Regularizing 3D medial axis using medial scaffold transforms

This paper addresses a key bottleneck in the use of the 3D medial axis (MA) representation, namely, how the complex MA structure can be regularized so that similar, within-category 3D shapes yield similar 3D MA that are distinct from the non-category shapes. We rely on previous work which (i) constructs a hierarchical MA hypergraph, the medial scaffold (MS), and (ii) the theoretical classification of the instabilities of this structure, or transitions (sudden topological changes due to a small perturbation). The shapes at transition point are degenerate. Our approach is to recognize the transitions which are close-by to a given shape and transform the shape to this transition point, and repeat until no close-by transitions exists. This move towards degeneracy is the basis of simplification of shape. We derive 11 transforms from 7 transitions and follow a greedy scheme in applying the transform. The results show that the simplified MA preserves with-in-category similarity, thus indicating its potential use in various applications including shape analysis, manipulation, and matching.