A topological approach to watersheds

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Presentation transcript:

A topological approach to watersheds G. Bertrand Laboratoire A2SI, ESIEE www.esiee.fr/a2si Paris 27/05/2003

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) Paris 27/05/2003

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 Paris 27/05/2003

Flooding paradigm Paris 27/05/2003

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 Paris 27/05/2003

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 Paris 27/05/2003

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) Paris 27/05/2003

Destructible point Paris 27/05/2003

Destructible point Paris 27/05/2003

Destructible point Paris 27/05/2003

Destructible point Paris 27/05/2003

Destructible point Paris 27/05/2003

Destructible point Paris 27/05/2003

Destructible point Paris 27/05/2003

Destructible point Paris 27/05/2003

Destructible point Paris 27/05/2003

Destructible point Paris 27/05/2003

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) Paris 27/05/2003

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) Paris 27/05/2003

Collapsing paradigm Paris 27/05/2003

Topological watershed 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 Paris 27/05/2003

Topological watershed 40 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 Paris 27/05/2003

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}. Paris 27/05/2003

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’. Paris 27/05/2003

k-separation y x x and y are 20-separated 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 Paris 27/05/2003

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

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). Paris 27/05/2003

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. Paris 27/05/2003

2D case Any connected object without hole reduces to one point Paris 27/05/2003

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 Paris 27/05/2003

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). Paris 27/05/2003

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? Paris 27/05/2003

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}. Paris 27/05/2003

Ordered minima F(.,<)=8 5 3 2 F(.,<)=22 F(.,<)=30 1 4 40 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 Paris 27/05/2003

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,<). Paris 27/05/2003

<-map F(.,<)=8 5 3 2 F(.,<)=22 F(.,<)=30 1 4 F(.,<)=0 40 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 Paris 27/05/2003

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. Paris 27/05/2003

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) Paris 27/05/2003

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. Paris 27/05/2003

Dynamics: counter-example Paris 27/05/2003

Dynamics: counter-example Paris 27/05/2003

Ordered minima 2 3 1 Paris 27/05/2003

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) Paris 27/05/2003

Results of watershed algorithms 4 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 Paris 27/05/2003

Results of watershed algorithms 30 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 Paris 27/05/2003

Results of watershed algorithms 2 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 Paris 27/05/2003 Meyer Topological

Homotopy: an illustration F(x,y) G(x,y) F1 G1 x x Paris 27/05/2003

Homotopy: an illustration F(x,y) G(x,y) x x F2 G2 F1 G1 Paris 27/05/2003

Watershed transform Paris 27/05/2003

Flooding paradigm Paris 27/05/2003

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 Paris 27/05/2003

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 Paris 27/05/2003

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 Paris 27/05/2003

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 Paris 27/05/2003

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 Paris 27/05/2003

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 Paris 27/05/2003

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 Paris 27/05/2003

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 Paris 27/05/2003

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 Paris 27/05/2003

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 Paris 27/05/2003

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 Paris 27/05/2003

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 Paris 27/05/2003

Topological watershed 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 Paris 27/05/2003

Topological watershed 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 Paris 27/05/2003

Topological watershed 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 Paris 27/05/2003

Topological watershed 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 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 Paris 27/05/2003

Topological watershed 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 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 Paris 27/05/2003

Topological watershed 40 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 Paris 27/05/2003

Topological watershed 40 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 Paris 27/05/2003

Topological watershed 40 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 Paris 27/05/2003

Topological watershed 40 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 Paris 27/05/2003

Topological watershed 40 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 Paris 27/05/2003

Topological watershed 40 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 Paris 27/05/2003

Topological watershed 40 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 Paris 27/05/2003

Topological watershed 40 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 Paris 27/05/2003

Topological watershed 40 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 Paris 27/05/2003

Topological watershed 40 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 Paris 27/05/2003

Topological watershed 40 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 Paris 27/05/2003

Topological watershed 40 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 Paris 27/05/2003

Topological watershed 40 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 Paris 27/05/2003

k-separation y x x and y are 8-separated 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 Paris 27/05/2003

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

Homotopic thinning Paris 27/05/2003

Homotopic thinning Paris 27/05/2003

Homotopic thinning Paris 27/05/2003

Homotopic thinning Paris 27/05/2003

Homotopic thinning Paris 27/05/2003

Homotopic thinning Paris 27/05/2003

Homotopic thinning Paris 27/05/2003

Homotopic thinning Paris 27/05/2003

Homotopic thinning Paris 27/05/2003

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

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

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

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

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