1. A Simple and Robust Thinning Algorithm on Cell Complexes
Lu Liu+, Erin Wolf Chambers*, David Letscher*, Tao Ju+
+ Washington University in St. Louis
* St. Louis University

2. Background
Thinning: a widely used approach in discrete domain to compute skeleton
Say what thinning is, and then present the example

3. Background Applications of skeletons Hand writing recognition Shape
Shape segmentation
Hand
writing
recognition
Shape
matching and
retrieval
Animation

4. Motivation
Problems
Thinning: sensitive to perturbation
Goals
Robust

5. Motivation Problems Goals Thinning: sensitive to perturbation
Pruning: complex
Goals
Robust
Simple
The area constraint
(global)
The angle constraint
(local)
Don’t go to details, say what pruning is
[Sud 05]
[Shaked 98]

6. Motivation Problems Goals Thinning: sensitive to perturbation
Pruning: complex
Hard to control
Goals
Robust
Simple
Controllable
Curve skeleton
1d components of skeleton are curves
2d components of skeleton are surfaces
Surface skeleton
Shape descriptor
Animation

7. Our Thinning Algorithm – 2D
Input
Output
2nd round thinning
1st round thinning
Measure you compute it somehow, and you use it for what purpose
measure

8. Our Thinning Algorithm – 2D
Input
Output
2nd round thinning
1st round thinning
measure

9. Cell Complexes A closed set of cells at various dimensions
0-cell (point), 1-cell (edge), 2-cell (face), 3-cell (cube), etc.
Why cell complexes:
Has explicit geometry
Easy to maintain topology during thinning
Removing simple pairs
Simple pair removal, don’t start with example,
Just say what it is and what it is good for
Simple pair: (σ, δ) where δ is the only higher-dimensional cell adjacent to σ

10. Our Thinning Algorithm – 2D
Input
Output
2nd round thinning
1st round thinning
measure

11. A Naïve Thinning Process
Peel off layer by layer by removing simple pairs
Boundary element s->?> simple pair

12. Our Observation I = 6, R = 20, R >> I Highlighted medial edge 1
Isolated in iteration 6
Highlighted medial edge
I = 6, R = 20, R >> I 1 5
Neighboring faces 6 10 11 15
Add faces  arrow
Removed in iteration 20 20 16

13. Our Observation I = 2, R = 4, R ≈ I Highlighted medial edge 1 5 6 10
Isolated in iteration 2
Highlighted medial edge
I = 2, R = 4, R ≈ I 1 5
Removed in iteration 4
Neighboring faces 6 10 11 15 20 16

14. Medial Persistence Measure (MP)
Low
High
Delta removed

15. Geometric Explanation
I and R approximate different shape measures
I: Radius of largest inscribing disc – “Thickness”
R: Half-length of longest inscribing tube – “Length”
MP captures tubular-ness:
R-I: “Scale”
1-I/R: “Sharpness” I R

16. Our Thinning Algorithm – 2D
Preserving the medial edges with measures larger than thresholds
Input
Output
2nd round thinning
1st round thinning
Complete sentence fro threshold
measure

17. Medial Persistence (3D)
Same computation
Get isolation (I) and removal (R) iterations for each edge and face
Compute absolute (R-I) and relative (1-I/R) medial persistence
Simple computation
Higher MP means:
Edges: more significant tubular-ness
Faces: more significant plate-likeness
Absolute/Relative MP measures the scale/sharpness of feature
Robust to boundary perturbation
just

18. Our Thinning Algorithm – 3D
Output
Input
2nd round thinning
for color, for Size
Play Video
1st round thinning
Thresholding

19. Input
MP of faces
MP of edges
Mixed dimensional skeletons
Curve skeletons only (infinity thresholds for faces)

20. Input
MP of faces
MP of edges
Mixed dimensional skeletons
Curve skeletons only (infinity thresholds for faces)

21. Input
MP of faces
MP of edges
Mixed dimensional skeletons
Curve skeletons only (infinity thresholds for faces)

22. Strength of Our Algorithm
Robust to noise and cell shapes
Cubic
Noisy
Tetrahedral

23. Strength of Our Algorithm
Robust to noise and cell shapes
Cubic
Noisy
Tetrahedral

24. Strength of Our Algorithm
Robust to different resolutions

25. Summary Proposed a thinning algorithm on cell complexes
Simple: 2 rounds of thinning, multiple dimensions
Robust: stable medial persistence measure (MP)
Noise
Different cell shapes
Different resolutions
Controllable: different thresholds for medial geometry in different dimensions

26. Limitations and Future Work
Skeletons vary with the structure of the cell complex
Medial persistence can be biased by grid directions
Future work
Continuous formulation of thinning and skeleton measures
cubic
tetrahedral
diagonal bias
Smoother skeleton with resolution increase

27. Check out our project page (program, data, video, and more)
Google (Keywords)
Cell complex, skeleton, project

29. Beta sheets
Alpha helix
Protein
(Cryo-EM volume)
Secondary structure

30. R I Scale dependent Scale independent
T(Mabs)= 0.05L, T(Mrel) = 0.5 for both k = 1,2 (faces, edge)
L is the width of the bounding box

31. Discussion & Future work
Artifacts
Measure is anisotropic on isotropic shapes
Rely on regular grid
Future: distance guided thinning, octree

32. Discussion & Future work
Artifacts
Measure is anisotropic on isotropic shapes
Rely on regular grid
Future: distance guided thinning, octree
Observations
Smoother skeleton with the increase of resolution
Future: continuous definition

33. Discussion & Future work
Artifacts
Measure is anisotropic on isotropic shapes
Different representatin: octree
Remedy: distance based thinning
Observations:
Different resolutionsL
Continuous definition

34. Our thinning algorithm – 2D
Low
High
2D model in
cell complex
representation
thinning
thinning
Intermediate
measure
The stable part 34