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

Hypercomplex mathematical morphology

Jesús Angulojesus.angulo@ensmp.fr ; 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

1 / 56

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

2 / 56

Two basic operators

δB(f )(x) = f (y) : f (y) =∨

[f (z)], z ∈ Bx,

εB(f )(x) = f (y) : f (y) =∧

[f (z)], z ∈ Bx.

2 / 56

Two basic operators

δB(f )(x) = f (y) : f (y) =∨

[f (z)], z ∈ Bx,

εB(f )(x) = f (y) : f (y) =∧

[f (z)], z ∈ Bx.

2 / 56

Two basic operators

δB(f )(x) = f (y) : f (y) =∨

[f (z)], z ∈ Bx,

εB(f )(x) = f (y) : f (y) =∧

[f (z)], z ∈ Bx.

2 / 56

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 ))

3 / 56

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)

5 / 56

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

6 / 56

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

7 / 56

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

8 / 56

State-of-the art,

8 / 56

State-of-the art,

8 / 56

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

9 / 56

Introduction

1 Introduction

10 / 56

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).

11 / 56

Introduction

f (t) 7→ fA(t) = f (t) + ifHi (t)

11 / 56

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.

11 / 56

Introduction

f (t) 7→ fA(t) = f (t) + ifHi (t)

11 / 56

Introduction

f (t) 7→ fA(t) = f (t) + ifHi (t)

11 / 56

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.

12 / 56

Introduction

f (x) 7→ fC (x) = f (x) + iψB(f )(x)

12 / 56

Introduction

f (x) 7→ fC (x) = f (x) + iψB(f )(x)

12 / 56

Complex representation and associated total orderings

1 Introduction

13 / 56

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 ).

14 / 56

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.

15 / 56

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.

15 / 56

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

cn ≤Ωθ01

cm ⇔ρn < ρm or

ρn = ρm and θn θ0 θm

Ordering ≤Ω

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.

16 / 56

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

17 / 56

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

fC (y) ≤Ω

fC (z) =⇒ f (y) 4Ω

f (z),

where 4Ω

allows to compute the supremum∨

Ωθ01

and the inmum∧Ω

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

Formally, we have

fC (y) ≤Ω

fC (z) =⇒ f (y) 4Ω

f (z),

where 4Ω

Ωθ01

and the inmum∧Ω

fC (y) =∨

Ωθ01 ,z∈Z

[fC (z)] =⇒ f (y) =∨

Ωθ01 ,z∈Z

[f (z)] .

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

18 / 56

Formally, we have

fC (y) ≤Ω

fC (z) =⇒ f (y) 4Ω

f (z),

where 4Ω

Ωθ01

and the inmum∧Ω

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

19 / 56

γ-complex image

Opening: γB(f ) = δB (εB(f ))

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

20 / 56

γ-complex image

Opening: γB(f ) = δB (εB(f ))

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

21 / 56

ϕ-complex image

Closing: ϕB(f ) = εB (δB(f ))

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

22 / 56

γ-complex dilations and ϕ-complex erosions

π/21 , γ

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

δ〈1,γBC,B〉(f )(x) = f (y) : fC (y) =

[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) =

[fC (z)], z ∈ B(x)).

23 / 56

γ-complex dilations and ϕ-complex erosions

The equivalent(Ω

π/22 , γ

)-complex dilation and

π/22 , ϕ

)-complex

erosion are respectively

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

Ωπ/22

[fC (z)], z ∈ B(x)),

ε〈2,ϕBC,B〉(f )(x) = f (y) : fC (y) =

[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.

24 / 56

Properties

Proposition

The pair(δ〈1,γBC ,B〉, ε〈1,ϕBC

)is an adjunction, i.e.,

δ〈1,γBC ,B〉(f )(x) 4Ω

g(x) ⇔ f (x) 4Ω

ε〈1,ϕBC,B〉(g)(x),

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

)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.

26 / 56

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

27 / 56

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

28 / 56

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

29 / 56

τ+-complex dilations and τ−-complex erosions

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)).

30 / 56

τ+-complex dilations and τ−-complex erosions

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)).

31 / 56

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

32 / 56

Generalisation to multi-operator cases using real quaternions

1 Introduction

33 / 56

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...

34 / 56

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...

34 / 56

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.

35 / 56

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.

36 / 56

Quaternion polar form

Any quaternion may be represented in polar form as

q = ρeξθ

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

)is the angle, sometimes called eigenangle,

between the real part and the 3D imaginary part.

37 / 56

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),

V‖(q) =1

2(V (q)− puV (q)pu)

V⊥(q) =1

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

38 / 56

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.

39 / 56

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

40 / 56

Morphological hypercomplex image

~Ψ-hypercomplex image

Given the four-variate transformation

~Ψ =(ψ0B0, ψI

BI, ψJ

BJ, ψK

)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 .

41 / 56

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

is particularly interesting.

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

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

is particularly interesting.43 / 56

Comparative examples

1 Introduction

44 / 56

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

(a) Standard (b)

(c) (d)

(b)“Ω

π/21 , γBC

”-complex dilation,

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

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

(a) Standard (b)

(c) (d)

(b)“Ω

π/21 , γBC

π/21 , ϕBC

(c)“Ω

π/22 , γBC

π/22 , ϕBC

(d) ~Ψ+/− =“Id/Id, %Lx10

/− %Lx10, %

/− %Ly10

, 0”, Ω

q04 , q0 = i + j

47 / 56

Example: Contrast operator using erosion and dilation

(a) Standard (b)

(c) (d)

(b)“Ω

π/42 , τ+

“Ω−π/42 ,−[τ−

(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

, 0”, Ω

q04 , q0 = i + j

48 / 56

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

(a) Standard (b)

(c) (d)

(b)“Ω

π/42 , τ+

“Ω−π/42 ,−[τ−

(c) ~Ψ+/− =`γB0/ϕB0 , γBI /ϕBI

, γBJ /ϕBJ, 0

´, Ω

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

(d) ~Ψ+/− =“Id/Id, %Lx5

/− %Lx5, %

/− %Ly10

, 0”, Ω

q04 , q0 = i + j

49 / 56

Example: Edge detection

(a) Standard (b)

(c) (d)

(b)“Ω

π/42 , τ+

“Ω−π/42 ,−[τ−

(c) ~Ψ+/− =`γB0/ϕB0 , γBI /ϕBI

, γBJ /ϕBJ, 0

´, Ω

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

(d) ~Ψ+/− =“Id/Id, %Lx5

/− %Lx5, %

/− %Ly10

, 0”, Ω

q04 , q0 = i + j

50 / 56

Example: Opening by reconstruction

(a) Standard (b)

(c) (d)

(b)“Ω

π/21 , γBC

π/21 , ϕBC

(c)“Ω

π/42 , τ+

“Ω−π/42 ,−[τ−

(d) ~Ψ+/− =`γB0/ϕB0 , γBI /ϕBI

, γBJ /ϕBJ, 0

´, Ω

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

51 / 56

Example: Opening by reconstruction

f , σ = 30

(a) Standard (b)

(c) (d)

(b)“Ω

π/21 , γBC

π/21 , ϕBC

(c)“Ω

π/42 , τ+

“Ω−π/42 ,−[τ−

, γBJ /ϕBJ, 0

´, Ω

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

52 / 56

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

mrk = f ∗ Gσ

(a) Standard (b)

(c) (d)

(b)“Ω

π/21 , γBC

π/21 , ϕBC

(c) ~Ψ+/− =`Id/Id, γBI /ϕBI

, γBJ /ϕBJ, 0

´, Ω

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

, γBJ /ϕBJ, 0

´, Ω

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

53 / 56

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

mrk = f ∗ Gσ

(a) Standard (b)

(c) (d)

(b)“Ω

π/21 , γBC

π/21 , ϕBC

(c) ~Ψ+/− =`Id/Id, γBI /ϕBI

, γBJ /ϕBJ, 0

´, Ω

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

, γBJ /ϕBJ, 0

´, Ω

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

54 / 56

Conclusions and perspectives

1 Introduction

55 / 56

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.

56 / 56

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

Documents

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

Lecture: Binary Images and Morphologyclass.vision/96-97/04_Morphology.pdf · SRTTU –A.Akhavan Lecture 4-1 ۱۳۹۶ دنفسا۹- هبنشراهچ Lecture: Binary Images and Morphology

MENTION SCIENCES DU LANGAGE - ged.univ-lille3.fr · • exercices d¶application Bibliographie succincte : Anderson S.R. (1992), A-Morphous Morphology, Cambridge, Cambridge, University

Classiﬁcationdessurfacestopologiquescompactespol/memoirem1.pdfIntroduction This work concerns mathematical objects called manifolds. A n-dimensional manifold is a set whose local

Bibliography - Springer978-1-4020-2455-9/1.pdf · BIBLIOGRAPHY BARROW, 1., The Usefulness of Mathematical Learning Explained and Demonstrated: Being Mathematical Lectures Read in

Journal of Mathematical Systems, Estimation, and Conmitter/publications/79_linear_systems_JMS.pdf · 2013. 3. 12. · Journal of Mathematical Systems, Estimation, and Con trol c 1996

Maharaja Surajmal Brij Universitymsbrijuniversity.ac.in/assets//uploads/newsupdate... · 2021. 1. 17. · Phonetic Symbols, Transcription of Words ii. Elementary Morphology Section

hypertexte et hypermodernité - lucdall.free.frlucdall.free.fr/publicat/htxt_hmdt.pdf · 4 - H2PTM 2009 puissance notionnelle conférée aux mots-clés ou tags à l'oeuvre dans les

air.unimi.it...Digital Object Identiﬁer (DOI) Commun. Math. Phys. 377, 1883–1959 (2020) Communications in Mathematical Physics Non ...

Les paysages vosgiens et le plan de paysagepiece-jointe-carto.developpement-durable.gouv.fr/DEPT088A/publicat… · Côté massif, l'ouverture du paysage peut se maintenir grâce

Mathematical modelling for the transmission of dengue ...

Mathematical Modeling of Phase Change Materials: An ... paper.pdf · Mathematical Modeling of Phase Change Materials: An Updated Review ... Specific Heat [kJ/kg K] process 1 ... This

Traitements des images de couleur en représentation ...angulo/publicat/AnguloSerra_TS_05.pdf · Traitements des images de couleur en représentation luminance/saturation/teinte par

A Mathematical Un mathématicien Visit to Africa en Afrique A I

13 Nourissat - rhumatologie-bichat.comrhumatologie-bichat.com/PolyParis2013/13 Nourissat.pdf · and classiﬁed the morphology of the bicipital tu-berosity ridge as smooth (absent),

Experimental Study and Mathematical Modeling of Asphaltene … · 1051> Experimental Study and Mathematical Modeling of Asphaltene Deposition Mechanism in Core Samples Étude expérimentale

SYNODE - Diocèse de Toursdiocesedetours.catholique.fr/fileadmin/diocese/contenus/... · 2018. 1. 24. · En f évrier 2017, après la publicat ion du document préparat oire au synode

Morphological haplology in amazigh - · PDF fileMorphological haplology in amazigh ... allomorphy revolving around affixation, ... standing problem in the treatment of the verb morphology

du signe à l’interface ou la problématique du néodesignlucdall.free.fr/publicat/signe_interface.pdf · approches structurales de « méta-design » des nouveaux architectes des

Nicolas, neveu exemplaire - European Mathematical Society

ONTOLOGY-BASED LYMPHOCYTE POPULATION ...Using tools from mathematical morphology for processing peripheral blood colour images, we have developed an image-based approach, to provide