Contract Transforms

1. Input: Polygon Cloud Point Cloud 2D: 3D: curve Sampling node sheet Full MS computation Full Medial Scaffold (FMS) curve elements MS segregation node element Object Surface + Pruned Fine-scale Medial Scaffold (PMS) sheet elements Splice regularization Coarse-scale MS extraction Object Surface + Medial Scaffold Hypergraph (MSH) All-transition regularization Contract Transforms Regularized Object surface + Regularized MS Hypergraph Make sheets geometry implicit Medial Scaffold Graph (MSG) parallel plane with a bump prism squished tube twisted parabolic gutter squished parabolic gutter parabolic gutter with a bump A15 A14 A1A3-I A5 A1A3-II A12A3-I A12A3-II C1a C2a N1 N2 N C1 C1b C2 C2b S S T S3 N2 S3 S3 T1 T2 d T1 T2 T S1 S1 C S1 N C N1 N2 S S N N1 N S2 S2 S2 C1 C T T T d C1 N C4 T N p p S S C2 S1 C2 S1 C3 S S S2 N N q S2 N q R2 R1 v u R p p p C m e e S2 A3 R1 R2 S S S Next transition with lowest cost v N1 q q T T T T T1 T2 N q N N d C N2 C1 N C2 S S Regularizing 3D Medial Axis Using Medial Scaffold Transforms Ming-Ching Chang Benjamin B. Kimia Laboratory for Engineering Man/Machine Systems (LEMS), Brown University, Providence RI, USA Overview The Medial Scaffold (MS):Dual-Scale Representation Computational Scheme Result: the Stanford Dragon Head Coarse-scale: hypergraph Input: 127k points Re-meshed surface • Vertex: A14 or A1A3node • Link: A13 or A3curve • Hyperlink: A12sheet Fine-scale: (non-manifold) mesh Initial MS: ≈74,000 sheets Initial interior MS: ≈ 35,000 sheets • Vertex: A14node element • Edge: A13curve element • Face: A12sheet element Goal: Find a graph-based medial axis (MA) to qualitatively represent 3D shapes. Captures topology + geometry + dynamics + associated generatorsof the MS and approximates the true MA of the underlying shape. • The 3D MA generally consists of medial sheets, curves, and isolated points and can be organized into a hypergraph form --- the Medial Scaffold (MS). • The instability of the 3D MA is classified into 7 generic cases (the transitions, sudden topological changes). • A set of transforms is defined on the MS in a case-by-case analysis, to model the MA across all transitions. After splice regularization: 285 sheets, 1,695 curves, 1,446 nodes After all-transform regularization: 76 sheets, 262 curves, 219 nodes Seven Generic MA Transitions The MS graph (with sheets implicit) The manipulation of the medial structure (coupled with the 3D shape) toward a nearby degenerate transition point is the basis of simplification of shape. Background and Motivation Applications The MA is with great promise as a universal model for shape: Eleven MS Transforms (Splice, Contract, Merge) Shape Simplification Ridge Detection Animation / Shape Generation A1A3-I sheet-splice: A15 curve-contract: A1A3-II node-node merge: A12A3-II sheet-splice: A12A3-I node-curve merge: Advantages: • Intuitive, qualitative description of the essence structure. • Shape features: make explicit curvature extrema (ridges). • Built-in hierarchy of scale: coarse-to-fine, parts made explicit. • Complete: parameterize the whole embedded space of the shape, allow exact reconstruction. • Generative: model deformation, generation of shapes. A15 sheet-contract: A5 curve-contract: A14 curve-curve merge: Skeletal Representation Component Segmentation A14 sheet-contract: A12A3-I curve-contract: A1A3 -II curve-curve merge: Difficulties: • Difficult to represent and compute. • Instability: sensitive to small perturbations. The Regularization Capturing Qualitative Shape (for Matching) of carpal bones in medical imaging: Need a geodesic distance transform to detect the next merge transform. Analysis of the MA: Move toward a representative shape in each category. Notation Akn:k+1 degree of contact at n distinct points.