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1. 9 89 13 100 80 71 64 93 59 60 50 74 65 102 57 46 48 65 91 37 59 19 76 90 70 10 0 68 99 68 56 52 69 51 51 G: M: G: i j k slack i a a b c j k b c d slack slack d curve node sheet …… Measuring 3D Shape Similarity by Matching the Medial Scaffolds Ming-Ching Chang Benjamin B. Kimia Laboratory for Engineering Man/Machine Systems (LEMS), Brown University, Providence RI, USA Overview Graph Matching Approach to Match the MS Hypergraphs Define Compatibility Measures • Apply MS transforms to regularize the MS across low-cost transitions. match Extend the Graduated Assignment Scheme • Represent the match between graphs nodes in a matrix M. • Use slack row and column to handle missing/extra nodes. • Extend to hypergraph matching intuitively. Main idea: Measure 3D shape similarity by matching a medial axis (MA) based representation ─ the Medial Scaffold (MS). Node structural compatibility: 1− Curve parametric compatibility: Medial Representation 2D: shock graphs. Correspondence via dynamic programming • Energy minimization: • Shocks: medial axis points endowed with dynamics of flow. • Shape similarity is measured as the minimum extent of deformation necessary for one shape to match another. • Optimal deformation path as the transitions (discrete topological changes) between the two graphs. Virtual Links to Improve Robustness 3D: Medial Scaffold hypergraphs. • Vertex: A14 or A1A3node • Link: A13 or A3curve • Hyperlink: A12sheet Group two nodes into a virtual node. Add two virtual links to simulate such grouping. Experimental Results Matching carpal bones (hamates) across 6 patients. Similarity in a 25 shape database. Notation Akn:k+1 degree of contact at n distinct points. Key Novelties • The Medial Scaffold representation is: • Hierarchical: capturing an intrinsic graph-like structure of shape, and • Complete: allowing a full reconstruction of shape. • We extend a popular graph matching scheme to match the MS hypergraphs.