Technology Category
Machine Learning: Learning with ManifoldsSearch and Optimization: Mixed Discrete/Continuous SearchKnowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
The Mid-sphere Cousin of the Medial Axis Transform
🤖 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
Application Category
📝 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
Research questions and friction points this paper is trying to address.
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
Methods, ideas, or system contributions that make the work stand out.
Generalizes medial axis to mid-sphere axis
Uses persistent homology for saddle pairing
Provides discrete-algebraic multi-scale computation
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