A new definition for the dynamics

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Presentation transcript:

A new definition for the dynamics G. Bertrand Laboratoire A2SI, ESIEE Institut Gaspard Monge – UMR UMLV/ESIEE/CNRS ISMM 2005

A discrete approach Let G = (V,E) be an (undirected) graph. We denote by Func (V) the family composed of all maps from V to Z. ISMM 2005

Pass value Let F be in Func (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}. ISMM 2005

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 ISMM 2005

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 ISMM 2005

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 ISMM 2005

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 ISMM 2005 F(X,Y) = 31

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) ISMM 2005

Dynamics ISMM 2005

Dynamics ∞ ISMM 2005

Dynamics ∞ ISMM 2005

Dynamics ∞ ISMM 2005

Dynamics ∞ ISMM 2005

Dynamics ∞ ISMM 2005

Dynamics ∞ ISMM 2005

k-Separation Let F be in Func (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. ISMM 2005

x and y are not separated k-separation x and y are not separated 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 ISMM 2005

k-separation y x x and y are 20-separated ISMM 2005 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 ISMM 2005

Separation Let F and G be in Func (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. ISMM 2005

Separation F ISMM 2005 G

Separation F ISMM 2005 G

Separation F K ISMM 2005 G

Separation F ISMM 2005 G

Separation F K ISMM 2005 G

Separation F K ISMM 2005 G is a separation of F G

Dynamics and separation Let G ≤ F (G being a minima extension of F) If G is a separation of F, then the dynamics of a minimum of G is the same than the dynamics of the corresponding minimum of F ISMM 2005

Dynamics and separation Let G ≤ F (G being a minima extension of F). If G is a separation of F, then the dynamics of a minimum of G is the same than the dynamics of the corresponding minimum of F. The converse is not true ISMM 2005

Dynamics: counter-example F ∞ ISMM 2005

Dynamics: counter-example G ∞ ISMM 2005

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}. ISMM 2005

Ordered minima 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 ISMM 2005

Ordered minima F(.,<)=8 5 3 2 F(.,<)=20 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(.,<)=20 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 ISMM 2005

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) ISMM 2005

Ordered minima Dyn(.,<)=8-5 5 3 2 Dyn(.,<)=20-0 Dyn(.,<)=30-2 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 40 3 3 2 3 10 6 6 6 22 2 3 40 Dyn(.,<)=20-0 40 6 6 40 6 11 11 11 25 4 4 4 40 Dyn(.,<)=30-2 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 Dyn(.,<)=31-3 40 40 40 40 40 40 40 40 40 40 40 40 40 Dyn(.,<)=infty ISMM 2005

Theorem (ordered dynamics and separation) Let G ≤ F (G being a minima extension of F). Let < be a minima ordering for F. The map G is a separation of F if and only if, for each minimum X for F, we have Dyn(X; F, <) = Dyn(X; G, <) . ISMM 2005

Dynamics: counter-example ∞ ISMM 2005

Ordered minima F ISMM 2005

Ordered minima F 2 3 ISMM 2005 1

Ordered minima ∞ F 2 3 ISMM 2005 1

Ordered minima ∞ F 2 3 ISMM 2005 1

Ordered minima ∞ G 2 3 ISMM 2005 1

Ordered minima F 1 2 ISMM 2005 3

Ordered minima ∞ F 1 2 ISMM 2005 3

Ordered minima ∞ F 1 2 ISMM 2005 3

Ordered minima ∞ G 1 2 ISMM 2005 3

Remark If all the minima of a function F are distinct and if the ordering of the minima of F is made according to the altitudes of the minima of F, then the ordered dynamics of a minimum is equal to the unordered dynamics of this minimum. ISMM 2005

A tree associated to F and < 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(.,<)=20 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 ISMM 2005

Theorem (minimum spanning tree) Let F be in F (V) and let < be a minima ordering for F. Let T be a tree associated to F and <. Let G’ be the complete graph the vertices of which are the minima of F, an edge being labeled by the corresponding pass value. The tree T is a minimum spanning tree of G’. ISMM 2005

Conclusion ISMM 2005 Dyn > 22

Conclusion ISMM 2005 => Ordering the minima with arbitary criteria

Conclusion Preservation of the dynamics Equivalence between : Preservation of the dynamics Preservation of the contrast (separation) Preservation of an optimal spanning tree Preservation of the crests (topological watersheds) Preservation of the components of the cross-sections (extension) Preservation of the component tree ISMM 2005

Topological watershed ISMM 2005

Conclusion Preservation of the dynamics Equivalence between : Preservation of the dynamics Preservation of the contrast (separation) Preservation of an optimal spanning tree Preservation of the crests (topological watersheds) Preservation of the components of the cross-sections (extension) Preservation of the component tree ISMM 2005

Components of the cross-sections ISMM 2005 G

Components of the cross-sections ISMM 2005 G

Components of the cross-sections ISMM 2005 G

Conclusion Preservation of the dynamics Equivalence between : Preservation of the dynamics Preservation of the contrast (separation) Preservation of an optimal spanning tree Preservation of the crests (topological watersheds) Preservation of the components of the cross-sections (extension) Preservation of the component tree ISMM 2005

Components of the cross-sections ISMM 2005

Components of the cross-sections ISMM 2005

Components of the cross-sections ISMM 2005

Conclusion Preservation of the dynamics Equivalence between : Preservation of the dynamics Preservation of the contrast (separation) Preservation of an optimal spanning tree Preservation of the crests (topological watersheds) Preservation of the components of the cross-sections (extension) Preservation of the component tree ISMM 2005

Thank you for your attention ISMM 2005

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. ISMM 2005

Separation (sets) Let X be a subset of E and let x, y be in X. We say that x and y are separated for X if there is no path from x to y in X. Let X, Y be subsets of E such that Y is a subset of X. We say that Y is a separation of X if any x and y in X which are separated for X, are separated for Y. ISMM 2005

A subset X ISMM 2005

A separation ISMM 2005

Separation (maps) We denote by Func (V) the family composed of all maps from V to Z. Let F  Func (V), we set Fk = {x  V; F(x)  k}, Fk is the cross-section of F at level k Let F and G be both in Func(V) and such that G ≤ F. We say that G is a separation of F if, for any k, G[k] is a separation of F[k]. ISMM 2005

Strong separation F ISMM 2005 G

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) Watersheds ISMM 2005

Theorem (restriction to minima) Let F and G be in F (V) such that G  F. The map G is a separation of F if and only if, for all distinct minima X,Y for F, F(X,Y) = G(X,Y). ISMM 2005

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. ISMM 2005

Theorem (confluence) Let G be a W-thinning of F. If H is a W-thinning of F such that H >= G, then G is a W-thinning of H ISMM 2005

Strong separation F ISMM 2005 G is a strong separation of F G

Strong separation F destructible points may be lowered with an arbitrary order ISMM 2005 G

Theorem (restriction to minima) Let F and G be in F (V) such that G  F. The map G is a separation of F if and only if, for all distinct minima X,Y for F, F(X,Y) = G(X,Y). ISMM 2005

Theorem (restriction to minima) Let F and G be in F (V) such that G  F. The map G is a separation of F if and only if, for all distinct minima X,Y for F, F(X,Y) = G(X,Y). Is it possible to reduce the amount of information necessary to ‘‘encode’’ the topology of a W-thinning? ISMM 2005

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}. ISMM 2005

Theorem (ordered minima) Let F and G be in F (V) such that G <= F and let < be a minima ordering for F. The map G is a separation of F if and only if, for each minimum X for F, we have F(X,<) = G(X,<). ISMM 2005

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. ISMM 2005

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. ISMM 2005

Segmentation ISMM 2005

ISMM 2005

Watershed ISMM 2005

Segmentation based on dynamics ISMM 2005

Segmentation based on dynamics ISMM 2005

Minima ordering 10 9 8 6 7 5 2 1 3 4 ISMM 2005

Dynamics 10 9 8 6 7 5 2 1 3 4 ISMM 2005

Dynamics 10 9 8 6 7 5 2 1 3 4 ISMM 2005

Dynamics 6 1 3 ISMM 2005

Dynamics 6 1 3 ISMM 2005

Geodesic reconstruction 6 1 3 ISMM 2005

Watershed 6 1 3 ISMM 2005

ISMM 2005

ISMM 2005

Watershed ISMM 2005

ISMM 2005

Dyn > 9 ISMM 2005

Dyn > 9 ISMM 2005

Dyn > 22 ISMM 2005

Dyn > 22 ISMM 2005

‘Duality’ Let (V,E) be a connected graph and let E’ be a subset of E. We say that an edge u = {x,y} in E’ is destructible (for E’) if x and y belong to the same connected component of (V, E’\{u}) ISMM 2005

Homotopy: an illustration F(x,y) G(x,y) F1 G1 x x ISMM 2005

Homotopy: an illustration F(x,y) G(x,y) x x F2 G2 F1 G1 ISMM 2005

Watershed transform ISMM 2005

k-separation y x x and y are 8-separated ISMM 2005 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 ISMM 2005

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 ISMM 2005

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 ISMM 2005 31

ISMM 2005

ISMM 2005

ISMM 2005

Cross-sections, components ISMM 2005

Cross-sections, components ISMM 2005