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Spatially-variant anisotropic morphological filters driven by gradient fields Rafael Verd-Monedero 1 Jess Angulo 2 Jean Serra 3 1 Department of Information Technologies and Communications, Technical University of Cartagena, 30202, Cartagena,

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SLIDE 1

Spatially-variant anisotropic morphological filters driven by gradient fields

Rafael Verdú-Monedero1 Jesús Angulo 2 Jean Serra 3

1Department of Information Technologies and Communications,

Technical University of Cartagena, 30202, Cartagena, Spain, rafael.verdu@upct.es

2Centre de Morphologie Mathématique (CMM),

Ecole des Mines de Paris, Fontainebleau Cedex, France jesus.angulo@ensmp.fr

3Laboratoire A2SI - ESIEE, B.P

. 99, 93162 Noisy-le-Grand, France serraj@esiee.fr ISMM 2009, 9th International Symposium on Mathematical Morphology

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SLIDE 2

Outline

1

Introduction

2

Spatially-variant morphology Dilation/erosion and opening/closing Dilation for numerical functions

3

Directional field modelling Average squared gradient (ASG) Regularization of the ASG: ASGVF

4

Applications

5

Conclusions and perspectives

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 2 / 27

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SLIDE 3

Introduction

Outline

1

Introduction

2

Spatially-variant morphology Dilation/erosion and opening/closing Dilation for numerical functions

3

Directional field modelling Average squared gradient (ASG) Regularization of the ASG: ASGVF

4

Applications

5

Conclusions and perspectives

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 3 / 27

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SLIDE 4

Introduction

“Spatially variant” encompasses: Two level structure of

a space E, and of all subsets P(E), and functions on E.

Some variable processing over space E e.g. geometrical deformation of the Euclidean space

by perspective (e.g. actual application of traffic monitoring) by rotation invariance

Proposed approach Spatially variant morphological operators are computed from their direct geometric definitions. Structuring elements are linear orientated segments. Orientation information is extracted from the image under study itself.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 4 / 27

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SLIDE 5

Introduction

“Spatially variant” encompasses: Two level structure of

a space E, and of all subsets P(E), and functions on E.

Some variable processing over space E e.g. geometrical deformation of the Euclidean space

by perspective (e.g. actual application of traffic monitoring) by rotation invariance

Proposed approach Spatially variant morphological operators are computed from their direct geometric definitions. Structuring elements are linear orientated segments. Orientation information is extracted from the image under study itself.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 4 / 27

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SLIDE 6

Introduction

“Spatially variant” encompasses: Two level structure of

a space E, and of all subsets P(E), and functions on E.

Some variable processing over space E e.g. geometrical deformation of the Euclidean space

by perspective (e.g. actual application of traffic monitoring) by rotation invariance

Proposed approach Spatially variant morphological operators are computed from their direct geometric definitions. Structuring elements are linear orientated segments. Orientation information is extracted from the image under study itself.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 4 / 27

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SLIDE 7

Introduction

“Spatially variant” encompasses: Two level structure of

a space E, and of all subsets P(E), and functions on E.

Some variable processing over space E e.g. geometrical deformation of the Euclidean space

by perspective (e.g. actual application of traffic monitoring) by rotation invariance

Proposed approach Spatially variant morphological operators are computed from their direct geometric definitions. Structuring elements are linear orientated segments. Orientation information is extracted from the image under study itself.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 4 / 27

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SLIDE 8

Introduction

“Spatially variant” encompasses: Two level structure of

a space E, and of all subsets P(E), and functions on E.

Some variable processing over space E e.g. geometrical deformation of the Euclidean space

by perspective (e.g. actual application of traffic monitoring) by rotation invariance

Proposed approach Spatially variant morphological operators are computed from their direct geometric definitions. Structuring elements are linear orientated segments. Orientation information is extracted from the image under study itself.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 4 / 27

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SLIDE 9

Introduction

“Spatially variant” encompasses: Two level structure of

a space E, and of all subsets P(E), and functions on E.

Some variable processing over space E e.g. geometrical deformation of the Euclidean space

by perspective (e.g. actual application of traffic monitoring) by rotation invariance

Proposed approach Spatially variant morphological operators are computed from their direct geometric definitions. Structuring elements are linear orientated segments. Orientation information is extracted from the image under study itself.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 4 / 27

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SLIDE 10

Introduction

“Spatially variant” encompasses: Two level structure of

a space E, and of all subsets P(E), and functions on E.

Some variable processing over space E e.g. geometrical deformation of the Euclidean space

by perspective (e.g. actual application of traffic monitoring) by rotation invariance

Proposed approach Spatially variant morphological operators are computed from their direct geometric definitions. Structuring elements are linear orientated segments. Orientation information is extracted from the image under study itself.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 4 / 27

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SLIDE 11

Introduction

“Spatially variant” encompasses: Two level structure of

a space E, and of all subsets P(E), and functions on E.

Some variable processing over space E e.g. geometrical deformation of the Euclidean space

by perspective (e.g. actual application of traffic monitoring) by rotation invariance

Proposed approach Spatially variant morphological operators are computed from their direct geometric definitions. Structuring elements are linear orientated segments. Orientation information is extracted from the image under study itself.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 4 / 27

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SLIDE 12

Introduction

“Spatially variant” encompasses: Two level structure of

a space E, and of all subsets P(E), and functions on E.

Some variable processing over space E e.g. geometrical deformation of the Euclidean space

by perspective (e.g. actual application of traffic monitoring) by rotation invariance

Proposed approach Spatially variant morphological operators are computed from their direct geometric definitions. Structuring elements are linear orientated segments. Orientation information is extracted from the image under study itself.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 4 / 27

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SLIDE 13

Spatially-variant morphology

Outline

1

Introduction

2

Spatially-variant morphology Dilation/erosion and opening/closing Dilation for numerical functions

3

Directional field modelling Average squared gradient (ASG) Regularization of the ASG: ASGVF

4

Applications

5

Conclusions and perspectives

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 5 / 27

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SLIDE 14

Spatially-variant morphology

Operations with data-based variation The four basic operations must be expressed exclusively by means of the structuring function. It is not possible to resort to complement, or equivalently, to reciprocal dilation. Notation

E → arbitrary set (discrete or continuous space) or any graph x = (x, y) ∈ E → points of E X ⊆ E → subsets of E. P(E) is the set of all these subsets. A structuring function δ : E → P(E) is an arbitrary family {δ(x)} of sets indexed by the points of E. The transform of a point is a set. T → numerical axis [0, M] (e.g. [0, +∞], [0, 255], etc.) The family of all numerical functions f : E → T is denoted by F(E, T ). Both sets P(E) and F(E, T ) are complete lattices.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 6 / 27

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SLIDE 15

Spatially-variant morphology

Operations with data-based variation The four basic operations must be expressed exclusively by means of the structuring function. It is not possible to resort to complement, or equivalently, to reciprocal dilation. Notation

E → arbitrary set (discrete or continuous space) or any graph x = (x, y) ∈ E → points of E X ⊆ E → subsets of E. P(E) is the set of all these subsets. A structuring function δ : E → P(E) is an arbitrary family {δ(x)} of sets indexed by the points of E. The transform of a point is a set. T → numerical axis [0, M] (e.g. [0, +∞], [0, 255], etc.) The family of all numerical functions f : E → T is denoted by F(E, T ). Both sets P(E) and F(E, T ) are complete lattices.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 6 / 27

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SLIDE 16

Spatially-variant morphology

Operations with data-based variation The four basic operations must be expressed exclusively by means of the structuring function. It is not possible to resort to complement, or equivalently, to reciprocal dilation. Notation

E → arbitrary set (discrete or continuous space) or any graph x = (x, y) ∈ E → points of E X ⊆ E → subsets of E. P(E) is the set of all these subsets. A structuring function δ : E → P(E) is an arbitrary family {δ(x)} of sets indexed by the points of E. The transform of a point is a set. T → numerical axis [0, M] (e.g. [0, +∞], [0, 255], etc.) The family of all numerical functions f : E → T is denoted by F(E, T ). Both sets P(E) and F(E, T ) are complete lattices.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 6 / 27

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SLIDE 17

Spatially-variant morphology

Operations with data-based variation The four basic operations must be expressed exclusively by means of the structuring function. It is not possible to resort to complement, or equivalently, to reciprocal dilation. Notation

E → arbitrary set (discrete or continuous space) or any graph x = (x, y) ∈ E → points of E X ⊆ E → subsets of E. P(E) is the set of all these subsets. A structuring function δ : E → P(E) is an arbitrary family {δ(x)} of sets indexed by the points of E. The transform of a point is a set. T → numerical axis [0, M] (e.g. [0, +∞], [0, 255], etc.) The family of all numerical functions f : E → T is denoted by F(E, T ). Both sets P(E) and F(E, T ) are complete lattices.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 6 / 27

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Spatially-variant morphology Dilation/erosion and opening/closing

Dilation The two basic operations that map lattice P(E) into itself are those which preserve both union and intersection In mathematical morphology they are dilation δ and erosion ε: δ(∪Xi) = ∪δ(Xi) ; ε(∩Xi) = ∩ε(Xi) Xi ∈ P(E). X is the union of its singletons and dilation commutes under union δ(X) = ∪{δ{x}|{x}⊆X} = ∪{δ(x)| x ∈ X} = ∪{B(x)| x ∈ X}. A dilation may not be extensive, i.e. δ(X) ⊇ X. Extensivity is obtained iff for all x ∈ E we have x ∈ B(x)

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 7 / 27

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SLIDE 19

Spatially-variant morphology Dilation/erosion and opening/closing

Dilation The two basic operations that map lattice P(E) into itself are those which preserve both union and intersection In mathematical morphology they are dilation δ and erosion ε: δ(∪Xi) = ∪δ(Xi) ; ε(∩Xi) = ∩ε(Xi) Xi ∈ P(E). X is the union of its singletons and dilation commutes under union δ(X) = ∪{δ{x}|{x}⊆X} = ∪{δ(x)| x ∈ X} = ∪{B(x)| x ∈ X}. A dilation may not be extensive, i.e. δ(X) ⊇ X. Extensivity is obtained iff for all x ∈ E we have x ∈ B(x)

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 7 / 27

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SLIDE 20

Spatially-variant morphology Dilation/erosion and opening/closing

Dilation The two basic operations that map lattice P(E) into itself are those which preserve both union and intersection In mathematical morphology they are dilation δ and erosion ε: δ(∪Xi) = ∪δ(Xi) ; ε(∩Xi) = ∩ε(Xi) Xi ∈ P(E). X is the union of its singletons and dilation commutes under union δ(X) = ∪{δ{x}|{x}⊆X} = ∪{δ(x)| x ∈ X} = ∪{B(x)| x ∈ X}. A dilation may not be extensive, i.e. δ(X) ⊇ X. Extensivity is obtained iff for all x ∈ E we have x ∈ B(x)

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 7 / 27

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SLIDE 21

Spatially-variant morphology Dilation/erosion and opening/closing

Dilation The two basic operations that map lattice P(E) into itself are those which preserve both union and intersection In mathematical morphology they are dilation δ and erosion ε: δ(∪Xi) = ∪δ(Xi) ; ε(∩Xi) = ∩ε(Xi) Xi ∈ P(E). X is the union of its singletons and dilation commutes under union δ(X) = ∪{δ{x}|{x}⊆X} = ∪{δ(x)| x ∈ X} = ∪{B(x)| x ∈ X}. A dilation may not be extensive, i.e. δ(X) ⊇ X. Extensivity is obtained iff for all x ∈ E we have x ∈ B(x)

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 7 / 27

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SLIDE 22

Spatially-variant morphology Dilation/erosion and opening/closing

Dilation The two basic operations that map lattice P(E) into itself are those which preserve both union and intersection In mathematical morphology they are dilation δ and erosion ε: δ(∪Xi) = ∪δ(Xi) ; ε(∩Xi) = ∩ε(Xi) Xi ∈ P(E). X is the union of its singletons and dilation commutes under union δ(X) = ∪{δ{x}|{x}⊆X} = ∪{δ(x)| x ∈ X} = ∪{B(x)| x ∈ X}. A dilation may not be extensive, i.e. δ(X) ⊇ X. Extensivity is obtained iff for all x ∈ E we have x ∈ B(x)

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 7 / 27

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SLIDE 23

Spatially-variant morphology Dilation/erosion and opening/closing

Erosion The expression of erosion drawn from adjunction is ε(X) = {z | B(z) ⊆ X}, which is anti-extensive. The operation dual of dilation δ under complement is the erosion ε∗(X) = [δ(X c)]c, whose associated structuring function is the reciprocal version of δ y ∈ ζ(x) if and only if x ∈ δ(y) x, y ∈ E.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 8 / 27

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SLIDE 24

Spatially-variant morphology Dilation/erosion and opening/closing

Erosion The expression of erosion drawn from adjunction is ε(X) = {z | B(z) ⊆ X}, which is anti-extensive. The operation dual of dilation δ under complement is the erosion ε∗(X) = [δ(X c)]c, whose associated structuring function is the reciprocal version of δ y ∈ ζ(x) if and only if x ∈ δ(y) x, y ∈ E.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 8 / 27

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SLIDE 25

Spatially-variant morphology Dilation/erosion and opening/closing

Erosion The expression of erosion drawn from adjunction is ε(X) = {z | B(z) ⊆ X}, which is anti-extensive. The operation dual of dilation δ under complement is the erosion ε∗(X) = [δ(X c)]c, whose associated structuring function is the reciprocal version of δ y ∈ ζ(x) if and only if x ∈ δ(y) x, y ∈ E.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 8 / 27

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SLIDE 26

Spatially-variant morphology Dilation/erosion and opening/closing

Opening and closing The smallest inverse of ε(X) is the composition product γ(X) = δε(X), which is

increasing, X ⊆ Y ⇒ γ(X) ⊆ γ(Y) anti-extensive, γ(X) ⊆ X idempotent, γγ(X) = γ(X)

these three features define an opening. By inverting δ and ε, we obtain the closing ϕ = εδ, which is increasing, extensive, and idempotent.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 9 / 27

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SLIDE 27

Spatially-variant morphology Dilation/erosion and opening/closing

Opening and closing The smallest inverse of ε(X) is the composition product γ(X) = δε(X), which is

increasing, X ⊆ Y ⇒ γ(X) ⊆ γ(Y) anti-extensive, γ(X) ⊆ X idempotent, γγ(X) = γ(X)

these three features define an opening. By inverting δ and ε, we obtain the closing ϕ = εδ, which is increasing, extensive, and idempotent.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 9 / 27

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SLIDE 28

Spatially-variant morphology Dilation/erosion and opening/closing

Opening and closing The smallest inverse of ε(X) is the composition product γ(X) = δε(X), which is

increasing, X ⊆ Y ⇒ γ(X) ⊆ γ(Y) anti-extensive, γ(X) ⊆ X idempotent, γγ(X) = γ(X)

these three features define an opening. By inverting δ and ε, we obtain the closing ϕ = εδ, which is increasing, extensive, and idempotent.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 9 / 27

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SLIDE 29

Spatially-variant morphology Dilation/erosion and opening/closing

The four basic operations δ(X) = ∪{B(x)| x ∈ X}, ε(X) = {z | B(z) ⊆ X}, δε(X) = ∪{B(x)|B(x)⊆X}, εδ(X) = ∪{x |B(x)⊆ ∪ [B(x)| x ∈ X]}. They are completely determined by the structuring function x → B(x) = δ(x). Do not involve any reciprocal function. In particular, opening γ = δε and closing ϕ = εδ are not dual of each other for the complement.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 10 / 27

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SLIDE 30

Spatially-variant morphology Dilation/erosion and opening/closing

The four basic operations δ(X) = ∪{B(x)| x ∈ X}, ε(X) = {z | B(z) ⊆ X}, δε(X) = ∪{B(x)|B(x)⊆X}, εδ(X) = ∪{x |B(x)⊆ ∪ [B(x)| x ∈ X]}. They are completely determined by the structuring function x → B(x) = δ(x). Do not involve any reciprocal function. In particular, opening γ = δε and closing ϕ = εδ are not dual of each other for the complement.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 10 / 27

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SLIDE 31

Spatially-variant morphology Dilation/erosion and opening/closing

The four basic operations δ(X) = ∪{B(x)| x ∈ X}, ε(X) = {z | B(z) ⊆ X}, δε(X) = ∪{B(x)|B(x)⊆X}, εδ(X) = ∪{x |B(x)⊆ ∪ [B(x)| x ∈ X]}. They are completely determined by the structuring function x → B(x) = δ(x). Do not involve any reciprocal function. In particular, opening γ = δε and closing ϕ = εδ are not dual of each other for the complement.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 10 / 27

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SLIDE 32

Spatially-variant morphology Dilation/erosion and opening/closing

The four basic operations δ(X) = ∪{B(x)| x ∈ X}, ε(X) = {z | B(z) ⊆ X}, δε(X) = ∪{B(x)|B(x)⊆X}, εδ(X) = ∪{x |B(x)⊆ ∪ [B(x)| x ∈ X]}. They are completely determined by the structuring function x → B(x) = δ(x). Do not involve any reciprocal function. In particular, opening γ = δε and closing ϕ = εδ are not dual of each other for the complement.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 10 / 27

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SLIDE 33

Spatially-variant morphology Dilation for numerical functions

Dilation for numerical functions

Duality under complement works only for sets, whereas adjunction duality applies to any complete lattice. Associated with numerical function f : E → T and the set structuring function x → B(x), we introduce the pulse function: ix,t(x) = t Dilating ix,t by the structuring function B results in the cylinder CB(x),t f can be decomposed into the supremum of its pulses, f = ∨{ix,f(x), x ∈ E}, and dilation commutes under supremum.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 11 / 27

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SLIDE 34

Spatially-variant morphology Dilation for numerical functions

Dilation for numerical functions

Duality under complement works only for sets, whereas adjunction duality applies to any complete lattice. Associated with numerical function f : E → T and the set structuring function x → B(x), we introduce the pulse function: ix,t(x) = t Dilating ix,t by the structuring function B results in the cylinder CB(x),t f can be decomposed into the supremum of its pulses, f = ∨{ix,f(x), x ∈ E}, and dilation commutes under supremum.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 11 / 27

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SLIDE 35

Spatially-variant morphology Dilation for numerical functions

Dilation for numerical functions

Duality under complement works only for sets, whereas adjunction duality applies to any complete lattice. Associated with numerical function f : E → T and the set structuring function x → B(x), we introduce the pulse function: ix,t(x) = t Dilating ix,t by the structuring function B results in the cylinder CB(x),t f can be decomposed into the supremum of its pulses, f = ∨{ix,f(x), x ∈ E}, and dilation commutes under supremum.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 11 / 27

slide-36
SLIDE 36

Spatially-variant morphology Dilation for numerical functions

Dilation for numerical functions

Duality under complement works only for sets, whereas adjunction duality applies to any complete lattice. Associated with numerical function f : E → T and the set structuring function x → B(x), we introduce the pulse function: ix,t(x) = t Dilating ix,t by the structuring function B results in the cylinder CB(x),t f can be decomposed into the supremum of its pulses, f = ∨{ix,f(x), x ∈ E}, and dilation commutes under supremum.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 11 / 27

slide-37
SLIDE 37

Spatially-variant morphology Dilation for numerical functions

Dilation for numerical functions

Duality under complement works only for sets, whereas adjunction duality applies to any complete lattice. Associated with numerical function f : E → T and the set structuring function x → B(x), we introduce the pulse function: ix,t(x) = t Dilating ix,t by the structuring function B results in the cylinder CB(x),t f can be decomposed into the supremum of its pulses, f = ∨{ix,f(x), x ∈ E}, and dilation commutes under supremum.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 11 / 27

slide-38
SLIDE 38

Spatially-variant morphology Dilation for numerical functions

Dilation for numerical functions: dilation and erosion

The dilate of f by δ is given by the supremum of the dilates of its pulses: δ(f) = ∨{CB(x),f(x), x ∈ E}. The eroded ε(f) is the supremum of those pulses whose dilated cylinders are smaller than f: ε(f) = ∨{ix,t | CB(x),t ≤ f, x ∈ E}. The duality under adjunction does not coincide with that under the involution f → M − f. The operation ε∗ = M − δ(M − f) turns out to still be an erosion, but ε∗ is different from the ε.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 12 / 27

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SLIDE 39

Spatially-variant morphology Dilation for numerical functions

Dilation for numerical functions: dilation and erosion

The dilate of f by δ is given by the supremum of the dilates of its pulses: δ(f) = ∨{CB(x),f(x), x ∈ E}. The eroded ε(f) is the supremum of those pulses whose dilated cylinders are smaller than f: ε(f) = ∨{ix,t | CB(x),t ≤ f, x ∈ E}. The duality under adjunction does not coincide with that under the involution f → M − f. The operation ε∗ = M − δ(M − f) turns out to still be an erosion, but ε∗ is different from the ε.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 12 / 27

slide-40
SLIDE 40

Spatially-variant morphology Dilation for numerical functions

Dilation for numerical functions: dilation and erosion

The dilate of f by δ is given by the supremum of the dilates of its pulses: δ(f) = ∨{CB(x),f(x), x ∈ E}. The eroded ε(f) is the supremum of those pulses whose dilated cylinders are smaller than f: ε(f) = ∨{ix,t | CB(x),t ≤ f, x ∈ E}. The duality under adjunction does not coincide with that under the involution f → M − f. The operation ε∗ = M − δ(M − f) turns out to still be an erosion, but ε∗ is different from the ε.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 12 / 27

slide-41
SLIDE 41

Spatially-variant morphology Dilation for numerical functions

Dilation for numerical functions: dilation and erosion

The dilate of f by δ is given by the supremum of the dilates of its pulses: δ(f) = ∨{CB(x),f(x), x ∈ E}. The eroded ε(f) is the supremum of those pulses whose dilated cylinders are smaller than f: ε(f) = ∨{ix,t | CB(x),t ≤ f, x ∈ E}. The duality under adjunction does not coincide with that under the involution f → M − f. The operation ε∗ = M − δ(M − f) turns out to still be an erosion, but ε∗ is different from the ε.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 12 / 27

slide-42
SLIDE 42

Spatially-variant morphology Dilation for numerical functions

Dilation for numerical functions: dilation and erosion

The dilate of f by δ is given by the supremum of the dilates of its pulses: δ(f) = ∨{CB(x),f(x), x ∈ E}. The eroded ε(f) is the supremum of those pulses whose dilated cylinders are smaller than f: ε(f) = ∨{ix,t | CB(x),t ≤ f, x ∈ E}. The duality under adjunction does not coincide with that under the involution f → M − f. The operation ε∗ = M − δ(M − f) turns out to still be an erosion, but ε∗ is different from the ε.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 12 / 27

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SLIDE 43

Spatially-variant morphology Dilation for numerical functions

Dilation for numerical functions: opening and closing

The two products γ = δε and ϕ = εδ are opening and closing on F(E, T ) Opening γ admits the following expression γ(f) = ∨{CB(x),t ≤ f, x ∈ E}. In the product space E × T the subgraph of the opening γ(f) is generated by the zone swept by all cylinders CB(x),t smaller than f. The closing ϕ = εδ does not coincide with M − γ(M − f), obtained by replacing f by M − f.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 13 / 27

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SLIDE 44

Spatially-variant morphology Dilation for numerical functions

Dilation for numerical functions: opening and closing

The two products γ = δε and ϕ = εδ are opening and closing on F(E, T ) Opening γ admits the following expression γ(f) = ∨{CB(x),t ≤ f, x ∈ E}. In the product space E × T the subgraph of the opening γ(f) is generated by the zone swept by all cylinders CB(x),t smaller than f. The closing ϕ = εδ does not coincide with M − γ(M − f), obtained by replacing f by M − f.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 13 / 27

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SLIDE 45

Spatially-variant morphology Dilation for numerical functions

Dilation for numerical functions: opening and closing

The two products γ = δε and ϕ = εδ are opening and closing on F(E, T ) Opening γ admits the following expression γ(f) = ∨{CB(x),t ≤ f, x ∈ E}. In the product space E × T the subgraph of the opening γ(f) is generated by the zone swept by all cylinders CB(x),t smaller than f. The closing ϕ = εδ does not coincide with M − γ(M − f), obtained by replacing f by M − f.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 13 / 27

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SLIDE 46

Spatially-variant morphology Dilation for numerical functions

Dilation for numerical functions: opening and closing

The two products γ = δε and ϕ = εδ are opening and closing on F(E, T ) Opening γ admits the following expression γ(f) = ∨{CB(x),t ≤ f, x ∈ E}. In the product space E × T the subgraph of the opening γ(f) is generated by the zone swept by all cylinders CB(x),t smaller than f. The closing ϕ = εδ does not coincide with M − γ(M − f), obtained by replacing f by M − f.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 13 / 27

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SLIDE 47

Directional field modelling

Outline

1

Introduction

2

Spatially-variant morphology Dilation/erosion and opening/closing Dilation for numerical functions

3

Directional field modelling Average squared gradient (ASG) Regularization of the ASG: ASGVF

4

Applications

5

Conclusions and perspectives

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 14 / 27

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SLIDE 48

Directional field modelling

Orientation of the data

(a) G (b) GVF (c) ASG (d) ASGVF

(a) Gradient, G (b) Gradient Vector Flow, GVF (Xu & Prince 1998) (c) Average Squared Gradient, ASG (Kass & Witkin 1987) (d) Average Squared Gradient Vector Flow, ASGVF

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 15 / 27

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SLIDE 49

Directional field modelling

Orientation of the data

(a) G (b) GVF (c) ASG (d) ASGVF

(a) Gradient, G (b) Gradient Vector Flow, GVF (Xu & Prince 1998) (c) Average Squared Gradient, ASG (Kass & Witkin 1987) (d) Average Squared Gradient Vector Flow, ASGVF

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 15 / 27

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SLIDE 50

Directional field modelling

Orientation of the data

(a) G (b) GVF (c) ASG (d) ASGVF

(a) Gradient, G (b) Gradient Vector Flow, GVF (Xu & Prince 1998) (c) Average Squared Gradient, ASG (Kass & Witkin 1987) (d) Average Squared Gradient Vector Flow, ASGVF

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 15 / 27

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SLIDE 51

Directional field modelling

Orientation of the data

(a) G (b) GVF (c) ASG (d) ASGVF

(a) Gradient, G (b) Gradient Vector Flow, GVF (Xu & Prince 1998) (c) Average Squared Gradient, ASG (Kass & Witkin 1987) (d) Average Squared Gradient Vector Flow, ASGVF

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 15 / 27

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SLIDE 52

Directional field modelling

Orientation of the data

(a) G (b) GVF (c) ASG (d) ASGVF

(a) Gradient, G (b) Gradient Vector Flow, GVF (Xu & Prince 1998) (c) Average Squared Gradient, ASG (Kass & Witkin 1987) (d) Average Squared Gradient Vector Flow, ASGVF

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 15 / 27

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SLIDE 53

Directional field modelling Average squared gradient (ASG)

ASG uses the following definition of the gradient g = g1(x, y) g2(x, y)

  • = sign
  • ∂f(x,y)

∂x

∂f(x,y)

∂x ∂f(x,y) ∂y

  • .

Then the gradient is squared and averaged in neighborhood W gs = gs,1(x, y) gs,2(x, y)

  • =

W

  • g2

1(x, y) − g2 2(x, y)

  • W (2 g1(x, y) g2(x, y))
  • .

The directional field ASG is d = [d1(x, y), d2(x, y)]⊤, where its angle is obtained as ∠d = Φ 2 − sign(Φ)π 2, being Φ = ∠gs; and the magnitude of d can be left as the magnitude of gs, or the squared root of gs or it can be set to unity.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 16 / 27

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SLIDE 54

Directional field modelling Average squared gradient (ASG)

ASG uses the following definition of the gradient g = g1(x, y) g2(x, y)

  • = sign
  • ∂f(x,y)

∂x

∂f(x,y)

∂x ∂f(x,y) ∂y

  • .

Then the gradient is squared and averaged in neighborhood W gs = gs,1(x, y) gs,2(x, y)

  • =

W

  • g2

1(x, y) − g2 2(x, y)

  • W (2 g1(x, y) g2(x, y))
  • .

The directional field ASG is d = [d1(x, y), d2(x, y)]⊤, where its angle is obtained as ∠d = Φ 2 − sign(Φ)π 2, being Φ = ∠gs; and the magnitude of d can be left as the magnitude of gs, or the squared root of gs or it can be set to unity.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 16 / 27

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SLIDE 55

Directional field modelling Average squared gradient (ASG)

ASG uses the following definition of the gradient g = g1(x, y) g2(x, y)

  • = sign
  • ∂f(x,y)

∂x

∂f(x,y)

∂x ∂f(x,y) ∂y

  • .

Then the gradient is squared and averaged in neighborhood W gs = gs,1(x, y) gs,2(x, y)

  • =

W

  • g2

1(x, y) − g2 2(x, y)

  • W (2 g1(x, y) g2(x, y))
  • .

The directional field ASG is d = [d1(x, y), d2(x, y)]⊤, where its angle is obtained as ∠d = Φ 2 − sign(Φ)π 2, being Φ = ∠gs; and the magnitude of d can be left as the magnitude of gs, or the squared root of gs or it can be set to unity.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 16 / 27

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SLIDE 56

Directional field modelling Average squared gradient (ASG)

ASG uses the following definition of the gradient g = g1(x, y) g2(x, y)

  • = sign
  • ∂f(x,y)

∂x

∂f(x,y)

∂x ∂f(x,y) ∂y

  • .

Then the gradient is squared and averaged in neighborhood W gs = gs,1(x, y) gs,2(x, y)

  • =

W

  • g2

1(x, y) − g2 2(x, y)

  • W (2 g1(x, y) g2(x, y))
  • .

The directional field ASG is d = [d1(x, y), d2(x, y)]⊤, where its angle is obtained as ∠d = Φ 2 − sign(Φ)π 2, being Φ = ∠gs; and the magnitude of d can be left as the magnitude of gs, or the squared root of gs or it can be set to unity.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 16 / 27

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SLIDE 57

Directional field modelling Regularization of the ASG: ASGVF

The Average Squared Gradient Vector Flow (ASGVF) is the vector field v = [v1(x, y), v2(x, y)]⊤ that minimizes the energy functional: E(v) = D(v) + αS(v). D represents a distance measure D(v) = 1 2

  • E

||d||2||v − d||2 dx dy. S determines the smoothness of the directional field S(v) = 1 2

  • E

∂v1 ∂x 2 + ∂v1 ∂y 2 + ∂v2 ∂x 2 + ∂v2 ∂y 2 dx dy. ASGVF can be found by solving the following Euler equations (v − d)|d|2 − α∇2v = 0.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 17 / 27

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SLIDE 58

Directional field modelling Regularization of the ASG: ASGVF

The Average Squared Gradient Vector Flow (ASGVF) is the vector field v = [v1(x, y), v2(x, y)]⊤ that minimizes the energy functional: E(v) = D(v) + αS(v). D represents a distance measure D(v) = 1 2

  • E

||d||2||v − d||2 dx dy. S determines the smoothness of the directional field S(v) = 1 2

  • E

∂v1 ∂x 2 + ∂v1 ∂y 2 + ∂v2 ∂x 2 + ∂v2 ∂y 2 dx dy. ASGVF can be found by solving the following Euler equations (v − d)|d|2 − α∇2v = 0.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 17 / 27

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SLIDE 59

Directional field modelling Regularization of the ASG: ASGVF

The Average Squared Gradient Vector Flow (ASGVF) is the vector field v = [v1(x, y), v2(x, y)]⊤ that minimizes the energy functional: E(v) = D(v) + αS(v). D represents a distance measure D(v) = 1 2

  • E

||d||2||v − d||2 dx dy. S determines the smoothness of the directional field S(v) = 1 2

  • E

∂v1 ∂x 2 + ∂v1 ∂y 2 + ∂v2 ∂x 2 + ∂v2 ∂y 2 dx dy. ASGVF can be found by solving the following Euler equations (v − d)|d|2 − α∇2v = 0.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 17 / 27

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SLIDE 60

Directional field modelling Regularization of the ASG: ASGVF

The Average Squared Gradient Vector Flow (ASGVF) is the vector field v = [v1(x, y), v2(x, y)]⊤ that minimizes the energy functional: E(v) = D(v) + αS(v). D represents a distance measure D(v) = 1 2

  • E

||d||2||v − d||2 dx dy. S determines the smoothness of the directional field S(v) = 1 2

  • E

∂v1 ∂x 2 + ∂v1 ∂y 2 + ∂v2 ∂x 2 + ∂v2 ∂y 2 dx dy. ASGVF can be found by solving the following Euler equations (v − d)|d|2 − α∇2v = 0.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 17 / 27

slide-61
SLIDE 61

Directional field modelling Regularization of the ASG: ASGVF

The Average Squared Gradient Vector Flow (ASGVF) is the vector field v = [v1(x, y), v2(x, y)]⊤ that minimizes the energy functional: E(v) = D(v) + αS(v). D represents a distance measure D(v) = 1 2

  • E

||d||2||v − d||2 dx dy. S determines the smoothness of the directional field S(v) = 1 2

  • E

∂v1 ∂x 2 + ∂v1 ∂y 2 + ∂v2 ∂x 2 + ∂v2 ∂y 2 dx dy. ASGVF can be found by solving the following Euler equations (v − d)|d|2 − α∇2v = 0.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 17 / 27

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SLIDE 62

Applications

Outline

1

Introduction

2

Spatially-variant morphology Dilation/erosion and opening/closing Dilation for numerical functions

3

Directional field modelling Average squared gradient (ASG) Regularization of the ASG: ASGVF

4

Applications

5

Conclusions and perspectives

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 18 / 27

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SLIDE 63

Applications Synthetic image #1, 256×256 pixels

Spatially-variant operators for filtering gray-level images Structuring element of fixed length λ with variable orientation B(x) ≡ Lθ(x)

λ (a) f1(x) (b) ASGV field (c) εLθ(x)

λ

(f1)(x) (d) δLθ(x)

λ

(f1)(x)

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 19 / 27

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SLIDE 64

Applications Synthetic image #2, 256×256 pixels

Spatially-variant operators for filtering gray-level images Structuring element of fixed length λ with variable orientation B(x) ≡ Lθ(x)

λ (a) f2(x) (b) ASGV field (c) γLθ(x)

λ

(f2)(x) (d) ϕLθ(x)

λ

(f2)(x)

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 20 / 27

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SLIDE 65

Applications Real image #1, 447×447 pixels

(a) Original image f(x) (b) ASGV field, η=0.001

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 21 / 27

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SLIDE 66

Applications Real image #1, 447×447 pixels

(a) Original image f(x) (b) γLθ(x)

21

(f)(x) (c) ϕLθ(x)

21

(f)(x) (d) γLlines

21 (f)(x)

(e) ϕLlines

21 (f)(x)

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 22 / 27

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SLIDE 67

Applications Real image #1, 447×447 pixels

(a) Original image f(x) (b) ϕLθ(x)

11

γLθ(x)

11

(c) ϕ

Lθ(x) 21

γ

Lθ(x) 21

ϕ

Lθ(x) 11

γ

Lθ(x) 11

(d) γLθ(x)

11

ϕLθ(x)

11

(e) γ

Lθ(x) 21

ϕ

Lθ(x) 21

γ

Lθ(x) 11

ϕ

Lθ(x) 11

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 23 / 27

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SLIDE 68

Applications Real image #2, 256×256 pixels

(a) Original image f(x) (b) fγ(x) ≡ γLθ(x)

21

(f)(x) (c) fϕ(x) ≡ ϕLθ(x)

21

(f)(x) (d) (fγ(x) + fϕ(x)) /2

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 24 / 27

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SLIDE 69

Conclusions and perspectives

Outline

1

Introduction

2

Spatially-variant morphology Dilation/erosion and opening/closing Dilation for numerical functions

3

Directional field modelling Average squared gradient (ASG) Regularization of the ASG: ASGVF

4

Applications

5

Conclusions and perspectives

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 25 / 27

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SLIDE 70

Conclusions and perspectives

Conclusions Definitions of spatially-variant MM operators (dilation/erosion and

  • pening/closing) for gray-level images.

Structuring element can locally adapt its orientation. Orientation field obtained by regularizing the ASG field, which extends the orientation information to homogeneous areas. Image information → shape of structuring element → spatially variant filtering Results prove the validity of this novel approach for filtering anisotropic features in images Future works Spatially-variant geodesic operators. Full exploitation of the directional field (magnitude, angular coherence, ...) → more general anisotropic structuring elements. Extension to n-D spaces and application to 3-D images.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 26 / 27

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SLIDE 71

Conclusions and perspectives

Conclusions Definitions of spatially-variant MM operators (dilation/erosion and

  • pening/closing) for gray-level images.

Structuring element can locally adapt its orientation. Orientation field obtained by regularizing the ASG field, which extends the orientation information to homogeneous areas. Image information → shape of structuring element → spatially variant filtering Results prove the validity of this novel approach for filtering anisotropic features in images Future works Spatially-variant geodesic operators. Full exploitation of the directional field (magnitude, angular coherence, ...) → more general anisotropic structuring elements. Extension to n-D spaces and application to 3-D images.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 26 / 27

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SLIDE 72

Conclusions and perspectives

Conclusions Definitions of spatially-variant MM operators (dilation/erosion and

  • pening/closing) for gray-level images.

Structuring element can locally adapt its orientation. Orientation field obtained by regularizing the ASG field, which extends the orientation information to homogeneous areas. Image information → shape of structuring element → spatially variant filtering Results prove the validity of this novel approach for filtering anisotropic features in images Future works Spatially-variant geodesic operators. Full exploitation of the directional field (magnitude, angular coherence, ...) → more general anisotropic structuring elements. Extension to n-D spaces and application to 3-D images.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 26 / 27

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SLIDE 73

Conclusions and perspectives

Conclusions Definitions of spatially-variant MM operators (dilation/erosion and

  • pening/closing) for gray-level images.

Structuring element can locally adapt its orientation. Orientation field obtained by regularizing the ASG field, which extends the orientation information to homogeneous areas. Image information → shape of structuring element → spatially variant filtering Results prove the validity of this novel approach for filtering anisotropic features in images Future works Spatially-variant geodesic operators. Full exploitation of the directional field (magnitude, angular coherence, ...) → more general anisotropic structuring elements. Extension to n-D spaces and application to 3-D images.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 26 / 27

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SLIDE 74

Conclusions and perspectives

Conclusions Definitions of spatially-variant MM operators (dilation/erosion and

  • pening/closing) for gray-level images.

Structuring element can locally adapt its orientation. Orientation field obtained by regularizing the ASG field, which extends the orientation information to homogeneous areas. Image information → shape of structuring element → spatially variant filtering Results prove the validity of this novel approach for filtering anisotropic features in images Future works Spatially-variant geodesic operators. Full exploitation of the directional field (magnitude, angular coherence, ...) → more general anisotropic structuring elements. Extension to n-D spaces and application to 3-D images.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 26 / 27

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SLIDE 75

Conclusions and perspectives

Conclusions Definitions of spatially-variant MM operators (dilation/erosion and

  • pening/closing) for gray-level images.

Structuring element can locally adapt its orientation. Orientation field obtained by regularizing the ASG field, which extends the orientation information to homogeneous areas. Image information → shape of structuring element → spatially variant filtering Results prove the validity of this novel approach for filtering anisotropic features in images Future works Spatially-variant geodesic operators. Full exploitation of the directional field (magnitude, angular coherence, ...) → more general anisotropic structuring elements. Extension to n-D spaces and application to 3-D images.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 26 / 27

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SLIDE 76

Conclusions and perspectives

Conclusions Definitions of spatially-variant MM operators (dilation/erosion and

  • pening/closing) for gray-level images.

Structuring element can locally adapt its orientation. Orientation field obtained by regularizing the ASG field, which extends the orientation information to homogeneous areas. Image information → shape of structuring element → spatially variant filtering Results prove the validity of this novel approach for filtering anisotropic features in images Future works Spatially-variant geodesic operators. Full exploitation of the directional field (magnitude, angular coherence, ...) → more general anisotropic structuring elements. Extension to n-D spaces and application to 3-D images.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 26 / 27

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SLIDE 77

Conclusions and perspectives

Conclusions Definitions of spatially-variant MM operators (dilation/erosion and

  • pening/closing) for gray-level images.

Structuring element can locally adapt its orientation. Orientation field obtained by regularizing the ASG field, which extends the orientation information to homogeneous areas. Image information → shape of structuring element → spatially variant filtering Results prove the validity of this novel approach for filtering anisotropic features in images Future works Spatially-variant geodesic operators. Full exploitation of the directional field (magnitude, angular coherence, ...) → more general anisotropic structuring elements. Extension to n-D spaces and application to 3-D images.

Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 26 / 27

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SLIDE 78

Spatially-variant anisotropic morphological filters driven by gradient fields

Rafael Verdú-Monedero1 Jesús Angulo 2 Jean Serra 3

1Department of Information Technologies and Communications,

Technical University of Cartagena, 30202, Cartagena, Spain, rafael.verdu@upct.es

2Centre de Morphologie Mathématique (CMM),

Ecole des Mines de Paris, Fontainebleau Cedex, France jesus.angulo@ensmp.fr

3Laboratoire A2SI - ESIEE, B.P

. 99, 93162 Noisy-le-Grand, France serraj@esiee.fr ISMM 2009, 9th International Symposium on Mathematical Morphology