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Computing a Family of Skeletons of Volumetric Models for Shape Description Tao Ju Washington University in St. Louis
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Skeleton A medial representation of an object – Thin (dimension reduction) – Preserving shape and topology
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Where Skeletons Are Used Animating characters – Skeletal animation Shape analysis – Shape comparison – Character recognition Medical applications – Colon unwinding – Modeling blood vessels
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New Application – Protein Modeling Identifying tubular and plate-like shapes is the key in locating α-helices and β-sheets in Cryo-EM protein maps Atomic Model Secondary Structures Cryo-EM map at intermediate resolution α β Tube Plate
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Curvature Descriptors Depicting surface properties – Principle curvatures, shape index [Koenderink 92] – Cons: Easily disrupted by a bumpy surface Min Curvature Max Curvature Shape Index
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Intuition Represent tubes and plates as skeleton curves and surfaces. = = Skeleton
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Thinning Classical method for computing skeleton of a discrete image V. Iterative process – At each iteration, remove boundary points from V – Retain non-simple boundary points Topology preservation [Bertrand 94] – Retain curve-end or surface-end boundary points Shape preservation [Tsao 81] [Gong 90] [Lee 94] [Bertrand 94] [Bertrand 95] Curve thinning or surface thinning Result in curve skeleton or surface skeleton
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Problems Curve skeleton: containing mostly 1D edges Surface skeleton: contains mostly 2D faces Volume Image Curve Skeleton Surface Skeleton
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Goal Compute simple and descriptive skeletons – Consists of curves and surfaces corresponding to tubes and plates Solution – Alternate thinning and pruning
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Method Overview – Step 1 Surface Thinning Surface Pruning
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Method Overview – Step 2 Curve Thinning Curve Pruning
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End Points – A Geometric Definition Curves and surfaces – Consists of edges and faces Curve-end and surface-end points – Points not contained in any 1-manifold or 2-manifold 1-manifold2-manifold
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Theorem Let V be the set of object points. x is a curve-end point if and only if: x is a surface-end point if and only if: = 0 N k (x,V)=N k (x) V
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Pruning Coupling erosion and dilation – Erosion: removes all curve-end (surface-end) points. – Dilation: extends discrete 1-manifold (2-manifold) from curve- end (surface-end) points. – d rounds of erosion followed by d rounds of dilation Erode Dilate
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Surface Pruning Example d = 4 d = 7 d = 10
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Curve Pruning Example d = 5 d = 10 d = 20 [Mekada and Toriwaki 02] [Svensson and Sanniti di Baja 03]
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Results – 3D Models Original[Bertrand 95][Ju et al. 06]
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Results – 3D Models OriginalSkeletons with different pruning parameters
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Results – Protein Data Cryo-EM[Bertrand 95][Ju et al. 06]Actual Structure
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Visualization: UCSF Chimera Cryo-EMSkeletonActual StructureOverlay
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Collaboration and Outlook Future work – Descriptive skeleton of grayscale images – Descriptive skeleton on adaptive grids (octrees) – Protein model building Finding connectivity among α/β elements Using graph matching (Skeleton vs. protein sequence) Collaboration – National Center of Macromolecular Imaging (NCMI), Houston (M. Baker, S. Ludtke, W. Chiu)NCMI
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Thinning Example Original[Bertrand 95] Surface thinning Curve thinning
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