Computing a Family of Skeletons of Volumetric Models for Shape DescriptionComputing a Family of Skeletons of Volumetric Models for Shape Description

Tao Ju

Washington University in St. Louis

SkeletonSkeleton

• A medial representation of an object

– Thin (dimension reduction)

– Preserving shape and topology

Where Skeletons Are UsedWhere Skeletons Are Used

• Animating characters – Skeletal animation

• Shape analysis– Shape comparison

– Character recognition

• Medical applications– Colon unwinding

– Modeling blood vessels

New Application – Protein ModelingNew 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 DescriptorsCurvature Descriptors

• Depicting surface properties

– Principle curvatures, shape index [Koenderink 92]

– Cons: Easily disrupted by a bumpy surface

Min Curvature

Max Curvature

Shape Index

IntuitionIntuition

• Represent tubes and plates as skeleton curves and surfaces.

=

=

Skeleton

ThinningThinning

• 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

ProblemsProblems

• Curve skeleton: containing mostly 1D edges

• Surface skeleton: contains mostly 2D faces

Volume Image

Curve Skeleton

Surface Skeleton

GoalGoal

• Compute simple and descriptive skeletons

– Consists of curves and surfaces corresponding to tubes and plates

• Solution

– Alternate thinning and pruning

Method Overview – Step 1Method Overview – Step 1

Surface Thinning

Surface Pruning

Method Overview – Step 2Method Overview – Step 2

Curve Thinning

Curve Pruning

End Points – A Geometric DefinitionEnd 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

TheoremTheorem

• 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

PruningPruning

• 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 Erode Dilate Dilate

Surface Pruning ExampleSurface Pruning Example

d = 4 d = 7 d = 10

Curve Pruning ExampleCurve Pruning Example

d = 5 d = 10 d = 20

[Mekada and Toriwaki 02] [Svensson and Sanniti di Baja 03]

Results – 3D ModelsResults – 3D Models

Original [Bertrand 95] [Ju et al. 06]

Results – 3D ModelsResults – 3D Models

Original Skeletons with different pruning parameters

Results – Protein DataResults – Protein Data

Cryo-EM [Bertrand 95] [Ju et al. 06] Actual Structure

Visualization: UCSF ChimeraVisualization: UCSF Chimera

Cryo-EM Skeleton Actual Structure Overlay

Collaboration and OutlookCollaboration 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)

http://www.seas.wustl.edu/

Thinning ExampleThinning Example

Original [Bertrand 95]Surface thinning

Curve thinning