An overview is given to a new approach for obtaining generalized Fourier transforms in the context of hypercomplex analysis (or Clifford analysis). These transforms are applicable to higher-dimensional signals with several components and are different from the classical Fourier transform in that they mix the components of the signal. Subsequently, attention is focused on the special case of the so-called Clifford-Fourier transform where recently a lot of progress has been made. A fractional version of this transform is introduced and a series expansion for its integral kernel is obtained

De Bie, Hendrik

De Schepper, Nele

English

Ghent University Academic Bibliography

Fourier transforms of hypercomplex signalsHendrik De Bie1 and Nele De Schepper11Department of Mathematical AnalysisFaculty of Engineering and ArchitectureGhent UniversityGalglaan 2, 9000 Ghent, Belgium{hendrik.debie,nele.deschepper}@ugent.beAbstract — An overview is given to a new approach for obtaining generalizedFourier transforms in the context of hypercomplex analysis (or Clifford analysis).These transforms are applicable to higher-dimensional signals with several com-ponents and are different from the classical Fourier transform in that they mix thecomponents of the signal.Subsequently, attention is focused on the special case of the so-called Clifford-Fouriertransform where recently a lot of progress has been made. A fractional version of thistransform is introduced and a series expansion for its integral kernel is obtained.1 IntroductionRecently, there has been a lot of interest in the theory of hypercomplex signals (i.e. func-tions taking values in a Clifford algebra) and the possibility of defining and using Fouriertransforms that interact with the Clifford algebra structure. This has been investigatedfrom a practical engineering point of view (see e.g. [1, 2, 3, 4, 5]) but also from a purelymathematical point of view (see e.g. [6, 7, 8]) using the function theory of Clifford anal-ysis established in a.o. [9]. Also in applications, there is an increasing interest in havingavailable a good hypercomplex Fourier transform (e.g. in GIS research, see [10]).In the context of this paper, a hypercomplex signal is a function ofm variables x1, . . . , xmtaking values in a Clifford algebra Cl0,m. This means that our signals can equivalently berepresented by a vector[f1(x1, . . . , xm), . . . , f2m(x1, . . . , xm)]with at most 2m components fi(x1, . . . , xm) which are real-valued signals.There are quite a few drawbacks to most of the kernels proposed so far in the literature.First, several authors work only in low dimensions (dimension 3 or 4, enabling them touse quaternions instead of a full Clifford algebra) which is usually because they havea specific application in mind in these dimensions. Second, most authors use ad hocformulations for the kernel function of their transforms, after which the properties of therelated transform are studied in detail.DE BIE, DE SCHEPPERIn recent work (see [11, 12, 13, 14, 15]) we have developed a different methodology: westart from a list of properties or general mathematical principles we want a hypercomplexFourier transform (hypercomplex FT) to have, and then determine all kernels that satisfythese properties.In the present paper, after a short introduction to Clifford algebras and analysis, wegive an overview of the results obtained so far. In particular we devote our attention tothe so-called Clifford-Fourier transform (as studied in [15, 11]). Finally, we introduce anew fractional version of the Clifford-Fourier transform and obtain an expression for itsintegral kernel.2 PreliminariesClifford analysis (see e.g. [9]) is a theory that offers a natural generalization of complexanalysis to higher dimensions. To Rm, the Euclidean space in m dimensions, we firstassociate the Clifford algebra Cl0,m, generated by the canonical basis ei, i = 1, . . . ,m.These generators satisfy the multiplication rules eiej + ejei = −2δij .The Clifford algebra Cl0,m can be decomposed as Cl0,m = ⊕mk=0Clk0,m with Clk0,m thespace of k-vectors defined byClk0,m = span{ei1...ik = ei1 . . . eik , i1 < . . . < ik}.More precisely, we have that the space of 1-vectors is given by Cl10,m = span{ei, i =1, . . . ,m} and it is obvious that this space is isomorphic with Rm. The space of so-calledbivectors is given explicitly by Cl20,m = span{eij = eiej, i < j}.We identify the point (x1, . . . , xm) in Rm with the vector variable x given by x =∑mj=1 xjej . The Clifford product of two vectors splits into a scalar part and a bivectorpart:xy = x.y + x ∧ y,withx.y = −〈x, y〉 = −m∑j=1xjyj =12(xy + yx)andx ∧ y =∑j<kejk(xjyk − xkyj) = 12(xy − yx).Note that the square of a vector variable x is scalar-valued and equals the norm squared upto a minus sign: x2 = −〈x, x〉 = −|x|2. We also introduce a first order vector differentialoperator by∂x =m∑j=1∂xjej.This operator is the so-called Dirac operator. Its square equals, up to a minus sign, theLaplace operator in Rm: ∂2x = −∆.The Dirac operator ∂x acts on functions f taking values in Cl0,m. Such functions canalways be decomposed asf(x) = f0(x) +m∑i=1eifi(x) +∑i<jeiejfij(x) + . . .+ e1 . . . emf1...m(x)FOURIER TRANSFORMS OF HYPERCOMPLEX SIGNALSwith f0, fi, fij, . . . , f1...m all real-valued functions. As f has in total 2m components(which equals the dimension of Cl0,m), this is the maximal number of components ofthe signals that can be studied using the transforms in the next section.Finally, another important operator in Clifford analysis is the so-called Gamma opera-tor, defined byΓx = −x ∧ ∂x = −∑j<kejk(xj∂xk − xk∂xj).This operator is bivector-valued.3 Overview of recent resultsIn the paper [11] we started our analysis from 4 different, yet equivalent, definitions of theclassical Fourier transform in Rm. Each of these formulations allows for generalizationto hypercomplex FTs. We summarize them briefly. The first and most basic formulationis given by the integral transformF1 F [f ](y) = 1(2pi)m/2∫Rme−i〈x,y〉 f(x) dxwith 〈x, y〉 the standard inner product and dx the Lebesgue measure onRm. Alternatively,one can rewrite the transform asF2 F [f ](y) = 1(2pi)m/2∫RmK(x, y) f(x) dxwhere K(x, y) is, up to a multiplicative constant, the unique solution of the system ofPDEs∂yjK(x, y) = −ixjK(x, y), j = 1, . . . ,m.Yet another formulation is given byF3 F = e ipim4 e ipi4 (∆−|x|2)with ∆ the Laplacian in Rm. This expression connects the Fourier transform with theLie algebra sl2 generated by ∆ and |x|2 and with the theory of the quantum harmonicoscillator. Finally, the kernel can also be expressed as an infinite series in terms of specialfunctions asF4 K(x, y) = 2λΓ(λ)∞∑k=0(k + λ)(−i)k(|x||y|)−λJk+λ(|x||y|) Cλk (〈ξ, η〉), (1)where ξ = x/|x|, η = y/|y| and λ = (m− 2)/2. Here, Jν is the Bessel function and Cλkthe Gegenbauer polynomial.Each formulation has its specific advantages and uses. F1 immediately yields a boundof the kernel and is hence ideal to study the transform on L1 spaces or more generalfunction spaces. F2 gives the calculus properties of the transform. F3 emphasizes thestructural (Lie algebraic) properties of the Fourier transform and allows to compute itsDE BIE, DE SCHEPPEReigenfunctions and spectrum. Finally, F4 connects the Fourier transform with the theoryof special functions, and is the ideal formulation to obtain e.g. the Bochner identities.These 4 definitions serve as guidance in defining hypercomplex Fourier transforms, aseach definition gives access to a crucial piece of information about the transform. Asuitable hypercomplex transform should hence be expressible in these 4 different ways.So far, we have applied these ideas in 3 different directions of hypercomplex FTs,namely– k-vector Fourier transforms ([12])– radially deformed Fourier transforms ([13, 14])– Clifford-Fourier transforms ([15, 11]).We briefly discuss the first 2 directions, after which we focus on the Clifford-Fouriertransform. k-vector Fourier transforms are transforms with integral kernel K(x, y) a k-vector valued function (i.e. taking values in Clk0,m) which is moreover a solution to thesystem of PDEs (compare with F2)∂yK(x, y) = −iK(x, y)x. (2)Contrary to the classical case, the system (2) does not have a single unique solution butinstead a whole family of suitable solutions.Radially deformed Fourier transforms are obtained by adapting the formulation F3 ofthe classical Fourier transform toFD = eipi2 (12+ µ−12(1+c))e−ipi4(1+c)2(D2−(1+c)2x2).withD = ∂x + c|x|−2xm∑j=1xj∂xj ,a radial deformation (depending on the numerical parameter c ∈ R) of the Dirac operator∂x underlying the Laplace operator and µ = −D(x) ∈ R. So far, for this class of trans-forms also formulation F2 and F4 have been obtained, but the question to find an explicitanalytical expression for its kernel (formulation F1) is extremely hard.The situation is quite different for the Clifford-Fourier transform, where by now analmost complete treatment has been obtained. This transform was first introduced in [16]and further studied in [17, 18]. It was initially also defined by a generalization of F3,namelyF± = e ipim4 e∓ ipi2 Γe ipi4 (∆−|x|2) = e ipim4 e ipi4 (∆−|x|2∓2Γ).The motivation behind this definition was to find a couple of transforms F± such thatF+F− = F−F+ = F2.It turned out to be a difficult problem to find the other formulations for this particulartransform. A breakthrough was recently obtained in [15]. As an analog of formulation F4the following result was obtained:FOURIER TRANSFORMS OF HYPERCOMPLEX SIGNALSTheorem 3.1 The Clifford-Fourier transform F− = e ipim4 e ipi2 Γe ipi4 (∆−|x|2) is given by theintegral transform1(2pi)m/2∫RmK−(x, y) f(x) dxwith integral kernelK−(x, y) = Aλ +Bλ +(x ∧ y)CλwithAλ(w, z) = 2λ−1Γ(λ+ 1)∞∑k=0(i2λ+2 + (−1)k)z−λJk+λ(z) Cλk (w),Bλ(w, z) = − 2λ−1Γ(λ)∞∑k=0(k + λ)(i2λ+2 − (−1)k)z−λJk+λ(z) Cλk (w),Cλ(w, z) = − 2λΓ(λ+ 1)∞∑k=1(i2λ+2 + (−1)k)z−λ−1Jk+λ(z) Cλ+1k−1 (w),where z = |x||y|, w = 〈ξ, η〉 and λ = (m− 2)/2.Note that it can be computed that the kernel K+(x, y) of F+ is given by K+(x, y) =K−(x,−y) where the bar denotes complex conjugation.Moreover, it was then possible to obtain formulation F1 in the case where the dimensionm is even.Theorem 3.2 The kernel of the Clifford-Fourier transform in even dimension m > 2 isgiven byK−(x, y) = eipi2Γye−i〈x,y〉= (−1)m2(pi2) 12 (A∗(m−2)/2(s, t) +B∗(m−2)/2(s, t) + (x ∧ y) C∗(m−2)/2(s, t))where s = 〈x, y〉 and t = |x ∧ y| =√|x|2|y|2 − s2 andA∗(m−2)/2(s, t) =bm4 − 34c∑`=0sm/2−2−2`12``!Γ(m2)Γ(m2− 2`− 1) J˜(m−2`−3)/2(t),B∗(m−2)/2(s, t) = −bm4 − 12c∑`=0sm/2−1−2`12``!Γ(m2)Γ(m2− 2`) J˜(m−2`−3)/2(t),C∗(m−2)/2(s, t) = −bm4 − 12c∑`=0sm/2−1−2`12``!Γ(m2)Γ(m2− 2`) J˜(m−2`−1)/2(t)with J˜α(t) = t−αJα(t).DE BIE, DE SCHEPPERSubsequently, in [11] we studied the formulation F2 for the Clifford-Fourier transform,which determines the kernel of this transform as a solution of the following system ofPDEs:∂y[K+(x, y)] = (−i)m K−(x, y) x[K+(x, y)]∂x = (−i)m y K−(x, y).(3)Indeed, it is e.g. easy to check that the result of Theorem 3.2 satisfies (3). However, anew phenomenon appears here, as the solution given by Theorem 3.2 is not the uniquesolution of the system (contrary to the case of the ordinary Fourier transform or the Dunkltransform, see [19]).To obtain more insight in this phenomenon, we determined all solutions of (3) that areof the formK−(x, y) = f(s, t) + (x ∧ y) g(s, t)with s = 〈x, y〉, t = |x ∧ y| and f and g real-valued functions. In this way, we aimed tofind an explanation for the precise form of the kernel given in Theorem 3.2.As result, we obtained, as solutions of (3) in even dimension m, a set of m− 1 kernelsKj−,m(x, y), j = 0, . . . ,m− 2that can be organized in a diagram as followsm = 2 m = 4 m = 6 m = 8K6−,8K4−,655// K5−,8K2−,455// K3−,6 // K4−,8K0−,255))// K1−,4 // K2−,6 // K3−,8K0−,4))// K1−,6 // K2−,8K0−,6))// K1−,8K0−,8Here the arrows describe certain recursion relations (given explicitly in [11]).The middle line, namely the kernels K(m−2)/2−,m (x, y) correspond to the kernels obtainedin Theorem 3.2. The other kernels yield new integral transforms that have the same cal-culus properties as the original Clifford-Fourier transform but with different spectrum.Of all these transforms, only the transform given by Theorem 3.2 is unitary. We observeFOURIER TRANSFORMS OF HYPERCOMPLEX SIGNALSthat with each step in the dimension, 2 new kernels appear. In particular the kernelsK0−,m(x, y) on the lower diagonal are interesting, as they give us the Fourier-Bessel trans-form which was previously introduced in [20] for different reasons.For explicit formulas for the kernels Kj−,m(x, y) we refer the reader to formula (4.6)and Theorem 4.2 (where series representations as in F4 are obtained) in [11].4 Fractional version of the Clifford-Fourier transformIn this section, we show how a fractional version of the Clifford-Fourier transform can beobtained. To that end, we adapt the definition F3F± = e ipim4 e ipi4 (∆−|x|2∓2Γ)toF±,α = e iαm2 e iα2 (∆−|x|2∓2Γ)where α ∈ [−pi/2, pi/2]. Notice that we have immediatelyF+,αF−,α = F2αwith Fα = e iαm2 e iα2 (∆−|x|2) the fractional version of the ordinary Fourier transform, givenexplicitly by (see [21])Fα(f)(y) =(pi(1− e−2iα))−m/2 ∫Rme−i〈x,y〉/ sinαei2(cotα)(|x|2+|y|2)f(x)dxOur aim is to find an integral expression for F−,α:F−,α(f)(y) =(pi(1− e−2iα))−m/2 ∫RmK−,α(x, y) f(x) dx.We compute formallyF−,α = e iαm2 e iα2 (∆−|x|2+2Γ)= eiαm2 eiαΓeiα2(∆−|x|2)=(pi(1− e−2iα))−m/2 eiαΓy ∫Rme−i〈x,y〉/ sinαei2(cotα)(|x|2+|y|2)(.)dxhence the kernel is given byK−,α(x, y) = eiαΓy(e−i〈x,y〉/ sinαei2(cotα)(|x|2+|y|2))= ei2(cotα)(|x|2+|y|2)eiαΓy(e−i〈x,y〉/ sinα)where the last line follows because Γy commutes with |y|. Now using the series expansion(1), the term eiαΓy(e−i〈x,y〉/ sinα)can be computed. Indeed, we findeiαΓy(e−i〈x,y〉/ sinα)= eiαΓyΓ(λ)∞∑k=02λ(k + λ)(i sinα)−k(|x||y|)kJ˜k+λ( |x||y|sinα)Cλk (〈ξ, η〉)= Γ(λ)∞∑k=02λ(k + λ)(i sinα)−kJ˜k+λ( |x||y|sinα)eiαΓy((|x||y|)kCλk (〈ξ, η〉))DE BIE, DE SCHEPPERso we have reduced the problem to calculating eiαΓy((|x||y|)kCλk (〈ξ, η〉)). This can bedone in a manner analogous to Lemma 3.1 in [15]. The result is given in the followinglemma.Lemma 4.1 One haseiαΓy(|x||y|)kCλk (〈ξ, η〉) =12(eiα(k+m−2) + e−iαk)(|x||y|)kCλk (〈ξ, η〉)− λ2(k + λ)(eiα(k+m−2) − e−iαk) (|x||y|)kCλk (〈ξ, η〉)+λk + λx ∧ y(eiα(k+m−2) − e−iαk)(|x||y|)k−1Cλ+1k−1 (〈ξ, η〉).Using this lemma and the previous computation, we arrive at the following series rep-resentation for the kernel of the fractional Clifford-Fourier transform.Theorem 4.2 The fractional Clifford-Fourier transform F−,α = e iαm2 e iα2 (∆−|x|2+2Γ) isgiven by the integral transform(pi(1− e−2iα))−m/2 ∫RmK−,α(x, y) f(x) dxwith integral kernelK−,α(x, y) =(Aλ +Bλ +(x ∧ y)Cλ) e i2 (cotα)(|x|2+|y|2)withAλ(w, z) = − 2λ−1Γ(λ+ 1)∞∑k=0i−k(eiα(k+2λ) − e−iαk)z−λJk+λ(z) Cλk (w),Bλ(w, z) = 2λ−1Γ(λ)∞∑k=0(k + λ)i−k(eiα(k+2λ) + e−iαk)z−λJk+λ(z) Cλk (w),Cλ(w, z) =2λΓ(λ+ 1)sinα∞∑k=1i−k(eiα(k+2λ) − e−iαk)z−λ−1Jk+λ(z) Cλ+1k−1 (w),where z = (|x||y|)/ sinα, w = 〈ξ, η〉 and λ = (m− 2)/2.Finding an explicit analytic expression (as in F1) for this series representation is againa difficult problem. Hence we postpone its determination to a subsequent publication.References[1] T. Bu¨low and G. Sommer, Hypercomplex signals—a novel extension of the analytic signal to themultidimensional case. IEEE Trans. Signal Process. 49 (2001), 2844–2852.[2] J. Ebling and G. Scheuermann, Clifford Fourier transform on vector fields. IEEE Transactions onVisualization and Computer Graphics 11 (2005), 469–479.[3] T. Ell and S. Sangwine, Hypercomplex Fourier transforms of color images. IEEE Trans. Image Pro-cess. 16 (2007), 22–35.FOURIER TRANSFORMS OF HYPERCOMPLEX SIGNALS[4] C. Moxey, S. Sangwine and T. Ell, Hypercomplex correlation techniques for vector images. IEEETrans. Signal Process. 51 (2003), 1941-1953.[5] S. Said, N. Le Bihan and S. Sangwine, Fast complexified quaternion Fourier transform. IEEE Trans.Signal Process. 56 (2008), 1522-1531.[6] K. Kou and T. Qian, The Paley-Wiener theorem in Rn with the Clifford analysis setting. J. Funct.Anal. 189 (2002), 227–241.[7] K. Kou and T. 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Fourier transforms of hypercomplex signals

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Fourier transforms of hypercomplex signals

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