Introduction to grayscale image processing by mathematical morphology

Introduction to grayscale image processing by mathematical morphology

Jean Cousty MorphoGraph and Imagery

2011

J. Cousty : Morpho, graphes et imagerie 3D 1/15

Outline of the lecture

1 Grayscale images

2 Operators on grayscale images

Grayscale images

Images

Definition

Let V be a set of values

An image (on E with values in V ) is a map I from E into V I (x) is called the value of the point (pixel) x for I

Example

Images with values in R ⁺ : euclidean distance map D X to a set X ∈ P(E)

Images with values in Z ⁺ : distance map D _X for a geodesic distance in a uniform network

J. Cousty : Morpho, graphes et imagerie 3D 3/15

Images

Definition

Let V be a set of values

An image (on E with values in V ) is a map I from E into V I (x) is called the value of the point (pixel) x for I

Example

Images with values in R ⁺ : euclidean distance map D X to a set X ∈ P(E)

Images with values in Z ⁺ : distance map D _X for a geodesic

distance in a uniform network

Grayscale images

Grayscale images

We denote by I the set of all images with integers values on E An image in I is also called grayscale (or graylevel) image

We denote by I an arbitrary image in I

The value I (x) of a point x ∈ E is also called the gray level of x, or the gray intensity at x

J. Cousty : Morpho, graphes et imagerie 3D 4/15

Grayscale images

We denote by I the set of all images with integers values on E An image in I is also called grayscale (or graylevel) image We denote by I an arbitrary image in I

The value I (x) of a point x ∈ E is also called the gray level of x,

or the gray intensity at x

Grayscale images

Topographical interpretation

An grayscale image I can be seen as a topographical relief I (x) is called the altitude of x

Bright regions: mountains, crests, hills Dark regions: bassins, valleys

J. Cousty : Morpho, graphes et imagerie 3D 5/15

Topographical interpretation

An grayscale image I can be seen as a topographical relief I (x) is called the altitude of x

Bright regions: mountains, crests, hills

Dark regions: bassins, valleys

Grayscale images

Level set

Definition

Let k ∈ Z

The k-level set (or k-section, or k-threshold) of I, denoted by I _k , is the subset of E defined by:

I

= {x ∈ E | I (x ) ≥ k }

I I 80 I 150 I 220

J. Cousty : Morpho, graphes et imagerie 3D 6/15

Level set

Definition

Let k ∈ Z

The k-level set (or k-section, or k-threshold) of I, denoted by I _k , is the subset of E defined by:

I

= {x ∈ E | I (x ) ≥ k }

I I 80 I 150 I 220

Grayscale images

Reconstruction

Property

∀k, k ⁰ ∈ Z , k ⁰ > k = ⇒ I _k

⊆ I _k I (x) = max{k ∈ Z | x ∈ I _k }

J. Cousty : Morpho, graphes et imagerie 3D 7/15

Operators on grayscale images

grayscale operators

An operator (on I) is a map from I into I

Let γ be an increasing operator on E

The stack operator induced by γ is the operator on I, also denoted by γ, defined by:

∀I ∈ I, ∀k ∈ Z , [γ(I )]

= γ(I

)

Exercice. Show that a same construction cannot be used to derive an

operator on I from an operator on E that is not increasing.

Operators on grayscale images

grayscale operators

An operator (on I) is a map from I into I

Definition (flat operators)

Let γ be an increasing operator on E

The stack operator induced by γ is the operator on I, also denoted by γ, defined by:

∀I ∈ I, ∀k ∈ Z , [γ(I )]

= γ(I

)

Exercice. Show that a same construction cannot be used to derive an operator on I from an operator on E that is not increasing.

J. Cousty : Morpho, graphes et imagerie 3D 8/15

grayscale operators

An operator (on I) is a map from I into I

Definition (flat operators)

Let γ be an increasing operator on E

The stack operator induced by γ is the operator on I, also denoted by γ, defined by:

∀I ∈ I, ∀k ∈ Z , [γ(I )]

= γ(I

)

Exercice. Show that a same construction cannot be used to derive an

operator on I from an operator on E that is not increasing.

Operators on grayscale images

Characterisation of grayscale operators

Property

Let γ be an increasing operator on E [γ (I )](x) = max{k ∈ Z | x ∈ γ(I _k )}

Remark. Untill now, all operators seen in the MorphoGraph and Imagery course are increasing

J. Cousty : Morpho, graphes et imagerie 3D 9/15

Characterisation of grayscale operators

Property

Let γ be an increasing operator on E [γ (I )](x) = max{k ∈ Z | x ∈ γ(I _k )}

Remark. Untill now, all operators seen in the MorphoGraph and

Imagery course are increasing

Operators on grayscale images

Illustration: dilation on I by Γ

I I 80 I 150 I 220

δ _Γ (I) δ _Γ (I ) ₈₀ δ _Γ (I) ₁₅₀ δ _Γ (I ) ₂₂₀

J. Cousty : Morpho, graphes et imagerie 3D 10/15

Illustration: erosion on I by Γ

I I 80 I 150 I 220

_Γ (I ) _Γ (I ) _{80 Γ} (I ) _{150 Γ} (I ) ₂₂₀

Operators on grayscale images

Illustration: opening on I by Γ

I I 80 I 150 I 220

γ _Γ (I ) γ _Γ (I ) ₈₀ γ _Γ (I ) ₁₅₀ γ _Γ (I ) ₂₂₀

J. Cousty : Morpho, graphes et imagerie 3D 12/15

Illustration: closing on I by Γ

I I 80 I 150 I 220

φ _Γ (I ) φ _Γ (I) ₈₀ φ _Γ (I ) ₁₅₀ φ _Γ (I) ₂₂₀

Operators on grayscale images

Dilation/Erosion by a structuring element: characterisation

Property (duality)

Let Γ be a structuring element Γ (I ) = −δ _Γ

(−I )

Property

Let Γ be a structuring element [δ Γ (I)](x) = max{I(y) | y ∈ Γ ⁻¹ (x )} [ _Γ (I )](x) = min{I (y) | y ∈ Γ(x)}

J. Cousty : Morpho, graphes et imagerie 3D 14/15

Dilation/Erosion by a structuring element: characterisation

Property (duality)

Let Γ be a structuring element Γ (I ) = −δ _Γ

(−I )

Property

Let Γ be a structuring element [δ Γ (I)](x) = max{I(y) | y ∈ Γ ⁻¹ (x )}

[ Γ (I )](x) = min{I (y) | y ∈ Γ(x)}

Operators on grayscale images

Exercise

Write an algorithm whose data are a graph (E , Γ) and a grayscale image I on E and whose result is the image I ⁰ = δ _Γ (I ⁰ )

J. Cousty : Morpho, graphes et imagerie 3D 15/15