The Mid-sphere Cousin of the Medial Axis Transform

2025-04-20

AI Summary

This paper addresses the sensitivity to distance local minima and poor noise robustness of conventional medial axis transforms (MAT) in extracting ridge structures from smooth 3D surfaces. We propose the *mid-sphere axis*—a novel geometric skeleton grounded in the Euclidean distance function. Methodologically, we reinterpret saddle-point pairing and swapping in persistent homology as a generative paradigm for skeleton extraction, integrating differential geometry and critical point theory; we further introduce a staircase approximation algorithm—the first to enable discrete algebraization and multi-scale computation of this skeleton. Unlike MAT, the mid-sphere axis does not rely on local distance minima and exhibits topological stability. Experiments demonstrate its superior noise robustness and enhanced capability in capturing multi-scale ridge features compared to state-of-the-art methods. This work establishes the first computationally tractable and theoretically rigorous mid-sphere axis model for surface feature analysis.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

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.

Problem

• Generalize medial axis to mid-sphere axis using persistent homology
• Compute ridge-like structures on surfaces with discrete-algebraic approach
• Provide algorithm for stair-case approximations of mid-sphere axis

Innovation

• Generalizes medial axis to mid-sphere axis
• Uses persistent homology for saddle pairing
• Provides discrete-algebraic multi-scale computation