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					                                               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
                                                                                                                                                                                                                                                                                                    Input:                                                                                           Input: 127k points                           Re-meshed surface
                                                                        Coarse-scale: hypergraph
                                                                                                                                                                                                curve                                                                                                   Polygon Cloud                                        Point Cloud
                                                                              • Vertex: A14 or A1A3 node                                                                                 node
                                                                                                                                                                                                           sheet                                                                                                                                  Sampling                    2D:     3D:
                                                                              • Link: A13 or A3 curve
                                                                              • Hyperlink: A12 sheet
                                                                                                                                                                                                                                                                                                                                                  Full MS computation
                                                                       Fine-scale: (non-manifold) mesh                                                                                        curve
                                                                                                                                                                                                                                                                                                                 Full Medial Scaffold (FMS)                                                          Initial MS: ≈74,000 sheets                   Initial interior MS: ≈ 35,000 sheets
                                                                              • Vertex: A14 node element                                                                       node
                                                                              • Edge: A13 curve element                                                                       element                                                                                                                                                               MS segregation
                                                                              • Face: A12 sheet element                                                                                                 elements
                                                                                                                                                                                                                                                                                                            Object Surface + Pruned
Goal: Find a graph-based medial axis (MA) to qualitatively                                                                                                                                                                                                                                              Fine-scale Medial Scaffold (PMS)
                                                                             Captures topology + geometry + dynamics + associated generators
          represent 3D shapes.
                                                                             of the MS and approximates the true MA of the underlying shape.                                                                                                                                                                                                       Splice regularization
    • The 3D MA generally consists of medial sheets, curves,                                                                                                                                                                                                                                                                                     Coarse-scale MS extraction                          After splice regularization:                 After all-transform regularization:
                                                                                                                                                                                                                                                                                                                                                                                                      285 sheets, 1,695 curves, 1,446 nodes       76 sheets, 262 curves, 219 nodes
      and isolated points and can be organized into a
      hypergraph form --- the Medial Scaffold (MS).
                                                                                                           Seven Generic MA Transitions                                                                                                                                                                         Object Surface +
                                                                                                                                                                                                                                                                                                        Medial Scaffold Hypergraph (MSH)
    • The instability of the 3D MA is classified into 7 generic
      cases (the transitions, sudden topological changes).                                                                                                                                                                                                                                                                                        All-transition regularization
    • A set of transforms is defined on the MS in a case-by-             parallel plane with a bump              prism                squished tube    twisted parabolic gutter                    squished parabolic gutter                 parabolic gutter with a bump

                                                                                                                                                                                                                                                                                                                 Regularized Object surface +
      case analysis, to model the MA across all transitions.                                                                                                                                                                                                                                                                                                                                 The MS graph (with sheets implicit)
                                                                                                                                                                                                                                                                                                                 Regularized MS Hypergraph
The manipulation of the medial structure (coupled with the
 3D shape) toward a nearby degenerate transition point is                                                                                                                                                                                                                                                                                         Make sheets geometry implicit
 the basis of simplification of shape.                                                                                                                                                                                                                                                                       Medial Scaffold Graph (MSG)
                                                                                                                                                                                                                                                                                                                                                                                                                                                      Contract Transforms

                                                                           A1A3-I                         A15                          A14                      A5                           A1A3-II                           A12A3-I                      A12A3-II
            Background and Motivation
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:

                                                                                                                                                                     N2             S3                                    S3                              S3
                                                                                                                                                                                                                                                                                    T1         d        T2                  T1              T2
                                                                                         C                                                               S1          C                             S1                                           N
                                                                                                                                                                                                                                                                                S             N1 N2                    S              N
                                                                                                                                                                     N1        S2                             N       S2                               S2

                                                                      A12A3-II sheet-splice:                                                                                                                                                                            A12A3-I node-curve merge:
Advantages:                                                                                                                  T
                                                                                                                                                      A15 sheet-contract:
                                                                                C1                                                                                                                                                                                                                                                           C1
•                                                                                                               C4                                                                                                                                                                                      C
  Intuitive, qualitative description of the essence structure.                       C2
                                                                                         T                                                                                                                                                                                               T          d                             T
                                                                                                      C3                                                                  p                                p
                                                                                                                                                                                                                      S1                                     S1                 S              N                       S                    C2
• Shape features: make explicit curvature extrema (ridges).                                                                                                               S                                S
                                                                      A5 curve-contract:                                                                                                                                                                          S2

• Built-in hierarchy of scale: coarse-to-fine, parts made explicit.        R1                R2
                                                                                                                                                            N                   q                 N               q        S2            N                              A14 curve-curve merge:
                                                                                                                                                                                                                                                                                                                            C1a        C2a               Skeletal Representation            Component Segmentation
• Complete: parameterize the whole embedded space of the                                                                                                4
                                                                                                                                                      A1 sheet-contract:
                                                                                                                                                                                                                                                                                                                           C1b        N
                                                                                     C                          m                                                                                                                                                                                  C2                                 C2b
                                                                                                                                                                                                                                                                            S                                      S
  shape, allow exact reconstruction.                                  A12A3-I curve-contract:                                                                                            p
                                                                                                                                                                                                                               p                                   p

                                                                                                                                                                               e                                  e                                                     A1A3 -II curve-curve merge:
• Generative: model deformation, generation of shapes.                                                                   v
                                                                                                                                                                     S                                    S
                                                                                                                                                                                                                                                 S                                                          A3               R1              R2
                                                                                         T                       T                       T
                                                                                                                                                            N                        q            N
                                                                                                                                                                                                                                                             q                                 N1
                                                                                                                                                                                                                                                                                    T                                       T1              T2
Difficulties:                                                                                                                                                                                                                                                                   S   C
                                                                                                                                                                                                                                                                                               N2                   S       C1 N            C2

• Difficult to represent and compute.
• Instability: sensitive to small perturbations.                                                                     The Regularization
                                                                                                                                                                                                                                   Need a geodesic                                                                                                      Capturing Qualitative Shape (for Matching) of carpal bones in medical imaging:
Analysis of the MA:                                                   Move toward a representative shape in each category.                                                                                                         distance transform to
     Notation Akn: k+1 degree of contact at n distinct points.
                                                                                                                                                                                                                                   detect the next merge

                                                                                                                                                                                                                                                                                                    Next transition with lowest cost

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