Connected operators

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Connected operators

Jean Cousty MorphoGraphs and Imagery

2011

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Introduction: a problem seen during practical session

Problem: Fill the holes of the cells

Closing by a structuring element, Holes are closed

But contours “have moved”

A connected closing by reconstruction

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Introduction: a problem seen during practical session

Problem: Fill the holes of the cells Closing by a structuring element,

Holes are closed

But contours “have moved”

A connected closing by reconstruction

(4)

Introduction: a problem seen during practical session

Problem: Fill the holes of the cells Closing by a structuring element,

Holes are closed

But contours “have moved”

A connected closing by reconstruction

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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

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Connectivity: reminder

Connexit´ e

E is a set

G = (E, Γ) is a graph

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Connected components of a set

Definition

Let X ⊆ E

The subgraph of G induced by X is the graph (X , Γ _X ) such that

− →

Γ

X

= {(x, y) ∈ − →

Γ | x ∈ X , y ∈ X }

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

C _X denotes the set of all connected components of X

X ⊆ E (in black)

(E = Z ² et Γ = Γ 4 )

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Connected components of a set

Definition

Let X ⊆ E

The subgraph of G induced by X is the graph (X , Γ _X ) such that

− →

Γ

X

= {(x, y) ∈ − →

Γ | x ∈ X , y ∈ X }

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

C _X denotes the set of all connected components of X

X ⊆ E (in black)

(E = Z ² et Γ = Γ 4 )

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Connected components of a set

Definition

Let X ⊆ E

The subgraph of G induced by X is the graph (X , Γ _X ) such that

− →

Γ

X

= {(x, y) ∈ − →

Γ | x ∈ X , y ∈ X }

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

C _X denotes the set of all connected components of X

Partition of X into connected components

(E = Z ² et Γ = Γ 4 )

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Connected components of a set

Definition

Let X ⊆ E

The subgraph of G induced by X is the graph (X , Γ _X ) such that

− →

Γ

X

= {(x, y) ∈ − →

Γ | x ∈ X , y ∈ X }

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

C _X denotes the set of all connected components of X

(E = Z ² et Γ = Γ 4 )

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Connected openings/closings: the case of sets

Connected openings/closings

Definition

A filter γ on E is a connected opening if

∀X ∈ P(E ), C

_γ(X₎

⊆ C

X

A filter φ on E is a connected closing if

∀X ∈ P(E ), C

_γ(X₎

⊆ C

_X

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)

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Connected openings/closings

Definition

A filter γ on E is a connected opening if

∀X ∈ P(E ), C

_γ(X₎

⊆ C

X

A filter φ on E is a connected closing if

∀X ∈ P(E ), C

_γ(X₎

⊆ C

_X

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)

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Connected openings/closings

Definition

A filter γ on E is a connected opening if

∀X ∈ P(E ), C

_γ(X₎

⊆ C

X

A filter φ on E is a connected closing if

∀X ∈ P(E ), C

_γ(X₎

⊆ C

_X

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)

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Example: area connected opening

Definition

Let λ ∈ N

The area connected opening of parameter λ is the operator α _λ on E defined by

∀X ∈ P(E ), α

_λ

(X ) = ∪{C ∈ C

_X

| |C| ≥ λ}

X ⊆ E

(E = Z ² et Γ = Γ ₄ )

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Example: area connected opening

Definition

Let λ ∈ N

The area connected opening of parameter λ is the operator α _λ on E defined by

∀X ∈ P(E ), α

_λ

(X ) = ∪{C ∈ C

_X

| |C| ≥ λ}

X ⊆ E

(E = Z ² et Γ = Γ ₄ )

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Example: area connected opening

Definition

Let λ ∈ N

The area connected opening of parameter λ is the operator α _λ on E defined by

∀X ∈ P(E ), α

_λ

(X ) = ∪{C ∈ C

_X

| |C| ≥ λ}

α ₁ (X )

(E = Z ² et Γ = Γ ₄ )

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Example: area connected opening

Definition

Let λ ∈ N

The area connected opening of parameter λ is the operator α _λ on E defined by

∀X ∈ P(E ), α

_λ

(X ) = ∪{C ∈ C

_X

| |C| ≥ λ}

α ₂ (X )

(E = Z ² et Γ = Γ ₄ )

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Example: area connected opening

Definition

Let λ ∈ N

The area connected opening of parameter λ is the operator α _λ on E defined by

∀X ∈ P(E ), α

_λ

(X ) = ∪{C ∈ C

_X

| |C| ≥ λ}

α ₁₃ (X )

(E = Z ² et Γ = Γ ₄ )

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Example: connected opening by reconstruction

Definition

Let M ⊆ E

The opening by reconstruction of (the marker) M is the operator ψ _M on E defined by

∀X ∈ P(E ), ψ

_M

(X ) = ∪{C ∈ C

_X

| M ∩ C 6= ∅}

X ⊆ E et M (hatched dots)

(E = Z ² et Γ = Γ ₄ )

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Example: connected opening by reconstruction

Definition

Let M ⊆ E

The opening by reconstruction of (the marker) M is the operator ψ _M on E defined by

∀X ∈ P(E ), ψ

_M

(X ) = ∪{C ∈ C

_X

| M ∩ C 6= ∅}

X ⊆ E

et M (hatched dots)

(E = Z ² et Γ = Γ ₄ )

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Example: connected opening by reconstruction

Definition

Let M ⊆ E

The opening by reconstruction of (the marker) M is the operator ψ _M on E defined by

∀X ∈ P(E ), ψ

_M

(X ) = ∪{C ∈ C

_X

| M ∩ C 6= ∅}

0000 111100

11 00 11 00 11 0000 1111 0000 1111 00 11 00 11

0000 00 1111 11 0000 00 1111 11

00 11

X ⊆ E et M (hatched dots)

(E = Z ² et Γ = Γ ₄ )

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Example: connected opening by reconstruction

Definition

Let M ⊆ E

The opening by reconstruction of (the marker) M is the operator ψ _M on E defined by

∀X ∈ P(E ), ψ

_M

(X ) = ∪{C ∈ C

_X

| M ∩ C 6= ∅}

0000 111100

11 00 11 00 11 0000 1111 0000 1111 00 11 00 11

0000 00 1111 11 0000 00 1111 11

00 11

ψ _M (X )

(E = Z ² et Γ = Γ ₄ )

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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

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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

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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 , F

_k

⊆ H

_k

= ⇒ γ(F)

_k

⊆ γ(H)

_k

(increasing)

∀F ∈ I , ∀k ∈ Z , C

_[γ(F₎_k_]

⊆ C

_[F_k_]

(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 , F

_k

⊆ H

_k

= ⇒ γ(F )

_k

⊆ γ(H)

_k

(increasing)

∀F ∈ I , ∀k ∈ Z , C

_[γ(F₎

k]

⊆ C

_[F

k]

(connected, extensive)

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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 , F

_k

⊆ H

_k

= ⇒ γ(F)

_k

⊆ γ(H)

_k

(increasing)

∀F ∈ I , ∀k ∈ Z , C

_[γ(F₎_k_]

⊆ C

_[F_k_]

(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 , F

_k

⊆ H

_k

= ⇒ γ(F)

_k

⊆ γ(H)

_k

(increasing)

∀F ∈ I , ∀k ∈ Z , C

_[γ(F₎

k]

⊆ C

_[F

k]

(connected, extensive)

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Examples

The stack operator induced by α λ (where λ ∈ N) is a connected opening on I

The stack operator induced by ψ _M (where M ⊆ E ) is a connected opening on I

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

I Altitudes

E

∀x ∈ {x, − + 1}

Connected components of the level sets (red)

J. Cousty : MorphoGraph and Imagery 11/28

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Examples

The stack operator induced by α λ (where λ ∈ N) is a connected opening on I

The stack operator induced by ψ _M (where M ⊆ E ) is a connected opening on I

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

Altitudes

I

E

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

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Examples

The stack operator induced by α λ (where λ ∈ N) is a connected opening on I

The stack operator induced by ψ _M (where M ⊆ E ) is a connected opening on I

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

Altitudes

I

E

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

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Examples

The stack operator induced by α λ (where λ ∈ N) is a connected opening on I

The stack operator induced by ψ _M (where M ⊆ E ) is a connected opening on I

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

Altitudes

I

E

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

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Examples

The stack operator induced by α λ (where λ ∈ N) is a connected opening on I

The stack operator induced by ψ _M (where M ⊆ E) is a connected opening on I

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

Altitudes

I

E

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

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Examples

The stack operator induced by α λ (where λ ∈ N) is a connected opening on I

The stack operator induced by ψ _M (where M ⊆ E) is a connected opening on I

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

Altitudes

I

E

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

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Grayscale connected opening

Definition

Let F ∈ I

The opening by reconstruction of (the marker) F is the operator ψ _F on I defined by

∀H ∈ I , ∀k ∈ Z , [ψ

F

(H)]

k

= ψ

_[F_k_]

(H

k

)

E Altitudes

H F

H in black, F in blue

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Grayscale connected opening

Definition

Let F ∈ I

The opening by reconstruction of (the marker) F is the operator ψ _F on I defined by

∀H ∈ I , ∀k ∈ Z , [ψ

F

(H)]

k

= ψ

_[F_k_]

(H

k

)

E Altitudes

H F

H in black, F in blue

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Grayscale connected opening

Definition

Let F ∈ I

The opening by reconstruction of (the marker) F is the operator ψ _F on I defined by

∀H ∈ I , ∀k ∈ Z , [ψ

F

(H)]

k

= ψ

_[F_k_]

(H

k

)

E F

Altitudes

H

ψ F (H) (in black)

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Properties

Property (duality)

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

∀F ∈ I , ?φ(F) = −φ(−F )

Property (contour preservation)

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 )

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Properties

Property (duality)

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

∀F ∈ I , ?φ(F) = −φ(−F )

Property (contour preservation)

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 )

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Component tree

Probl` eme

How to implement a connected operator?

Data structure: component tree How to build other connected operators?

Criterion on the nodes of the component tree

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Component tree

Notation

In the following, the symbol I denotes a grayscale image

on E: I ∈ I

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Component tree

Connected component of an image

Definition

We denote by C _I the union of the sets of connected components of the level sets of I

C

_I

= ∪{C

_[I_k_]

| k ∈ Z }

Each element in C _I is called a connected component of I

Altitudes

I

E

Components of I (bold red lines)

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Component tree

Definition

Let C 1 and C 2 be two components of I in C _I We say that C ₁ is a child of C ₂ (for I) if

C

1

⊆ C

2

∀C

⁰

∈ C

I

, if C

1

⊆ C

⁰

⊆ C

2

then C

⁰

= C

2

or C

⁰

= C

1

The component tree of I is the graph A _I = (C _I , Γ _I ) such that (C

₁

, C

₂

) ∈ Γ

_I

if C

₂

is a child of C

₁

Altitudes

I

The relation “is child of” is

represented by red arrows

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Component tree

Definition

Let C 1 and C 2 be two components of I in C _I We say that C ₁ is a child of C ₂ (for I) if

C

1

⊆ C

2

∀C

⁰

∈ C

I

, if C

1

⊆ C

⁰

⊆ C

2

then C

⁰

= C

2

or C

⁰

= C

1

The component tree of I is the graph A _I = (C _I , Γ _I ) such that (C

₁

, C

₂

) ∈ Γ

_I

if C

₂

is a child of C

₁

Altitudes

I

The relation “is child of” is

represented by red arrows

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Component tree

Arborescence

Property

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

Altitudes

I

E

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Component tree

Simplifying an image by razings

A leaf of the component tree A _I 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 of the tree A _I

Such removal is called a razing

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Component tree

Razing: intuition

Altitudes

I

E

Successive razing steps

A maximum has been removed

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Component tree

Razing: intuition

Altitudes

I

E

Successive razing steps

A maximum has been removed

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Component tree

Razing: intuition

Altitudes

I

E

Successive razing steps

A maximum has been removed

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Component tree

Razing: intuition

Altitudes

I

E

Successive razing steps

A maximum has been removed

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Component tree

Razing: definition

Definition

Let x ∈ E

The elementary razing ν _x of I at x is defined b

[ν

_x

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

_x

(I )](y) = I (y) otherwise

An image obtained from I by composition of elementary razings is

called a razing of I

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Component tree

Simplifying an image by razings

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

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Component tree

Razing by area

Altitudes

I

E

surface

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Component tree

Razing by depth, or dynamics

Altitudes

I

E

profondeur

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Component tree

Razing by volume

volume

E Altitudes

I

(54)

Component tree

Flooding

The dual of a razing is called a flooding

A flooding removes minima by iteratively increasing their values

Property (contour preservation)

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)

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Component tree

Flooding

The dual of a razing is called a flooding

A flooding removes minima by iteratively increasing their values

Property (contour preservation)

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)

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Component tree

Flooding

The dual of a razing is called a flooding

A flooding removes minima by iteratively increasing their values

Property (contour preservation)

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)

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Component tree

Exercise

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

Compare the number of maxima of F and of I

Altitudes

I

E

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Component tree

Connected operators