Gierke & Peter's skeleton scale-spaces fix medial axis noise sensitivity

This paper from Gierke and Peter addresses a longstanding weakness of the Hamilton-Jacobi skeleton (medial axis): its extreme sensitivity to noise, where tiny boundary variations cause disproportionately large skeletal branches. Classical pruning methods remove branches sequentially but lack formal scale-space properties. The authors propose skeletonisation scale-spaces that embed shapes into a family of reconstructions from increasingly sparse (sparsification) or dense (densification) skeleton representations. The framework is built on rigorous theory in both continuous and discrete domains, guaranteeing hierarchical architecture, controllable simplification, and geometric equivariance.

Densification is a novel twist: instead of only pruning, the skeleton can be grown to produce overcomplete shape representations that go beyond the original medial axis. Proof-of-concept experiments demonstrate effectiveness for robust skeletonisation (less noise), shape compression (reduced data with preserved topology), and practical applications like stiffness enhancement in additive manufacturing (designing support structures). The work appears on arXiv (2509.21398) and opens new paths for shape analysis and geometry processing.