Hypercomplex mathematical morphologyangulo/publicat/... · 2009-12-13 · 1/56. Hypercomplex...

Hypercomplex mathematical morphology


Jesús [email protected] ; http://cmm.ensmp.fr/∼angulo

CMM-Centre de Morphologie Mathématique,Mathématiques et Systèmes, MINES Paristech

Mathematics and Image Analysis 2009 - MIA'09December 14-16, 2009 - Paris, France

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Mathematical Morphology

Nonlinear image processing methodology based on the application oflattice theory to dene spatial operators

Main ingredient: an appropriate partial order ≤ for the set of imagevalues; e.g., natural ordering for grey-level images

Two basic operators

Denition (Dilation and Erosion)

δB(f )(x) = f (y) : f (y) =∨

[f (z)], z ∈ Bx,

εB(f )(x) = f (y) : f (y) =∧

[f (z)], z ∈ Bx.

The structuring element B ⊂ E introduces the shape/size of thespatial eect

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Morphological operators

Image ltering using centre of size 3

f (x) ζ(f ) = [f ∨ (γϕγ(f ) ∧ ϕγϕ(f ))]∧ f − ζ(f )

(γϕγ(f ) ∨ ϕγϕ(f ))

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Image ltering using centre of size 3

f (x) ζ(f ) = [f ∨ (γϕγ(f ) ∧ ϕγϕ(f ))]∧ f − ζ(f )

(γϕγ(f ) ∨ ϕγϕ(f ))

bf (x), σ = 354 / 56



Image simplication using opening by reconstruction

f (x) m(x)

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f (x) m(x) γrec (f ,m) = δigeo(f ,m) ∧m (*)

(*) δigeo(f ,m) = δ1geo(f , δi−1geo (f ,m)); δ1geo(f ,m) = δB (m) ∧m

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f (x) m(x) γrec (f ,m) = δigeo(f ,m) ∧m (*)

(*) δigeo(f ,m) = δ1geo(f , δi−1geo (f ,m)); δ1geo(f ,m) = δB (m) ∧m

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Regularized mathematical morphology

Our fundamental objective: To introduce regularized dilations anderosion

State-of-the art,

n−rank pseudo-morphologySoft morphology and fuzzy logic morphologyStatistical physics-based morphologySelf-dual morphologyViscous and micro-viscous morphology

Our framework: Formulation using geometric algebra representations

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Plan

1 Introduction

2 Complex representation and associated total orderings

3 Complex dilations and erosions

4 Generalisation to multi-operator cases using real quaternions

5 Comparative examples

6 Conclusions and perspectives

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Introduction

1 Introduction






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Introduction

Introduction

Context Classical complex-valued signal processing model

The analytic signal for one-dimensional signals:

f (t) 7→ fA(t) = f (t) + ifHi (t)

where fHi (t) is the Hilbert transformation, i.e., fHi (t) = f (t) ∗ 1πt .

In the frequency domain: FA(u) = F (u) + iFHi (u) =F (u) (1 + sign(u)) (the Hilbert transform is a phase shifting thesignal by −π/2).The polar representation yields the local amplitude and the localphase.

Extensions to 2D signals:Quaternionic Fourier transform by (Bulow and Sommer, 2001);Monogenic signal (Riesz transform) by (Felsberg and Sommer, 2001);2D scalar-valued images are embedded into the geometric algebra ofthe Euclidean 4D space and then the image structures aredecomposed using monogenic curvature tensor (Zang and Sommer,2007).

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Introduction

Introduction



f (t) 7→ fA(t) = f (t) + ifHi (t)


In the frequency domain: FA(u) = F (u) + iFHi (u) =F (u) (1 + sign(u)) (the Hilbert transform is a phase shifting thesignal by −π/2).

The polar representation yields the local amplitude and the localphase.


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Introduction

Introduction



f (t) 7→ fA(t) = f (t) + ifHi (t)


In the frequency domain: FA(u) = F (u) + iFHi (u) =F (u) (1 + sign(u)) (the Hilbert transform is a phase shifting thesignal by −π/2).The polar representation yields the local amplitude and the localphase.


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Introduction

Introduction

ApproachWhat about a morphological complex representation?

f (x) 7→ fC (x) = f (x) + iψB(f )(x)

Aim To construct (hyper)-complex image representations which willbe endowed with total orderings and consequently, which will lead tocomplete lattices.

Rationale To obtain regularized morphological operators whoseresult depends not only on the sup/inf of the grey values, locallycomputed in the structuring element, but also on dierentialinformation or more regional information.

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Complex representation and associated total orderings

1 Introduction






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Morphological complex image

ψ-complex image

Mapping from the scalar image to the complex image

f (x) 7→ fC (x) = f (x) + iψB(f )(x),

where ψB(f )(x) is the transformation ψ applied to f (x) ∈ F(E , T )according to the shape and size associated to the structuringelement B,

with fC ∈ F(E , T × iT ).

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Morphological complex image

The data of the bivalued image are discrete complex numbers:fC (x) = cn = an + ibn, where an and bn are respectively the real andthe imaginary part of the complex of index n in the nite spaceT × iT ⊂ C.

Let us consider the polar representation, i.e., cn = ρn exp (iθn),where the modulus is given by

ρn = |cn| =√

a2n + b2n

and the phase is computed as

θn = arg (cn) = atan2 (bn, an) = sign(bn) atan (|bn|/an) ,

with atan2 (·) ∈ (−π, π]. The phase can be mapped to [0, 2π) byadding 2π to negative values.

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Total orderings for complex images

Working in the polar representation, two alternative total orderings basedon lexicographic cascades can be dened for complex numbers:

Ordering ≤Ω

θ01

cn ≤Ωθ01

cm ⇔ρn < ρm or

ρn = ρm and θn θ0 θm

and

Ordering ≤Ω

θ02

cn ≤Ωθ02

cm ⇔θn ≺θ0 θm or

θn =θ0 θm and ρn ≤ ρm

where θ0 depends on the angular dierence to a reference angle θ0 onthe unit circle.

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θ0-reference based phase ordering θ0

θn θ0 θm ⇔

(θn ÷ θ0) > (θm ÷ θ0) or

(θn ÷ θ0) = (θm ÷ θ0) and θn ≤ θm

such that

θp ÷ θq =

| θp − θq | if | θp − θq |≤ π

2π− | θp − θq | if | θp − θq |> π

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Given now a set Z ⊂ E of pixels of the initial image [f (z)]z∈Z , thebasic idea behind our approach is to use the morphological compleximage to calculate the indirectly the supremum and the inmum

Formally, we have

Indirect total ordering 4Ω

θ01

fC (y) ≤Ω

θ01

fC (z) =⇒ f (y) 4Ω

θ01

f (z),

where 4Ω

θ01

allows to compute the supremum∨

Ωθ01

and the inmum∧Ω

θ01

in the original scalar-valued image, i.e.,

fC (y) =∨

Ωθ01 ,z∈Z

[fC (z)] =⇒ f (y) =∨

Ωθ01 ,z∈Z

[f (z)] .

Alternatively, f (y) = Re fC (y)

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Given now a set Z ⊂ E of pixels of the initial image [f (z)]z∈Z , thebasic idea behind our approach is to use the morphological compleximage to calculate the indirectly the supremum and the inmum

Formally, we have

Indirect total ordering 4Ω

θ01

fC (y) ≤Ω

θ01

fC (z) =⇒ f (y) 4Ω

θ01

f (z),

where 4Ω

θ01

allows to compute the supremum∨

Ωθ01

and the inmum∧Ω

θ01

in the original scalar-valued image, i.e.,

fC (y) =∨

Ωθ01 ,z∈Z

[fC (z)] =⇒ f (y) =∨

Ωθ01 ,z∈Z

[f (z)] .

Alternatively, f (y) = Re fC (y)18 / 56


Complex dilations and erosions

1 Introduction






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γ-complex image

Opening: γB(f ) = δB (εB(f ))

γ-complex image: fC (x) = f (x) + iγBC(f )(x)

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γ-complex image

Opening: γB(f ) = δB (εB(f ))

γ-complex image: fC (x) = f (x) + iγBC(f )(x)

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ϕ-complex image

Closing: ϕB(f ) = εB (δB(f ))

ϕ-complex image: fC (x) = f (x) + iϕBC(f )(x)

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γ-complex dilations and ϕ-complex erosions

(Ω

π/21 , γ

)-complex dilation8<:fC (x) = f (x) + iγBC (f )(x),

δ〈1,γBC,B〉(f )(x) = f (y) : fC (y) =

WΩ

π/21

[fC (z)], z ∈ B(x)).

and(Ω

π/21 , ϕ

)-complex erosion8<:fC (x) = f (x) + iϕBC (f )(x),

ε〈1,ϕBC,B〉(f )(x) = f (y) : fC (y) =

VΩ

π/21

[fC (z)], z ∈ B(x)).

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γ-complex dilations and ϕ-complex erosions

The equivalent(Ω

π/22 , γ

)-complex dilation and

(Ω

π/22 , ϕ

)-complex

erosion are respectively

δ〈2,γBC ,B〉(f )(x) = f (y) : fC (y) =∨

Ωπ/22

[fC (z)], z ∈ B(x)),

and

ε〈2,ϕBC,B〉(f )(x) = f (y) : fC (y) =

∧Ω

π/22

[fC (z)], z ∈ B(x)),

where the complex part of the image fC (x) is again an opening for thedilation and a closing for the erosion.

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Properties

Proposition

The pair(δ〈1,γBC ,B〉, ε〈1,ϕBC

,B〉

)is an adjunction, i.e.,

δ〈1,γBC ,B〉(f )(x) 4Ω

π/21

g(x) ⇔ f (x) 4Ω

π/21

ε〈1,ϕBC,B〉(g)(x),

∀f , g ∈ F(E , T ). Similarly, the pair(δ〈2,γBC ,B〉, ε〈2,ϕBC

,B〉

)is also an

adjunction.

Proposition

The two pairs of γ−complex dilation and ϕ−complex erosion are dualoperators, i.e.,

δ〈1,γBC ,B〉(f ) =[ε〈1,ϕBC

,B〉(fc)

]c; δ〈2,γBC ,B〉(f ) =

[ε〈2,ϕBC

,B〉(fc)

]c25 / 56



Interpretation

Complex plane for points ∈ f (x) + iψB(f )(x).

By the anti-extensivity of opening, we have f (x) ≥ γB(f )(x) andhence 0 ≤ θ ≤ π/4.

By the extensivity of closing: ϕB(f )(x)/f (x) ≥ 1 ⇒ π/4 ≤ θ ≤ π/2.

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Example: γ, ϕ-complex operators

f δB (f ) εB (f )

δ〈1,γBC,B〉(f ) δ〈2,γBC

,B〉(f ) ε〈1,ϕBC,B〉(f ) ε〈2,ϕBC

,B〉(f )

B = 5, BC = 3

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Example: γ, ϕ-complex operators

f ϕB (f ) γB (f )

ϕ〈1,(ϕBC,γBC

),B〉(f ) ϕ〈2,(ϕBC,γBC

),B〉(f ) γ〈1,(γBC,ϕBC

),B〉(f ) γ〈2,(γBC,ϕBC

),B〉(f )

B = 5, BC = 3

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τ+-complex and τ−-complex images

White top-hat: τ+B (f ) = f − γB(f )

Black top-hat: τ−B (f ) = ϕB(f )− f

τ+-complex image: fC (x) = f (x) + iτ+BC

(f )(x)

τ−-complex image: fC (x) = f (x)− i [τ−BC(f )(x)]c

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τ+-complex dilations and τ−-complex erosions

(Ω

π/41 , τ+

)-complex dilation8><>:

fC (x) = f (x) + iτ+BC

(f )(x),

δ〈1,τ+BC

,B〉(f )(x) = f (y) : fC (y) =W

Ωπ/41

[fC (z)], z ∈ B(x)).

and(Ω−π/41 ,−[τ−]c

)-complex erosion8><>:

fC (x) = f (x)− i [τ−BC

(f )(x)]c ,

ε〈1,−[τ−BC

]c ,B〉(f )(x) = f (y) : fC (y) =V

Ω−π/41

[fC (z)], z ∈ B(x)).

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τ+-complex dilations and τ−-complex erosions

(Ω

π/42 , τ+

)-complex dilation8><>:

fC (x) = f (x) + iτ+BC

(f )(x),

δ〈2,τ+BC

,B〉(f )(x) = f (y) : fC (y) =W

Ωπ/42

[fC (z)], z ∈ B(x)).

and(Ω−π/42 ,−[τ−]c

)-complex erosion8><>:

fC (x) = f (x)− i [τ−BC

(f )(x)]c ,

ε〈2,−[τ−BC

]c ,B〉(f )(x) = f (y) : fC (y) =V

Ω−π/42

[fC (z)], z ∈ B(x)).

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Example: τ+, τ−-complex representation

f (x) τ+B (f )(x) τ−

B(f )(x)

ρ(x), f (x) + iτ+B (f )(x) θ(x), f (x) + iτ+

B (f )(x) θ(x)÷ π/4

ρ(x), f (x)− i [τ−B

(f )(x)]c θ(x), f (x)− i [τ−B

(f )(x)]c θ(x)÷−π/4

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Generalisation to multi-operator cases using real quaternions

1 Introduction






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Natural hypercomplex generalisation: Quaternions

We generalise the previous ideas by the extension to imagerepresentations based on hypercomplex numbers or real quaternions.

Before that, we remind the foundations of quaternions...

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Reminder on quaternions

Quaternion

A quaternion q ∈ H may be represented in hypercomplex form as

q = a + bi + cj + dk ,

where a, b, c and d are real.

A quaternion has a real part or scalar part, S(q) = a, and animaginary part or vector part, V (q) = bi + cj + dk :

q = S(q) + V (q)

A quaternion with a zero scalar part is called a pure quaternion.

Addition of quaternions

The addition of two quaternions, q,q′ ∈ H, is dened as follows q+ q′ =(a + a′) + (b + b′)i + (c + c ′)j + (d + d ′)j . The addition is commutativeand associative.

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Reminder on quaternions

Product of two quaternions

The product of two quaternions, q,q′ ∈ H, is dened as follows

q′′ = qq′ = (aa′ − bb′ − cc ′ − dd ′) + (ab′ + ba′ + cd ′ − dc ′)i+(ac ′ + ca′ + db′ − bd ′)j + (ad ′ + a′d + bc ′ − cb′)k

In terms of dot and cross product of vectors:q′′ = qq′ = S(q′′) + V (q′′), with

S(q′′) = S(q)S(q′)− V (q) · V (q′),

andV (q′′) = S(q)V (q′) + S(q′)V (q) + V (q)× V (q′),

where · and × represent the dot product vector and the crossproduct vector respectively.

The multiplication of quaternions is not commutative, i.e.,qq′ 6= q′q; but it is associative.

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Quaternion polar form

Any quaternion may be represented in polar form as

q = ρeξθ

with

ρ =√a2 + b2 + c2 + d2 is the modulus of q;

ξ = bi+cj+dk√b2+c2+d2

is the pure unitary quaternion associated to q (by

the normalisation, the pure unitary quaternion discards intensityinformation, but retains orientation information), sometimes calledeigenaxis;

θ = arctan(√

b2+c2+d2

a

)is the angle, sometimes called eigenangle,

between the real part and the 3D imaginary part.

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Quaternion parallel/perpendicular decomposition

A full quaternion q may be decomposed about a pure unit quaternion pu:

q = q⊥ + q‖,

the parallel part of q according to pu, also called the projection part, isgiven by

q‖ = S(q) + V‖(q),

and the perpendicular part, also named the rejection part, is obtained as

q⊥ = V⊥(q),

where

V‖(q) =1

2(V (q)− puV (q)pu)

and

V⊥(q) =1

2(V (q) + puV (q)pu).

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Total orderings for quaternion images

Working in the polar representation, three alternative total orderings forhypercomplex numbers:

Ordering ≤Ωq01

qn ≤Ωq01

qm ⇔

8<:ρn < ρm or

ρn = ρm and θn ≺θ0 θm or

ρn = ρm and θn =θ0 θm and ‖ξn − ξ0‖ ≥ ‖ξm − ξ0‖

Ordering ≤Ωq02

qn ≤Ωq02

qm ⇔

8<:θn ≺θ0 θm or

θn =θ0 θm and ρn < ρm or

θn =θ0 θm and ρn = ρm and ‖ξn − ξ0‖ ≥ ‖ξm − ξ0‖

Ordering ≤Ωq03

qn ≤Ωq03

qm ⇔

8<:‖ξn − ξ0‖ > ‖ξm − ξ0‖ or

‖ξn − ξ0‖ = ‖ξm − ξ0‖ and ρn < ρm or

‖ξn − ξ0‖ = ‖ξm − ξ0‖ and ρn = ρm and θn θ0 θm

where q0 is the quaternion of reference.

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Total orderings for quaternion images

We can also introduce another total ordering based on the ‖ / ⊥decomposition along reference quaternion q0 as follows,

Ordering ≤Ωq04

qn ≤Ωq04

qm ⇔

8>>>>>><>>>>>>:

|q‖ n| < |q‖ m| or|q‖ n| = |q‖ m| and |q⊥ n| > |q⊥ m| or|q‖ n| = |q‖ m| and |q⊥ n| = |q⊥ m| and8<:

bn < bm or

bn = bm and cn < cm or

bn = bm and cn = cm and dn ≤ dm

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Morphological hypercomplex image

~Ψ-hypercomplex image

Given the four-variate transformation

~Ψ =(ψ0B0, ψI

BI, ψJ

BJ, ψK

BK

)applied to the scalar image f (x) ∈ F(E , T ), the ~Ψ-hypercomplex imageis dened by the mapping

f (x) 7→ fH(x) = ψ0B0

(f )(x) + iψIBI

(f )(x) + jψJBJ

(f )(x) + kψKBK

(f )(x).

where the four transformations are scalar operators, i.e., ψB : T → T .

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Hypercomplex dilations and hypercomplex erosions

(~Ψ+,Ωq0

+

)-hypercomplex dilation

A generic pair of(~Ψ+,Ωq0

+

)-hypercomplex dilation is given by the

expression8><>:~Ψ+ = (ψ0+

B0, ψI+

BI, ψJ+

BJ, ψK+

BK) : f (x) 7→ fH(x),

δ〈Ψ+,Ωq0+ ,B〉(f )(x) = f (y) : fH(y) =

WΩq0+

[fH(z)], z ∈ B(x))

(~Ψ−,Ωq0

−

)-hypercomplex erosion

A generic pair of(~Ψ−,Ωq0

−

)-hypercomplex erosion is given by the

expression8><>:~Ψ− = (ψ0−

B0, ψI−

BI, ψJ−

BJ, ψK−

BK) : f (x) 7→ fH(x),

ε〈Ψ−,Ωq0+ ,B〉(f )(x) = f (y) : fH(y) =

VΩq0−

[fH(z)], z ∈ B(x))42 / 56




Two kinds of degrees of freedom must be set up to have totallystated the operator:

the hypercomplex transformation, which includes the set ofstructuring elements B0,BI ,BJ ,BK and,the quaternionic ordering, which includes the choice of the referencequaternion.

The hypercomplex transformation allows to introduce directionaleects according to the main grid directions in 3D images:

fH(x) = ψ0B0

(f )(x) + iψILx

(f )(x) + jψJLy

(f )(x) + kψKLz

(f )(x).

where B0 is an isotropic (disk) structuring element of size s (whichcan be s = 0 so the transformation is the identity) and where Lx isan linear structuring element of size x according the direction x .For the transformations ψB , the openings/closings and the top-hatsare useful, but also the internal/external gradients, which favor thedilation/erosion of points close to the object contours.Any of the quaternionic orderings can be used; but the ordering Ωq0

4

is particularly interesting.

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fH(x) = ψ0B0

(f )(x) + iψILx

(f )(x) + jψJLy

(f )(x) + kψKLz

(f )(x).

where B0 is an isotropic (disk) structuring element of size s (whichcan be s = 0 so the transformation is the identity) and where Lx isan linear structuring element of size x according the direction x .

For the transformations ψB , the openings/closings and the top-hatsare useful, but also the internal/external gradients, which favor thedilation/erosion of points close to the object contours.Any of the quaternionic orderings can be used; but the ordering Ωq0

4


43 / 56







fH(x) = ψ0B0

(f )(x) + iψILx

(f )(x) + jψJLy

(f )(x) + kψKLz

(f )(x).

where B0 is an isotropic (disk) structuring element of size s (whichcan be s = 0 so the transformation is the identity) and where Lx isan linear structuring element of size x according the direction x .For the transformations ψB , the openings/closings and the top-hatsare useful, but also the internal/external gradients, which favor thedilation/erosion of points close to the object contours.

Any of the quaternionic orderings can be used; but the ordering Ωq04


43 / 56







fH(x) = ψ0B0

(f )(x) + iψILx

(f )(x) + jψJLy

(f )(x) + kψKLz

(f )(x).

where B0 is an isotropic (disk) structuring element of size s (whichcan be s = 0 so the transformation is the identity) and where Lx isan linear structuring element of size x according the direction x .For the transformations ψB , the openings/closings and the top-hatsare useful, but also the internal/external gradients, which favor thedilation/erosion of points close to the object contours.Any of the quaternionic orderings can be used; but the ordering Ωq0

4

is particularly interesting.43 / 56


Comparative examples

1 Introduction






44 / 56



Example: Center ζ(f ) = [f ∨ (γϕγ(f ) ∧ ϕγϕ(f ))] ∧ (γϕγ(f ) ∨ ϕγϕ(f ))

f

(a) Standard (b)

(c) (d)

(b)“Ω

π/21 , γBC

”-complex dilation,

“Ω

π/21 , ϕBC

”-complex erosion, BC = D5

(c) ~Ψ+/− =“

γD10/ϕD10 , γLx10/ϕLx10

, γLy10

/ϕLy10

, 0”, Ω

q04 , q0 = i + j

(d) ~Ψ+/− =“


, γLy10

/ϕLy10

, 0”, Ω

q04 , q0 = i

45 / 56



Example: Center ζ(f ) = [f ∨ (γϕγ(f ) ∧ ϕγϕ(f ))] ∧ (γϕγ(f ) ∨ ϕγϕ(f ))

bf , σ = 35

(a) Standard (b)

(c) (d)

(b)“Ω

π/21 , γBC


“Ω

π/21 , ϕBC


(c) ~Ψ+/− =“


, γLy10

/ϕLy10

, 0”, Ω

q04 , q0 = i + j

(d) ~Ψ+/− =“


, γLy10

/ϕLy10

, 0”, Ω

q04 , q0 = i

46 / 56



Example: Top-hat τ+B

f

(a) Standard (b)

(c) (d)

(b)“Ω

π/21 , γBC


“Ω

π/21 , ϕBC


(c)“Ω

π/22 , γBC


“Ω

π/22 , ϕBC


(d) ~Ψ+/− =“Id/Id, %Lx10

/− %Lx10, %

Ly10

/− %Ly10

, 0”, Ω

q04 , q0 = i + j

47 / 56



Example: Contrast operator using erosion and dilation

f

(a) Standard (b)

(c) (d)

(b)“Ω

π/42 , τ+

BC


“Ω−π/42 ,−[τ−

BC]c


(c) ~Ψ+/− =

„τ+D3

/− [τ−D3

]c , τ+Lx3

/− [τ−Lx3

]c , τ+

Ly3/− [τ−

Ly3]c , 0

«, Ω

q04 , q0 = i + j

(d) ~Ψ+/− =“Id/Id, %Lx10

/− %Lx10, %

Ly10

/− %Ly10

, 0”, Ω

q04 , q0 = i + j

48 / 56



Example: Gradient %B(f ) = δB(f )− εB(f )

f

(a) Standard (b)

(c) (d)

(b)“Ω

π/42 , τ+

BC


“Ω−π/42 ,−[τ−

BC]c


(c) ~Ψ+/− =`γB0/ϕB0 , γBI /ϕBI

, γBJ /ϕBJ, 0

´, Ω

q04 , q0 = i + j, B0 = D5, BI = Lx5 , BJ = L

y5

(d) ~Ψ+/− =“Id/Id, %Lx5

/− %Lx5, %

Ly10

/− %Ly10

, 0”, Ω

q04 , q0 = i + j

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Example: Edge detection

f

(a) Standard (b)

(c) (d)

(b)“Ω

π/42 , τ+

BC


“Ω−π/42 ,−[τ−

BC]c


(c) ~Ψ+/− =`γB0/ϕB0 , γBI /ϕBI

, γBJ /ϕBJ, 0

´, Ω

q04 , q0 = i + j, B0 = D5, BI = Lx5 , BJ = L

y5

(d) ~Ψ+/− =“Id/Id, %Lx5

/− %Lx5, %

Ly10

/− %Ly10

, 0”, Ω

q04 , q0 = i + j

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Example: Opening by reconstruction

f

(a) Standard (b)

(c) (d)

(b)“Ω

π/21 , γBC


“Ω

π/21 , ϕBC


(c)“Ω

π/42 , τ+

BC


“Ω−π/42 ,−[τ−

BC]c


(d) ~Ψ+/− =`γB0/ϕB0 , γBI /ϕBI

, γBJ /ϕBJ, 0

´, Ω

q04 , q0 = i + j, B0 = D5, BI = Lx5 , BJ = L

y5

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Example: Opening by reconstruction

f , σ = 30

(a) Standard (b)

(c) (d)

(b)“Ω

π/21 , γBC


“Ω

π/21 , ϕBC


(c)“Ω

π/42 , τ+

BC


“Ω−π/42 ,−[τ−

BC]c



, γBJ /ϕBJ, 0

´, Ω

q04 , q0 = i + j, B0 = D5, BI = Lx5 , BJ = L

y5

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Example: Levelling λ(f ,m) (Cartoon + Texture/noiseDecomposition)

f

mrk = f ∗ Gσ

(a) Standard (b)

(c) (d)

(b)“Ω

π/21 , γBC


“Ω

π/21 , ϕBC


(c) ~Ψ+/− =`Id/Id, γBI /ϕBI

, γBJ /ϕBJ, 0

´, Ω

q04 , q0 = i + j, B0 = D5, BI = Lx5 , BJ = L

y5


, γBJ /ϕBJ, 0

´, Ω

q04 , q0 = i + j, B0 = D5, BI = Lx5 , BJ = L

y5

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Example: Levelling λ(f ,m) (Cartoon + Texture/noiseDecomposition)

f

mrk = f ∗ Gσ

(a) Standard (b)

(c) (d)

(b)“Ω

π/21 , γBC


“Ω

π/21 , ϕBC


(c) ~Ψ+/− =`Id/Id, γBI /ϕBI

, γBJ /ϕBJ, 0

´, Ω

q04 , q0 = i + j, B0 = D5, BI = Lx5 , BJ = L

y5


, γBJ /ϕBJ, 0

´, Ω

q04 , q0 = i + j, B0 = D5, BI = Lx5 , BJ = L

y5

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Conclusions and perspectives

1 Introduction






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Conclusions and perspectives

Conclusions and Perspectives

Morphological operators for grey-level images based on indirect totalorderings using hypercomplex image representations.

The motivation was to introduce in the basic erosion/dilationoperators some information on size invariance or on relative contrastof structures.

Other representations using upper dimensional Cliord Algebras canbe foreseen in order to have a more generic framework not limited tofour-variables image representations.

Approach can also be extended to already natural multivariateimages (i.e., multispectral images) → tensor representations andassociated total orderings.

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Hypercomplex mathematical morphologyangulo/publicat/... · 2009-12-13 · 1/56. Hypercomplex...

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Transcript of Hypercomplex mathematical morphologyangulo/publicat/... · 2009-12-13 · 1/56. Hypercomplex...