3D Thinning on Cell Complexes for Computing Curve and Surface Skeletons

1. 3D Thinning on Cell Complexes for Computing Curve and Surface Skeletons Lu Liu Advisor: Tao Ju Master Thesis Defense Dec 18th, 2008

2. Outline • Motivation • Goal and Rationale • Cell Complex • Our thinning algorithm • Conclusion & Future work

3. Skeleton as a Shape Descriptor • Thin geometric structure lying in the center Tube Curve (1D) Elongated Part Curve (1D) Plate Surface (2D) 2D Object 1D Skeleton Dimension Reduction 3D Object 1D/2D Skeleton

4. Skeleton as a Shape Descriptor • Thin geometric structure lying in the center • Applications • Handwritten character recognition • Shape matching and retrieval • Shape segmentation • Shape deformation • Medical image visualization • homepages.inf.ed.ac.uk/rbf/HIPR2/thin.htm

5. Skeleton as a Shape Descriptor • Thin geometric structure lying in the center • Applications • Handwritten character recognition • Shape matching and retrieval • Shape segmentation • Shape deformation • Medical image visualization http://www.cs.brown.edu/research/projects/shape-based_image_retrieval.html

6. Skeleton as a Shape Descriptor • Thin geometric structure lying in the center • Applications • Handwritten character recognition • Shape matching and retrieval • Shape segmentation • Shape deformation • Medical image visualization http://cheng.zhiquan.googlepages.com/publication

7. Skeleton as a Shape Descriptor • Thin geometric structure lying in the center • Applications • Handwritten character recognition • Shape matching and retrieval • Shape segmentation • Shape deformation • Medical image visualization http://www.emeraldinsight.com/Insight/viewContentItem.do?contentType=Article&hdAction=lnkhtml&contentId=1532798

8. Skeleton as a Shape Descriptor • Thin geometric structure lying in the center • Applications • Handwritten character recognition • Shape matching and retrieval • Shape segmentation • Shape deformation • Medical image visualization Vessels Nerve cells Bone Matrix Protein

9. Goal • Thinness • Definition of skeleton (1-dimension reduction) • Easy to detect curve and surface components • Topology preservation • Genus, connectivity • Handwritten character recognition, shape matching • Shape preservation • Curve skeleton for tube-like shape components • Surface skeleton forplate-like shape components • Shape segmentation, shape deformation

10. Computing Skeletons • On continuous models • As simplified Medial Axes • On digital models • As a subset of lattice points 1. In many applications, such as medical imaging, data come as a collection of digital points 2. Computing skeleton on digital model is simple to implement and stable to perform [Sud et. al., 2005]

11. Computing Skeletons • Digital model is represented as a set of points on a spatial grid • Geometry and topology • Adjacency relation 2D 4-connectivity 2D 8-connectivity 3D 6-connectivity 3D 26-connectivity

12. Computing Skeletons • Thinning on point based representation • Peeling off boundary points • Topology preservation:simple points • Shape preservation:curve/surface enpoints • Local operations: simple

13. Computing Skeletons • Obstacles • Topology preservation under parallel thinning • Thinness • 4 points joints • Shape preservation • Endpoints detection is sensitive to noise A fundamental different representation

14. Cell Complexes • In N-D, a set of k-cells (k<=N) • A closed set: the facets of each k-cell (e.g., edges of a square) also belong to the same set Point (0-cell) Edge (1-cell) Square (2-cell) Cube (3-cell)

15. Cell Complexes • Construction: • All those k-cells whose boundary points are in the “points on a spatial grid” representation • Result in a closed set • Any grid, any dimension

16. δ σ Cell Complexes – Simplicial Collapse • Removal of k-simple pair • Dimension , • is only on the boundary of • Topology preserving • Local operation A k-cell σ and a (k-1)-cell δ, so that δ is not contained in another k-cell than σ.

17. Cell Complexes – Simplicial Collapse Proposition 1(Topology-preservation): Simultaneous removal of multiple simple pairs preserves the homotopy of a cell complex. Proposition 2(Thinness): Removal of all simple N-simple pairs deletes all N-cells in a for a N dimensional cell complex.

18. Our Thinning Algorithm • Simplicial collapse • Topology preservation • Thinness • Significance measures • Shape preservation • Our thinning algorithm

19. Significance Measures • Shape elongation(dimension awareness): • k-D skeleton is elongated in k directions: curve-skeleton: 1; surface skeleton: 2 • D, d measures, significance measures S1, S2 • cells with large significance measures are preserved 2-D shape S1 = d – D (2) S2 = 1 – D/d 1-D skeleton

20. Significance Measures – Approximation • Significance measures computation • Approximation of D,d

21. Significance measures – approximation D of a cell is the index of the iteration in which the cell becomes isolated d of a cell is the index of iteration in which the cell becomes simple

22. Significance Measures – Approximation D measure d measure S1 measure S2 measure

23. Significance Measure – Approximation S1 measure S2 measure T1 = 5; T2 = 0.5

24. Our Thinning Algorithm Significance measures Approximate d Approximate D Parallel thinning

25. Our Thinning Algorithm • Algorithm is simple to implement • Skeleton is thin, topology preserving, and shape preserving

26. Results - T Shape Model t1 = 5, t2 = 0.5

27. Results – Rocker Arm Model t1 = 5, t2 = 0.5

28. Results – Hip Bone Model t1 = 5, t2 = 0.5

29. Results – Hip Bone Model t1 = 9, t2 = 0.5

30. Results – Fertility Model t1 = 5, t2 = 0.5

31. Results – Dragon Model t1 = 5, t2 = 0.5

32. Results – Protein timModel t1 = 5, t2 = 0.5

33. Performance X 3 X 3 X 1 128 * 128 * 128 uniform grid

34. Performance

35. Future Work • Queue structure for outmost layer in thinning • To overcome time consuming • Adaptive thinning algorithm on octree grid • To overcome memory consuming • Other topology preserving operators • ? Growing operator: skinning • Growing operator + simplicial collapse: topology preserving and volume preserving deformation

36. Conclusion • Present and prove two properties of simplicialcollapse on cell complexes • Thinness • Topology preservation under parallel thinning • Propose two significant measures • Shape preservation • Develop a simple thinning algorithm

37. Acknowledgement • Great thanks goes to • My advisor: Professor Tao Ju • My committee members:Professor Cindy GrimmProfessor Robert Pless

38. Q & A