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

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118 F. Brackx et al.

Fig. 3 The real part of the

cylindrical Fourier spectrum

of the characteristic function

of a geodesic triangle on S

Fig. 4 The e

-component

of the cylindrical Fourier

spectrum of the characteristic

function of a geodesic

triangle on S

(i) ξ =η ∈S

m−1

, leading to an integral transform from S

m−1

to S

m−1

(ii) ξ

= ξ

+···+ξ

m−1

, i.e., ξ belongs to the afﬁne subspace given by

=0.

To evaluate the cylindrical Fourier transform explicitly, it sufﬁces in both cases to

express the function f(ω

) as a series of spherical monogenics and to apply a Funk–

Hecke argument on the spherical monogenics. This may lead to correspondences

between function spaces on S

m−1

and isomorphisms between them including inver-

sion methods. The establishment of direct inversion formulae remains an indepen-

dent and interesting problem for future research.

As an example (see Figs. 3, 4, 5, and 6 ), we have computed directly the cylin-

drical Fourier image of the characteristic function of a geodesic triangle on the two

sphere S

that may be expressed in spherical coordinates by the integral

(2π)

3/2



π/2



π/2

exp (ω ∧ξ ) sin (θ ) dθ dφ

with ω

=sin (θ) cos (φ)e

+sin (θ) sin (φ)e

+cos (θ)e

and ξ =ae

+be

6Conclusion

There is a recent increasing interest in integral transforms and in particular Fourier

transforms which take advantage of the algebraic structure inherent in hypercom-

plex function theories, especially quaternionic and Clifford analysis. In this pa-

The Cylindrical Fourier Transform 119

Fig. 5 The e

-component

of the cylindrical Fourier

spectrum of the characteristic

function of a geodesic

triangle on S

Fig. 6 The e

-component

of the cylindrical Fourier

spectrum of the characteristic

function of a geodesic

triangle on S

per we have shown that the recently developed cylindrical Fourier transform of

Clifford analysis in Euclidean space of arbitrary dimension is a promising higher-

dimensional integral transform with application potential. We have introduced a few

methods for the practical computation of the corresponding spectra and illustrated

one of these methods by working out an explicit example. For the theory underlying

the cylindrical Fourier transform and similar integral transforms in Clifford analy-

sis, we refer the reader to, e.g., [2–4] and in particular to the survey paper [5], and

the references contained therein.

References

1. Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis. Pitman Publishers, London (1982)

2. Brackx, F., De Schepper, N., Sommen, F.: The Clifford–Fourier transform. J. Fourier Anal.

Appl. (2005). doi:10.1007/s00041-005-4079-9

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

J. Math. Imaging Vis. (2006). doi:10.1007/s10851-006-3605-y

4. Brackx, F., De Schepper, N., Sommen, F.: The cylindrical Fourier spectrum of an L

-basis

consisting of generalized Clifford–Hermite functions. In: Simos, T.E., Psihoyios, G., Tsi-

touras, Ch. (eds.) Numerical Analysis and Applied Mathematics, AIP Conference Proceed-

ings, Kos, Greece, pp. 686–690 (2008)

5. Brackx, F., De Schepper, N., Sommen, F.: The Fourier transform in Clifford analysis. Adv.

Imaging Electron Phys. (2009). doi:10.1016/S1076-5670(08)01402-x

6. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions.

vol. 1. McGraw-Hill, New York (1953)

7. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Academic Press, San

Diego (1980)

Analyzing Real Vector Fields with Clifford

Convolution and Clifford–Fourier Transform

Wieland Reich and Gerik Scheuermann

Abstract Postprocessing in computational ﬂuid dynamics and processing of ﬂuid

ﬂow measurements need robust methods that can deal with scalar and vector ﬁelds.

While image processing of scalar data is a well-established discipline, there is a

lack of similar methods for vector data. This paper surveys a particular approach

deﬁning convolution operators on vector ﬁelds using geometric algebra. This in-

cludes a corresponding Clifford–Fourier transform including a convolution theo-

rem. Finally, a comparison is tried with related approaches for a Fourier transform

of spatial vector or multivector data. In particular, we analyze the Fourier series

based on quaternion holomorphic functions of Gürlebeck et al. (Funktionentheorie

in der Ebene und im Raum, Birkhäuser, Basel, 2006), the quaternion Fourier trans-

form of Hitzer (Proceedings of Function Theories in Higher Dimensions, 2006) and

the biquaternion Fourier transform of Sangwine et al. (IEEE Trans. Signal Process.

56(4),1522–1531, 2007).

1 Fluid Flow Analysis

Fluid ﬂow, especially of air and water, is usually modeled by the Navier–Stokes

equations or simpliﬁcations like the Euler equations [1]. The physical ﬁelds in this

model include pressure, density, velocity, and internal energy [22]. These variables

depend on space and often also on time. While there are mainly scalar ﬁelds, veloc-

ity is a vector ﬁeld and of high importance for any analysis of numerical or physical

experiments. Some numerical simulations use a discretization of the spatial domain

and calculate the variables at a ﬁnite number of positions on a regular lattice (ﬁnite

difference methods). Other methods split space into volume elements and assume

a polynomial solution of a certain degree in each volume element (ﬁnite element

methods or ﬁnite volume methods). These numerical methods create a large amount

W. Reich (



)

Institut fuer Informatik, Universität Leipzig, 04009 Leipzig, Germany

e-mail: reich@informatik.uni-leipzig.de

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

DOI 10.1007/978-1-84996-108-0_7, © Springer-Verlag London Limited 2010

121

122 W. Reich and G. Scheuermann

of data and its analysis, i.e., postprocessing, usually uses computer graphics to cre-

ate images. Research about this process is an important part of scientiﬁc visualiza-

tion [21].

Besides numerical simulations, modern measurement techniques like particle im-

age velocimetry (PIV) [16] create velocity vector ﬁeld measurements on a regular

lattice using laser sheets and image processing. Therefore, simulations and experi-

ments in ﬂuid mechanics create discretized vector ﬁelds as part of their output. The

analysis of these ﬁelds is an important postprocessing task. In the following, we will

assume that there is a way to continuously integrate the data to simplify the notation.

Of course, one can also discretize the integrals conversely.

Flow visualization knows a lot of direct methods like hedgehogs, streamlines,

or streamsurfaces [21] and also techniques based on mathematical data analysis

like topology or feature detection methods [10, 17, 20]. Since these methods are

often not very robust, a transfer of image processing to vector data is an attractive

approach. A look into any image processing book, e.g., by Jähne [15], reveals the

importance of shift-invariant linear ﬁlters based on convolution. Of course, there

is vector data processing in image processing, but the usual techniques for optical

ﬂow, which is the velocity ﬁeld warping one image into another, do not really help

as they concentrate mainly on noncontinuities in the ﬁeld which is not a typical

event in ﬂuid ﬂow velocity ﬁelds.

2 Geometric Algebra

In classical linear algebra, there are several multiplications involving vectors. Some

multiplications have led to approaches for a convolution on vector ﬁelds. Scalar mul-

tiplication can be easily applied and can be seen as a special case of component-wise

multiplication of two vectors which has been used by Granlund and Knutsson [8].

The scalar product has been used by Heiberg et al. [11] as a convolution operator.

Obviously, one would like a uniﬁed convolution operator that incorporates these ap-

proaches and can be applied several times, in contrast to the scalar product version

that creates a scalar ﬁeld using two vector ﬁelds. Furthermore, one looks for all the

nice theorems like convolution theorem with a suitable generalized Fourier trans-

form, derivation theorem, shift theorem, and Parseval’s theorem. Geometric algebra

allows such a convolution [5, 6].

Let R

,d=2, 3, be the Euclidean space with the orthonormal basis

} resp. {e

}. (1)

We use the 2

-dimensional real geometric algebras G

,d =2, 3,whichhaveaas-

sociative, bilinear geometric product satisfying

= e

,j=1,...,d, (2)

= 1,j=1,...,d, (3)

=−e

,j,k=1,...,d, j =k. (4)

Analyzing Real Vector Fields 123

Their basis is given by

{1,e

:=e

} resp. {1,e

:=e

}. (5)

It can be veriﬁed quickly that the squares of i

, i

and those of the bivectors

, and e

are −1. General elements in G

are called multivectors, while

elements of the form

v =



j=1

(6)

are called vectors, i.e., v ∈ R

⊂ G

. The geometric product of two vectors a,b ∈

results in

ab =a ·b +a ∧b, (7)

where · is the inner product, and ∧ is the outer product. We have

a ·b =|a||b|cos(α), (8)

|a ∧b|=|a||b|sin(α) (9)

with the usual norm for vectors and the angle α from a to b.

Let F be a multivector-valued function (ﬁeld) of a vector variable x deﬁned on

some region G of the Euclidean space R

; compare (19)ford = 3 and (23)for

d =2. We deﬁne the Riemann integral of F by



F(x)|dx|= lim

|Δx|→0,n→∞



j=1

F(x

)|Δx

|. (10)

We deﬁne Δx = dx

∧ dx

−1

(d = 2) resp. Δx = dx

∧ dx

−1

(d = 3)

as the dual oriented scalar magnitude. The quantity |Δx| is used here to make the

integral grade preserving since dx is a vector within geometric algebra in general.

The directional derivative of F in direction r is

(x) = lim

h→0

F(x +hr) −F(x)

(11)

with r ∈R

, h ∈ R. With the vector derivative

∇=



j=1

∂

∂e

(12)

(vector valued), the complete (left) derivative of F is deﬁned as

∇F(x)=



j=1

(x). (13)

124 W. Reich and G. Scheuermann

Similarly, we deﬁne the complete right derivative as

F(x)∇=



j=1

(x)e

. (14)

The curl and divergence of a vector ﬁeld f can be computed as scalar and bivector

parts of (12):

curlf =∇∧f =

(∇f −f ∇), divf =∇·f =

(∇f +f ∇). (15)

The readers interested in the basics and more applications of geometric algebra are

also referred to [12] and [4]. That ﬂuid ﬂow dynamics is accessible by geometric

algebra methods and is also discussed in [3].

3 Clifford Convolution

Let V,H : R

→G

be two multivector ﬁelds. We deﬁne the Clifford convolution

(H ∗V )(r) :=



H(ξ)V(r −ξ)|dξ|. (16)

If both ﬁelds are scalar ﬁelds, this is the usual convolution in image processing. If

H is a scalar ﬁeld, e.g., a Gaussian kernel, and V is a vector ﬁeld, we get a scalar

multiplication and can model smoothing of a vector ﬁeld. If H is a vector ﬁeld, and

V a is multivector ﬁeld, the simple relation

H(ξ)V(r −ξ)=H(ξ)·V(r−ξ)+H(ξ)∧V(r−ξ) (17)

shows that the scalar part of the result is Heiberg’s convolution, while the bivector

part contains additional information. General multivector ﬁelds allow a closed op-

eration in G

, so that several convolutions can be combined. We have shown [5, 7]

that this convolution can be used for the analysis of velocity vector ﬁelds from com-

putational ﬂuid dynamics (CFD) simulations and PIV measurements.

4 Clifford–Fourier Transform

Our group has found a generalization of the Fourier transform of complex signals to

multivector ﬁelds [6] that allows the generalization of the well-known theorems like

the convolution theorem for the convolution deﬁned in the previous section. There

are different approaches of transforming multivector valued data, e.g., in [2, 14],

and [18]. In Sect. 5 we discuss the relation to ours.

Analyzing Real Vector Fields 125

Let F : R

→G

be a multivector ﬁeld. We deﬁne

F{F }(u) :=



F(x)exp(−2πi

x ·u)|dx| (18)

as the Clifford–Fourier transform (CFT) with the inverse

−1

{F }(x) :=



F(u)exp(2πi

x ·u)|du|. (19)

In three dimensions, the CFT is a linear combination of four classical complex

Fourier transforms as can be seen by looking at the real components. Since

F(x) = F

(x) +F

(x)e

123

(x)e

(20)

= F

(x) +F

(x)e

(x)i

123

(x)i

(21)



(x) +F

123

(x)i



1 +



(x) +F

(x)i





(x) +F

(x)i





(x) +F

(x)i



, (22)

we get

F{F }(u) =



F{F

123

}(u)



1 +



F{F

}(u)





F{F

}(u)





F{F

}(u)



. (23)

We have proven earlier [6] that the convolution, derivative, shift, and Parseval the-

orem hold. For vector ﬁelds, we can see that the CFT treats each component as a

real signal that is transformed independently from the other components. Various

examples of 3D patterns and their CFT are shown in Fig. 1.

In two dimensions, the CFT is a linear combination of two classical complex

Fourier transforms. We have

F(x) = F

(x) +F

(x)e

(24)

= F

(x) +F

(x)e

(x)i

(25)

= 1



(x) +F

(x)i





(x) +F

(x)i



(26)

and obtain

F{F }(u) =1



F{F

}(u)





F{F

}(u)



. (27)

Regarding the convolution theorem, one has to separate the vector and spinor parts

since i

does not commute with all algebra elements. With this restriction, the the-

orem holds again [6]. Figure 2 shows a turbulent ﬂuid in 2D, in Figs. 3 and 4 its

discrete CFT is visualized.

126 W. Reich and G. Scheuermann

Fig. 1 Top: Various 3D patterns. Middle: The vector part of their DCFT. Bottom: The bivector part

of their DCFT, displayed as normal vector of the plane. Left:3×3 ×3 rotation in one coordinate

plane. Middle:3×3 ×3 convergence. Right:3×3 ×3 saddle line. The mean value of the discrete

Clifford–Fourier transform (DCFT) is situated in the center of the ﬁeld. In 3D, the waves forming

the patterns can be easily seen in the frequency domain. The magnitude of the bivectors of the

DCFT is only half the magnitude of the corresponding vectors, though both are displayed with

same length

5 Relation to Other Fourier Transforms

In recent years, other deﬁnitions of a Fourier transform have appeared in the liter-

ature that can be applied to vector ﬁelds. In this section, we compare our approach

with the Fourier series based on quaternion analysis by Gürlebeck, Habetha, and

Sprößig [9], the biquaternion Fourier transform by Sangwine et al. [19], and the

quaternion Fourier transform by Hitzer in [13]. Therefore we use three important

isomorphisms. First of all, quaternions are isomorphic to the even subalgebra G

of G

i →e

,j→e

,k→e

, (28)

biquaternions are isomorphic to G

by additionally

I →e

. (29)

Analyzing Real Vector Fields 127

Fig. 2 A 2D slice of a turbulent swirling jet entering a ﬂuid at rest. The image shows color coding

of the absolute magnitude of the vectors. The colors are scaled from zero (blue) to the maximal

magnitude (red)

Fig. 3 This image shows a (fast) discrete Clifford–Fourier transform of the data set. Zero fre-

quency is located in the middle of the image. Vectors are treated as rotors when using Clifford

algebra in the frequency domain, thus color coding is based on the magnitude of the transformed

rotor. The scaling of the colors is the same as the last image. We zoomed in to get more information