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Computing a Family of Skeletons of Volumetric Models for Shape Description

Computing a Family of Skeletons of Volumetric Models for Shape Description. Tao Ju Washington University in St. Louis. Skeleton. A medial representation of an object Thin (dimension reduction) Preserving shape and topology. Where Skeletons Are Used. Animating characters

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Computing a Family of Skeletons of Volumetric Models for Shape Description

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  1. Computing a Family of Skeletons of Volumetric Models for Shape Description Tao Ju Washington University in St. Louis

  2. Skeleton • A medial representation of an object • Thin (dimension reduction) • Preserving shape and topology

  3. Where Skeletons Are Used • Animating characters • Skeletal animation • Shape analysis • Shape comparison • Character recognition • Medical applications • Colon unwinding • Modeling blood vessels

  4. Cryo-EM map at intermediate resolution Plate β Tube α New Application – Protein Modeling Atomic Model Secondary Structures • Identifying tubular and plate-like shapes is the key in locating α-helices and β-sheets in Cryo-EM protein maps

  5. Curvature Descriptors • Depicting surface properties • Principle curvatures, shape index [Koenderink 92] • Cons: Easily disrupted by a bumpy surface Min Curvature Max Curvature Shape Index

  6. Intuition • Represent tubes and plates as skeleton curves and surfaces.  = =  Skeleton

  7. 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

  8. Problems • Curve skeleton: containing mostly 1D edges • Surface skeleton: contains mostly 2D faces Volume Image Curve Skeleton Surface Skeleton

  9. Goal • Compute simple and descriptive skeletons • Consists of curves and surfacescorresponding to tubesandplates • Solution • Alternate thinning and pruning

  10. Method Overview – Step 1 Surface Thinning Surface Pruning

  11. Curve Pruning Method Overview – Step 2 Curve Thinning

  12. 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-manifold 2-manifold

  13. 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 Nk(x,V)=Nk(x)  V

  14. Erode 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 Dilate

  15. Surface Pruning Example d = 7 d = 4 d = 10

  16. Curve Pruning Example d = 10 d = 5 d = 20 [Mekada and Toriwaki 02] [Svensson and Sanniti di Baja 03]

  17. Results – 3D Models Original [Bertrand 95] [Ju et al. 06]

  18. Results – 3D Models Original Skeletons with different pruning parameters

  19. Results – Protein Data Cryo-EM [Bertrand 95] [Ju et al. 06] Actual Structure

  20. Visualization: UCSF Chimera Cryo-EM Skeleton Actual Structure Overlay

  21. 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)

  22. Thinning Example Surface thinning Curve thinning Original [Bertrand 95]

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