In this paper, it is shown that there exists a Hermite basis for the two-sided quaternionic Fourier transform. This basis is subsequently used to give an alternative proof for the inversion theorem and to give insight in translation and convolution for the quaternionic Fourier transform

De Bie, Hendrik

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Ghent University Academic Bibliography

New techniques for the two-sided quaternionicFourier transformHendrik De BieAbstract In this paper, it is shown that there exists a Hermite basis for the two-sidedquaternionic Fourier transform. This basis is subsequently used to give an alternativeproof for the inversion theorem and to give insight in translation and convolution forthe quaternionic Fourier transform.1 IntroductionThe topic of hypercomplex Fourier transforms has recently lead to various new in-sights, in two more or less separated communities of mathematicians on the onehand and engineers on the other hand. For the mathematical point of view, we referto e.g. [1, 4, 5, 6, 8]. For the engineering point of view, we refer to [2, 3, 10, 12, 14]and the references mentioned therein.The goals in both communities are quite different as well. Whereas in mathemat-ics one is in general interested in structural properties of such transforms and theunderlying algebraic ideas, in engineering one really wants to construct and studytransforms that are applicable in, say, image processing (with better results than theordinary Fourier transform).The main aim of the present paper is to bridge the gap between both approaches,at least for one particular transform, namely the two-sided quaternionic Fouriertransform (qFT). For this particular transform (see e.g. [10, 12, 14]), which is veryimportant in image processing, we will adapt the typical techniques used in themathematical community. This will allow for a new proof of the inversion theoremand new insights in the concept of convolution for the qFT.The paper is organized as follows. In section 2 we discuss the Hermite basisfor the ordinary Fourier transform. In section 3 we define the qFT and generalizeHendrik De BieDepartment of Mathematical Analysis, Faculty of Engineering and Architecture, Ghent University,Galglaan 2, 9000 Gent, Belgium, e-mail: Hendrik.DeBie@UGent.be12 Hendrik De Biethe Hermite basis. As a consequence, an inversion theorem is obtained. Finally, insection 4 we consider the translation operator for the qFT and discuss the impact onconvolution.2 Classical Fourier transformThe classical Fourier transform, defined over the real line, is given byF ( f )(y) := (2pi)−12∫ +∞−∞e−ixy f (x)dxfor f ∈ L1(R).Let us now assume that f is either in the Schwartz space of rapidly decreasingfunctionsS (R) or in the space of square integrable functions L2(R). In the formercase, the integral definition is still valid as S (R) ⊂ L1(R). In the latter case, theFourier transform needs to be extended via a density argument. For both functionspaces, an interesting basis exists, given by the Hermite functions. These functionsare defined byψk :=(x− ddx)ke−x2/2= Hk(x)e−x2/2for k = 0,1,2, . . . and are orthogonal with respect to the L2 inner product. The nor-malization can be computed explicitly, but will not be used in the present paper.Note that Hk(x) is a polynomial of degree k, the so-called Hermite polynomial (seee.g. [18]).The functions ψk diagonalize the Fourier transform. In particular, one hasF (ψk)(y) = (−i)kψk(y), (1)which follows immediately from the operational calculus. As arbitrary functions inS (R) or L2(R) can be expanded in terms of the Hermite basis:f =+∞∑k=0akψk, ak ∈ C,we immediately obtain the action of the Fourier transform on f viaF ( f ) =+∞∑k=0(−i)kakψk (2)For numerical computations, it is in general not efficient to decompose a functioninto its Hermite components and apply formula (2). FFT methods will work faster.New techniques for the two-sided qFT 3Nevertheless, it is possible to generalize the Hermite basis to the quaternionic set-ting, where it allows to prove several theoretical results in a straight-forward way.3 The two-sided quaternionic Fourier transform3.1 The algebra of quaternionsThe quaternion algebra H is defined over R with three imaginary units i, j and ksatisfyingij=−ji= ki2 = j2 = k2 =−1.Every quaternion can be written explicitly asq= q0 +q1i+q2j+q3kwith q0, . . . ,q3 real numbers. The quaternionic conjugate is given byq= q0−q1i−q2j−q3kand satisfies|q|2 = qq= q20 +q21 +q22 +q23.A pure quaternion q is a quaternion with real part q0 = 0. It satisfiesq2 =−(q21 +q22 +q23).3.2 Definition of the qFTLet µ,ν ∈ H be pure quaternions with µ2 = ν2 = −1. Then, following [14], wedefine the two-sided qFT asF µ,ν(h)(y1,y2) := (2pi)−1∫R2e−µx1y1h(x1,x2)e−νx2y2dx1dx2, h ∈ L1(R2;H)where we have introduced a different normalization (2pi)−1. The first definition ofthis two-sided transform, with µ = j and ν = k, was introduced in the Ph.D. thesis[10], see also [11]. In earlier work, a one sided version was given by Ernst et al. [13]and by Delsuc [9], although these authors use an adaptation of the quaternion alge-bra. The applicability of the qFT to color image processing was first demonstratedin [16] using a discrete version. At that point, the switch was made to general axes µ4 Hendrik De Bieand ν instead of j and k. Indeed, for color image processing there is an arbitrary butpreferred axis of the grey-line in the color space, so the transform kernel axes aregenerally aligned to or perpendicular to this axis. At the same time ([17]), a changewas again made to one-sided transforms, mostly driven by the complexity of theresulting operational formula when using the two-sided qFT definition. For a recentreview on the use of the qFT in image processing, we refer the reader to [12].In the sequel we will concentrate on the two-sided qFT, but the techniques devel-oped are also applicable to one-sided versions.Note that we will often use the shorthand x := (x1,x2), y := (y1,y2).3.3 Basis and inversion theoremWe are now interested in the function spacesS (R2;H) and L2(R2;H), to which theqFT is immediately extended.Again there exists an interesting basis for these spaces,{ψk,`}k,`∈Nwithψk,` := Hk(x1)H`(x2)e−x21/2e−x22/2with Hk,H` the Hermite polynomial of degree k, resp. `. This basis is the tensorproduct basis, as also used in [7]. Every function h in S (R2;H) or L2(R2;H) cannow be decomposed ash=+∞∑k,`=0ak,`ψk,`, ak,` ∈H.It is important to note that, although the basis functions are real-valued, the coeffi-cients in the expansion are quaternions.Let us compute the qFT of the basis functions ψk,`. We obtain, using (1)F µ,ν(ψk,`)(y1,y2) = (2pi)−1∫R2e−µx1y1ψk,`(x1,x2)e−νx2y2dx1dx2= (2pi)−1∫R2e−µx1y1Hk(x1)H`(x2)e−x21/2e−x22/2e−νx2y2dx1dx2=((2pi)−1/2∫ +∞−∞e−µx1y1Hk(x1)e−x21/2dx1)×((2pi)−1/2∫ +∞−∞H`(x2)e−x22/2e−νx2y2dx2)= (−µ)kHk(y1)H`(y2)e−y21/2e−y22/2(−ν)`= (−µ)kψk,`(y1,y2)(−ν)`.New techniques for the two-sided qFT 5Hence we have obtained the following result:Theorem 1. The eigenfunctions of the qFT are given by the functions ψk,` with fol-lowing (quaternionic) eigenvaluesF µ,ν(ψk,`)(y1,y2) = (−µ)k(−ν)`ψk,`(y1,y2).One has to be careful in applying this result to a function h expanded in theHermite basis. In that case we obtainF µ,ν(h)(y1,y2) =+∞∑k,`=0F µ,ν(ak,`ψk,`)(3)=+∞∑k,`=0(−µ)kak,`(−ν)`ψk,`,as the expansion coefficients ak,` do not necessarily commute with µ and ν .Using this result, we can give an alternative proof of the inversion theorem. Theoriginal result was obtained in [10], page 59.Theorem 2. The inverse transform for the qFT F µ,ν acting on the function spaceS (R2;H) is given byF−µ,−ν .Proof. We need to show thatF−µ,−ν (F µ,ν(h)) = hfor all h ∈S (R2;H). Now applying formula (3) we obtainF−µ,−ν (F µ,ν(h)) = F−µ,−ν(+∞∑k,`=0(−µ)kak,`(−ν)`ψk,`)=+∞∑k,`=0F−µ,−ν((−µ)kak,`(−ν)`ψk,`)=+∞∑k,`=0µk(−µ)kak,`(−ν)`ν`ψk,`=+∞∑k,`=0ak,`ψk,`= hyielding the desired result. uunionsq6 Hendrik De Bie4 Translation and convolution for the qFTTying in with recent advances in pure mathematics (see e.g. [19] for the case ofthe so-called Dunkl transform), it is possible to connect a generalized translationoperator to a hypercomplex Fourier transform. We will do this for the qFT andprove that for this particular transform, the generalized translation coincides withordinary translation.Let us first explain the concept for the ordinary FT. Define the translation over yof a function f byτy f (x) := f (x− y).Then we computeF (τz f )(y) = (2pi)−12∫ +∞−∞e−ixy f (x− z)dx= e−izy(2pi)−12∫ +∞−∞e−ixy f (x)dx= e−izyF ( f )(y).This means that the ordinary translation is recovered viaτz f (u) =F−1(e−izyF ( f )(y)). (4)This formula can immediately be generalized to any invertible hypercomplexFourier transform. This has already been done for the Clifford-Fourier transform,see [8]. Now we want to apply this formula to the two-sided qFT. To that aim, wedefine τ˜zg, with g a quaternionic function, viaF µ,ν(τ˜zg)(y) := e−µz1y1F µ,ν(g)(y)e−νz2y2 .Applying the inverse qFT then yields:τ˜zg(u) = (2pi)−1∫R2eµu1y1e−µz1y1F µ,ν(g)(y)e−νz2y2eνu2y2dy1dy2.Let us expand the right-hand side:τ˜zg(u) = (2pi)−2∫R4eµy1(u1−z1−x1)g(x)eνy2(u2−z2−x2)dx1dx2dy1dy2= (2pi)−2∫R2(∫ +∞−∞eµy1(u1−z1−x1)dy1)g(x)(∫ +∞−∞eνy2(u2−z2−x2)dy2)dx1dx2=∫R2δ (u1− z1− x1)g(x)δ (u2− z2− x2)dx1dx2= g(u1− z1,u2− z2)= τzg(u).New techniques for the two-sided qFT 7Here we have used the well-known integral representation of the Delta function:δ (x) = (2pi)−1∫ +∞−∞eixydy.We hence obtain that the generalized translation for the qFT coincides with theordinary translation. The reader familiar with the work on generalized translationsin e.g. [19, 8] may find this a surprising result. However, note that the qFT wasoriginally designed in [10] precisely in order to satisfy this property.As a consequence, we find that convolution for the qFT may be defined byf ∗g(x) :=∫R2f (y)τyg(x)dy, with τyg(x) := g(x− y). (5)This is of course nothing but ordinary convolution.Note that this convolution product does not satisfy the typical convolution prop-erty, i.e.F µ,ν ( f ∗g) 6=F µ,ν( f )F µ,ν(g).Alternatively, based on an idea of Mustard [15], one may define convolution forthe qFT as followsf ∗q g(x) :=F−µ,−ν (F µ,ν( f )F µ,ν(g)) (6)which clearly satisfiesF µ,ν ( f ∗q g) =F µ,ν( f )F µ,ν(g).The differences and analogies between the two convolution products (5) and (6) canagain be studied using the Hermite basis ψk,`. This problem will be tackled in asubsequent publication.Acknowledgements The author would like to thank Todd Ell and Stephen Sangwine for emailcommunication about the content of the present paper.References1. F. Brackx, N. De Schepper and F. Sommen, The Clifford-Fourier transform. J. Fourier Anal.Appl. 11 (2005), 669–681.2. R. Bujack, G. Scheuermann and E. Hitzer, A general geometric Fourier transform. In 9th In-ternational Conference on Clifford Algebras and their Applications in Mathematical Physics.K. Gu¨rlebeck (ed.). Weimar, Germany, 15–20 July 2011. 19 pages3. T. Bu¨low and G. Sommer, Hypercomplex signals—a novel extension of the analytic signal tothe multidimensional case. IEEE Trans. Signal Process. 49 (2001), 2844–2852.4. H. De Bie and N. De Schepper, Fractional Fourier transforms of hypercomplex signals.Accepted for publication in Signal, Image and Video Proc.8 Hendrik De Bie5. H. De Bie, N. De Schepper and F. Sommen, The class of Clifford-Fourier transforms. J.Fourier Anal. Appl. 17 (2011), 1198-1231.6. H. De Bie, B. Ørsted, P. Somberg and V. Soucˇek, The Clifford deformation of the Hermitesemigroup. arXiv:1101.5551, 27 pages.7. H. De Bie and F. Sommen, Vector and bivector Fourier transforms in Clifford analysis. In18th International Conference on the Application of Computer Science and Mathematics inArchitecture and Civil Engineering. K. Gurlebeck and C. Ko¨nke (eds.), Weimar, Germany,07-09 July 2009, 11 pages.8. H. De Bie and Y. Xu, On the Clifford-Fourier transform. Int. Math. Res. Not. IMRN (2011),Art. ID rnq288, 41 pages.9. M. A. Delsuc, Spectral representation of 2D NMR spectra by hypercomplex numbers. J.Magn. Reson. 77 (1988), 119-124.10. T.A. Ell, Hypercomplex Spectral Transformations, Ph.D. Thesis. University of Minnesota,University Microfilms International Number 9231031, June 1992.11. T.A. Ell, Quaternion-Fourier transforms for analysis of two-dimensional linear time-invariantpartial differential systems, In Proc. 32nd IEEE Conf. on Decision and Control. San Antonio,TX, December 1993. vol.1–4, pp. 1830-1841.12. T.A. Ell and S.J. Sangwine, Hypercomplex Fourier transforms of color images, IEEE Trans.Image Process. 16 (2007), 22-35.13. R. R. Ernst, G. Bodenhausen and A. Wokaun, Principles of Nuclear Magnetic Resonancein One and Two Dimensions. International Series of Monographs on Chemistry, OxfordUniversity Press, 1987.14. E. Hitzer and S.J. Sangwine, The orthogonal planes split of quaternions. In 9th Interna-tional Conference on Clifford Algebras and their Applications in Mathematical Physics. K.Gu¨rlebeck (ed.). Weimar, Germany, 15–20 July 2011. 12 pages15. D. Mustard, Fractional convolution. J. Austral. Math. Soc. Ser. B 40 (1998), 257–265.16. S. J. Sangwine, Fourier transforms of colour images using quaternion, or hypercomplex,numbers. Electron. Lett. 32 (1996), 1979-1980.17. S. J. Sangwine and T. A. Ell, The discrete Fourier transform of a colour image. In ImageProcessing II Mathematical Methods, Algorithms and Applications. J. M. Blackledge and M.J. Turner, Eds., Chichester, 2000, pp. 430-441.18. G. Szego˝, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. Vol. 23, Providence, 4thedition, 1975.19. S. Thangavelu and Y. Xu, Convolution operator and maximal function for the Dunkl trans-form. J. Anal. Math. 97 (2005), 25–55.

New techniques for the two-sided quaternionic fourier transform

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New techniques for the two-sided quaternionic fourier transform

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