Bayro-Corrochano E., Scheuermann G. Geometric Algebra Computing: in Engineering and Computer Science

Подождите немного. Документ загружается.

158 T. Batard et al.

Then, if we note e the neutral element of G, we have



exp

(X)



=exp



φ(X)



. (5)

The map T

φ is a Lie algebra morphism, i.e., it satisﬁes



[X, Y ]





φ(X),T

φ(Y)



∀X, Y ∈g.

From (5) we deduce that if the group G is connected and the exponential map of G

is onto, then the Lie group morphisms from G to H are determined by Lie algebras

morphisms from g to h.

A.2 Clifford Algebras

Let V be a vector space of ﬁnite dimension n over R equipped with a quadratic

form Q. Formally speaking, the Clifford algebra Cl(V , Q) is the solution of the

following universal problem. We search a couple (Cl(V, Q), i

) where Cl(V , Q) is

an R-algebra and i

:V −→ Cl(V , Q) is R-linear satisfying



(v)



=Q(v).1

for all v in V (1 denotes the unit of Cl(V , Q)) such that, for each R-algebra A and

each R-linear map f :V −→ A with



f(v)



=Q(v).1

for all v in V (1 denotes the unit of A), then there exists a unique morphism

g :Cl(V , Q) −→ A

of R-algebras such that f =g ◦i

The solution is unique up to isomorphisms and is given as the (noncommutative)

quotient

T(V)/



v ⊗v −Q(v).1



of the tensor algebra of V by the ideal generated by v ⊗v −Q(v).1, where v belongs

to V (see [12] for a proof).

It is well known that there exists a unique anti-automorphism t on Cl(V, Q) such

that



(v)



(v)

for all v in V . It is called reversion and usually denoted by x −→x

†

, x in Cl(V , Q).

In the same way there exists a unique automorphism α on Cl(V, Q) such that



(v)



=−i

(v)

Clifford–Fourier Transform for Color Image Processing 159

for all v in V . In this paper we write v for i

(v) (according to the fact that i

embeds V in Cl(V , Q)).

As a vector space, Cl(V, Q) is of dimension 2

on R and a basis is given by the

set



···e

< ···<i

,k∈{1,...,n}



and the unit 1. An element of degree k



<···<i

...i

···e

is called a k-vector. A 0-vector is a scalar, and e

···e

is called the pseudoscalar.

We denote x

the component of degree k of an element x of Cl(V , Q).

The inner product of x

of degree r and y

of degree s is deﬁned by

·y

=x



|r−s|

if r and s are positive and by

·y

otherwise.

The outer product of x

of degree r and y

of degree s is deﬁned by

∧y

=x



r+s

These products extend by linearity on Cl(V , Q). Clearly, if a and b are vectors of

V , then the inner product of a and b coincides with the scalar product deﬁned by Q.

When it is deﬁned (for example, when x is a versor and Q is positive), we denote

x=



†

and say that x is a unit if xx

†

=±1.

In this paper, we deal in particular with the Clifford algebra of the Euclidean R

denoted by R

n,0

. R

n,0

is the subspace of elements of degree k, and R

∗

n,0

is the group

of elements that admit an inverse in R

n,0

. We denote by S

n,0

the set of elements of

n,0

of norm 1.

Let a be a vector in R

n,0

, and B be the k-vector a

∧ a

∧···∧a

. Then the

orthogonal projection of a on the k-plane generated by the a

’s is the vector

(a) =(a ·B)B

−1

The vector

a −(a ·B)B

−1

=(a ∧B)B

−1

is called the rejection of a on B.

160 T. Batard et al.

A.3 The Spinor Group Spin(n)

It is deﬁned by

Spin(n) =





i=1

∈R

n,0

, a

=1



or equivalently

Spin(n) =



x ∈R

n,0

,α(x)=x, xx

†

=1,xvx

−1

∈R

n,0

∀v ∈R

n,0



It is well known that Spin(n) is a connected compact Lie group that universally

covers SO(n)(n ≥3). One can verify that Spin(3) is the group



a1 +be

+ce

+de



and is isomorphic to the group H

of unit quaternions. It is also a classical result

that Spin(4) is isomorphic to Spin(3) × Spin(3) (see [9] for more information on

spinors in R

and R

The Lie algebra of Spin(n)isR

n,0

with Lie bracket

A ×B =AB −BA.

As the exponential map from its Lie algebra to Spin(n) is onto (see [7] for a proof),

every spinor can be written as

S =

∞



i=0

for some bivector A.

From Hestenes and Sobczyk [8] we know that every A in R

n,0

can be written as

A =A

+···+A

where m ≤n/2, and

=A

a

,j∈{1,...,m}

with

,...,a

,...,b

}

a set of orthonormal vectors. Thus,

∧A

whenever j =k and

=−A



< 0.

Clifford–Fourier Transform for Color Image Processing 161

This means that the planes encoded by A

and A

are orthogonal and implies that

+···+A

σ(1)

σ(2)

...e

σ(m)

for all σ in the permutation group S(m). Actually, as A

is negative, we have

=cos



A





+sin



A





A



The corresponding rotation

:x −→e

−A

acts in the oriented plane deﬁned by A

as a plane rotation of angle 2A

.The

vectors orthogonal to A

are invariant under R

It then appears that any element R of SO(n) is a composition of commuting

simple rotations, in the sense that they have only one invariant plane. The vectors

left invariant by R are those of the orthogonal subspace to A.Ifm =n/2, this latter

is trivial. The previous decomposition is not unique if A

=A

 for some j and

k with j =k. In this case inﬁnitely many planes are left invariant by R.

References

1. Brackx, F., De Schepper, N., Sommen, F.: The two-dimensional Clifford–Fourier transform.

J. Math. Imaging Vis. 26, 5–18 (2006)

2. Bülow, T.: Hypercomplex spectral signal representations for the processing and analysis of

images. Ph.D. thesis, Kiel (1999)

3. Bülow, T., Sommer, G.: The hypercomplex signal—a novel approach to the multidimensional

analytic signal. IEEE Trans. Signal Process. 49(11), 2844–2852 (2001)

4. Ebling, J., Scheuermann, G.: Clifford Fourier transform on vector ﬁelds. IEEE Trans. Vis.

Comput. Graph. 11(4), 2844–2852 (2005)

5. Ell, T.A., Sangwine, S.: Hypercomplex Fourier transforms of color Images. IEEE Trans. Image

Process. 16(1), 5–18 (2007)

6. Felsberg, M.: Low-level image processing with the structure multivector. Ph.D. thesis, Kiel

(2002)

7. Helgason, S.: Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press, San

Diego (1978)

8. Hestenes, D., Sobczyk, G.: Clifford Algebra to Geometric Calculus. Reidel, Dordrecht (1984)

9. Lounesto, P.: Clifford Algebras and Spinors. London Mathematical Society Lecture Notes

Series. Cambridge University Press, Cambridge (1997)

10. Martin, D., Fowlkes, C., Tal, D., Malik, J.: A database of human segmented natural images

and its application to evaluating segmentation algorithms and measuring ecological statistics.

In: Proc. 8th Intl. Conf. Computer Vision, vol. 2, pp. 416–423 (2001)

11. Mawardi, B., Hitzer, E.: Clifford Fourier transformation and uncertainty principle for the Clif-

ford algebra cl

3,0

. Adv. App. Clifford Algebr. 16(1), 41–61 (2006)

12. Postnikov, M.: Leçons de géométrie. Groupes et algèbres de Lie. Mir (1982)

13. Sangwine, S., Ell, T.A.: Hypercomplex Fourier transforms of color images. In: IEEE Inter-

national Conference on Image Processing (ICIP), vol. 1, pp. 137–140. Thessaloniki, Greece

(2001)

162 T. Batard et al.

14. Sangwine, S., Ell, T.A., Gatsheni, B.: Colour-dependent linear vector image ﬁltering. In:

Twelfth European Signal Processing Conference, EUSIPCO 2004, pp. 585–588. Vienna, Aus-

tria (2004)

15. Smach, F., Lemaître, C., Gauthier, J.P., Miteran, J., Atri, M.: Generalized Fourier descriptors

with applications to objects recognition in SVM context. J. Math. Imaging Vis. 30(1), 43–71

(2008)

16. Vilenkin, N.J.: Special Functions and the Theory of Group Representations. Translations of

Mathematical Monographs, vol. 22. Am. Math. Soc., Providence (1968)

Hilbert Transforms in Clifford Analysis

Fred Brackx, Bram De Knock,

and Hennie De Schepper

Abstract The Hilbert transform on the real line has applications in many ﬁelds.

In particular in one-dimensional signal processing, the Hilbert operator is used to

extract global and instantaneous characteristics, such as frequency, amplitude, and

phase, from real signals. The multidimensional approach to the Hilbert transform

usually is a tensorial one, considering the so-called Riesz transforms in each of

the cartesian variables separately. In this paper we give an overview of generalized

Hilbert transforms in Euclidean space developed within the framework of Clifford

analysis. Roughly speaking, this is a function theory of higher-dimensional holo-

morphic functions particularly suited for a treatment of multidimensional phenom-

ena since all dimensions are encompassed at once as an intrinsic feature.

1 Introduction: The Hilbert Transform on the Real Line

The Hilbert transform is named after D. Hilbert, who, in his studies of integral equa-

tions, was the ﬁrst to observe what is nowadays known as the Hilbert transform pair.

However, the Hilbert transform theory was developed mainly by E.C. Titchmarsh

and G.H. Hardy. It was Hardy who named it after Hilbert. The Hilbert transform

is applied in the theoretical description of many devices and has become an indis-

pensable tool for both global and local descriptions of a signal. It has been directly

implemented in the form of Hilbert analogue or digital ﬁlters which allow one to

distinguish different frequency components and therefore locally reﬁne the struc-

ture analysis. Those ﬁlters are essentially based on the notion of analytic signal,

which consists of the linear combination of a bandpass ﬁlter, selecting a small part

of the spectral information and its Hilbert transform, the latter basically being the

result of a phase shift by

on the original ﬁlter (see, e.g., [33]).

F. Brackx (



)

Clifford Research Group, Faculty of Engineering, Ghent University, Galglaan 2, 9000 Gent,

Belgium

e-mail: Freddy.Brackx@UGent.be

E. Bayro-Corrochano, G. Scheuermann (eds.), Geometric Algebra Computing,

DOI 10.1007/978-1-84996-108-0_9, © Springer-Verlag London Limited 2010

163

164 F. Brackx et al.

For a real one-dimensional ﬁnite energy signal f , i.e., f ∈ L

(R), its Hilbert

transform on the real line is given by

S[f ](x) =



+∞

−∞

f(t)

x −t

dt, (1)

where Pv denotes the Cauchy principal value, meaning that in the integral the sin-

gularity at t = x is approached in a symmetrical way. Inﬁnite energy signals, such

as (piecewise) constant functions and sines and cosines, should be interpreted as

tempered distributions for which the Hilbert transform is deﬁned as the convolution

S[f ](x) =



∗f(t)



(x), (2)

where Pv

is the Principal Value distribution satisfying, in the distributional sense,

ln|t|=Pv

and t Pv

=1.

In order to recall the fundamental properties of the Hilbert transform on the real

line, we introduce the Cauchy integral of a function f ∈L

(R):

C[f ](x, y) =−

2πi



+∞

−∞

f(t)

(x −t)+iy

dt, y =0. (3)

This Cauchy integral is, as a function of the complex variable z = x + iy, holo-

morphic in the upper and lower halves of the complex plane and decays to zero for

y →±∞. In other words, for f ∈ L

(R), its Cauchy integral C[f ](x, y) belongs

to the Hardy spaces H

), respectively deﬁned by







F holomorphic in C

such that sup

y≷0



+∞

−∞



F(z)



dx < +∞



. (4)

Proposition 1 The Hilbert operator S :L

(R) →L

(R),(1), enjoys the following

properties:

P(1) S is translation invariant, i.e.,



S[f ]





[f ]



with τ

[f ](t) =f(t −a).

P(2) S is dilation invariant, i.e.,



S[f ]



=sgn(a)S



[f ]



with d

[f ](t) =f(t/a)/

√

|a|.

P(3) S is a linear, bounded, and norm-preserving operator.

P(4) S is invertible with S

−1

=−S, and thus S

=−1.

Hilbert Transforms in Clifford Analysis 165

P(5) S is unitary, i.e., S

∗

S =SS

∗

=1.

P(6) S commutes with differentiation, i.e.,



S[f ](t)





f(t)



P(7) S arises in a natural way by considering the nontangential boundary limits

(in L

sense) of the Cauchy integral (3), i.e.,

[f ](x) = lim

y→

0±

C[f ](x, y) =±

f(x)+

i S[f ](x), x ∈R. (5)

The operators C

are usually called the Cauchy transforms, and formulae (5) and

P(7) are the Plemelj–Sokhotzki formulae in Clifford analysis.

Thus putting H = iS we obtain an involutive, norm-preserving, and bounded

linear operator H : L

(R) → L

(R), which may be used to deﬁne the Hardy

space H

(R) as the closed subspace of L

(R) consisting of functions g for which

H[g]=g. We call those functions g ∈ H

(R) analytic signals, inspired by the

fact that the nontangential boundary limit C

[f ] of the holomorphic (or analytic)

Cauchy integral C[f ],(3), indeed belongs to the Hardy space H

(R). The real and

imaginary parts u = Re[g] and v =Im[g] of an analytic signal g satisfy the Hilbert

formulae

H[u]=iv and H[iv]=u. (6)

It follows that

g =(1 +H)[u] and g = (1 +H)[iv], (7)

showing that an analytic signal contains redundant information since it can be re-

covered from its real (or its imaginary) part solely. Note that the Hardy spaces

(R) and H

),(4), are isomorphic, since the nontangential boundary limit

for y → 0+ of F(z) ∈ H

) exists a.e. and belongs to H

(R), and the Cauchy

integral in C

of F(x +i0) precisely is F(z).

In the frequency space the Hilbert transform, which is convolutional in nature,

takes the form of a multiplication operator. Denoting by F the usual Fourier trans-

form, for a function f ∈L

(R),wehave



H[f ]



(ω) =sgn ωF [f ](ω) and



F [f ]



(ω) =−F



sgntf (t )



(ω).

(8)

In particular the Fourier spectrum of an analytic signal g ∈ H

(R) is a causal func-

tion with only positive frequencies, and conversely; more explicitly, it reads:

F [g](ω) = F (1 +H)[u](ω) =(1 +sgn ω)F [u](ω)



2F [u](ω), ω > 0,

0,ω<0.

(9)

166 F. Brackx et al.

2 Hilbert Transforms in Euclidean Space

The Hilbert transform was ﬁrst generalized to m-dimensional Euclidean space by

means of the Riesz transforms R

in each of the cartesian coordinates x

,j =

1,...,m, given by

[f ](x) = lim

ε→0+

m+1



\B(x,ε)

−y

|x −y|

m+1

f(y)dV(y), (10)

where a

m+1

2π

(m+1)/2

Γ ((m+1)/2)

denotes the area of the unit sphere S

in R

m+1

.Itwas

Horváth who, already in his 1953 paper [29], introduced the Clifford vector-valued

Hilbert operator

S =



j=1

. (11)

The multidimensional Hilbert transform was taken up again in the 1980s and further

studied in, e.g., [21, 22, 27, 32, 36] in the Clifford analysis setting.

Clifford analysis is a function theory which offers an elegant and powerful gener-

alization to higher dimension of the theory of holomorphic functions in the complex

plane. In its most simple but still useful setting, ﬂat m-dimensional Euclidean space,

Clifford analysis focusses on so-called monogenic functions, i.e., null solutions of

the Clifford vector-valued Dirac operator

∂



j=1

∂

, (12)

where (e

,...,e

) forms an orthonormal basis for the quadratic space R

underly-

ing the construction of the Clifford algebra R

0,m

, and where the basis vectors satisfy

the multiplication rules

=−2 δ

j,k

,j,k=1,...,m. (13)

Monogenic functions have a special relationship with harmonic functions of sev-

eral variables: they are reﬁning their properties, since the Dirac operator factorizes

the m-dimensional Laplacian ∂

=−Δ

. Euclidean space R

is embedded in the

Clifford algebra R

0,m

by identifying the point (x

,...,x

) ∈ R

with the vector

variable



j=1

. (14)

For more details on Clifford analysis and its applications, we refer to, e.g.,

[2, 20, 23].

Hilbert Transforms in Clifford Analysis 167

2.1 Deﬁnition and Properties

In the framework of Euclidean Clifford analysis, the (Clifford–)Hilbert transform

for a suitable function or distribution f is given by

H[f ](x

) =

m+1



x −y

|x −y|

m+1

f(y)dV(y)

m+1

lim

ε→0+



|x−y|>ε

x −y

|x −y|

m+1

f(y)dV(y). (15)

In the above expression, e

is an additional basis vector for which also

=−1 and e

=−2 δ

0,j

,j=1,...,m. (16)

Furthermore,

· stands for the usual conjugation in the Clifford algebra R

0,m+1

i.e., the main anti-involution for which

=−e

, j = 0,...,m. As in the one-

dimensional case, there is a strong relationship between the Hilbert transform and

the Cauchy integral of a function f ∈ L

). The functions considered here take

their values in the Clifford algebra R

0,m+1

. The space L

) is equipped with the

0,m+1

-valued inner product and corresponding squared norm:

f, g=



f(x)g(x)dV(x), f 



f, f 



, (17)

where [λ]

denotes the scalar part of the Clifford number λ. The Cauchy integral of

f ∈L

) is deﬁned by

C[f ](x) =C[f ](x

,x) =

m+1



(x −y)

+x −y|

m+1

f(y)dV(y), x

=0.

(18)

Observe the formal similarity with the Cauchy integral (3)off ∈ L

(R), x

taking

theroleofy, and the vector y

taking the role of t. It is a (left-)monogenic function in

the upper and lower half spaces R

m+1

={x

+x :x ∈R

≷ 0}. By a mono-

genic function in R

m+1

is meant a function annihilated by the Cauchy–Riemann

operator,

∂

+∂

) =∂

∂

, (19)

which decomposes the Laplace operator in R

m+1

, D

=Δ

m+1

Moreover the Cauchy integral decays to zero as x

→±∞. Summarizing, for a

function f ∈ L

), its Cauchy integral C[f ](x

,x),(18), belongs to the Hardy

spaces H

m+1

), respectively deﬁned by



m+1





F =0inR

m+1

such that sup

≷0



+∞

−∞



F(x

+x)



dx < +∞



(20)