1. CSE554SkeletonsSlide 1 CSE 554 Lecture 2: Shape Analysis (Part I) Fall 2015

2. CSE554SkeletonsSlide 2 Review Binary pictures – Tresholding grayscale images – Basic operations Connected component labeling Morphological operators

3. CSE554SkeletonsSlide 3 Shape analysis Questions about shapes: – Metrics: length? Width? orientation? – What are the parts? – How similar are two shapes? Microtubules on the cell surface Sperms of fruit fliesCerebral artery aneurysms Monkey skulls

4. CSE554SkeletonsSlide 4 Skeletons Geometry at the center of the object – Compact, and capturing protruding shape parts Skeleton of 2D shapes: 1D curves Skeleton of 3D shapes: 1D curves and 2D surfaces

5. CSE554SkeletonsSlide 5 Applications Computer graphics and vision – Optical character recognition (a) – Shape retrieval (b) – Animating articulated shapes (c) Bio-medical image analysis – Vessel network analysis (d) – Virtual colonoscopy (e) – Protein modeling (f) (d) (b)(c) (e)(f) (a)

6. CSE554SkeletonsSlide 6 Medial Axes (MA) Interior points with multiple closest points on the boundary Object MA

7. CSE554SkeletonsSlide 7 Medial Axes (MA) Properties Thin MA are curves (1D) in a 2D object, and surfaces (2D) in a 3D object. 2D MA 3D MA

8. CSE554SkeletonsSlide 8 Medial Axes (MA) Properties Preserves object’s shape The object can be reconstructed from MA and its distances to the boundary

9. CSE554SkeletonsSlide 9 Medial Axes (MA) Properties Preserves object’s topology 2D: # of connected components of object and background 3D: # of connected components of object and background, and # of tunnels A 2D shape with 1 object component and 2 background components A 3D shape with 5 tunnels

10. CSE554SkeletonsSlide 10 Medial Axes (MA) Properties  Not stable under boundary perturbation

11. CSE554SkeletonsSlide 11 Skeletons Approximation of medial axes – Roughly corresponds to the stable parts of the medial axes – No unique or precise definition (e.g., application dependent)

12. CSE554SkeletonsSlide 12 Computing Skeletons A classical method: thinning – Relatively simple to implement – Can create curve skeletons in 2D and curve or surface skeletons in 3D What we will cover: – Thinning on binary pictures (this lecture) – Thinning on cell complexes (next lecture)

13. CSE554SkeletonsSlide 13 Medial Axes (MA) Grassfire analogy: – Let the object represent a field of grass. A fire starts at the field boundary, and burns across the field at uniform speed. – MA are where the fire fronts meet. Object MA

14. CSE554SkeletonsSlide 14 Medial Axes (MA) Grassfire analogy: – Let the object represent a field of grass. A fire starts at the field boundary, and burns across the field at uniform speed. – MA are where the fire fronts meet.

15. CSE554SkeletonsSlide 15 2D Thinning Iterative process that reduces a binary picture to a skeleton – Simulating the “grassfire burning” that defines MA Thinning on a binary picture

16. CSE554SkeletonsSlide 16 2D Thinning Iterative process that reduces a binary picture to a skeleton – Simulating the “grassfire burning” that defines MA Thinning on a binary picture

17. CSE554SkeletonsSlide 17 2D Thinning Iterative erosion will remove all pixels – We need to retain “important” pixels, where different erosion fronts meet Thinning on a binary picture

18. CSE554SkeletonsSlide 18 2D Thinning Iteratively remove pixels on the object border – Protecting end pixels or interior pixels End pixel Interior pixel

19. CSE554SkeletonsSlide 19 2D Thinning Iteratively remove pixels on the object border – Protecting end pixels or interior pixels End pixel Interior pixel

20. CSE554SkeletonsSlide 20 2D Thinning Border pixels – Pixels lying on the border of the object – To be considered for removal at each thinning iteration

21. CSE554SkeletonsSlide 21 2D Thinning Border pixels criteria – Object pixel p is on the border if and only if p is connected to some background pixel A border pixel and its 8-connected background pixels b

22. CSE554SkeletonsSlide 22 2D Thinning Border pixels criteria – Object pixel p is on the border if and only if p is connected to some background pixel bbb bbb bbb b Border pixels for 8-connectivity

23. CSE554SkeletonsSlide 23 2D Thinning Border pixels criteria – Object pixel p is on the border if and only if p is connected to some background pixel – Pick a connectivity rule and stay with it! bb bbb bbb b Border pixels for 4-connectivity bbb bbb bbb b Border pixels for 8-connectivity

24. CSE554SkeletonsSlide 24 2D Thinning Curve-end pixels – Object pixels lying at the ends of curves, whose removal would shrink the skeleton (and hence losing shape information). c Curve-end pixel

25. CSE554SkeletonsSlide 25 2D Thinning Curve-end pixels criteria – Object pixel c is a curve-end pixel if and only if c is connected to exactly one object pixel. c Curve-end pixel and its connected pixel

26. CSE554SkeletonsSlide 26 2D Thinning Simple pixels – Object pixels whose removal from the object does not change topology (i.e., # of components of object and background) 123 4 O:1 B:1 (using 8-connectivity for object, 4-connectivity for background) O:1 B:1 Simple! O:2 B:1 O:1 B:2O:1 B:1 1 2 3 4

27. CSE554SkeletonsSlide 27 2D Thinning Simple pixels criteria – Object pixel p is simple if and only if setting p to background does not change the # of components of either the object or background in the 3x3 neighborhood of p. 11 O:1 B:1 22 O:1 B:2O:2 B:1 O:1 B:2O:1 B:3O:1 B:1 33 44 123 4 (using 8-connectivity for object, 4-connectivity for background) Simple!

28. CSE554SkeletonsSlide 28 2D Thinning Simple pixels criteria – Object pixel p is simple if and only if setting p to background does not change the # of components of either the object or background in the 3x3 neighborhood of p. s sss sss s All simple pixels

29. CSE554SkeletonsSlide 29 2D Thinning Putting together: Removable pixels – Border pixels that are simple and not curve-end s sss sss s c bbb bbb bbb b Border pixelsSimple pixelsCurve-end pixels Removal pixels (8-conn for object)

30. CSE554SkeletonsSlide 30 2D Thinning Algorithm (attempt) 1 – Simultaneous removal of all removable points (“Parallel thinning”) // Parallel thinning on a binary image I 1.Repeat: 1.Collect all removable pixels as S 2.If S is empty, Break. 3.Set all pixels in S to be background in I 2.Output I

31. CSE554SkeletonsSlide 31 2D Thinning Algorithm (attempt) 1 – Simultaneous removal of all removable points (“Parallel thinning”)

32. CSE554SkeletonsSlide 32 2D Thinning Why does parallel thinning breaks topology? – Simple pixels, when removed together, may change topology ssss ssss

33. CSE554SkeletonsSlide 33 2D Thinning Algorithm 2 – Sequentially visit each removable pixel and check its simple-ness before removing the pixel. (“Serial Thinning”) // Serial thinning on a binary image I 1.Repeat: 1.Collect all border pixels as S 2.If S is empty, Break. 3.Repeat for each pixel x in S (in certain order): 1.If x is simple and not curve-end, set x to be background in I 2.Output I

34. CSE554SkeletonsSlide 34 2D Thinning Algorithm 2 – Sequentially visit each removable pixel and check its simple-ness before removing the pixel. (“Serial Thinning”) Serial thinning

35. CSE554SkeletonsSlide 35 2D Thinning Algorithm 2 – Sequentially visit each removable pixel and check its simple-ness before removing the pixel. (“Serial Thinning”) – Result is affected by the visiting “sequence” Serial thinning with two different visiting sequences of removable pixels

36. CSE554SkeletonsSlide 36 3D Thinning Identifying removable voxels – Border voxels Similar to 2D: object voxels connected to at least one background voxel – Simple voxels Harder to characterize than 2D: Maintaining # of connected components is not sufficient (need to consider # of tunnels too) – Curve-end and surface-end voxels Curve-end criteria same as in 2D Surface-end criteria are much harder to describe (e.g., requires a table look-up) x Setting voxel x to background creates a “tunnel” in the object (using 26-conn for object)

37. CSE554SkeletonsSlide 37 3D Thinning Two kinds of skeletons – Curve skeletons: only curve-end voxels are preserved during thinning – Surface skeletons: both curve-end and surface-end voxels are preserved (see further readings) Method of [Palagyi and Kuba, 1999] Object Curve skeleton Surface skeleton

38. CSE554SkeletonsSlide 38 Skeleton Pruning Thinning is sensitive to boundary noise – Due to the instability of medial axes Skeleton pruning – During thinning E.g., using more selective criteria for end pixels (voxels) – After thinning E.g., based on branch length – See Further Readings Object with boundary noise Resulting skeleton

39. CSE554SkeletonsSlide 39 Further Readings on: Binary Pictures, MA and Thinning Books – “Digital Geometry: geometric methods for digital picture analysis”, by Klette and Rosenfeld (2004) – “Medial representations: mathematics, algorithms and applications”, by Siddiqi and Pizer (2008) Papers – “Digital topology: introduction and survey”, by Kong and Rosenfeld (1989) Theories of binary pictures – “Thinning methodologies - a comprehensive survey”, by Lam et al. (1992) A survey of 2D methods – “A Parallel 3D 12-Subiteration Thinning Algorithm”, by Palagyi and Kuba (1999) Includes a good survey of 3D thinning methods – “Pruning medial axes”, by Shaked and Bruckstein (1998) A survey of MA and skeleton pruning methods