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 – Skeletal animation Shape analysis – Shape comparison – Character recognition Medical applications – Colon unwinding – Modeling blood vessels

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

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

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

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

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

Goal Compute simple and descriptive skeletons – Consists of curves and surfaces corresponding to tubes and plates Solution – Alternate thinning and pruning

Method Overview – Step 1 Surface Thinning Surface Pruning

Method Overview – Step 2 Curve Thinning Curve Pruning

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

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

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

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

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

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

Results – 3D Models OriginalSkeletons with different pruning parameters

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

Visualization: UCSF Chimera Cryo-EMSkeletonActual StructureOverlay

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

Thinning Example Original[Bertrand 95] Surface thinning Curve thinning