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128 W. Reich and G. Scheuermann

Fig. 4 In that image we zoom in further and take a look at the direction of the “vectors” in a

neighborhood of zero frequency

Further the quaternions are isomorphic to the Clifford Algebra Cl

0,2

of the Anti-

Euclidean Space R

0,2

i →e

,i→e

,k→e

, (30)

which we will only use in for the deﬁnition of holomorphicity in Sect. 5.1.

5.1 H-Holomorphic Functions and Fourier Series

We follow the deﬁnitions by Gürlebeck et al. [9] and identify the quaternions as

in (29). Let f : H → H be a function with real partial derivatives ∂

∂

∂q

.We

deﬁne the complete real differential as

df =



k=0

∂

fdq

(31)

and set

dq =dq



k=1

,d¯q =dq

−



k=1

. (32)

Analyzing Real Vector Fields 129

This leads to

df =





k=0

∂



d ¯q +



∂

fdq−



k=1

∂

fdqe



. (33)

A real C

-function f is right H-holomorphic in G ⊂ H if, for every q ∈ G and

h →0, there exist a

(q) ∈H with

f(q+h) =f(q)+



k=1

(q) (h

−h

) +o(h) (34)

and left H-holomorphic if

f(q+h) =f(q)+



k=1

−h

(q) +o(h) (35)

with Landau symbol o(h).Theh

are the 4D coordinates of h ∈ H. With the

operator

∂ :=

∂

∂q



k=1

∂

∂q

, (36)

we have

f is left H-holomorphic ⇐⇒

∂f =0, (37)

f is right H-holomorphic ⇐⇒ f

∂ =0. (38)

Let B

:={q ∈H||q|=1} be the unit sphere in H.LetL

) be the functions on

with existing integral of the squared function. Then one can write

) ∩ker

∂ =

∞



k=0

, (39)

where H

are homogenous H-holomorphic polynomials. There is an orthogonal

basis for this space that allows a Fourier series approximation [9].

If we look at a vector ﬁeld

v :R

→R

⊂G

, (40)

we have to ﬁnd a related H-holomorphic function f : B

→ H ⊂ G

to apply the

construction above. We tried

v(x) =f(x)e

f(x) (41)

and

v(x) =f(x)e

,v(x)=e

f(x), (42)

130 W. Reich and G. Scheuermann

as well as

f(x)=v

(x)e

(43)

with e

∈G

In all examined cases, general linear vector ﬁelds v do not generate a H-

holomorphic function f , which makes applying the Fourier series expansion to our

purpose inconvenient.

5.2 Biquaternion Fourier Transform

Let H

be the biquaternions, i.e.,

={q

i +q

j +q

k |q

=(q

) +I (q

) ∈C} (44)

with the algebra isomorphism H

→ G

as in (28)–(29). Sangwine et al. [19]

choose a μ ∈ G

with μ

=−1 and deﬁne the right biquaternion Fourier trans-

form (BiQFT) for a signal F :R

→G

{F }(u) =



F(x)exp(−2πμx ·u)|dx| (45)

and the left biquaternion Fourier transform by

{F }(u) =



exp(−2πμx ·u)F (x)|dx|. (46)

For μ =i

, this is the 3D-CFT. But for a pure bi-quarternion, i.e., a bivector, one can

choose an orthogonal basis μ,ν, ξ =μν, with {μ,ν, ξ} being quaternionic roots of

−1 such that any q ∈G

can be written as

q =q

= q

+˜q

μ +˜q

ν +˜q

= (q

+˜q

μ) +( ˜q

+˜q

μ)ν

= Simp(q) +Perp(q)ν (47)

with Simp(q) and Perp(q) denoting the so-called simplex and perplex of q.Fora

pure bi-quarternion μ, the corresponding BiQFT fulﬁlls

−1

T (48)

with the linear operator T(1) = 1,T(e

) = μ, T (e

) = ν, T (e

) = ξ ,soany

two BiQFTs with pure bi-quarternion μ differ just by an orthogonal transforma-

tion. The BiQFT splits like the CFT in four independent classical complex Fourier

Analyzing Real Vector Fields 131

transforms:

{F }(u) =



F(x)exp(−2πμx ·u)|dx|





(x) +

(x)μ



exp



−2πμx ·u



|dx|





(x) +

(x)μ



ν exp(−2πμx ·u)|dx|









(x)



+



(x)



exp



−2πμx ·u



|dx|









(x)



+



(x)



exp



−2πμx ·u



|dx|i









(x)



+



(x)



ν exp



−2πμx ·u



|dx|









(x)



+



(x)



ν exp



−2πμx ·u



|dx|i

. (49)

For a real vector ﬁeld

v :R

→R

⊂G

,x→



k=1

(x)e

, (50)

we have

v(x) =



−v

(x)e

−v

(x)e



(51)



−v

(x)e





−v

(x) +v

(x)e



. (52)

We get for μ =e

{v}(u) =





−v

(x)e



exp



−2πe

x ·u



|dx|i





−v

(x) +v

(x)e



exp(−2πe

x ·u)|dx|i

, (53)

which means that the vector ﬁeld is split in a purely complex signal, −v

(x)e

and a complex signal −v

(x) +v

(x)e

, which are transformed independently by

two classical Fourier transforms.

In essence, the BiQFT means choosing a planar direction μ in R

, transforming

the planar part of the vector ﬁeld with a 2D-CFT in each plane parallel to μ, and

transforming the scalar part orthogonal to μ as an independent real signal. One can

say that the BiQFT is the direct generalization of the 2D-CFT to three dimensions,

while the 3D-CFT looks at a vector ﬁeld as three independent real signals.

132 W. Reich and G. Scheuermann

5.3 Two-Sided Quaternion Fourier Transform

Another approach for transforming a function F including the main QFT Theorems

can be found in [2], and its generalization in [13]. Hitzer also stated a Plancherel

Theorem for the QFT and, together with Mawardi, extended the theory to higher-

dimensional Clifford algebras in [14]. Let F : R

→ G

; then the QFT is deﬁned

F{F }(u) =



−2πe

F(x

−2πe

|dx|. (54)

The inverse QFT is given by

−1

{F }(x) =



2πe

F(u

2πe

|du|. (55)

Though the usual decomposition of F into four real-valued resp. two complex-

valued signals is possible via

F =F

+(F

, (56)

and there are several options to embed two real variables in a quaternion for applying

the transform to a real vector ﬁeld, the Two-Sided QFT is different from the 2D-

CFT. Not only the multiplication from two sides and using two distinct axes of

transformation at once lead to different numerical results; even if the Fourier kernel

is all right-sided, we cannot merge the two exponentials, because the functional

equation does not hold for arbitrary quaternions.

Investigating the precise relationship of both transforms is left for future work.

6Conclusion

It has been shown that a convolution of vector ﬁelds is a nice asset for the analysis

of ﬂuid ﬂow simulations or physical velocity measurements. Geometric algebra al-

lows a formulation of a suitable convolution as closed operation. Furthermore, one

can deﬁne a Clifford–Fourier transform in two- and three- dimensional Euclidean

spaces that allows the well-known theorems like convolution theorem, derivative

theorem, and Parseval’s theorem. Looking into the two CFT transforms reveals that

they look at the vector ﬁeld in a totally different manner, i.e., the 2D-CFT transforms

the vector ﬁeld as one complex signal, while the 3D-CFT transforms the vector ﬁeld

as three independent real signals. This mismatch can be interpreted by the BiQFT of

Sangwine et al. which needs an element μ ∈G

with μ

=−1. For a pure bivector,

this means choosing a planar direction in which the vector ﬁeld is transformed as

complex signal. The perpendicular part of the vector ﬁeld is independently trans-

formed as real signal. For μ = i

, one gets the 3D-CFT. We have also shown that

several constructions do not allow a use of the Fourier series approach based on [9].

Analyzing Real Vector Fields 133

Acknowledgements We want to thank our former colleague Julia Ebling for her work. We also

thank Heinz Krüger, TU Kaiserslautern, for many valuable hints and comments.

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Birkhäuser, Basel (2006)

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ings of Function Theories in Higher Dimensions (2006)

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Clifford–Fourier Transform for Color Image

Processing

Thomas Batard, Michel Berthier,

and Christophe Saint-Jean

Abstract The aim of this paper is to deﬁne a Clifford–Fourier transform that is

suitable for color image spectral analysis. There have been many attempts to deﬁne

such a transformation using quaternions or Clifford algebras. We focus here on a

geometric approach using group actions. The idea is to generalize the usual deﬁ-

nition based on the characters of abelian groups by considering group morphisms

from R

to spinor groups Spin(3) and Spin(4). The transformation we propose is

parameterized by a bivector and a quadratic form, the choice of which is related to

the application to be treated. A general deﬁnition for 4D signal deﬁned on the plane

is also given; for particular choices of spinors, it coincides with the deﬁnitions of

S. Sangwine and T. Bülow.

1 Introduction

During the last years several attempts have been made to generalize the classical ap-

proach of scalar signal processing with the Fourier transform to higher-dimensional

signals. The reader will ﬁnd a detailed overview of the related works at the beginning

of [1]. We only mention in this introduction some of the approaches investigated by

several authors.

Motivated by the spectral analysis of color images, S. Sangwine and T. Ell have

proposed in [13] and [5] a generalization based on the use of quaternions: a color

corresponds to an imaginary quaternion, and the imaginary complex i is replaced by

the unit quaternion μ coding the grey axis. A quaternionic deﬁnition is also given

by T. Bülow and G. Sommer in the context of analytic signals, for signals deﬁned on

the plane and with values in the algebra H of quaternions [3]. Concerning analytic

signals, M. Felsberg makes use of the Clifford algebras R

2,0

and R

3,0

to deﬁne an

appropriate Clifford–Fourier transform [6].

T. Batard (



)

Lab. MIA, La Rochelle University, La Rochelle, France

e-mail: tbatar01@univ-lr.fr

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

135

136 T. Batard et al.

A generalization in the Clifford algebras context appears also in J. Ebling and

G. Scheuermann [4]. The authors mainly use their transformation to analyze fre-

quencies of vector ﬁelds. Using the same Fourier kernel, B. Mawardi and E. Hitzer

obtain an uncertainty principle for R

3,0

multivector functions [11]. The reader may

ﬁnd in [1] deﬁnitions of Clifford–Fourier transform and Clifford–Gabor ﬁlters based

on the Dirac operator and Clifford analysis.

One could ask the reason why to propose a new generalization. An important

thing when dealing with Fourier transform is its link with group representations.

We then recall in Sect. 2 the usual deﬁnition of the Fourier transform of a function

deﬁned on an abelian Lie group by means of the characters of the group. The deﬁni-

tion we propose in Sect. 3 relies mainly on the generalization of the notion of char-

acters; that is why we study the group morphisms from R

to Spin(3) and Spin(4).

These morphisms help to understand the behavior of the Fourier transform with re-

spect to well chosen spinors. We treat in Sect. 4 three applications corresponding to

speciﬁc bivectors of R

4,0

. They consist in ﬁltering frequencies according to color,

hue, and chrominance part of a given color. In Sect. 5, we show that for particular

choices of group morphisms and under well-chosen identiﬁcation with quaternions,

the Clifford–Fourier transform we propose coincides with the deﬁnitions of S. Sang-

wine [5] and T. Bülow [2].

2 Fourier Transform and Group Actions

Let us recall brieﬂy some basic ideas related to the group approach of the deﬁnition

of the Fourier transform. Details can be found in the Appendix; see also [15]for

examples of applications to Fourier descriptors.

Let G be a Lie group. The Pontryagin dual of G, denoted



G, is the set of equiv-

alence classes of unitary irreducible representations of G. It appears that if G is

abelian, every irreducible unitary representation of G is of dimension 1, i.e., is a

continuous group morphism from G to S

. This is precisely the deﬁnition of a char-

acter. It is well known that the characters of R

are given by

,...,x

) −→e

i(u

+···+u

)

with real u

,...,u

. This shows that



. The characters of SO(2) are the

group morphisms

θ −→e

inθ

for n ∈Z, and the corresponding Pontryagin dual is Z. The characters of Z/nZ are

the group morphisms

u −→e

2πku

for k ∈Z/nZ, from which we deduce that Z/nZ is its own Pontryagin dual.

Clifford–Fourier Transform for Color Image Processing 137

In the general case (provided that G is unimodular), the Fourier transform of a

function f ∈L

(G;C) is deﬁned on



G by



f(ϕ)=



f(x)ϕ



−1



dν(x)

(for ν a well-chosen invariant measure on G). Applying this formula to the case G =

, resp. G =SO(2), resp. G = Z/nZ leads to the usual deﬁnition of the Fourier

transform, resp. Fourier coefﬁcients, resp. discrete Fourier transform.

Traditionally, the Fourier transform in L

,(R

, 

)) is deﬁned by n stan-

dard Fourier transforms in L

, R) on each one of the components, embedding R

into C. Using group representations theory, we are able to deﬁne Fourier transforms

that treat jointly the different components.

From now on, we deal with the abelian group G = (R

, +) since this paper is

devoted to image processing applications.

Let us make some crucial remarks about the case n =2.

Let f be a real- or complex-valued function deﬁned on R

. Its Fourier transform

is given by



f(a,b)=



f(x,y)e

−i(ax+by)

dx dy.

Identifying C with (R

, 

),wehaveS

= SO(2), and the action of S

on C,

given by the complex multiplication, corresponds to the action of the group SO(2)

on (R

, 

). Hence, we can deﬁne a Fourier transform in L

,(R

, 

)) using

the action of group morphisms from R

to SO(2) on (R

, 

). These ones are real

unitary representations of the group R

of dimension 2.

The Fourier transform of f ∈L

,(R

, 

)) deﬁned above can be written in

the Clifford algebra language. Indeed, from the embedding of (R

, 

) into R

2,0

f maybeviewedasanR

2,0

-valued function

f(x,y) =f

(x, y)e

where e

=1 and e

=−e

. From this point of view, the Fourier transform

of f is given by



f(a,b)=





cos



(ax +by)/2



1 +sin



(ax +by)/2







(x, y)e





cos



−(ax +by)/2



1 +sin



−(ax +by)/2





dx dy

using the fact that the action of Spin(2) on R

2,0

corresponds to the action of SO(2)

on (R

, 

) (see Appendix). We can write this last formula in the following form:



f(a,b)=





(x, y)e



⊥ϕ

a,b

(−x,−y)dx dy,