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

Подождите немного. Документ загружается.

88 H. Li

generates A by the Cayley transform. So the Cayley transform provides a quarter-

distance bivector representation of the translation.

Example 7 Let A =e

e∧a

be a rotor realizing a dilation (or scaling), where θ ∈R,

a is a null vector representing a point, and a·e =−1. Rotor A generates the dilation

centering at point a and with scale e

−θ

. Denote I

=e ∧a. Then

−e

−I

tanh

, B

−1

/ tanh

, (53)

and both generate A by the Cayley transform. So the Cayley transform provides a

quarter-scale bivector representation of the dilation.

All orientation-preserving similarity transformations in R

can be induced by

bivectors in Λ

4,1

) through the Cayley transform and adjoint action. A translation

is induced by the Cayley transform of a unique bivector. Any other orientation-

preserving similarity transformation is induced by the Cayley transform of exactly

two bivectors.

When choosing between the two bivector preimages B

and B

−1

of a rotor, since

||B

−1

|=1, one of |B

| and |B

−1

| is greater than or equal to 1. By (50),



−B

−1



|A|

|A

≥2. (54)

So for two rotors that are close to each other, we can always choose their bivector

preimages to be close to each other. If their magnitudes are greater than 1, we can

choose their inverses so that the magnitudes become smaller than 1.

Then what is the use of having two bivector preimages for the same rotor? Take,

as an example, the 2D rotation in Example 5. It is well known that SO(2) is a cir-

cle which has the following rational parameterization induced by the stereographic

projection from the north pole N:

=cos θ +I

sin θ =

1 −t

1 +t

1 −t

, (55)

where, as shown in Fig. 2, t = tan(θ/2) is half the signed distance from the south

pole S to point X in the horizontal direction, and point e

iθ

in the complex plane is

represented by the intersection R of line NX with the unit circle.

Since the map θ → e

is an isometric immersion, the Jacobian of the rational

parameterization (55) equals

dθ





dθ



=2 cos

. (56)

When point R moves from the south pole S to point T , the Jacobian decreases from

2 to 1, while when point R continues to move from point T to the north pole N,the

Parametrization of 3D Conformal Transformations 89

Fig. 2 Rational parametrization of a 2D rotation

Jacobian decreases from 1 to 0. The parameterization becomes practically useless

nearby the north pole.

The two preimages in (51) are both mapped to the rotor e

θ/2

by the Cayley

transform. So the Cayley transform provides a generalization of the rational param-

eterization of a circle taken as the 2D rotation group. In fact, it serves as two rational

parameterizations of the circle derived from two different stereographic projections:

one from the north pole, and the other from the south pole.

Since the two maps

θ →I

tan

and θ →−I

ctan

(57)

have Jacobians 1/(4 cos

(θ/4)) and 1/(4sin

(θ/4)), respectively, the maps

t =tan

Cayley

−→ e

θ/2

t =−ctan

Cayley

−→ e

θ/2

(58)

have Jacobians J

=2 cos

and J

=2sin

, respectively.

• When (4k −1)π ≤θ ≤(4k +1)π , then 1 ≤J

≤2 and 0 ≤J

≤1.

• When (4k +1)π ≤θ ≤(4k +3)π , then 1 ≤J

≤2 and 0 ≤J

≤1.

Hence the two maps in (58) cover different zones of the parameter θ ∈R for the Ja-

cobians to be effective between 1 and 2. They serve as two different local coordinate

charts whose union covers the whole Lie group.

The Cayley transform as a polynomial map of degree 4 is computationally su-

perior; its inverse map has two branches and involves only one square-root com-

puting. Its domain of deﬁnition and its image space, when restricted to orientation-

preserving conformal transformations, are both larger than those of exterior differ-

ential and quaternionic Vahlen parameterization. Thus the Cayley transform and its

inverse are an ideal tool for motion planning, interpolating, and ﬁtting in the Lie

algebra of 3D conformal transformations.

90 H. Li

5Conclusion

This chapter explores the issue of parameterizing 3D conformal transformations.

Two new results are presented, one on quaternionic Vahlen parameterization, the

other on the polynomial 3D Cayley transform. They provide compact representa-

tions of 3D conformal transformations and should prove to be useful in geometric

applications.

References

1. Ahlfors, L.V.: Möbius transformations in R

expressed through 2 × 2 matrices of Clifford

numbers. Complex Var. 5, 215–224 (1986)

2. Angles, P.: Conformal Groups in Geometry and Spin Structures. Birkhäuser, Basel (2008)

3. Bayro-Corrochano, E., Reyes-Lozano, L., Zamora-Esquivel, J.: Conformal geometric algebra

for robotic vision. J. Math. Imaging Vis. 24(1), 55–81 (2006)

4. Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra for Computer Science. Morgan Kauf-

mann, San Mateo (2007)

5. Helmstetter, J., Micali, A.: Quadratic Mappings and Clifford Algebras. Birkhäuser, Basel

(2000)

6. Hestenes, D., Sobczyk, G.: Clifford Algebra to Geometric Calculus. Kluwer, Dordrecht

(1984)

7. Hildenbrand, D.: Geometric computing in computer graphics using conformal geometric al-

gebra. Comput. Graph. 29(5), 795–803 (2005)

8. Lasenby, A.: Recent applications of conformal geometric algebra. In: Li, H., et al. (eds.)

Computer Algebra and Geometric Algebra with Applications. LNCS, vol. 3519, pp. 298–328.

Springer, Berlin (2005)

9. Li, H.: Invariant Algebras and Geometric Reasoning. World Scientiﬁc, Singapore (2008)

10. Li, H., Hestenes, D., Rockwood, A.: Generalized homogeneous coordinates for computational

geometry. In: Sommer, G. (ed.) Geometric Computing with Clifford Algebras, pp. 27–60.

Springer, Heidelberg (2001)

11. Lounesto, P.: Clifford Algebras and Spinors. Cambridge University Press, Cambridge (1997)

12. Maks, J.: Clifford algebras and Möbius transformations. In: Micali, A., et al. (eds.) Clifford Al-

gebras and Their Applications in Mathematical Physics, pp. 57–63. Kluwer, Dordrecht (1992)

13. Mourrain, B., Stolﬁ, N.: Computational symbolic geometry. In: White, N.L. (ed.) Invariant

Methods in Discrete and Computational Geometry, pp. 107–139. Reidel, Dordrecht (1995)

14. Riesz, M.: Clifford Numbers and Spinors, 1958. Kluwer, Dordrecht (1993). From lecture notes

made in 1957–1958, edited by Bolinder, E. and Lounesto, P.

15. Rosenhahn, B., Sommer, G.: Pose estimation in conformal geometric algebra I, II. J. Math.

Imaging Vis. 22, 27–70 (2005)

16. Ryan, J.: Conformal Clifford manifolds arising in Clifford analysis. Proc. R. Ir. Acad. A 85,

1–23 (1985)

17. Selig, J.M.: Geometrical Methods in Robotics. Springer, New York (1996)

18. Sommer, G., Rosenhahn, B., Perwass, C.: Twists—an operational representation of shape. In:

Li, H., et al. (eds.) Computer Algebra and Geometric Algebra with Applications. LNCS, vol.

3519, pp. 278–297. Springer, Berlin (2005)

19. White, N.: Grassmann–Cayley algebra and robotics applications. In: Bayro-Corrochano, E.

(ed.) Handbook of Geometric Computing, pp. 629–656. Springer, Heidelberg (2005)

Part II

Clifford Fourier Transform

Two-Dimensional Clifford Windowed Fourier

Transform

Mawardi Bahri, Eckhard M.S. Hitzer,

and Sriwulan Adji

Abstract Recently several generalizations to higher dimension of the classical

Fourier transform (FT) using Clifford geometric algebra have been introduced, in-

cluding the two-dimensional (2D) Clifford–Fourier transform (CFT). Based on the

2D CFT, we establish the two-dimensional Clifford windowed Fourier transform

(CWFT). Using the spectral representation of the CFT, we derive several important

properties such as shift, modulation, a reproducing kernel, isometry, and an orthog-

onality relation. Finally, we discuss examples of the CWFT and compare the CFT

and CWFT.

1 Introduction

One of the basic problems encountered in signal representations using the conven-

tional Fourier transform (FT) is the ineffectiveness of the Fourier kernel to represent

and compute location information. One method to overcome such a problem is the

windowed Fourier transform (WFT). Recently, some authors [4, 7] have extensively

studied the WFT and its properties from a mathematical point of view. In [6, 8]they

applied the WFT as a tool of spatial-frequency analysis which is able to characterize

the local frequency at any location in a fringe pattern.

On the other hand, Clifford geometric algebra leads to the consequent generaliza-

tion of real and harmonic analysis to higher dimensions. Clifford algebra accurately

treats geometric entities depending on their dimension as scalars, vectors, bivec-

tors (oriented plane area elements), and trivectors (oriented volume elements), etc.

Motivated by the above facts, we generalize the WFT in the framework of Clifford

geometric algebra.

In the present paper we study the two-dimensional Clifford windowed Fourier

transform (CWFT). A complementary motivation for studying this topic comes from

M. Bahri (



)

School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia

e-mail: mawardibahri@gmail.com

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

DOI 10.1007/978-1-84996-108-0_5, © Springer-Verlag London Limited 2010

94 M. Bahri et al.

the understanding that the 2D CWFT is in fact intimately related with Clifford–

Gabor ﬁlters [1] and quaternionic Gabor ﬁlters [2, 3]. This generalization also en-

ables us to establish the two-dimensional Clifford–Gabor ﬁlters.

2 Real Clifford Algebra G

Let us consider an orthonormal vector basis {e

, e

} of the real 2D Euclidean vector

space R

2,0

. The geometric algebra over R

denoted by G

then has the graded

four-dimensional basis

{1, e

, e

}, (1)

where 1 is the real scalar identity element (grade 0), e

, e

∈ R

are vectors

(grade 1), and e

= e

= i

deﬁnes the unit oriented pseudoscalar

(grade 2),

i.e., the highest grade blade element in G

The associative geometric multiplication of the basis vectors obeys the following

basic rules:

=1, e

=−e

. (2)

The general elements of a geometric algebra are called multivectors. Every mul-

tivector f ∈G

can be expressed as

f = α



scalar part

+α

  

vector part

+ α



bivector part

∀α

,α

∈R. (3)

The grade selector is deﬁned as f 

for the k-vector part of f . We often write

...=...

. Then (3) can be expressed as

f =f +f 

+f 

. (4)

The multivector f is called a parabivector if the vector part of (4) is zero, i.e.,

f =α

+α

. (5)

The reverse

f of a multivector f ∈G

is an anti-automorphism given by



f =f +f 

−f 

, (6)

which fulﬁlls



fg =˜g

f for every f,g ∈G

. In particular,

=−i

The scalar product of two multivectors f, ˜g is deﬁned as the scalar part of the

geometric product f ˜g,

f ∗˜g =f ˜g=α

+α

, (7)

Other names in use are bivector or oriented area element.

Note that (4)and(6) show grade selection and not component selection.

Two-Dimensional Clifford Windowed Fourier Transform 95

which leads to the cyclic product symmetry

pqr=qrp∀p,q, r ∈G

. (8)

For f =g in (7), we obtain the modulus (or magnitude) |f | of a multivector f ∈G

deﬁned as

|f |

=f ∗

f =α

+α

. (9)

It is convenient to introduce an inner product for two multivector-valued func-

tions f, g :R

→G

as follows:

(f, g)

)



f(x)



g(x)d

x. (10)

One can check that this inner product satisﬁes the following rules:

(f, g +h)

)

=(f, g)

)

+(f, h)

)

(f, λg)

)

=(f, g)

)

λ,

(f λ, g)

)

=(f, g

λ)

)

(f, g)

)



(g, f )

)

(11)

where f, g ∈ L

), and λ ∈ G

is a multivector constant. The scalar part of

the inner product gives the L

-norm

f 

)



(f, f )

)



. (12)

Deﬁnition 1 (Clifford module) Let G

be the real Clifford algebra of 2D Euclidean

space R

. The Clifford algebra module L

) is deﬁned by







f :R

−→ G

|f 

)

< ∞



. (13)

3 Clifford Fourier Transform (CFT)

It is natural to extend the FT to the Clifford algebra G

. This extension is often called

the Clifford–Fourier transform (CFT). For detailed discussions on the properties of

the CFT and their proofs, see, e.g., [1, 5]. In the following we brieﬂy review the 2D

CFT.

Deﬁnition 2 The CFT of f ∈ L

) ∩ L

) is the function F {f }:

→G

given by

F {f }(ω) =



f(x)e

−i

ω·x

x, (14)

96 M. Bahri et al.

where we can write ω =ω

+ω

and x =x

. Note that

x =

∧dx

(15)

is scalar valued (dx

= dx

, k =1, 2, no summation). Notice that the Clifford–

Fourier kernel e

−i

ω·x

does not commute with every element of the Clifford alge-

bra G

. Furthermore, the product has to be performed in a ﬁxed order.

Theorem 1 Suppose that f ∈ L

) and F {f }∈L

). Then the CFT

is an invertible transform, and its inverse is calculated by

−1



F {f }(ω)



(x) =f(x) =

(2π)



F {f }(ω)e

ω·x

ω. (16)

4 2D Clifford Windowed Fourier Transform

In [1, 5] the 2D CFT has been introduced. This enables us to establish the 2D CWFT.

We will see that several properties of the WFT can be established in the new con-

struction with some modiﬁcations. We begin with the deﬁnition of the 2D CWFT.

4.1 Deﬁnition of the CWFT

Deﬁnition 3 A Clifford window function is a function φ ∈L

) \{0} such

that |x|

1/2

φ(x) ∈L

ω,b

(x) =

ω·x

φ(x −b)

(2π)

(17)

denote the so-called Clifford window daughter functions.

Deﬁnition 4 (Clifford windowed Fourier transform) The Clifford windowed

Fourier transform (CWFT) G

f of f ∈L

) is deﬁned by

f(x) −→ G

f(ω,b) =(f, φ

ω,b

)

(2π)



f(x)



ω·x

φ(x −b)



∼

(2π)



f(x)



φ(x −b)e

−i

ω·x

x. (18)

This shows that the CWFT can be regarded as the CFT of the product of a

Clifford-valued function f and a shifted and reversed Clifford window function φ,

Two-Dimensional Clifford Windowed Fourier Transform 97

or as an inner product (10) of a Clifford-valued function f and the Clifford window

daughter functions φ

Taking the Gaussian function as the window function of (17) with ﬁxed ω =

=ω

0,1

+ω

0,2

, we obtain Clifford Gabor ﬁlters, i.e.,

(x,σ

,σ

) =

(2π)

·x

−[(x

/σ

)

+(x

/σ

)

]/2

, (19)

where σ

and σ

are standard deviations of the Gaussian functions, and the transla-

tion parameters are b

=0.

In terms of the G

Clifford–Fourier transform, (19) can be expressed as

F {g

}(ω) =

πσ

−

[(σ

(ω

−ω

0,1

)

+σ

(ω

−ω

0,2

)

]

. (20)

From (19) and (20) we see that Clifford–Gabor ﬁlters are well localized in the spatial

and Clifford–Fourier domains.

The energy density is deﬁned as the square modulus of the CWFT (18) given by



f(ω, b)



(2π)





f(x)



φ(x −b)e

−i

ω·x



. (21)

Equation (21) is often called a spectrogram which measures the energy of a Clifford-

valued function f in the position–frequency neighborhood of (b, ω).

In particular, when the Gaussian function (19) is chosen as the Clifford window

function, the CWFT (18) is called the Clifford–Gabor transform.

4.2 Properties of the CWFT

We will discuss the properties of the CWFT. We ﬁnd that many of the properties of

the WFT are still valid for the CWFT, however, with certain modiﬁcations.

Theorem 2 (Left linearity) Let φ ∈L

) be a Clifford window function. The

CWFT of f, g ∈L

) is a left linear operator,

which means that



(λf +μg)



(ω, b) =λG

f(ω, b) +μG

g(ω, b) (22)

with Clifford constants λ,μ ∈G

Proof Using the deﬁnition of the CWFT, the proof is obvious. 

Remark 1 Since the geometric multiplication is noncommutative, the right-linearity

property of the CWFT does not hold in general.

The CWFT of f is a linear operator for real constants μ, λ ∈R.