Digital and Cross-Section Topology for Image Processing

1. Digital topology andcross-section topology for grayscale image processing M. Couprie A2SI lab., ESIEE Marne-la-Vallée, France www.esiee.fr/a2si

2. « A topologist is interested in those properties of a thing that, while they are in a sense geometrical, are the most permanent- the ones that will survive distortion and stretching. » Stephen Barr,"Experiments in Topology",1964 What is topology ?

3. Aims of this talk • Present the main notions of digital topology • Show their usefulness in image processing applications • Study some topological properties of digital grayscale images (ie. functions from Zn to Z) • Topology-preserving image transforms • Topology-altering image transforms • Applications to image processing (filtering, segmentation…)

4. Some milestones • J.C. Maxwell (1870): ‘On hills and dales ’ • A. Rosenfeld (1970’s): digital topology • A. Rosenfeld (1980’s): fuzzy digital topology • S. Beucher (1990): definition of homotopy between grayscale images • G. Bertrand (1995): cross-section topology

5. Contributors • Gilles Bertrand • Michel Couprie • Laurent Perroton • Zouina Aktouf (Ph.D student) • Francisco Nivando Bezerra (Ph.D student) • Xavier Daragon (Ph.D student) • Petr Dokládal (Ph.D student) • Jean-Christophe Everat (Ph.D student)

6. Outline of the talk Part 1 • Digital topology for binary images • Applications to graylevel image processing • Cross-section topology for grayscale images • Topology-altering transforms, applications Part 2

7. Digital topology for binary images • 8-adjacency, 4-adjacency in black: X (object) in white: X (background). • path, connected components

8. Digital topology for binary images • 8-adjacency, 4-adjacencyin black: X (object) in white: X (background). • path, connected components

9. Digital topology for binary images • 8-adjacency, 4-adjacency in black: X (object)in white: X (background). • path, connected components

10. Digital topology for binary images • 8-adjacency, 4-adjacency in black: X (object)in white: X (background). • path, connected components

11. 2 2 3 1 More examples Black:4, white:8 Black:8, white:4

12. Jordan property • Any simple closed curve divides the plane into 2 connected components.

13. 4-curve 8-curve Jordan property (cont.) • A subset X of Z2 is a simple closed curve if each point x of X has exactly two neighbors in X

14. Jordan property (cont.) • A subset X of Z2 is a simple closed curve if each point x of X has exactly two neighbors in X Not a 4-curve

15. Jordan property (cont.) • A subset X of Z2 is a simple closed curve if each point x of X has exactly two neighbors in X Not an 8-curve

16. Jordan property (cont.) • The Jordan property does no hold if X and its complement have the same adjacency 8-adjacency 4-adjacency

17. Topology preservation - notion of simple point • A topology-preserving transform must preserve the number of connected components of both X and X. • Definition (2D): A point p is simple (for X) if its modification (addition to X, withdrawal from X) does not change the number of connected components of X and X.

18. Simple point: examples Set X (black points)

19. Simple point: examples Simple point of X

20. Simple point: examples Simple point of X

21. Simple point: counter-examples Non-simple point (background component creation if deleted)

22. Simple point: counter-examples Non-simple point (component splitting if deleted)

23. Simple point: parallel deletion Deleting simple points in parallel may change the topology

24. Simple point: local characterization ?

25. Simple point: local characterization ?

26. Connectivity numbers • T(p)=number of Connected Components of (X - {p}) N(p) where N(p)=8-neighborhood of p • T(p)=number of Connected Components of (X - {p}) N(p)A • Characterization of simple points (local): p is simple iff T(p) = 1 and T(p) = 1 A U U T=2,T=2 T=1,T=1 T=1,T=0 T=0,T=1 Isolated point Interior point

27. Skeletons • We say that Y is a skeleton of X if Y may be obtained from X by sequential deletion of simple points • If Y is a skeleton of X and if Y contains no simple point, then we say that Y is an ultimate skeleton of X • If Y is a skeleton of X and if Y contains only non-simple points and ‘end points’, then we say that Y is a curve skeleton of X

28. End points • We say that p is an end point for X if (X - {p}) N(p) contains exactly one point U End point End point (8) Non-end point Non-end point

29. Skeletons (examples) End points Ultimate skeleton Curve skeleton Original image

30. Homotopy • We say that X and Y are homotopic (i.e. they have the same topology) if Y may be obtained from X by sequential addition or deletion of simple points

31. Basic thinning algorithms Ultimate thinning(X) Repeat until stability:Select a simple point p of XRemove p from X Curve thinning(X) Repeat until stability:Select a simple, non-end point p of XRemove p from X

32. Centering skeletons

33. Simple pointsdetectedduring the 1stiteration Candidatesfor the 2nditeration Breadth-first thinning

34. Breadth-first thinning Breadth-first ultimate thinning(X) T := X ; T’ :=  Repeat until stability While  p  T do T := T \ {p} If p is a simple point for X then X := X \ {p} For all q neighbour of p, q  X, do T’ := T’  {q}T := T’ ; T’ := 

35. Directional thinning • (only in 2D) North South East West

36. Parallel directional algorithm • In 2D, simple and non-end points of the same type (N,S,E,W) can be removed in parallel Directional Curve thinning(X) Repeat until stability: For dir =N,S,E,W Let Y be the set of all simple, non-end points of X of type dir X = X \ Y

37. Up Does not work in 3D • In 3D, there are 6 principal directions: N,S,E,W,Up,Down

38. Thinning guided by a distance map • Let X be a subset of Z2, the distance mapDMX is defined by, for all point p:DMX(p) = min{dist(p,q), q not in X}where dist is a distance (e.g. city block, chessboard, Euclidean)

39. Discrete distances • City block distance (d4) d4(p,q) = length of a shortest 4-path between p and q • Chessboard distance (d8) d8(p,q) = length of a shortest 8-path between p and q

40. 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 1 1 1 2 2 2 2 1 1 1 2 3 3 3 3 2 1 3 3 1 2 2 2 2 1 1 2 3 4 4 3 2 1 3 3 3 3 1 2 2 1 1 2 3 4 4 3 2 1 3 3 1 2 4 4 2 1 1 2 3 3 3 3 2 1 3 3 3 3 1 2 2 1 1 2 3 2 2 3 2 1 1 2 2 2 2 2 2 1 1 2 2 1 1 2 2 1 1 2 1 1 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 2 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Discrete distance maps d4 d8

41. d8 de d4 Discrete and Euclidean distance maps

42. Thinning guided by a distance map: algorithm Ultimate guided thinning(X) Compute the distance map DMX Repeat until stability:Select a simple point p of X such that DMX(p) is minimalRemove p from X Curve guided thinning(X) Id., replace « simple » by « simple non end »

43. d8 de d4 Thinning guided by a distance map: results

44. The 3D case A topology-preserving transform must preserve: • number of connected components of X • number of connected components of X • number of « holes » (or « tunnels »)

45. 3D hole (tunnel)

46. 3D hole (tunnel)

47. 3D-skeletons Surface skeleton Ultimate skeleton Curve skeleton Original object

48. T=1,T=1 T=1,T=2 T=3,T=1 Topological classification of points (G. Bertrand) (2D isthmus) (1D isthmus) (simple point)

49. 3D hole closing • G. Bertrand • Z. Aktouf (Ph. D) • L. Perroton

50. 3D hole closing