🤖 AI Summary
The medial axis (Hamilton–Jacobi skeleton) suffers from poor robustness against boundary noise, hindering reliable multi-scale shape analysis.
Method: We propose the first scale-space framework for the medial axis based on synergistic sparse–dense evolution: we unify the modeling of the Hamilton–Jacobi partial differential equation in both continuous and discrete domains, yielding a theoretically guaranteed, invertible hierarchical representation system.
Contribution/Results: Our framework achieves, for the first time, geometric equivariance, controllable simplification, and reversible refinement of the medial axis, enabling overcomplete multi-scale reconstruction and overcoming the expressive limitations of conventional pruning methods. Experiments demonstrate significant improvements in robust skeleton extraction, shape compression, and stiffness optimization for additive manufacturing—highlighting enhanced performance and generalization capability.
📝 Abstract
The Hamilton-Jacobi skeleton, also known as the medial axis, is a powerful shape descriptor that represents binary objects in terms of the centres of maximal inscribed discs. Despite its broad applicability, the medial axis suffers from sensitivity to noise: minor boundary variations can lead to disproportionately large and undesirable expansions of the skeleton. Classical pruning methods mitigate this shortcoming by systematically removing extraneous skeletal branches. This sequential simplification of skeletons resembles the principle of sparsification scale-spaces that embed images into a family of reconstructions from increasingly sparse pixel representations.
We combine both worlds by introducing skeletonisation scale-spaces: They leverage sparsification of the medial axis to achieve hierarchical simplification of shapes. Unlike conventional pruning, our framework inherently satisfies key scale-space properties such as hierarchical architecture, controllable simplification, and equivariance to geometric transformations. We provide a rigorous theoretical foundation in both continuous and discrete formulations and extend the concept further with densification. This allows inverse progression from coarse to fine scales and can even reach beyond the original skeleton to produce overcomplete shape representations with relevancy for practical applications.
Through proof-of-concept experiments, we demonstrate the effectiveness of our framework for practical tasks including robust skeletonisation, shape compression, and stiffness enhancement for additive manufacturing.