Frontiers of Mathematical Morphology April 17-20, 2000 Strasbourg, France. Viscous Lattices

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1 Frontiers of Mathematical Morphology April 17-20, 2000 Strasbourg, France Viscous Lattices Definition Connections geodesic Reconstruction Application to watershed By J. Serra Ecole des Mines de Paris J. Serra, Ecole des Mines de Paris ( 2000 ) Treillis visqueux 1

2 Bibliography Initial Idea :: F.Meyer, La La viscosité en en ligne de de partage des eaux. Note CMM 1994 Applications :: 1-1-Ph. Degrize, Reconstruction d'images IRM ;; Analyse automatique d'images par Morphologie Mathématique. Phd Thesis, Université Paris 7, 7, 1994 ;; 2-2-C. Vachier, F. F. Meyer et et R. R. Lamara :: Segmentation d'image par simulation d'une inondation visqueuse. To To be be published in in RFIA Paris ;; Current Development :: J. J. Serra,, Viscous Lattices Note CMM D J. Serra, Ecole des Mines de Paris ( 2000 ) Treillis visqueux 2

3 Objectives (1) Purpose : To swell the space so that the Lion be wedged in in its cage.. J. Serra, Ecole des Mines de Paris ( 2000 ) Treillis visqueux 3

4 Objectives (2) Method : To generate viscous reconstruction, i.e. with a given meniscus. Means : 1-1-To build the convenient lattice Λ :: it it is is called viscous, and generated by the dilates of of the points by a given dilation δ (the viscosity) ;; 2-2-To define connections on Λ and to to use the associated connected openings. An example : Contour of of the heart muscle. J. Serra, Ecole des Mines de Paris ( 2000 ) Treillis visqueux 4

5 Notation and Reminder Notation : E :: an arbitrary set ;; Π(E): the lattice of of all subsets of of E; E; δ :: dilation Π(E) Π(E), of of adjoint erosion.. Dilation δ is is determined by the images of of the singletons { } de Π(E) :: δ (X) = { δ ( ), ε X } X ε Π(E) (1); Β = {δ( ),, ε E} :: class of of the dilates of of the singletons; γ = δ = opening adjoint to to dilation δ. δ. Reminder : The family Λ = { δ(x), X ε Π(E)} of of the dilates of of the elements of of Π(E) is is also the image of of Π(E) under the opening γ = δ,, adjoint to to dilation δ. δ. J. Serra, Ecole des Mines de Paris ( 2000 ) Treillis visqueux 5

6 Viscous Lattices Proposition 1 : Set Λ has a structure of of complete lattice for the ordering of of the inclusion. In In this lattice, the supremum coincides with the set union, although the infimum is is the opening of of the intersection by γ = δ {X i i,, iε iε I} I} = γ ( {X i i,, iε iε I }) }) { X i i,, iε iε I } ε Λ(2) The extreme elements of of Λ are E and the empty set. set Λ is is said to to be the viscous lattice of of dilation δ. δ. J. Serra, Ecole des Mines de Paris ( 2000 ) Treillis visqueux 6

7 Atoms et Sup-generators Sup-generators : The class Β of of the singletons dilates is is a sup-generator of of lattice Λ.. Atoms : But the δ( ) themselves are not atoms in in general. However, when E = n or or n,, and for δ invariant under translation, the elements of of Β are the translates of of the transform B = δ(0) of of the origin. In In such a case, and for B a compact set, the associated viscous lattice is is atomic, of of atoms the translates de B.. J. Serra, Ecole des Mines de Paris ( 2000 ) Treillis visqueux 7

8 A few Counter-performances (1) Proposition 2 : The viscous lattice of of dilation δis generally neither distributive, nor co-prime, and does not admit unique complements. An example of of the lack of of distributivity: Y X X' X Y' X' (X (X X') Y = Y Y (X (X Y) (( X' Y) = (X (X Y) (X (X X') Y' Y' = Y' Y' (X (X Y') ( ( X' X' Y') =.. J. Serra, Ecole des Mines de Paris ( 2000 ) Treillis visqueux 8

9 A few Counter-performances (2) Complement : A complement to to X ε Λ is is every set Y ε Λ such that Y X cc An example : and γ (Y X) = What is is left : is is Galois involution between adjoint dilation and erosion, i.e δ(y) X (X) Y X,Y ελ (3) Y X J. Serra, Ecole des Mines de Paris ( 2000 ) Treillis visqueux 9

10 Dilation and Erosion in Λ Links between a dilation viewed in in Λ and in in Π(E) :: Dilation erosion α:π( α:π( Π((E) Π((E) α 11 :: Π((E) Π( Π(Π( Π(E) Π(Π( α: α: Λ Λ β: β: Λ Λ Identity of of α acting in in Π(E) or or in in Λ... Links between an an erosion dilation viewed in in Λ and in in Π(E) :: β(x) = { { δ ( ) ( ),, δ ( ) ( ) α α 11 (X) }= }= = β(x) β = δ α α 11 (X) When α and δ commute, then erosion β,, in in Λ,, equals the the opening by by γ =δ =δ of of erosion α 1 1,, adjoint to to dilatation α in in Π((E). J. Serra, Ecole des Mines de Paris ( 2000 ) Treillis visqueux 10

11 Connection (Reminder) Set case :: Every set set family Χ Π(E) that satisfies the the 3 following axioms i i // ε Χ ii/ ii/ ε Ε { } ε Χ (( class Χ is is sup generating )) iii/ iii/ { X i, i, i i ε Ι} Ι} ε Χ et et X ii X ii ε Χ (( class Χ is is conditionally closed under union). Generalisation :: The definition of of connectivity extends to to any sup generating lattice, by by replacing by by the the zero of of the the lattice, and the the set set and by by the the sup et et l'inf of of the the lattice. Axiom ii/ ii/ just means that class Χ is is sup-generating. J. Serra, Ecole des Mines de Paris ( 2000 ) Treillis visqueux 11

12 First Connections on Λ (1) Definition : Let Λ be a viscous lattice of of dilation δon Π (E). A class Χ of of Λ defines a connection on Λ when i i // εχ ε ii/ ii/ ε Ε δ{ }ε Χ Χ (5) (5) iii/ { X i, i, i i ε Ι} Ι} ε Χ Χ and X i i X ii ε Χ Χ Proposition 3 : A class Χ Λis a connection on Λ. Λ. if if it it is is the restriction to to Λ of of the union of of a connection Χ on Π (E), and of of the image Β of of the singletons of of Π(E) under δ,, i.e. if if Χ '' = (Χ Λ) Β= ( Χ Β) Λ J. Serra, Ecole des Mines de Paris ( 2000 ) Treillis visqueux 12

13 First Connections on Λ (2) An Example : Χ= Arcwise connection Λ = set set of of the the dilates by by the the unit disc B. B. The union of of the the three lobes belongs to to both Χ and Λ, Λ, hence to to Χ.. However, the the three lobes are are disjoint in in Λ because their erosions by by B are are disjoint in in Π(E). If If we we want to to get get separated particles here (for connections on on Λ ), ), we we have to to take into account the the status of of connectivity (( in in Π(E) )) before dilation by by δ. δ. J. Serra, Ecole des Mines de Paris ( 2000 ) Treillis visqueux 13

14 Second Connections on Λ (1) Theorem 1 : Let Χ be a connection on Π(E) and δ : Π(E) Π(E) be a dilation, of of adjoint erosion,, which generates the viscous lattice Λ. Then the image Χ = δ (Χ) of of connection Χ turns out to to be a connection on Λ.. Comments : these new connections on Λ, are sensibly more restrictive than those of of prop. 4.. For example, the three lobes of of the figure are now three disjoint connected components. J. Serra, Ecole des Mines de Paris ( 2000 ) Treillis visqueux 14

15 Second Connections on Λ (2) Generality of of the theorem : We did not assume that the the dilates of of the the elements de de Χ are are still Χ- Χ- connected. For example, in in 22,, with the the usual arcwise connection, take for for δ the the dilation by by a doublet of of points from h apart. Then the the left two discs of of the the figure form a connected set, although the the group of of the the three discs is is no no longer connected h However, if if ε ε E δ { }ε Χ, Χ, we we have Χ' Χ' = δ (Χ (Χ )) )Χ and the the elements of of Χ' Χ' are are then connected in in both lattices Λ and Π(E) :: Dilation δ preserves connection Χ.. J. Serra, Ecole des Mines de Paris ( 2000 ) Treillis visqueux 15

16 Geodesic Operators (1) Theorem 2 : When dilation δ preserves connection Χ,, then connection Χ '' induced on Λ,, in in the sense du theorem 1, 1, is is also preserved by δ.. In In addition, if if γ et et γ γ δ( ) δ( ) stand for the elementary connected openings on Π(E) and on Λ respectively, then γ γ δ( ) δ( ) = δ γ = γ δ ε E.. Comment : Every Χ'-particle is is the opening by δ of of the corresponding Χ- particle. That allows to to extract the Χ'-particles. Alternatively, can we build directly geodesic dilations in in Λ? J. Serra, Ecole des Mines de Paris ( 2000 ) Treillis visqueux 16

17 Geodesic Operators (2) Proposition 4 :: Let dilation δ be be extensive and preserving connection Χ Π((E). Given A;Z ε Π((E), with A Ζ and Χ -connected, the the conditional dilate ζ (A (A)) = δ (A) Z is is Χ '- '-connected and included in in Z. Z. N. N. B. B. :: This does not mean that the the iterated versions of of ζ tend to to a Χ '- '- component of of Z.. Here is is a counter-example in in 22 :: E := := square D of of side a and of of centre y ;; B(y) := := disc of of diameter < a centred in in y Dilation := := δ ( ) = { } ε D // y ;; δ (y) (y) = B (y) (y) J. Serra, Ecole des Mines de Paris ( 2000 ) Treillis visqueux 17

18 Geodesic Operators (3) Definition: Dilation δ is is said to to be completely extensive when for any point ε E (E topological space) and for any compact set K E, E, there exists an integer n such that the n th th iteration of of δ ( ) covers K. Proposition 5 : When dilation δ is is completely extensive, then the Χ '-connected component Z of of Λ marked by A ε Χ '(Λ ),, if if it it can be covered by a compact set Z 0 of of Π(E), is is equal to to Z = ζ (n) (n) (A) for some integer n. n. J. Serra, Ecole des Mines de Paris ( 2000 ) Treillis visqueux 18

19 An Example of Binary Geodesy (1) a) b) c) Binary example :: in in white, marker A ;; in in black, complement of of mask Z; Z; dilation δ is is the the Minkowski addition by by the the disc of of radius r r :: a) a) step of of reconstruction for for r r = ;; b) b) reconstruction for for r r = ;; c) c) maximum reconstruction, corresponding to to r r = J. Serra, Ecole des Mines de Paris ( 2000 ) Treillis visqueux 19

20 An Example of Binary Geodesy (2) a) b) c) Binary example (followed) :: a) a) maximum reconstruction from the the edges of of the the field b) b) partition of of Z into two Χ'-connected components for for δ max max c) c) corresponding median element.. J. Serra, Ecole des Mines de Paris ( 2000 ) Treillis visqueux 20

21 An Example of Numerical Geodesy (1) a) b) c) Numerical example :: a) a) positron image of of the the heart muscle (copyright CEA-ARMINES) ;; b) b) Watershed line of of the the gradient of of fig.a c) c) optimum reconstruction of of the the Χ'-components internal to to zone b. b. J. Serra, Ecole des Mines de Paris ( 2000 ) Treillis visqueux 21

22 An Example of Numerical Geodesy (2) a) b) c) Numerical example (followed) :: a) a) external surface of of maximum viscosity ;; b) b) external and internal contours ;; c) c) median element between the the two contours. J. Serra, Ecole des Mines de Paris ( 2000 ) Treillis visqueux 22