between the critical values of the height function on the torus-with-a-nose. The lowest and highest of the three saddles both give birth, while the middle saddle gives death. On the right, we see the barcode that shows for which value a gap, loop, or closed surface exists in the sublevel set (see Section \ref{['sec:5']} for definitions).
  • Figure 4: The ordered sequences of critical points before and after a Faustian interchange of two saddles. Note that the two saddles swap their partners under the pairing.
  • Figure 5: The pink medial, black mid-sphere, and blue circum-sphere axes of an ellipsoid on the left and an elliptically thickened $(3,7)$-torus knot on the right.
  • ...and 1 more figures

Theorems & Definitions (4)

  • definition 1
  • definition 2
  • theorem 1
  • proof