1. Quasi-linear algorithms for the topological watershed
Michel Couprie

2. Contributors to this work
Gilles Bertrand
Michel Couprie
Laurent Najman

3. Prelude

4. Watersheds
Powerful segmentation operator from the field of Mathematical Morphology
Introduced as a tool for segmenting grayscale images by S. Beucher, H. Digabel and C. Lantuejoul in the 70s
Efficient algorithms based on immersion simulation were proposed by L. Vincent, F. Meyer, P. Soille (and others) in the 90s

5. Watershed transform

6. Flooding paradigm

7. Flooding paradigm

8. Meyer’s flooding algorithm
Label the regional minima with different colors
Repeat
Select a pixel p, not colored, not watershed, adjacent to some colored pixels, and having the lowest possible gray level
If p is adjacent to exactly one color then label p with this color
If p is adjacent to more than one color then label p as watershed
Until no such pixel exists

9. Flooding algorithm 3 3 3 5 5 5 10 10 10 1 10 15 1 1 20 1 20 3 3 3 5 5 5 10 10 10 1 10 1 15 1 20 1 20 3 3 3 5 5 30 30 30 10 15 1 15 20 1 1 20 3 3 3 5 30 20 20 20 30 1 15 1 15 20 1 1 20 40 40 40 40 40 20 20 20 40 40 40 40 40 10 10 10 10 40 20 20 20 40 1 10 10 1 1 10 1 10 5 5 5 5 10 40 20 40 10 10 1 5 1 5 5 1 1 1 3 5 10 15 1 20 1 15 10 5 1 1 1 1 1 1 1 3 5 10 15 20 15 10 5 1 1 1 3 5 10 15 20 15 10 5 1

10. Flooding algorithm 3 3 3 3 3 3 5 5 5 10 10 10 1 10 1 15 1 20 20 1 3 3 3 3 3 3 5 5 5 10 10 10 10 1 15 1 20 1 1 20 3 3 3 3 3 3 5 5 30 30 30 10 1 15 15 20 1 1 20 3 3 3 3 3 3 5 30 20 20 20 30 15 1 1 15 20 1 20 1 40 40 40 40 40 20 20 20 40 40 40 40 40 10 10 10 10 40 20 20 20 40 1 10 1 10 1 10 10 1 5 5 5 5 10 40 20 40 10 10 1 5 5 1 5 1 1 1 1 1 3 5 10 1 15 20 1 15 10 5 1 1 1 1 1 1 1 1 1 3 5 10 15 20 15 10 5 1 1 1 1 1 3 5 10 15 20 15 10 5 1

11. Flooding algorithm 3 3 3 3 3 3 3 3 3 5 5 5 10 10 10 10 1 15 1 1 20 20 1 3 3 3 3 3 3 3 3 3 5 5 5 10 10 10 10 1 15 1 20 1 20 1 3 3 3 3 3 3 3 3 3 5 5 30 30 30 10 15 1 15 1 20 20 1 3 3 3 3 3 3 3 3 3 5 30 20 20 20 30 15 1 15 1 1 20 20 1 40 40 40 40 40 20 20 20 40 40 40 40 40 10 10 10 10 40 20 20 20 40 10 1 10 1 1 10 10 1 5 5 5 5 10 40 20 40 10 10 1 5 1 5 1 5 1 1 3 5 10 15 1 20 1 15 10 5 1 1 1 1 1 1 1 3 5 10 15 20 15 10 5 1 1 1 3 5 10 15 20 15 10 5 1

12. Flooding algorithm 3 3 3 5 5 5 10 10 10 1 10 15 1 1 20 1 20 3 3 3 5 5 5 10 10 10 1 10 1 15 1 20 1 20 3 3 3 5 5 30 30 30 10 15 1 15 20 1 1 20 3 3 3 5 30 20 20 20 30 1 15 1 15 20 1 1 20 40 40 40 40 40 20 20 20 40 40 40 40 40 10 10 10 10 40 20 20 20 40 1 10 10 1 1 10 1 10 5 5 5 5 10 40 20 40 10 10 1 5 1 5 5 1 1 1 3 5 10 15 1 20 1 15 10 5 1 1 1 1 1 1 1 3 5 10 15 20 15 10 5 1 1 1 3 5 10 15 20 15 10 5 1

13. Flooding algorithm 3 3 3 5 5 5 10 10 10 1 10 15 1 1 20 1 20 3 3 3 5 5 5 10 10 10 1 10 1 15 1 20 1 20 3 3 3 5 5 30 30 30 10 15 1 15 20 1 1 20 3 3 3 5 30 20 20 20 30 1 15 1 15 20 1 1 20 40 40 40 40 40 20 20 20 40 40 40 40 40 10 10 10 10 40 20 20 20 40 1 10 10 1 1 10 1 10 5 5 5 5 10 40 20 40 10 10 1 5 1 5 5 1 1 1 3 5 10 15 1 20 1 15 10 5 1 1 1 1 1 1 1 3 5 10 15 20 15 10 5 1 1 1 3 5 10 15 20 15 10 5 1

14. Flooding algorithm 3 3 3 5 5 5 10 10 10 1 10 15 1 1 20 1 20 3 3 3 5 5 5 10 10 10 1 10 1 15 1 20 1 20 3 3 3 5 5 30 30 30 10 15 1 15 20 1 1 20 3 3 3 5 30 20 20 20 30 1 15 1 15 20 1 1 20 40 40 40 40 40 20 20 20 40 40 40 40 40 10 10 10 10 40 20 20 20 40 1 10 10 1 1 10 1 10 5 5 5 5 10 40 20 40 10 10 1 5 1 5 5 1 1 1 3 5 10 15 1 20 1 15 10 5 1 1 1 1 1 1 1 3 5 10 15 20 15 10 5 1 1 1 3 5 10 15 20 15 10 5 1

15. Flooding algorithm 3 3 3 5 5 5 10 10 10 10 1 15 1 1 20 20 1 3 3 3 5 5 5 10 10 10 10 1 15 1 20 1 20 1 3 3 3 5 5 30 30 30 10 1 15 15 20 1 1 20 3 3 3 5 30 20 20 20 30 15 1 1 15 1 20 20 1 40 40 40 40 40 20 20 20 40 40 40 40 40 10 10 10 10 40 20 20 20 40 10 1 10 1 1 10 1 10 5 5 5 5 10 40 20 40 10 10 1 5 5 1 5 1 1 1 3 5 10 15 1 20 1 15 10 1 5 1 1 1 1 1 1 3 5 10 15 20 15 10 5 1 1 1 3 5 10 15 20 15 10 5 1

16. Flooding algorithm 3 3 3 5 5 5 10 10 10 10 1 1 15 1 20 20 1 3 3 3 5 5 5 10 10 10 1 10 15 1 20 1 1 20 3 3 3 5 5 30 30 30 30 30 10 15 1 15 20 1 1 20 3 3 3 5 30 30 20 20 20 30 30 1 15 15 1 20 1 20 1 40 40 40 40 40 20 20 20 40 40 40 40 40 10 10 10 10 40 20 20 20 40 10 1 10 1 1 10 1 10 5 5 5 5 10 40 20 40 10 10 5 1 5 1 5 1 1 1 3 5 10 1 15 20 15 1 10 1 5 1 1 1 1 1 1 3 5 10 15 20 15 10 5 1 1 1 3 5 10 15 20 15 10 5 1

17. Flooding algorithm 3 3 3 5 5 5 10 10 10 10 1 1 15 20 1 20 1 3 3 3 5 5 5 10 10 10 10 1 15 1 20 1 1 20 3 3 3 5 5 30 30 30 10 15 1 15 20 1 1 20 3 3 3 5 30 20 20 20 30 15 1 1 15 1 20 1 20 40 40 40 40 40 20 20 20 40 40 40 40 40 10 10 10 10 40 20 20 20 40 1 10 10 1 10 1 1 10 5 5 5 5 10 40 20 40 10 10 1 5 1 5 5 1 1 1 3 5 10 1 15 20 15 1 10 5 1 1 1 1 1 1 1 3 5 10 15 20 15 10 5 1 1 1 3 5 10 15 20 15 10 5 1
There is no descending path from the 20s to the minimum 3
The contrast between minima is not preserved

18. Flooding algorithm
Efficient implementation thanks to a priority queue (non-monotonic)
Preserves the connectivity of the binary set constituted by the regional minima of the original image
Uses the graylevels as a « heuristics » to guide the propagation of the colors associated to the different minima

19. Topological watershed

20. Discrete sets and W-destructible points
Let (V,E) be a (undirected) graph and let X be a subset of V.

22. Graph

23. Graph

24. Discrete sets and W-destructible points
Let (V,E) be a (undirected) graph and let X be a subset of V.
We say that a point x  X is W-destructible for X if x is adjacent to exactly one connected component of X.
M. Couprie and G. Bertrand (1997)

25. W-destructible point

26. Non W-destructible point

27. W-destructible point

28. W-destructible point

29. Topological watersheds
Let X and Y be two subsets of V. We say that Y is a W-thinning of X if Y may be obtained from X by iteratively removing W-destructible points.
Let X and Y be subsets of V. We say that Y is a watershed of X if Y is a W-thinning of X and if there is no W-destructible point for Y.
M. Couprie and G. Bertrand (1997)

31. W-thinning

32. W-thinning

33. W-thinning

34. W-thinning

35. W-thinning

36. W-thinning

37. Watershed

38. Watershed 1

39. Watershed 2

40. Discrete maps and W-destructible points
Let (V,E) be a connected (undirected) graph. We denote by F (V) the family composed of all maps from V to Z.
Let F  F (V), we set Fk = {x  V; F(x)  k},
Fk is the cross-section of F at level k
Let x  V and let k = F(x). We say that x is W-destructible (for F) if x is adjacent to exactly one connected component of Fk
M. Couprie and G. Bertrand (1997)

41. W-destructible points3 3 3 5 5 5 10 10 10 1 10 15 1 1 20 1 20 3 3 3 5 5 5 10 10 10 1 10 1 15 1 20 1 20 3 3 3 5 5 30 30 30 10 15 1 15 20 1 1 20 3 3 3 5 30 20 20 20 30 1 15 1 15 20 1 1 20 40 40 40 40 40 20 20 20 40 40 40 40 40 10 10 10 10 40 20 20 20 40 1 10 10 1 1 10 1 10 5 5 5 5 10 40 20 40 10 10 1 5 1 5 5 1 1 1 3 5 10 15 1 20 1 15 10 5 1 1 1 1 1 1 1 3 5 10 15 20 15 10 5 1 1 1 3 5 10 15 20 15 10 5 1

42. W-destructible points3 3 3 5 5 5 10 10 10 1 10 15 1 1 20 1 20 3 3 3 5 5 5 10 10 10 1 10 1 15 1 20 1 20 3 3 3 5 5 30 30 30 10 15 1 15 20 1 1 20 3 3 3 5 30 20 20 20 30 1 15 1 15 20 1 1 20 40 40 40 40 40 20 20 20 40 40 40 40 40 10 10 10 10 40 20 20 20 40 1 10 10 1 1 10 1 10 5 5 5 5 10 40 20 40 10 10 1 5 1 5 5 1 1 1 3 5 10 15 1 20 1 15 10 5 1 1 1 1 1 1 1 3 5 10 15 20 15 10 5 1 1 1 3 5 10 15 20 15 10 5 1

43. W-destructible points3 3 3 5 5 5 10 10 10 1 10 15 1 1 20 1 20 3 3 3 5 5 5 10 10 10 1 10 1 15 1 20 1 20 3 3 3 5 5 30 30 30 10 15 1 15 20 1 1 20 3 3 3 5 30 20 20 20 30 1 15 1 15 20 1 1 20 40 40 40 40 40 20 20 20 40 40 40 40 40 10 10 10 10 40 20 20 20 40 1 10 10 1 1 10 1 10 5 5 5 5 10 40 20 40 10 10 1 5 1 5 5 1 1 1 3 5 10 15 1 20 1 15 10 5 1 1 1 1 1 1 1 3 5 10 15 20 15 10 5 1 1 1 3 5 10 15 20 15 10 5 1 F1

44. W-destructible points3 3 3 5 5 5 10 10 10 1 10 15 1 1 20 1 20 3 3 3 5 5 5 10 10 10 1 10 1 15 1 20 1 20 3 3 3 5 5 30 30 30 10 15 1 15 20 1 1 20 3 3 3 5 30 20 20 20 30 1 15 1 15 20 1 1 20 40 40 40 40 40 20 20 20 40 40 40 40 40 10 10 10 10 40 20 20 20 40 1 10 10 1 1 10 1 10 5 5 5 5 10 40 20 40 10 10 1 5 1 5 5 1 1 1 3 5 10 15 1 20 1 15 10 5 1 1 1 1 1 1 3 5 10 15 20 15 10 5 1 1 1 3 5 10 15 20 15 10 5 1

45. Topological watershed
Let F and G be in F (V).
We say that G is a W-thinning of F if G may be obtained from F by iteratively lowering W-destructible points (by 1).
If G is a W-thinning of F which has been obtained by lowering a single point p (by 1 or by more than 1) we say that p has been W-lowered.
We say that G is a (topological) watershed of F if G is a W-thinning of F and if there is no W-destructible point for G.
M. Couprie and G. Bertrand (1997)

46. Topological watershed3 3 3 5 5 5 10 10 10 1 10 15 1 1 20 1 20 3 3 3 5 5 5 10 10 10 1 10 1 15 1 20 1 20 3 3 3 5 5 30 30 30 10 15 1 15 20 1 1 20 3 3 3 5 30 20 20 20 30 1 15 1 15 20 1 1 20 40 40 40 40 40 20 20 20 40 40 40 40 40 10 10 10 10 40 20 20 20 40 1 10 10 1 1 10 1 10 5 5 5 5 10 40 20 40 10 10 1 5 1 5 5 1 1 1 3 5 10 15 1 20 1 15 10 5 1 1 1 1 1 1 3 5 10 15 20 15 10 5 1 1 1 3 5 10 15 20 15 10 5 1

47. Topological watershed3 3 3 3 3 3 3 3 3 1 3 1 3 3 1 3 1 3 3 3 3 3 3 3 3 3 1 3 3 1 1 3 3 1 3 3 3 3 3 30 30 30 3 1 3 3 3 1 3 3 3 3 3 30 20 20 20 30 1 3 3 1 1 3 1 3 40 40 40 40 40 20 20 20 40 40 40 40 40 1 1 1 1 40 20 20 20 40 1 1 1 1 1 1 1 1 1 40 20 40 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 20

48. Topological watershed3 3 3 3 3 3 3 3 3 1 3 1 3 3 1 3 1 3 3 3 3 3 3 3 3 3 1 3 3 1 1 3 3 1 3 3 3 3 3 30 30 30 3 1 3 3 3 1 3 3 3 3 3 30 20 20 20 30 1 3 3 1 1 3 1 3 40 40 40 40 40 20 20 20 40 40 40 40 40 1 1 1 1 40 20 20 20 40 1 1 1 1 1 1 1 1 1 40 20 40 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 20

49. Topological watershed3 3 3 3 3 3 3 3 3 1 3 1 3 3 1 3 1 3 3 3 3 3 3 3 3 3 1 3 3 1 1 3 1 3 3 3 3 3 3 30 30 30 3 3 1 3 3 1 3 3 3 3 3 30 20 20 20 30 1 3 3 1 3 1 3 1 40 40 40 40 40 20 20 20 40 40 40 40 40 1 1 1 1 40 20 20 20 40 1 1 1 1 1 1 1 1 1 40 20 40 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 20
F40

50. Topological watershed3 3 3 3 3 3 3 3 3 1 3 1 3 3 1 3 1 3 3 3 3 3 3 3 3 3 1 3 3 1 1 3 3 1 3 3 3 3 3 30 30 30 3 1 3 3 3 1 3 3 3 3 3 30 20 20 20 30 1 3 3 1 1 3 1 3 40 40 40 40 40 20 20 20 40 40 30 40 40 1 1 1 1 40 20 20 20 40 1 1 1 1 1 1 1 1 1 40 20 40 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 20

51. Topological watershed3 3 3 3 3 3 3 3 3 1 3 1 3 3 1 3 1 3 3 3 3 3 3 3 3 3 1 3 3 1 1 3 1 3 3 3 3 3 3 30 30 30 3 3 1 3 3 1 3 3 3 3 3 30 20 20 20 30 1 3 3 1 3 1 3 1 40 40 40 40 40 20 20 20 40 40 30 40 40 1 1 1 1 40 20 20 20 40 1 1 1 1 1 1 1 1 1 40 20 40 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 20
F30

52. Topological watershed3 3 3 3 3 3 3 3 3 1 3 1 3 3 1 3 1 3 3 3 3 3 3 3 3 3 1 3 3 1 1 3 3 1 3 3 3 3 3 30 30 30 3 1 3 3 3 1 3 3 3 3 3 30 20 20 20 30 1 3 3 1 1 3 1 3 40 40 40 40 40 20 20 20 40 40 30 40 40 1 1 1 1 40 20 20 20 40 1 1 1 1 1 1 1 1 1 40 20 40 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 20

53. Topological watershed3 1 30 30 30 30 20 20 20 30 40 40 40 40 40 20 20 20 40 40 30 40 40 1 40 20 20 1 20 40 40 20 40 20 20 20
F20

54. Topological watershed3 3 3 3 3 3 3 3 3 1 3 3 1 1 3 3 1 3 3 3 3 3 3 3 3 3 1 3 3 1 3 1 1 3 3 3 3 3 3 30 30 30 3 3 1 3 3 1 3 3 3 3 3 30 20 30 3 1 3 1 3 1 1 3 40 40 40 40 40 40 40 30 40 40 1 1 1 1 40 40 1 1 1 1 1 1 1 1 1 40 40 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

55. Topological watershed3 20 1 3 3 3 3 3 3 3 3 3 1 3 3 1 3 1 3 1 3 3 3 3 3 3 3 3 3 3 1 3 1 1 3 1 3 3 3 3 3 3 30 30 30 3 3 1 3 3 1 3 3 3 3 3 30 30 3 1 1 3 3 1 3 1 40 40 40 40 40 40 40 30 40 40 1 1 1 1 40 40 1 1 1 1 1 1 1 1 1 40 40 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
F30

56. Topological watershed3 3 3 3 3 3 3 3 3 1 3 1 3 3 1 3 1 3 3 3 3 3 3 3 3 3 1 3 3 1 1 3 3 1 3 3 3 3 3 30 30 30 3 1 3 3 3 1 3 3 3 3 3 30 20 20 20 30 1 3 3 1 1 3 1 3 40 40 40 40 40 20 20 20 40 40 30 40 40 1 1 1 1 40 20 20 20 40 1 1 1 1 1 1 1 1 1 40 20 40 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 20

57. Topological watershed3 3 3 3 3 3 3 3 3 1 3 1 3 3 1 3 1 3 3 3 3 3 3 3 3 3 1 3 3 1 1 3 1 3 3 3 3 3 3 30 30 30 3 3 1 3 3 1 3 3 3 3 3 30 20 20 20 30 1 3 3 1 3 1 3 1 40 40 40 40 40 20 20 20 40 40 40 40 40 1 1 1 1 40 20 20 20 40 1 1 1 1 1 1 1 1 1 40 20 40 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 20
F40

58. Topological watershed3 3 3 3 3 3 3 3 3 1 3 1 3 3 1 3 1 3 3 3 3 3 3 3 3 3 1 3 3 1 1 3 1 3 3 3 3 3 3 30 30 30 3 3 1 3 3 1 3 3 3 3 3 30 20 20 20 30 1 3 3 1 3 1 3 1 40 40 40 40 40 20 20 20 40 40 40 40 40 1 1 1 1 30 20 20 20 30 1 1 1 1 1 1 1 1 1 30 20 30 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 20
F30

59. Topological watershed3 3 3 3 3 3 3 3 3 1 3 1 3 3 1 3 1 3 3 3 3 3 3 3 3 3 1 3 3 1 1 3 1 3 3 3 3 3 3 30 30 30 3 3 1 3 3 1 3 3 3 3 3 30 20 20 20 30 1 3 3 1 3 1 3 1 40 40 40 40 40 20 20 20 40 40 40 40 40 1 1 1 1 20 20 20 20 20 1 1 1 1 1 1 1 1 1 20 20 20 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 20
F20

60. Topological watershed3 3 3 3 3 3 3 3 3 1 3 1 3 3 1 3 1 3 3 3 3 3 3 3 3 3 1 3 3 1 1 3 3 1 3 3 3 3 3 30 30 30 3 1 3 3 3 1 3 3 3 3 3 30 20 20 20 30 1 3 3 1 1 3 1 3 40 40 40 40 40 20 20 20 40 40 30 40 40 1 1 1 1 1 20 20 20 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 20

61. Topological watershed3 3 3 3 3 3 3 3 3 3 1 1 3 3 1 3 1 3 3 3 3 3 3 3 3 3 3 1 3 1 1 3 3 1 3 3 3 3 3 30 30 30 3 1 3 3 3 1 3 3 3 3 3 30 20 20 20 30 3 1 3 1 1 3 1 3 40 40 40 40 40 20 20 20 40 40 30 40 40 1 1 1 1 1 20 20 20 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 20

62. Topological watershed3 3 3 3 3 3 3 3 3 1 3 3 1 1 3 3 1 3 3 3 3 3 3 3 3 3 1 3 3 1 3 1 3 1 3 3 3 3 3 30 30 30 3 3 1 3 3 1 3 3 3 3 3 30 1 20 30 3 1 3 1 1 3 1 3 30 30 30 30 1 1 20 30 30 30 30 1 1 1 1 1 1 20 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 20

63. Topological watershed3 3 3 3 3 3 3 3 3 1 3 3 1 1 3 1 3 3 3 3 3 3 3 3 3 3 3 1 3 1 3 1 1 3 3 3 3 3 3 30 30 30 3 3 1 3 3 1 3 3 3 3 3 30 1 20 30 3 1 1 3 3 1 3 1 30 30 30 30 1 1 20 30 30 30 30 1 1 1 1 1 1 20 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 20
The watershed is located on the crest lines of the original image
The contrast between minima is preserved

64. W-thinning paradigm

65. Algorithms

66. Naive watershed algorithm
Repeat
For each point p in V n
If p is W-destructible n2
F(p) = F(p)-1
Until stability g.n2

67. How can we reduce the complexity ?
Two fundamental ideas:
1/ Find a characterization of W-destructible points which can be checked in constant time
2/ Find strategies which guarantee that each point is examined at most once

68. Different types of points (1)9 1 8 6 5 7 2 3
Separating point (crest)
A point p is separating for F if it is adjacent to at least two connected components of Fk, with k = F(p)

69. Different types of points (2)9 1 8 6 5 7 2 3
Inner point, minimal (flat)
A point p is an inner point for F if it is not adjacent to any connected component of Fk , with k = F(p)
A point p is a minimal point for F if it belongs to a regional minimum of F

70. Different types of points (3)9 1 8 6 5 7 2 3
Inner point, non minimal (flat)
A point p is an inner point for F if it is not adjacent to any connected component of Fk , with k = F(p)
A point p is a minimal point for F if it belongs to a regional minimum of F

71. Different types of points (4)9 1 8 6 5 7 2 3
Pre-separating point (crest)
A point p is pre-separating for F if it is W-destructible and if there is a value v such that W-lowering p until v makes p be a separating point

72. Different types of points (5)9 1 8 6 5 7 2 3
Pre-inner point, M-destructible (flat)
A point p is pre-inner for F if it is W-destructible and if there is a value v such that W-lowering p until v makes p be an inner point
A point p is M-destructible for F if it is pre-inner and it may become a minimal point after W-lowering

73. Different types of points (6)9 1 8 6 5 7 2 3
Pre-inner point, non M-destructible (flat)
A point p is pre-inner for F if it is W-destructible and if there is a value v such that W-lowering p until v makes p be an inner point
A point p is M-destructible for F if it is pre-inner and it may become a minimal point after W-lowering

74. Classification of points
crest
flat
W-destructible
separating
pre-separating
pre-inner
inner
M-destructible
non-M-destructible pre-inner
minimal
non-minimal inner

75. Transitions (1) 9 1 8 6 5 7 2 3 5
pre-separating
separating

76. pre-inner, M-destructible
Transitions (2) 9 1 8 6 5 7 2 3
pre-inner, M-destructible
inner, minimal

77. pre-inner, non M-destructible
Transitions (3) 9 1 8 6 5 7 2 3 8
pre-inner, non M-destructible
inner, non minimal

78. Transitions (4) 9 1 8 6 5 7 2 3
pre-separating
separating

79. pre-inner, M-destructible
Transitions (5) 9 1 8 6 5 7 2 3 2
pre-inner, M-destructible
pre-separating

80. pre-inner, M-destructible
Transitions (6) 9 1 8 6 5 7 2 3
pre-inner, M-destructible
separating

81. Transitions (7) pre-inner, non M-destructible9 1 8 6 5 7 2 3 1
pre-inner, non M-destructible
pre-inner, M-destructible

82. pre-inner, non M-destructible
Transitions (8) 9 1 8 6 5 7 2 3 2
pre-inner, non M-destructible
pre-separating

83. pre-inner, non M-destructible
Transitions (9) 9 1 8 6 5 7 2 3
pre-inner, non M-destructible
separating

84. pre-inner, non M-destructible
Transitions (10) 9 1 8 6 5 7 2 3 7
inner, non minimal
pre-inner, non M-destructible

85. pre-inner, M-destructible
Transitions (11) 9 1 8 6 5 7 2 3 2
inner, non minimal
pre-inner, M-destructible

86. Classification of transitions
Theorem: the only possible transitions for a point during a W-thinning are the 11 following ones:
crest
separating
pre-separating
pre-inner, non M-destructible
pre-inner, M-destructible
inner, non minimal
inner, minimal

87. Pass value
Let F be in F (V). If п is a path, we set F(п) = Max{F(x); x  п}.
Let x, y in V We set F(x,y) = Min {F(п); п  п(x,y)}, F(x,y) is the pass value between x and y.
Let X and Y be two subsets of V. We set F(X,Y) = Min{F(x,y); x  X and y  Y}.

88. Pass value
F(x,y) = 20 40 40 40 40 40 40 40 40 40 40 40 40 40 40 1 1 2 3 10 5 25 5 4 4 4 40 x y 40 1 2 8 6 5 5 20 3 2 3 40 40 3 3 2 3 10 6 6 6 22 2 3 40 40 6 6 40 6 11 11 11 25 4 4 4 40 40 40 35 10 30 15 15 15 35 31 36 10 40 40 10 8 5 10 32 33 34 10 10 15 38 40 40 8 5 1 1 15 40 10 6 3 15 20 40 40 10 8 5 10 15 35 15 6 6 15 35 40 40 40 40 40 40 40 40 40 40 40 40 40 40

89. Separation
Let F be in F (V) and let x and y be in V. We say that x and y are separated (for F) if F(x,y) > Max{F(x),F(y)}. We say that x and y are k-separated (for F) if x and y are separated and F(x,y) = k.
Let F and G be in F (V) such that G  F. We say that G is a separation of F if, for all x,y in V, if x and y are k-separated for F, then x and y are k-separated for G.

90. k-separation y x x and y are 20-separated 40 40 40 40 40 40 40 40 401 1 2 3 10 5 25 5 4 4 4 40 x y 40 1 2 8 6 5 5 20 3 2 3 40 40 3 3 2 3 10 6 6 6 22 2 3 40 40 6 6 40 6 11 11 11 25 4 4 4 40 40 40 35 10 30 15 15 15 35 31 36 10 40 40 10 8 5 10 32 33 34 10 10 15 38 40 40 8 5 1 1 15 40 10 6 3 15 20 40 40 10 8 5 10 15 35 15 6 6 15 35 40 40 40 40 40 40 40 40 40 40 40 40 40 40

91. x and y are NOT separated (they are linked)
k-separation
F(x,y) = 35 = F(y)
x and y are NOT separated (they are linked) 40 40 40 40 40 40 40 40 40 40 40 40 40 40 1 1 2 3 10 5 25 5 4 4 4 40 40 1 2 8 6 5 5 20 3 2 3 40 x y 40 3 3 2 3 10 6 6 6 22 2 3 40 40 6 6 40 6 11 11 11 25 4 4 4 40 40 40 35 10 30 15 15 15 35 31 36 10 40 40 10 8 5 10 32 33 34 10 10 15 38 40 40 8 5 1 1 15 40 10 6 3 15 20 40 40 10 8 5 10 15 35 15 6 6 15 35 40 40 40 40 40 40 40 40 40 40 40 40 40 40

92. Theorem (strong separation)
Let F and G be in F (V) such that G  F. The map G is a strong separation of F if and only if G is a W-thinning of F.
G. Bertrand (2003)

93. Property (characterization of W-destructible points)
Let F be in F (V), let p be in V.
We denote by N-(p) the set of points adjacent to p and with a strictly lower altitude (lower neighbours).
The point p is W-destructible for F if and only if N-(p) is not empty and for all x,y in N-(p), we have F(x,y) < F(p).

94. Illustration x y F(x,y) = 20 3 3 3 5 5 5 10 10 10 1 10 1 15 20 1 1 2030 30 30 10 1 15 15 20 1 20 1 3 3 3 5 30 20 20 20 30 1 15 1 15 1 20 20 1 40 40 40 40 40 20 20 20 40 40 40 40 40 10 10 10 10 40 20 20 20 40 10 1 10 1 10 1 10 1 5 5 5 5 10 40 20 40 10 10 5 1 1 5 5 1 1 1 3 5 10 15 20 15 10 1 5 1 1 1 1 1 1 3 5 10 x 15 20 15 y 10 5 1 1 1 3 5 10 15 20 15 10 5 1
F(x,y) = 20

95. Illustration 3 3 3 5 5 5 10 10 10 10 1 1 15 20 1 1 20 3 3 3 5 5 5 10 10 10 10 1 15 1 20 1 1 20 3 3 3 5 5 30 30 30 10 1 15 15 1 20 1 20 3 3 3 5 30 20 20 20 30 1 15 1 15 1 20 1 20 40 40 40 40 40 20 20 20 40 40 40 40 40 10 10 10 10 40 20 20 20 40 10 1 1 10 10 1 10 1 5 5 5 5 10 40 20 40 10 10 5 1 5 1 1 5 1 1 3 5 10 1 15 20 1 15 10 5 1 1 1 1 1 1 1 3 5 10 15 20 15 10 5 1 1 1 3 5 10 15 20 15 10 5 1
F(x,y) = 20

96. Illustration 3 3 3 5 5 5 10 10 10 10 1 1 15 20 1 1 20 3 3 3 5 5 5 10 10 10 10 1 15 1 20 1 1 20 3 3 3 5 5 30 30 30 10 1 15 15 1 20 1 20 3 3 3 5 30 20 20 20 30 1 15 1 15 1 20 1 20 40 40 40 40 40 20 20 20 40 40 40 40 40 10 10 10 10 40 20 20 20 40 10 1 1 10 10 1 10 1 5 5 5 5 10 40 20 40 10 10 5 1 5 1 1 5 1 1 3 5 10 1 15 20 1 15 10 5 1 1 1 1 1 1 1 3 5 10 15 20 15 10 5 1 1 1 3 5 10 15 20 15 10 5 1
F(x,y) = 10

97. W-destructible point
Definition: involves all the connected components of a lower cross-section
Chacterization: involves all the paths between the lower neighbors of the point
What have we gained ?

98. Cross-sections, componentsk
Fk = {x E, F(x) < k}

99. Component tree
Components + inclusion relation  tree structure

100. Component mapping 

101. Illustration 3,g 1,b T 4,i 1,a F 3,g 3,h 2,f 2,c 2,d 2,e 2,f 1,a 1,b 
3,g 3
1,b T
4,i
1,a 2 3 3 3 3 F
3,g
3,h 3 1
2,f 1 3 3 1 3 3
2,c
2,d
2,e
2,f c c c c i b b b b
1,a
1,b a a a i g i b b b a a i g g g i b b  a i f i g i e i b a a i a g b g b b a a a a g b b b b

102. Property
For any weighted graph F, the component tree of F and the component mapping of F can be computed in quasi-linear time, more precisely, in O(n  e(n)) where n = |V| and e(n) is a function which grows extremely slowly with n.
This result can be obtained thanks to the Tarjan’s Union-Find algorithm for the merging of disjoint sets.
Tarjan, 1984
Najman and Couprie, 2004

103. (proper lca: different from both A and B)
Least common ancestor
lca(A,B) A B
(proper lca: different from both A and B)

104. Least common ancestor E
lca(E,F) F
(not a proper lca)

105. Property
Any tree T can be represented in such a way that for any two nodes A, B of T, it is possible to find the least common ancestor of A and B in constant time (with a linear time pre-processing).
Harel and Tarjan, 1984
Bender and Farach-Colton, 2000

106. Least common ancestor 1 2 3 4 5 6 1 11 15 33 32 2 10 12 14 16 3 5 9 25 31 17 13 18 22 24 28 30 26 6 8 4 19 21 23 27 29 7 20

107. Highest fork Let T be a tree, let S be a set of n nodes of T.
For any pair (A,B) of nodes in S, take the proper least common ancestor of A and B if it exists.
The highest one among these ancestors (if it exists) is called the highest fork of S.

108. Highest fork B A C D
hfork(A,B,C,D)

109. No highest fork for (A,B,D)

110. Property Let T be a tree, let S be a set of n nodes of T.
The highest fork of the nodes in S (or an indicator saying that no highest fork exists) can be computed by n -1 binary least common ancestor queries.

111. Property Let F be in F (V), let p,q be in V.
Let T be the component tree of F and let  be the component mapping of F.
The points p,q are k-separated if and only if the nodes (p) and (q) have a highest fork H in T, and if H has the level k+1.
The points p,q are linked if and only if the nodes (p) and (q) have no highest fork in T.

112. Corollary (characterization of W-destructible points)
Let F be in F (V), let p be in V.
Let T be the component tree of F and let  be the component mapping of F.
Thanks to T and , it is possible to check whether p is W-destructible or not in constant time.
It is also possible to know in constant time the type of p and the lowest value v such that p can be W-lowered down to v.

113. Illustration 1 Highest fork = 3,g Not W-destructible T 4,i F 3,g 3,h2 1 3 T
4,i 3 3 3 2 2 3 F
3,g
3,h 3 1 3 2 1 3 1 3 3 2 2 2
2,c
2,d
2,e
1,b
2,e
2,f 2 c c c c i b b b b
1,a
1,b a a a i g i b b b
Highest fork = 3,g a a i g g g i b b  a i f i g i e i b
Not W-destructible a a i a g b g b b a a a a g b b b b

114. Illustration 2 Highest fork = 3,g W-destructible Lowest value = 2 T1 1 1 2 1 3 T
4,i 3 1 2 3 3 2 2 3 3 F
3,g
3,g
2,e
1,b
3,h 3 1 3 2 3 1 3 3 2 2
2,c
2,d
2,e
2,f 2 c c c c i b b b b
1,a
1,b a a a i g i b b b
Highest fork = 3,g a a i g g g i b b  a i f i g i e i b
W-destructible a a i a g b g b b
Lowest value = 2 a a a a g b b b b

115. Illustration 3 No highest fork W-destructible Lowest value = 0 T 4,i F1 1 1 2 1 2 1 3 T
4,i 3 3 3 3 2 2 3 F
3,g
2,c
1,a
3,g
3,h 3 1 3 2 3 1 3 3 2 2
2,c
2,d
2,e
2,f 2 c c c c i b b b b
1,a
1,b a a a i g i b b b
No highest fork a a i g g g i b b  a i f i g i e i b
W-destructible a a i a g b g b b
Lowest value = 0 a a a a g b b b b

116. How can we reduce the complexity ?
Two fundamental ideas:
1/ Find a characterization of W-destructible points which can be checked in constant time
2/ Find strategies which guarantee that each point is examined at most once

117. W-thinning algorithm
The outline of a W-thinning algorithm is:
Repeat until stability Select a W-destructible point p, according to a certain criterion Lower the value of p
In order to ensure a linear time complexity, we must avoid multiple selections of the same point.

118. W-thinning algorithm

119. M-thinning
Recall that a point p is M-destructible for F if it is W-destructible for F and if it may become a minimal point after W-lowering.
If an M-destructible point p is W-lowered down to the value of an adjacent minimum, then we say that p is M-lowered.
Obviously, a minimal point will never become W-destructible in further steps of a W-thinning process (recall the transition graph).

120. M-thinning Let F, G in F(V).
We say that G is an M-thinning of F if G may be obtained from F by iteratively M-lowering some M-destructible points.
We say that G is an M-watershed of F if G is an M-thinning of F and if G has no M-destructible point.

121. Property Let F in F(V), let G be a an M-watershed of F.
Any W-thinning of G has exactly the same regional minima as G.

122. Illustration Minima ( , , ) Crest ( , , ) Other ( , ) 9 3 9 3 2 3 1 93 9 3 2 3 1 9 4 3 1 9 5 4 7 7 1 1 9 7 2 2 1 1 9 2 2 2
Minima ( , , ) 1 2
Crest ( , , ) 9 7 3
Other ( , ) 5 4

123. Classification of transitions
separating
pre-separating
pre-inner, non M-destructible
pre-inner, M-destructible
inner, non minimal
inner, minimal

124. Illustration F Watershed of F M-watershed of F 1 2 9 8 5 7 3 4 6 1 7 61 2 9 8 5 7 3 4 6 F 1 7 6
Watershed of F 9 8 7 6 1
M-watershed of F

125. Illustration 9 8 7 6 1 1 1 2 9 8 5 7 3 4 6

126. Priority function 1 10 1 2 3 4 1 4 5 6 7 8 9 10

127. M-watershed algorithm

128. Watershed algorithm
Repeat until stability Select a W-destructible point p, according to a certain criterion Lower the value of p
If we give the greatest priority to a W-destructible point which may be W-lowered down to the lowest possible value, we can prove that the same point cannot be selected twice by the algorithm.

129. Watershed algorithm

130. Conclusion For a binary watershed :
Component tree building
Minima labelling
M-thinning
O(n  e(n))
O(n)
O(n)
For a (grayscale) topological watershed :
Component tree building
M-thinning
W-thinning
O(n  e(n))
O(n)
O(n)

131. Conclusion First algorithm Proved complexity and correctness Fast
General (weighted graph)
The implementation is achievable
The pre-processing step (component tree building) is also useful for filtering purposes

132. Perspectives Saliency (L. Najman) Integration of the filtering step
Parallelism ?
Architecture ?

134. Algorithm (Najman and Couprie)

135. Type checking

136. Type checking

137. « Light » component tree
« Branch » vertices are deleted Only forks and leaves are kept

138. Component mapping

139. Least common ancestor
lca(A,B,C,D) D A C B

140. Perspectives Saliency (L. Najman) Integration of the filtering step
Drop of water paradigm
Parallelism ?
Architecture ?

141. Quasi-linear algorithms for the topological watershed
Michel Couprie

142. Contributors to this work
Gilles Bertrand
Michel Couprie
Laurent Najman