Mixed 1D/2D Simplicial Approximation of Volumetric Medial Axis by Direct Palpation of Shape Diameter Function

In the domain of Shape Encoding, the approximation of the Medial Axis of a solid region in R3 with Boundary Representation M, is relevant because the Medial Axis is an efficient encoding for M in Design, Manufacturing, and Shape Learning. Existing Medial Axis approximations include (a) full Voronoi and (b) and partial Shape Diameter Function (SDF)-based ones. Methods (a) produce large high-frequency data, which must then be pruned. Methods (b) reduce computing expenses at the price of not handling some shapes (e.g., prismatic), and currently, they only synthesize 1D Medial Axes. To partially overcome these limitations, this investigation performs a direct synthesis of a 1D and 2D simplex-based Medial Axis approximation by a combination of stochastic geometric reasoning and graph operations on the SDF-originated point cloud. Our method covers one- and two-dimensional Simplicial Complex Medial Axes, thus improving on 1D Medial Axes approximation methods. Our approach avoids the expensive full computing plus pruning of Medial Axis based on Voronoi methods. Future work is needed in the synthesis of Medial Axis approximation for high-frequency neighborhoods of mesh M.