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A Simple and Robust Thinning Algorithm on Cell Complexes

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. Background. Thinning : a widely used approach in discrete domain to compute skeleton. Background.

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A Simple and Robust Thinning Algorithm on Cell Complexes

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  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

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

  4. Motivation • Problems • Thinning: sensitive to perturbation • Goals • Robust

  5. Motivation • Problems • Thinning: sensitive to perturbation • Pruning: complex • Goals • Robust • Simple The area constraint (global) The angle constraint (local) [Sud 05] [Shaked 98]

  6. Motivation • Problems • Thinning: sensitive to perturbation • Pruning: complex • Hard to control • Goals • Robust • Simple • Controllable Curve skeleton Surface skeleton Shape descriptor Animation

  7. Our Thinning Algorithm – 2D Input Output 2nd round thinning 1st round thinning 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: (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

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

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

  14. Medial Persistence Measure (MP) Low High

  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 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

  18. Our Thinning Algorithm – 3D Input Output 1st round thinning 2nd round thinning Play Video for color, for Size 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 • Limitations • 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) • Project page: • http://www.cse.wustl.edu/~ll10/paper/pgcc/pgcc.html • Google (Keywords) • Cell complex, skeleton, project

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

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

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

  31. 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 ofresolution • Future: continuous definition

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

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

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