jean cousty morphographs and imagery 2011

Connected operators

Jean Cousty

MorphoGraphs and Imagery2011

J. Cousty : MorphoGraph and Imagery 1/28

Introduction: a problem seen during practical session

Problem: Fill the holes of the cells

Closing by a structuring element,

Holes are closedBut contours “have moved”

A connected closing by reconstruction


Outline of the lecture

1 Connectivity: reminder

2 Connected openings/closings: the case of sets

3 Connected opening: the case of grayscale images

4 Component tree


Connectivity: reminder

Connexite

E is a set

G = (E , Γ) is a graph



Connected components of a set

Definition

Let X ⊆ E

The subgraph of G induced by X is the graph (X , ΓX ) such that−→ΓX = {(x , y) ∈

−→Γ | x ∈ X , y ∈ X}

A connected component of the subgraph of G induced by X issimply called a connected component of X

CX denotes the set of all connected components of X

X ⊆ E (in black)

(E = Z2 et Γ = Γ4)




Definition

Let X ⊆ E


−→Γ | x ∈ X , y ∈ X}



X ⊆ E (in black)

(E = Z2 et Γ = Γ4)J. Cousty : MorphoGraph and Imagery 5/28



Definition

Let X ⊆ E


−→Γ | x ∈ X , y ∈ X}



Partition of X into connectedcomponents




Definition

Let X ⊆ E


−→Γ | x ∈ X , y ∈ X}




Connected openings/closings: the case of sets

Connected openings/closings

Definition

A filter γ on E is a connected opening if

∀X ∈ P(E ), Cγ(X ) ⊆ CXA filter φ on E is a connected closing if

∀X ∈ P(E ), Cγ(X ) ⊆ CX

Property

A connected opening γ is an opening

∀X ,Y ∈ P(E ), X ⊆ Y =⇒ γ(X ) ⊆ γ(Y ) (increasing)∀X ∈ P(E ), γ(γ(X )) = γ(X ) (idempotent)∀X ∈ P(E ), γ(X ) ⊆ X (anti-axtensive)



Connected openings/closings

Definition

A filter γ on E is a connected opening if

∀X ∈ P(E ), Cγ(X ) ⊆ CXA filter φ on E is a connected closing if

∀X ∈ P(E ), Cγ(X ) ⊆ CX

Property

A connected closing γ is a closing

∀X ,Y ∈ P(E ), X ⊆ Y =⇒ γ(X ) ⊆ γ(Y ) (increasing)∀X ∈ P(E ), γ(γ(X )) = γ(X ) (idempotence)∀X ∈ P(E ), X ⊆ γ(X ) (extensive)



Example: area connected opening

Definition

Let λ ∈ NThe area connected opening of parameter λ is the operator αλ

on E defined by

∀X ∈ P(E ), αλ(X ) = ∪{C ∈ CX | |C | ≥ λ}

X ⊆ E

(E = Z2 et Γ = Γ4)




Definition


on E defined by

∀X ∈ P(E ), αλ(X ) = ∪{C ∈ CX | |C | ≥ λ}

α1(X )

(E = Z2 et Γ = Γ4)




Definition


on E defined by

∀X ∈ P(E ), αλ(X ) = ∪{C ∈ CX | |C | ≥ λ}

α2(X )

(E = Z2 et Γ = Γ4)




Definition


on E defined by

∀X ∈ P(E ), αλ(X ) = ∪{C ∈ CX | |C | ≥ λ}

α13(X )

(E = Z2 et Γ = Γ4)



Example: connected opening by reconstruction

Definition

Let M ⊆ E

The opening by reconstruction of (the marker) M is theoperator ψM on E defined by

∀X ∈ P(E ), ψM(X ) = ∪{C ∈ CX | M ∩ C 6= ∅}

X ⊆ E et M (hatched dots)

(E = Z2 et Γ = Γ4)




Definition

Let M ⊆ E


∀X ∈ P(E ), ψM(X ) = ∪{C ∈ CX | M ∩ C 6= ∅}

X ⊆ E

et M (hatched dots)

(E = Z2 et Γ = Γ4)




Definition

Let M ⊆ E


∀X ∈ P(E ), ψM(X ) = ∪{C ∈ CX | M ∩ C 6= ∅}

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X ⊆ E et M (hatched dots)

(E = Z2 et Γ = Γ4)




Definition

Let M ⊆ E


∀X ∈ P(E ), ψM(X ) = ∪{C ∈ CX | M ∩ C 6= ∅}

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ψM(X )

(E = Z2 et Γ = Γ4)



Properties of connected filters

Property

γ is a connected opening ⇔ ?γ is a connected closing

Property (contour preservation)

If γ is a connected opening or closing, then

∀X ∈ P(E ), ∀(x , y) ∈−→Γ ,

|{x , y} ∩ γ(X )| = 1 =⇒ |{x , y} ∩ X | = 1


Connected opening: the case of grayscale images

Grayscale connected opening/closing

Definition (connected opening)

An operator γ on I is a connected opening (on I) if

∀F ∈ I, γ(γ(F )) = γ(F ) (idempotence)∀F ,H ∈ I,∀k ∈ Z, Fk ⊆ Hk =⇒ γ(F )k ⊆ γ(H)k (increasing)∀F ∈ I, ∀k ∈ Z, C[γ(F )k ] ⊆ C[Fk ] (connected, anti-extensive)

Definition (connected closing)

An operator γ on I is a connected closing (on I) if

∀F ∈ I, γ(γ(F )) = γ(F ) (idempotence)∀F ,H ∈ I,∀k ∈ Z, Fk ⊆ Hk =⇒ γ(F )k ⊆ γ(H)k (increasing)∀F ∈ I, ∀k ∈ Z, C[γ(F )k ]

⊆ C[Fk ](connected, extensive)



Examples

The stack operator induced by αλ (where λ ∈ N) is a connectedopening on I

The stack operator induced by ψM (where M ⊆ E ) is a connectedopening on IThe stack operator induced by any connected opening on E is aconnected opening on I

I

Altitudes

EE = Z, ∀x ∈ E , Γ(x) = {x , x − 1, x + 1}

Connected components of the level sets (red)



Examples



Altitudes

I

EE = Z, ∀x ∈ E , Γ(x) = {x , x − 1, x + 1}

Connected components of the level sets (red)



Examples



Altitudes

I

EE = Z, ∀x ∈ E , Γ(x) = {x , x − 1, x + 1}

Connected components of area greater than 3 (blue)



Examples



Altitudes

I

EE = Z, ∀x ∈ E , Γ(x) = {x , x − 1, x + 1}

α3(I )



Examples

The stack operator induced by αλ (where λ ∈ N) is a connectedopening on IThe stack operator induced by ψM (where M ⊆ E ) is a connectedopening on I

The stack operator induced by any connected opening on E is aconnected opening on I

Altitudes

I

EE = Z, ∀x ∈ E , Γ(x) = {x , x − 1, x + 1}

α3(I )



Examples

The stack operator induced by αλ (where λ ∈ N) is a connectedopening on IThe stack operator induced by ψM (where M ⊆ E ) is a connectedopening on IThe stack operator induced by any connected opening on E is aconnected opening on I

Altitudes

I

EE = Z, ∀x ∈ E , Γ(x) = {x , x − 1, x + 1}

α3(I )



Grayscale connected opening

Definition

Let F ∈ IThe opening by reconstruction of (the marker) F is theoperator ψF on I defined by

∀H ∈ I, ∀k ∈ Z, [ψF (H)]k = ψ[Fk ](Hk)

E

Altitudes

H

F

H in black, F in blue




Definition


∀H ∈ I, ∀k ∈ Z, [ψF (H)]k = ψ[Fk ](Hk)

E

Altitudes

H

F

H in black, F in blueJ. Cousty : MorphoGraph and Imagery 12/28



Definition


∀H ∈ I, ∀k ∈ Z, [ψF (H)]k = ψ[Fk ](Hk)

E

F

Altitudes

H

ψF (H) (in black)J. Cousty : MorphoGraph and Imagery 12/28


Properties

Property (duality)

An operator φ is a connected closing on I if and only if its dual ?φis a connected opening on I, where

∀F ∈ I, ?φ(F ) = −φ(−F )


If γ is a connected opening or closing on I, then

∀F ∈ I,∀(x , y) ∈−→Γ , γ(F )(x) 6= γ(F )(y) =⇒ F (x) 6= F (y)


Component tree

Component tree

Probleme

How to implement a connected operator?

Data structure: component tree

How to build other connected operators?

Criterion on the nodes of the component tree


Component tree

Notation

In the following, the symbol I denotes a grayscale imageon E : I ∈ I


Component tree

Connected component of an image

Definition

We denote by CI the union of the sets of connected components ofthe level sets of I

CI = ∪{C[Ik ] | k ∈ Z}Each element in CI is called a connected component of I

Altitudes

I

E

Components of I (bold red lines)J. Cousty : MorphoGraph and Imagery 16/28

Component tree

Component tree

Definition

Let C1 and C2 be two components of I in CIWe say that C1 is a child of C2 (for I ) if

C1 ⊆ C2

∀C ′ ∈ CI , if C1 ⊆ C ′ ⊆ C2 then C ′ = C2 or C ′ = C1

The component tree of I is the graph AI = (CI , ΓI ) such that

(C1,C2) ∈ ΓI if C2 is a child of C1

Altitudes

I

E

The relation “is child of” isrepresented by red arrows


Component tree

Arborescence

Property

If the graph (E , Γ) is connected, then the component tree of I isan arborescence of root E

Altitudes

I

E


Component tree

Simplifying an image by razings

A leaf of the component tree AI of I is called

a regional maximum of I

A leaf of the component tree A−I of −I is called

a regional minimum of I

Given a criterion, it is possible to iteratively remove the leaves ofthe tree AI

Such removal is called a razing


Component tree

Razing: intuition

Altitudes

I

E

Successive razing steps

A maximum has been removed


Component tree

Razing: intuition

Altitudes

I

E

Successive razing stepsA maximum has been removed


Component tree

Razing: definition

Definition

Let x ∈ E

The elementary razing νx of I at x is defined b

[νx(I )](y) = I (y)− 1 if x and y belong to a same maximum of I[νx(I )](y) = I (y) otherwise

An image obtained from I by composition of elementary razings iscalled a razing of I


Component tree

Simplifying an image by razings

What criterion (attribute) for selecting a maxima to raze?


Component tree

Razing by area

Altitudes

I

E

surface


Component tree

Razing by depth, or dynamics

Altitudes

I

E

profondeur


Component tree

Razing by volume

volume

E

Altitudes

I


Component tree

Flooding

The dual of a razing is called a flooding

A flooding removes minima by iteratively increasing their values


If F is a razing or a flooding of I , then

∀(x , y) ∈−→Γ , F (x) 6= F (y) =⇒ I (x) 6= I (y)

Remark. An operator produces razings is not necessarily an opening(counter-examples: razing by dynamics or volume)


Component tree

Exercise

Draw the image F obtained by razings all components of depth isless than 12

Compare the number of maxima of F and of I

Altitudes

I

E


Component tree

Exercise

Draw the image H obtained by the dual of the previous one, (i.e.,the flooding that removes all components of depth is less than 12)Help

1 Draw −I2 Apply a razing to −I3 Inverse the obtained result

Compare the numbers of minima of H and of I

Altitudes

I

E


jean cousty morphographs and imagery 2011

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