Computing a Family of Skeletons of Volumetric Models for Shape Description

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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
Atomic Model
Secondary Structures

• Identifying tubular and plate-like shapes is the
  key in locating a-helices and ß-sheets in Cryo-EM
  protein maps

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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.

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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
Volume Image
Curve Skeleton
Surface Skeleton

• Curve skeleton containing mostly 1D edges
• Surface skeleton contains mostly 2D faces

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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
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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-manifold
2-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

Nk(x,V)Nk(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
Dilate
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Surface Pruning Example
d 7
d 4
d 10
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Curve Pruning Example
d 10
d 5
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
Original
Skeletons 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-EM
Skeleton
Actual Structure
Overlay
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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 a/ß elements
• Using graph matching (Skeleton vs. protein
  sequence)
• Collaboration
• National Center of Macromolecular Imaging (NCMI),
  Houston (M. Baker, S. Ludtke, W. Chiu)

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Thinning Example
Surface thinning
Curve thinning
Original
Bertrand 95