1. A topological approach to watersheds
G. Bertrand
Laboratoire A2SI, ESIEE
Paris 27/05/2003

2. To our knowledge there is no framework in which general properties for watersheds can be derived
Our goal is to show that the topological approach proposed previously* provides such a framework *M. Couprie and G. Bertrand (1997)
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3. 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 flooding simulation were proposed by F. Meyer, P. Soille, L. Vincent (and others) in the 90s
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4. Flooding paradigm
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5. Flooding paradigm 40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 5 5 5 10 10 10 1 10 1 15 1 20 40 40 3 3 5 5 30 30 30 10 15 1 15 1 20 40 40 3 3 5 30 20 20 20 30 15 1 15 1 20 1 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 10 10 10 40 20 20 20 40 10 1 10 1 10 1 40 40 5 5 5 10 40 20 40 10 10 1 5 5 1 40 40 1 3 5 10 1 15 20 1 15 10 1 5 1 1 1 40 40 1 3 5 10 15 20 15 10 5 1 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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6. Flooding paradigm 40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 5 5 5 10 10 10 1 10 1 15 20 1 40 40 3 3 5 5 30 30 30 10 15 1 15 20 1 40 40 3 3 5 30 20 20 20 30 15 1 15 1 1 20 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 10 10 10 40 20 20 20 40 10 1 1 10 10 1 40 40 5 5 5 10 40 20 40 10 10 1 5 5 1 40 40 1 3 5 10 15 1 20 15 1 10 5 1 1 1 1 40 40 1 3 5 10 15 20 15 10 5 1 40 40 40 40 40 40 40 40 40 40 40 40 40 40
There is no descending path from the 20s to the minimum 3
The contrast between minima is not preserved
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7. Discrete sets and destructible points
Let G = (V,E) be a (undirected) graph and let X be a subset of V.
We say that a point x  X is destructible for X if x is adjacent to exactly one connected component of X.
M. Couprie and G. Bertrand (1997)
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8. Destructible point
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9. Destructible point
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10. Destructible point
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11. Destructible point
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12. Destructible point
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13. Destructible point
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14. Destructible point
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15. Destructible point
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16. Destructible point
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17. Destructible point
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18. Discrete maps and destructible points
Let G = (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 destructible (for F) if x is adjacent to exactly one connected component of Fk
M. Couprie and G. Bertrand (1997)
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19. Topological watershed
Let F and F’ be in F (V). We say that F’ is a thinning of F if F’ may be obtained from F by iteratively lowering destructible points (by 1).
Let F and F’ be in F (V). We say that F’ is a watershed of F if F’ is a thinning of F and if there is no destructible point for F’.
M. Couprie and G. Bertrand (1997)
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20. Collapsing paradigm
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21. Topological watershed40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 5 5 5 10 10 10 1 10 1 15 1 20 40 40 3 3 5 5 30 30 30 10 15 1 15 1 20 40 40 3 3 5 30 20 20 20 30 15 1 15 1 20 1 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 10 10 10 40 20 20 20 40 10 1 10 1 10 1 40 40 5 5 5 10 40 20 40 10 10 1 5 5 1 40 40 1 3 5 10 1 15 20 1 15 10 1 5 1 1 1 40 40 1 3 5 10 15 20 15 10 5 1 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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22. Topological watershed40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 3 3 3 3 3 3 3 3 3 40 40 3 3 3 3 30 30 30 3 3 3 3 40 40 3 3 3 30 1 20 30 3 3 3 40 40 30 30 30 1 1 20 30 30 30 40 40 1 1 1 1 1 20 40 40 1 1 1 1 1 20 40 40 1 1 1 1 1 20 40 40 1 1 1 1 1 20 40 40 40 40 40 40 40 40 40 40 40 40 40 40
The watershed is located on the crest lines of the original image
The contrast between minima is preserved
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23. 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}.
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24. 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 F’ be in F (V) such that F’  F. We say that F’ 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 F’.
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25. 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
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26. x and y are NOT separated (they are linked)
k-separation
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
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27. Theorem (restriction to minima)
Let F and F’ be in F (V) such that F’  F. The map F’ is a separation of F if and only if, for all distinct minima X,Y for F, F(X,Y) = F’(X,Y).
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28. Theorem (strong separation)
Let F and F’ be in F (V) such that F’  F. The map F’ is a thinning of F if and only if F’ is a strong separation of F.
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29. 2D case Any connected object without hole reduces to one point
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30. Bing’s house with two rooms
3D case
Some connected objects without holes and cavities DO NOT reduce to one point
Bing’s house with two rooms
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31. Theorem (restriction to minima)
Let F and F’ be in F (V) such that F’  F. The map F’ is a separation of F if and only if, for all distinct minima X,Y for F, F(X,Y) = F’(X,Y).
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32. Theorem (restriction to minima)
Let F and F’ be in F (V) such that F’  F. The map F’ is a separation of F if and only if, for all distinct minima X,Y for F, F(X,Y) = F’(X,Y).
Is it possible to reduce the amount of information necessary to encode the  "topology" of a thinning?
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33. Ordered minima
Let F be in F (V). A minima ordering (for F) is a strict total order relation < on the minima of F.
Let X be a minimum for F. The pass value of X for (F,<) is the number F(X,<) such that: i) if X = Xmin, then F(X,<) = infinity; ii) otherwise, F(X,<) = Min {F(X,Y); for all minima Y such that Y < X}.
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34. Ordered minima F(.,<)=8 5 3 2 F(.,<)=22 F(.,<)=30 1 440 40 40 40 40 40 40 40 40 40 40 40 40 5 40 3 1 1 2 3 10 5 25 5 4 4 4 40 2 40 1 2 8 6 5 5 20 3 2 3 40
F(.,<)=22 40 3 3 2 3 10 6 6 6 22 2 3
F(.,<)=30 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 1 4 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
F(.,<)=infty
F(.,<)=31 40 40 40 40 40 40 40 40 40 40 40 40 40
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35. Theorem (ordered minima)
Let F and F’ be in F (V) such that F’ <= F and let < be a minima ordering for F. The map F’ is a separation of F if and only if, for each minimum X for F, we have F(X,<) = F’(X,<).
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36. <-map F(.,<)=8 5 3 2 F(.,<)=22 F(.,<)=30 1 4 F(.,<)=040 40 40 40 40 40 40 40 40 40 40 40 40 5 40 3 1 1 2 3 10 5 25 5 4 4 4 40 2 40 1 2 8 6 5 5 20 3 2 3 40
F(.,<)=22 40 3 3 2 3 10 6 6 6 22 2 3
F(.,<)=30 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 1 4 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
F(.,<)=0
F(.,<)=31 40 40 40 40 40 40 40 40 40 40 40 40 40
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37. Theorem (reconstruction from ordered pass values)
Let F be in F (V) and let < be a minima ordering for F. Let T be a tree associated to F(X,<). The pass values between all minima of F may be reconstructed from T.
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38. Dynamics (M. Grimaud,1992)
Let X be a minimum for F Let G(X) be the number such that: i) if X = Xmin, then G(X) = infinity; ii) otherwise, G(X) = Min {F(X,Y); for all minima Y such that F(Y) < F(X)}.
The dynamics of a minimum X is the number Dyn(X) = G(X) – F(X)
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39. Ordered dynamics
The notion of ordered pass values leads to a new definition of the dynamics of a minimum: Dyn(X; F, <) = F(X, <) – F(X)
This new definition of dynamics fully agrees with the notion of separation.
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40. Dynamics: counter-example
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41. Dynamics: counter-example
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42. Ordered minima 2 3 1
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43. Conclusion Basins, component tree Duality: minimum spanning trees
Comparison of existing algorithms (L. Najman and M. Couprie)
Algorithmic issues (M. Couprie and L. Najman)
Saliency (L. Najman)
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44. Results of watershed algorithms4 5 B 6 3 1 C 2
Topographical 3 4 5 6 2 1
Original image A B 6 1 C 3 5
Vincent-Soille, Meyer
and Topological
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45. Results of watershed algorithms30 3 31 4
255 2 1 5 30 C 31 D E A
255 B F
Original image
Vincent-Soille 30 C E D A
255 B F 30 C 31 D
255 A E B F
Meyer
Topological
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46. Results of watershed algorithms2 1 30 20 40 A B 1 C 20 40
Original image
Vincent-Soille A B 1 20 40 A A A A A A 1 A A A 30 30 30 A A 1 A 30 B 20 A 30 A 1 40 B B 20 A A 40 B B B 20 A A 1 A B B B 20 A A A B B 1 B 20 1 A A 1 A
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Meyer
Topological

47. Homotopy: an illustration
F(x,y)
G(x,y) F1 G1 x x
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48. Homotopy: an illustration
F(x,y)
G(x,y) x x F2 G2 F1 G1
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49. Watershed transform
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50. Flooding paradigm
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51. Flooding paradigm 40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 5 5 5 10 10 10 1 10 1 15 1 20 40 40 3 3 5 5 30 30 30 10 15 1 15 1 20 40 40 3 3 5 30 20 20 20 30 15 1 15 1 20 1 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 10 10 10 40 20 20 20 40 10 1 10 1 10 1 40 40 5 5 5 10 40 20 40 10 10 1 5 5 1 40 40 1 3 5 10 1 15 20 1 15 10 1 5 1 1 1 40 40 1 3 5 10 15 20 15 10 5 1 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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52. Flooding paradigm 40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 5 5 5 10 10 10 1 10 15 1 20 40 40 3 3 5 5 30 30 30 10 1 15 15 20 40 40 3 3 5 30 20 20 20 30 15 1 1 15 20 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 10 10 10 40 20 20 20 40 1 10 10 1 10 1 40 40 5 5 5 10 40 20 40 10 10 5 1 5 1 40 40 1 3 5 10 15 1 20 1 15 10 5 1 1 1 40 40 1 3 5 10 15 20 15 10 5 1 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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53. Flooding paradigm 40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 5 5 5 10 10 10 1 10 15 1 20 40 40 3 3 5 5 30 30 30 10 1 15 15 20 40 40 3 3 5 30 20 20 20 30 15 1 15 1 20 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 10 10 10 40 20 20 20 40 1 10 10 1 1 10 40 40 5 5 5 10 40 20 40 10 10 1 5 5 1 40 40 1 3 5 10 1 15 20 15 1 10 1 5 1 1 40 40 1 3 5 10 15 20 15 10 5 1 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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54. Flooding paradigm 40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 5 5 5 10 10 10 1 10 15 1 20 40 40 3 3 5 5 30 30 30 10 1 15 15 20 40 40 3 3 5 30 20 20 20 30 15 1 15 1 20 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 10 10 10 40 20 20 20 40 1 10 10 1 1 10 40 40 5 5 5 10 40 20 40 10 10 1 5 5 1 40 40 1 3 5 10 1 15 20 15 1 10 1 5 1 1 40 40 1 3 5 10 15 20 15 10 5 1 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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55. Flooding paradigm 40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 5 5 5 10 10 10 1 10 15 1 20 40 40 3 3 5 5 30 30 30 10 1 15 15 20 40 40 3 3 5 30 20 20 20 30 15 1 15 1 20 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 10 10 10 40 20 20 20 40 1 10 10 1 1 10 40 40 5 5 5 10 40 20 40 10 10 1 5 5 1 40 40 1 3 5 10 1 15 20 15 1 10 1 5 1 1 40 40 1 3 5 10 15 20 15 10 5 1 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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56. Flooding paradigm 40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 5 5 5 10 10 10 1 10 15 1 20 40 40 3 3 5 5 30 30 30 10 1 15 15 20 40 40 3 3 5 30 20 20 20 30 15 1 15 1 20 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 10 10 10 40 20 20 20 40 1 10 10 1 1 10 40 40 5 5 5 10 40 20 40 10 10 1 5 5 1 40 40 1 3 5 10 1 15 20 15 1 10 1 5 1 1 40 40 1 3 5 10 15 20 15 10 5 1 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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57. Flooding paradigm 40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 5 5 5 10 10 10 1 10 15 1 20 40 40 3 3 5 5 30 30 30 10 1 15 15 20 40 40 3 3 5 30 20 20 20 30 15 1 15 1 20 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 10 10 10 40 20 20 20 40 1 10 10 1 1 10 40 40 5 5 5 10 40 20 40 10 10 1 5 5 1 40 40 1 3 5 10 1 15 20 15 1 10 1 5 1 1 40 40 1 3 5 10 15 20 15 10 5 1 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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58. Flooding paradigm 40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 5 5 5 10 10 10 1 10 15 1 20 40 40 3 3 5 5 30 30 30 10 1 15 15 20 40 40 3 3 5 30 20 20 20 30 15 1 15 1 20 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 10 10 10 40 20 20 20 40 1 10 10 1 1 10 40 40 5 5 5 10 40 20 40 10 10 1 5 5 1 40 40 1 3 5 10 1 15 20 15 1 10 1 5 1 1 40 40 1 3 5 10 15 20 15 10 5 1 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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59. Flooding paradigm 40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 5 5 5 10 10 10 1 10 15 1 20 40 40 3 3 5 5 30 30 30 10 1 15 15 20 40 40 3 3 5 30 20 20 20 30 15 1 15 1 20 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 10 10 10 40 20 20 20 40 1 10 10 1 1 10 40 40 5 5 5 10 40 20 40 10 10 1 5 5 1 40 40 1 3 5 10 1 15 20 15 1 10 1 5 1 1 40 40 1 3 5 10 15 20 15 10 5 1 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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60. Flooding paradigm 40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 5 5 5 10 10 10 1 10 15 1 20 40 40 3 3 5 5 30 30 30 10 1 15 15 20 40 40 3 3 5 30 20 20 20 30 15 1 15 1 20 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 10 10 10 40 20 20 20 40 1 10 10 1 1 10 40 40 5 5 5 10 40 20 40 10 10 1 5 5 1 40 40 1 3 5 10 1 15 20 15 1 10 1 5 1 1 40 40 1 3 5 10 15 20 15 10 5 1 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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61. Flooding paradigm 40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 5 5 5 10 10 10 1 10 15 1 20 40 40 3 3 5 5 30 30 30 10 1 15 15 20 40 40 3 3 5 30 20 20 20 30 15 1 15 1 20 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 10 10 10 40 20 20 20 40 1 10 10 1 1 10 40 40 5 5 5 10 40 20 40 10 10 1 5 5 1 40 40 1 3 5 10 1 15 20 15 1 10 1 5 1 1 40 40 1 3 5 10 15 20 15 10 5 1 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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62. Flooding paradigm 40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 5 5 5 10 10 10 1 10 15 1 20 40 40 3 3 5 5 30 30 30 10 1 15 15 20 40 40 3 3 5 30 20 20 20 30 15 1 1 15 20 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 10 10 10 40 20 20 20 40 1 10 10 1 10 1 40 40 5 5 5 10 40 20 40 10 10 1 5 5 1 40 40 1 3 5 10 15 1 20 1 15 10 5 1 1 1 40 40 1 3 5 10 15 20 15 10 5 1 40 40 40 40 40 40 40 40 40 40 40 40 40 40
There is no descending path from the 20s to the minimum 3
Any path from 0 to the minimum 3 must climb at least at 30
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63. Topological watershed40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 5 5 5 10 10 10 1 10 1 15 1 20 40 40 3 3 5 5 30 30 30 10 15 1 15 1 20 40 40 3 3 5 30 20 20 20 30 15 1 15 1 20 1 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 10 10 10 40 20 20 20 40 10 1 10 1 10 1 40 40 5 5 5 10 40 20 40 10 10 1 5 5 1 40 40 1 3 5 10 1 15 20 1 15 10 1 5 1 1 1 40 40 1 3 5 10 15 20 15 10 5 1 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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64. Topological watershed40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 5 5 5 10 10 10 1 10 1 15 1 20 40 40 3 3 5 5 30 30 30 10 15 1 15 1 20 40 40 3 3 5 30 20 20 20 30 15 1 15 1 20 1 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 10 10 10 40 20 20 20 40 10 1 10 1 10 1 40 40 5 5 5 10 40 20 40 10 10 1 5 5 1 40 40 1 3 5 10 1 15 20 1 15 10 1 5 1 1 1 40 40 1 3 5 10 15 20 15 10 5 1 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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65. Topological watershed40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 5 5 5 10 10 10 1 10 1 15 1 20 40 40 3 3 5 5 30 30 30 10 15 1 15 1 20 40 40 3 3 5 30 20 20 20 30 15 1 15 1 20 1 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 10 10 10 40 20 20 20 40 10 1 10 1 10 1 40 40 5 5 5 10 40 20 40 10 10 1 5 5 1 40 40 1 3 5 10 1 15 20 1 15 10 1 5 1 1 1 40 40 1 3 5 10 15 20 15 10 5 1 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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66. Topological watershed40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 5 5 5 10 10 10 1 10 1 15 20 1 40 40 3 3 5 5 30 30 30 10 15 1 15 20 1 40 40 3 3 5 30 20 20 20 30 15 1 1 15 1 20 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 10 10 10 40 20 20 20 40 1 10 1 10 10 1 40 40 5 5 5 10 40 20 40 10 10 5 1 1 5 40 40 1 3 5 10 1 15 20 1 15 10 5 1 1 1 40 40 1 3 5 10 15 20 15 10 5 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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67. Topological watershed40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 5 5 5 10 10 10 1 10 1 15 20 1 40 40 3 3 5 5 30 30 30 10 15 1 15 20 1 40 40 3 3 5 30 20 20 20 30 15 1 1 15 1 20 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 10 10 10 40 20 20 20 40 1 10 1 10 10 1 40 40 5 5 5 10 40 20 40 10 10 5 1 1 5 40 40 1 3 5 10 1 15 20 1 15 10 5 1 1 1 40 40 1 3 5 10 15 20 15 10 5 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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68. Topological watershed40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 3 3 3 3 3 3 3 3 3 40 40 3 3 3 3 30 30 30 3 3 3 3 40 40 3 3 3 30 20 20 20 30 3 3 3 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 1 1 1 40 20 20 20 40 40 40 1 1 1 1 40 20 40 40 40 1 1 1 1 1 20 40 40 1 1 1 1 1 20 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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69. Topological watershed40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 3 3 3 3 3 3 3 3 3 40 40 3 3 3 3 30 30 30 3 3 3 3 40 40 3 3 3 30 20 20 20 30 3 3 3 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 1 1 1 40 20 20 20 40 40 40 1 1 1 1 40 20 40 40 40 1 1 1 1 1 20 40 40 1 1 1 1 1 20 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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70. Topological watershed40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 3 3 3 3 3 3 3 3 3 40 40 3 3 3 3 30 30 30 3 3 3 3 40 40 3 3 3 30 20 20 20 30 3 3 3 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 1 1 1 40 20 20 20 40 40 40 1 1 1 1 40 20 40 40 40 1 1 1 1 1 20 40 40 1 1 1 1 1 20 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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71. Topological watershed40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 3 3 3 3 3 3 3 3 3 40 40 3 3 3 3 30 30 30 3 3 3 3 40 40 3 3 3 30 20 20 20 30 3 3 3 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 1 1 1 40 20 20 20 40 40 40 1 1 1 1 40 20 40 40 40 1 1 1 1 1 20 40 40 1 1 1 1 1 20 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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72. Topological watershed40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 3 3 3 3 3 3 3 3 3 40 40 3 3 3 3 30 30 30 3 3 3 3 40 40 3 3 3 30 20 20 20 30 3 3 3 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 1 1 1 40 20 20 20 40 40 40 1 1 1 1 40 20 40 40 40 1 1 1 1 1 20 40 40 1 1 1 1 1 20 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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73. Topological watershed40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 3 3 3 3 3 3 3 3 3 40 40 3 3 3 3 30 30 30 3 3 3 3 40 40 3 3 3 30 20 20 20 30 3 3 3 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 1 1 1 40 20 20 20 40 40 40 1 1 1 1 40 20 40 40 40 1 1 1 1 1 20 40 40 1 1 1 1 1 20 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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74. Topological watershed40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 3 3 3 3 3 3 3 3 3 40 40 3 3 3 3 30 30 30 3 3 3 3 40 40 3 3 3 30 20 20 20 30 3 3 3 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 1 1 1 40 20 20 20 40 40 40 1 1 1 1 40 20 40 40 40 1 1 1 1 1 20 40 40 1 1 1 1 1 20 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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75. Topological watershed40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 3 3 3 3 3 3 3 3 3 40 40 3 3 3 3 30 30 30 3 3 3 3 40 40 3 3 3 30 20 20 20 30 3 3 3 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 1 1 1 40 20 20 20 40 40 40 1 1 1 1 1 20 40 40 40 1 1 1 1 1 20 40 40 1 1 1 1 1 20 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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76. Topological watershed40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 3 3 3 3 3 3 3 3 3 40 40 3 3 3 3 30 30 30 3 3 3 3 40 40 3 3 3 30 20 20 20 30 3 3 3 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 1 1 1 40 20 20 20 40 40 40 1 1 1 1 1 20 40 40 40 1 1 1 1 1 20 40 40 1 1 1 1 1 20 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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77. Topological watershed40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 3 3 3 3 3 3 3 3 3 40 40 3 3 3 3 30 30 30 3 3 3 3 40 40 3 3 3 30 20 20 20 30 3 3 3 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 1 1 1 40 20 20 20 40 40 40 1 1 1 1 1 20 40 40 40 1 1 1 1 1 20 40 40 1 1 1 1 1 20 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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78. Topological watershed40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 3 3 3 3 3 3 3 3 3 40 40 3 3 3 3 30 30 30 3 3 3 3 40 40 3 3 3 30 20 20 20 30 3 3 3 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 1 1 1 40 20 20 20 40 40 40 1 1 1 1 1 20 40 40 1 1 1 1 1 20 40 40 1 1 1 1 1 20 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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79. Topological watershed40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 3 3 3 3 3 3 3 3 3 40 40 3 3 3 3 30 30 30 3 3 3 3 40 40 3 3 3 30 20 20 20 30 3 3 3 40 40 40 40 40 40 20 20 20 40 40 40 40 40 40 1 1 1 40 20 20 20 40 40 40 1 1 1 1 1 20 40 40 1 1 1 1 1 20 40 40 1 1 1 1 1 20 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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80. Topological watershed40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 3 3 3 3 3 3 3 3 3 3 40 40 3 3 3 3 30 30 30 3 3 3 3 40 40 3 3 3 30 1 20 30 3 3 3 40 40 30 30 30 1 1 20 30 30 30 40 40 1 1 1 1 1 20 40 40 1 1 1 1 1 20 40 40 1 1 1 1 1 20 40 40 1 1 1 1 1 20 40 40 40 40 40 40 40 40 40 40 40 40 40 40
The watershed is located on the crest lines of the original image
The contrast between minima is preserved
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81. k-separation y x x and y are 8-separated 40 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
Paris 27/05/2003

82. x and y are NOT separated (they are linked)
k-separation
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
Paris 27/05/2003

83. Homotopic thinning
Paris 27/05/2003

84. Homotopic thinning
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85. Homotopic thinning
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86. Homotopic thinning
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87. Homotopic thinning
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88. Homotopic thinning
Paris 27/05/2003

89. Homotopic thinning
Paris 27/05/2003

90. Homotopic thinning
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91. Homotopic thinning
Paris 27/05/2003

92. Pass value 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 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
Paris 27/05/2003

93. Pass value 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 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
Paris 27/05/2003

94. Pass value 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 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
Paris 27/05/2003

95. Pass value 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 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
F(X,Y) = 31
Paris 27/05/2003

96. Pass value 20 40 40 40 40 40 40 40 40 40 40 40 40 40 40 1 1 2 8 3 10 5 25 5 20 4 4 4 40 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 30 31 40 6 6 40 6 11 11 11 25 4 4 4 40 31 30 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
Paris 27/05/2003 31