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

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98 M. Bahri et al.

Theorem 3 (Reversion) Let f ∈ L

) be a parabivector-valued function.

For a parabivector-valued window function φ, we have



f(ω, b) =



f(−ω, b)



∼

. (23)

Proof Application of Deﬁnition 4 to the left-hand side of (23)gives



f(ω, b) =

(2π)





f(x)φ(x −b)e

−i

ω·x

(2π)





ω·x



φ(x −b)f (x)d



∼

(2π)





f(x)



φ(x −b)e

ω·x



∼

. (24)

This ﬁnishes the proof of the theorem. 

Theorem 4 (Switching) If |x|

1/2

f(x) ∈ L

) and |x|

1/2

φ(x) ∈ L

)

are parabivector-valued functions, then we obtain

f(ω, b) =e

−i

ω·b



φ(−ω, −b)



∼

. (25)

Proof We have, by the CWFT deﬁnition,

f(ω, b) =

(2π)



f(x)



φ(x −b)e

−i

ω·x

(2π)





φ(x −b)



f(x)e

ω·x



∼

. (26)

The substitution y =x −b into the above expression gives

f(ω, b) =

(2π)





φ(y)



f(y +b)e

ω·(y+b)



∼

(2π)

−i

ω·b





φ(y)



f(y +b)e

ω·y



∼

(2π)

−i

ω·b





φ(y)





y −(−b)



−i

(−ω)·y



∼

, (27)

which proves the theorem. 

Theorem 5 (Parity) Let φ ∈ L

) be a Clifford window function. If P is the

parity operator deﬁned as Pφ(x) =φ(−x), then we have

Pφ

{Pf }(ω, b) =G

f(−ω, −b). (28)

Two-Dimensional Clifford Windowed Fourier Transform 99

Proof Direct calculations give, for every f ∈L

Pφ

{Pf }(ω, b) =

(2π)



f(−x)



φ(−x +b)



∼

−i

(−ω)·(−x)

(2π)



f(−x)





−x −(−b)



∼

−i

(−ω)·(−x)

(2π)



f(x)





x −(−b)



∼

−i

(−ω)·x

x, (29)

which completes the proof. 

Theorem 6 (Shift in space domain, delay) Let φ be a Clifford window function.

Introducing the translation operator T

f(x) =f(x −x

), we obtain

f }(ω, b) =



f(ω, b −x

)



−i

ω·x

. (30)

Proof We have by using (18)

f }(ω, b) =

(2π)



f(x −x

)



φ(x −b)e

−i

ω·x

x. (31)

We substitute t =x −x

into the above expression and get, with d

x =d

f }(ω, b) =

(2π)



f(t)





t −(b −x

)



∼

−i

ω·(t+x

)

(2π)





f(t)





t −(b −x

)



∼

−i

ω·t



−i

ω·x

(32)

This ends the proof of (30). 

Theorem 7 (Shift in frequency domain, modulation) Let φ be a parabivector-

valued Clifford window function. If ω

∈R

and f

(x) =f(x)e

·x

, then

(ω, b) =G

f(ω −ω

, b). (33)

Proof Using Deﬁnition 4 and simplifying it, we get

(ω, b) =

(2π)



f(x)e

·x



φ(x −b)e

−i

ω·x

(2π)



f(x)



φ(x −b)e

−i

(ω−ω

)·x

x, (34)

which proves the theorem. 

100 M. Bahri et al.

Theorem 8 (Reconstruction formula) Let φ be a Clifford window function. Then

every 2D Clifford signal f ∈L

) can be fully reconstructed by

f(x) =(2π)



f(ω, b)φ

ω,b

(x)



φ,



−1

)

ω. (35)

Proof It follows from the CWFT deﬁned by (18) that

f(ω, b) =

(2π)



f(x)



φ(x −b)



(ω). (36)

Taking the inverse CFT of both sides of (36), we obtain

f(x)



φ(x −b) = (2π)

−1



f(ω, b)



(x)

(2π)



f(ω, b)e

ω·x

ω. (37)

Multiplying both sides of (37)byφ(x −b) and then integrating with respect to d

we get

f(x)





φ(x −b)φ(x −b)d

b =



f(ω, b)e

ω·x

φ(x −b)d

ω d

(38)

or, equivalently,

f(x)



φ,



)

=(2π)



f(ω, b)φ

ω,b

(x)d

ω d

b, (39)

which gives (35). 

It is worth noting here that if the Clifford window function is a parabivector-

valued function, then the reconstruction formula (35) can be written in the form

f(x) =

(2π)

φ

)



f(ω, b)φ

ω,b

(x)d

ω. (40)

Theorem 9 (Orthogonality relation) Assume that the Clifford window function φ

is a parabivector-valued function. If two Clifford functions f,g ∈ L

), then

we have



(f, φ

ω,b

)



(g, φ

ω,b

)

ω d

φ

)

(2π)

(f, g)

)

. (41)

Two-Dimensional Clifford Windowed Fourier Transform 101

Proof By inserting (18) into the left side of (41), we obtain



(f, φ

ω,b

)



(g, φ

ω,b

)

ω d



(f, φ

ω,b

)





(2π)

ω·x

φ(x −b)



g(x)d



ω d







(2π)

f(x



)



φ(x



−b)e

ω·(x−x



)

ω d





×φ(x −b)



g(x)d

(2π)







f(x



)



φ(x



−b)δ(x −x



)φ(x −b)d







g(x)d

(2π)



f(x)





φ(x −b)φ(x −b)



 

φ parabiv. funct.



g(x)d

(2π)

φ

)



f(x)



g(x)d

x, (42)

which completes the proof of (41). 

Theorem 10 (Reproducing kernel) For a parabivector-valued Clifford window

function |x|

1/2

φ ∈L

), if



ω, b;ω



, b



(2π)

φ

)

(φ

ω,b

,φ



)

, (43)

then K

(ω, b;ω



, b



) is a reproducing kernel, i.e.,





, b





f(ω, b)K



ω, b;ω



, b



ω d

b. (44)

Proof By inserting the inverse CWFT (40) into the deﬁnition of the CWFT (18)we

easily obtain





, b





f(x)





(x)d





(2π)

φ

)



f(ω, b)φ

ω,b

(x)d







(x)d

102 M. Bahri et al.



f(ω, b)

(2π)

φ

)





ω,b

(x)





(x)d





f(ω, b)K



ω, b;ω



, b



ω, (45)

which ﬁnishes the proof. 

Remark 2 Formulas (40), (41), and (43) also hold if the Clifford window function

is a vector-valued function, i.e., φ(x) =φ

(x)e

+φ

(x)e

The above properties of the CWFT are summarized in Table 1.

Table 1 Properties of the CWFT of f, g ∈ L

), L

= L

),whereλ, μ ∈ G

are

constants, ω

=ω

0,1

+ω

0,2

∈R

,andx

∈R

Property Clifford-valued function 2D CWFT

Left linearity λf (x) +μg(x)λG

f(ω, b) +μG

g(ω, b)

Delay f(x −x

)(G

f(ω, b −x

)) e

−i

ω·x

Modulation f(x)e

·x

f(ω −ω

, b) if φ is a

parabivector-valued function

Formulas

Reversion G



f(ω, b) ={G

f(−ω, b)}

∼

if f and φ are parabivector-valued functions

Switching G

f(ω, b) = e

−i

ω·b

φ(−ω, −b)}

∼

if f and φ are parabivector-valued functions

Parity G

Pφ

{Pf }(ω, b) = G

f(−ω, −b)

Orthogonality

(2π)

φ

(f, g)



(f, φ

ω,b

)



(g, φ

ω,b

)

ω d

if φ is a parabivector-valued function

Reconstruction f(x) = (2π)



f(ω, b)φ

ω,b

(x)

×(

φ,

φ)

−1

)

ω,

(2π)

φ

)



f(ω, b)

×φ

ω,b

(x)d

ω,

if φ is a parabivector-valued function

Reproducing kernel G

f(ω



, b



) =



f(ω, b)K

(ω, b;ω



, b



ω d

(ω, b;ω



, b



)

(2π)

φ

)

(φ

ω,b

,φ



)

if φ is a parabivector-valued function

Two-Dimensional Clifford Windowed Fourier Transform 103

4.3 Examples of the CWFT

For illustrative purposes, we shall discuss examples of the CWFT. We then compute

their energy densities.

Example 1 Consider the Clifford–Gabor ﬁlters (see Fig. 1) deﬁned by (σ

=1/

√

f(x) =

(2π)

−x

·x

. (46)

Obtain the CWFT of f with respect to the Gaussian window function φ(x) =e

−x

By the deﬁnition of the CWFT (18), we have

f(ω, b) =

(2π)



−x

·x

−(x−b)

−i

ω·x

x. (47)

Substituting x =y +b/2, we can rewrite (47)as

f(ω, b) =

(2π)



−(y+b/2)

·(y+b/2)

−(y−b/2)

−i

ω·(y+b/2)

−b

(2π)



−2y

−i

ω·y

·y

−i

(ω−ω

)·b/2

−b

(2π)



−2y

−i

(ω−ω

)·y

−i

(ω−ω

)·b/2

−b

(2π)

−(ω−ω

)

−i

(ω−ω

)·b/2

−b

32π

−(ω−ω

)

−i

(ω−ω

)·b/2

. (48)

The energy density is given by



f(ω, b)



−b

(32π

)

−(ω−ω

)

. (49)

Example 2 Consider the ﬁrst-order two-dimensional B-spline window function de-

ﬁned by

φ(x) =



1if0≤x

≤1 and 0 ≤ x

≤1,

0 otherwise.

(50)

Obtain the CWFT of the function deﬁned as follows:

f(x) =



x if 0 ≤x

≤1 and 0 ≤ x

≤1,

0 otherwise.

(51)

104 M. Bahri et al.

Fig. 1 The scalar part (left) and bivector part (right) of Clifford–Gabor ﬁlter for the parameters

0,1

=ω

0,2

=1,b

=0,σ

=σ

=1/

√

2 in the spatial domain using Mathematica 6.0

Fig. 2 Plot of the CWFT of

Clifford–Gabor ﬁlter of

Example 1 using

Mathematica 6.0. Note that it

is scalar valued for the

parameters b

Applying Deﬁnition 4 and simplifying it, we obtain

f(ω, b)

(2π)



1+b



1+b

−i

ω·x

(2π)



1+b



1+b

)



−i



(2π)



1+b

−i



1+b

−i

(2π)



1+b

−i



1+b

−i





(1 +i

)



−i

−1



−i



−i

−1



−e



(1 +i

)



−i

−1



−i



−i

−1



−i

ω·b

(2πω

)

(52)

with

b =b

Two-Dimensional Clifford Windowed Fourier Transform 105

Fig. 3 Representation of the CFT basis for ω

= ω

= 1 with scalar part (left) and bivector part

(right) using Mathematica 6.0

Fig. 4 Representation of the CWFT basis of a Gaussian window function for the parameters

0,1

= ω

0,2

= 1,b

= b

= 0.2 with scalar part (left) and bivector part (right) using Mathemat-

ica 6.0

5 ComparisonofCFTandCWFT

Since the Clifford–Fourier kernel e

−i

ω·x

is a global function, the CFT basis has

an inﬁnite spatial extension as shown in Fig. 3. In contrast, the CWFT basis

φ(x −b)e

−i

ω·x

has a limited spatial extension due to the local Clifford window

function φ(x − b) (see Fig. 4). This means that the CFT analysis cannot provide

information about the signal with respect to position and frequency, so that we need

the CWFT to fully describe the characteristics of the signal simultaneously in both

spatial and frequency domains.

6Conclusion

Using the basic concepts of Clifford geometric algebra and the CFT, we introduced

the CWFT. Important properties of the CWFT were demonstrated. This general-

ization enables us to work with 2D Clifford–Gabor ﬁlters, which can extend the

applications of the 2D complex Gabor ﬁlters.

Because the CWFT represents a signal in a joint space–frequency domain, it

can be applied in many ﬁelds of science and engineering, such as image analysis

and image compression, object and pattern recognition, computer vision, optics and

ﬁlter banks.

106 M. Bahri et al.

Acknowledgements The authors would like to thank the referees for critically reading the

manuscript. The authors further acknowledge R.U. Gobithaasan’s assistance in producing the ﬁg-

ures using Mathematica 6.0. This research was supported by the Malaysian Research Grant (Fun-

damental Research Grant Scheme) from the Universiti Sains Malaysia.

References

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

J. Math. Imaging Vis. 26(1), 5–18 (2006)

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images. Ph.D. thesis, University of Kiel, Germany (1999)

3. Bülow, T., Felsberg, M., Sommer, G.: Non-commutative hypercomplex Fourier transforms of

multidimensional signals. In: Sommer, G. (ed.) Geom. Comp. with Cliff. Alg., Theor. Found.

and Appl. in Comp. Vision and Robotics, pp. 187–207. Springer, Berlin (2001)

4. Gröchenig, K., Zimmermann, G.: Hardy’s Theorem and the short-time Fourier transform of

Schwartz functions. J. Lond. Math. Soc. 2(63), 205–214 (2001)

5. Hitzer, E., Mawardi, B.: Clifford Fourier transform on multivector ﬁelds and uncertainty prin-

ciple for dimensions n = 2(mod4)andn = 3 (mod 4). Adv. Appl. Clifford Algebr. 18(3–4),

715–736 (2008)

6. Kemao, Q.: Two-dimensional windowed Fourier transform for fringe pattern analysis: princi-

ples, applications, and implementations. Opt. Laser Eng. 45, 304–317 (2007)

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pattern. Appl. Opt. 46(14), 2670–2675 (2007)

The Cylindrical Fourier Transform

Fred Brackx, Nele De Schepper,

and Frank Sommen

Abstract The aim of this paper is to show the application potential of the cylin-

drical Fourier transform, which was recently devised as a new integral transform

within the context of Clifford analysis. Next to the approximation approach where,

using density arguments, the spectrum of various types of functions and distribu-

tions may be calculated starting from the cylindrical Fourier images of the L

-basis

functions in R

, direct computation methods are introduced for speciﬁc distribu-

tions supported on the unit sphere, and an illustrative example is worked out.

1 Introduction

The Fourier transform is by far the most important integral transform. Since its

introduction by Fourier in the early 1800s, it has remained an indispensable and

stimulating mathematical concept that is at the core of the highly evolved branch of

mathematics called Fourier analysis.

The second subject of great relevance for the paper is Clifford analysis, an elegant

and powerful higher-dimensional generalization of the theory of holomorphic func-

tions, which is moreover closely related but complementary to harmonic analysis.

Clifford analysis also offers the possibility to generalize one-dimensional mathe-

matical analysis to higher dimension in a rather natural way by encompassing all

dimensions at once, as opposed to the usual tensorial approaches.

It is precisely this last qualiﬁcation which has been exploited in [2] and [3] to con-

struct a genuine multidimensional Fourier transform within the context of Clifford

analysis. This so-called Clifford–Fourier transform is brieﬂy discussed in Sect. 3.

In [4] and [5] we devised and thoroughly studied the so-called cylindrical Fourier

transform within the Clifford analysis setting. The idea is the following: for a ﬁxed

N. De Schepper (



)

Clifford Research Group, Department of Mathematical Analysis, Ghent University, Galglaan 2,

9000 Gent, Belgium

e-mail: nds@cage.ugent.be

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

107