RELATIONSHIP BETWEEN QUATERNION LINEAR CANONICAL AND QUATERNION FOURIER TRANSFORMS

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1 Proceedings of the 04 Internationa Conference on Waveet Anaysis and Pattern ecognition, Lanzhou, 3-6 Juy, 04 ELATIONSHIP BETWEEN QUATENION LINEA CANONICAL AND QUATENION FOUIE TANSFOMS MAWADI BAHI, YUICHI ASHINO Department of Mathematics, Hasanuddin University, Indonesia Mathematica Sciences, Osaka Kyoiku University, Japan Abstract: We briefy introduce the quaternion inear canonica transform QLCT), which is a generaization of the quaternion Fourier transform QFT) to the inear canonica transform LCT) domain We show that the QLCT can be reduced to the quaternion Fourier transform QFT) We then derive the inverse transform of the QLCT using the properties of the QFT We finay provide an exampe which describes the reationship between the QFT and QLCT Keywords: inear canonica transform; quaternion Fourier transform Introduction The inear canonica transform LCT) pays an important roe in various fieds of optics and signa processing In some papers, the LCT is aso known as the affine Fourier, the ABCD, and the Moshinsky-queue transforms see [6] and references therein) The LCT can be considered as a generaization of many mathematica transforms, such as Fourier, Lapace, fractiona Fourier, Fresne transforms and so on Many fundamenta properties of the LCT are aready known, incuding shift, moduation, convoution, correation, and uncertainty principe [5, 8] ecenty, the reationship between the Fourier transform FT) and LCT was demonstrated by [7] It is shown that some properties of the LCT, such as inversion formua, Hibert transform, and convoution theorems, can re-derived using the properties of the FT In this paper, we deveop this idea to estabish the reationship between the QLCT and QFT It is shown that the QLCT can be reduced to the QFT and that the inversion formua, which is an fundamenta property of the QLCT, can be obtained from the properties the QFT with a simpe change of variabe Quaternion The quaternions, a generaization of compex numbers, are members of a noncommutative division agebra The set of quaternions is denoted by H Every eement of H can be written in the foowing form H q q 0 + i q + j q + k q 3 ; q 0, q, q, q 3 }, which obeys the foowing mutipication rues: ij ji k, jk kj i, ki ik j, i j k ijk ) For a quaternion q q 0 +i q +j q +k q 3 H, q 0 is caed the scaar or rea) part of q denoted by Scq) and i q +j q +k q 3 is caed the vector or pure) part of q The vector part of q is conventionay denoted by q or Vecq) iq + jq + kq 3 Let p, q H and p, q be their vector parts, respectivey Equation ) yieds the quaternionic mutipication qp as where qp q 0 p 0 + q p + q 0 p + p 0 q + q p, ) q p q p + q p + q 3 p 3 ), q p i q p 3 q 3 p ) + j q 3 p q p 3 ) + k q p q p ) The conjugate q of the quaternion q is the quaternion given by q q 0 i q j q k q 3, q 0, q, q, q 3 3) It is an anti-invoution, that is, qp p q /4/$ IEEE 6

2 Proceedings of the 04 Internationa Conference on Waveet Anaysis and Pattern ecognition, Lanzhou, 3-6 Juy, 04 From 3) we obtain the norm or moduus of q H defined as q q q q0 + q + q + q 3 It is not difficut to see that qp q p, p, q H Using the conjugate 3) and the moduus of q, we can define the inverse of q H \ 0} as q q q, which shows that H is a normed division agebra By ), f and g L ; H) can be represented as foows: f f 0 + f, f if + jf + kf 3, ḡ g 0 g, g ig + jg + kg 3, where f, g, 0,,, 3, are rea-vaued functions of x Then, by ), we have fḡ f 0 g 0 f g f 0 g + g 0 f f g, where the scaar and vector parts are respectivey Note that Scfḡ) f 0 g 0 f g, Vecfḡ) f 0 g + g 0 f f g, Scfḡ) Scg f), Vecfḡ) Vecg f) It is convenient to introduce the inner product: f, g) fx)gx) dx, Scfḡ) dx + Vecfḡ) dx, where dx dx dx To introduce a norm, we define the symmetric rea scaar part: f, g [f, g) + g, f)] Scfḡ) dx Then, the L ; H)-norm is defined by f f, f ) / fx) dx, and the quaternion modue L ; H) is defined by L ; H) f f : H, f < } Muti-indices and derivatives A coupe α α, α ) of nonnegative integers is caed a muti-index We sha denote and for x, α α + α, α! α! α!, x α x α xα Derivatives are convenienty expressed by muti-indices: α α x α xα Denote by e, e } the standard basis of The vector differentia a aong the direction a is defined by where e + e a a + a, 3 QFT and its properties Definition The QFT of f L ; H) is the transform F q f} L ; H) given by the integra F q e iωx fx) e jωx dx 4) π Here F q is caed the quaternion Fourier transform operator or the quaternion Fourier transform emark Using the Euer formua for the quaternion Fourier kerne e iωx e jωx, we can rewrite 4) in the fo- 7

3 Proceedings of the 04 Internationa Conference on Waveet Anaysis and Pattern ecognition, Lanzhou, 3-6 Juy, 04 owing form F q fx) cosω x ) cosω x ) dx π ifx) sinω x ) cosω x ) dx π fx)j cosω x ) sinω x ) dx π + ifx)j sinω x ) sinω x ) dx π Definition 3 The inverse QFT of g L ; H) is the transform Fq g} L ; H) given by the integra Fq [g]x) e iωx gω) e jωx dω π Here F q is caed the inverse QFT operator Some important properties of the QFT are provided in the foowing emmas proved in [] Lemma 4 Let f L ; H) L ; H) and F q α f} L ; H) Then F q α f} ω) iω ) α F q jω ) α emark 5 Moreover, if F q,0) f } L ; H), then F q,0) f } ω) iω ) F q, and if F q 0,) f } L ; H), then F q 0,) f } ω) F q jω ) Lemma 6 Scaar QFT Parseva) The inner product of f, g L ; H) and the inner product of their QFTs satisfy f, g) L ;H) F q f}, F q g}) L ;H) In particuar, when f g, we have Panchere s identity: f L ;H) F qf} L ;H) This shows that the tota signa energy computed in the spatia domain is equa to the tota signa energy computed in the frequency domain As for other properties of the QFT, such as convoution theorems, see [] Main esuts Based on the definition of the two-sided) quaternion Fourier transform QFT), we obtain a definition of the quaternion inear canonica transform QLCT) by repacing the kerne of the FT with the kerne of the QFT in the LCT definition compare to [3]) Denote by SL, ), the specia inear group of degree over, that is, the group of a rea matrices with determinant one Let A s as b s SL, ), s, c s d s When b b 0, we define the kernes K As, s, of the QLCT by K A x, ω ) K A x, ω ) a πb i ei b x b x ω + d b ω, a πb j ej b x b x ω + d b ω Definition QLCT) The QLCT of f L ; H) is defined by L H A,A K A x, ω )fx)k A x, ω ) dx, b b 0, d d e i c d ω fd ω, d ω )e j c d ω, b b 0 Here e i c d ω and e j c d ω are caed chirp signas in signa processing From hence we wi dea with the case when b b 0, because L H A,A f} is trivia for b b 0 For p H \ 0} and q H, the quaternion power function p q is defined by p q e np)q But p q p q p q+q in genera, uness np)q and np)q commute [4, 57] We have the foowing Lemma Lemma Let represent one of the imaginary units i, j, k For θ, we have e θ e θ 8

4 Proceedings of the 04 Internationa Conference on Waveet Anaysis and Pattern ecognition, Lanzhou, 3-6 Juy, 04 Proof Since, we have e θ cos θ sin θ) cos θ sin θ e θ emark 3 When A A to the QFT, that is, 0, the LCT reduces 0 L H A,A e iωx fx) e jωx dx πi πj i F q j, 5) where we use Lemma a c For a matrix A b d with bd 0, define the operator G A on L ; H) by G A f}x) e i ab x/b) fx) e j cd x/d), f L ; H) Then, its inverse G A is given by G A G Σ 3A, Σ 3 0, 0 where Σ 3 is one of the so-caed Paui matrices For A s SL, ), s, with b b 0, put a a, a ), b b, b ), d d, d ), B a; b ) a a, b b B d; b ) d d b b For ω ω, ω ), we define eement wise mutipication and division / by ω b ω b, ω b ), ω/b ω /b, ω /b ), respectivey Then, we have a simiar resut beow as emark 3 for A s SL, ), s, with b b 0 Theorem 4 For f L ; H), denote f G B f} Then, the QLCT of f can be reduced to the QFT of f, that is, L H A,A GB F q πb i f} ) ω/b) πb j 6) Proof Denote f G B f} By Definition and Lemma, simpe computations impy that L H A,A πb i e i πb j ej πb i ei d b ω a b x b x ω + d b ω fx) a b x b x ω + d b ω e ix ω b e i a ω πb j e jx b e j d b ω dx πb i ei d b ω πb j ej d b ω e ix dx b x fx)e j a ω b fx)e jx ω b dx b x d b ω ) πb i ei b F q f}ω/b)e j d b ω ) b πb j GB F q πb i f} ) ω/b) πb j Theorem 5 The inverse transform of the QLCT can be derived from that of the QFT, that is, for a geven g L H A,A f}, f can be recovered by f πb i G Σ3B F q G Σ3B gω b)} ) πb j Proof Let gω) L H A,A be given By 6), we have Appy G B Hence, GB F q f} ) ω) πb i gω b) πb j 7) G Σ3B to 7), then we have F q f}ω) G Σ3B πb i gω b) πb j } πb i G Σ3B gω b)} πb j f Fq πb i G Σ3B gω b)} πb j πb i F q G Σ3B gω b)} ) πb j, ) 9

5 Proceedings of the 04 Internationa Conference on Waveet Anaysis and Pattern ecognition, Lanzhou, 3-6 Juy, 04 which impies f G Σ3B πb i Fq G Σ3B gω b)} ) ) πb j πb i G Σ3B F q G Σ3B gω b)} ) πb j It is interesting to describe the reationship between the QLCT and LCT as shown in the foowing exampe Exampe 6 Let us now compute the QLCT of the twodimensiona Gaussian function fx) e kx +kx ) with k, k > 0 From the definition of the QLCT 5) we easiy obtain L H A,A πb i e i a b x b x ω+ d b ω ) e j πb i πb j a b x b x ω+ d b ω ) fx) e kx e i e kx e j d πb i ei b ω d πb j ej b ω πb j dx a b x b x ω+ d b ω ) dx a b x b x ω + d b ω ) dx e b k b ia )x e i ω b x dx e b k b ja )x e j ω b x dx Using the QFT of the Gaussian function see []), that is, F q e iωx e kx +kx ) e jωx dx π e ω 4k + ω 4k ) k k We immediatey obtain L H A,A d πb i ei b ω d πb j ej b ω ω a + k b i e πb k b ia ) e πb k b ja ) e c +k id k b ia ) ω b k b ia ) ω b k b ja ) ω a + k b j e c +k jd k b ja ) 8) It is easiy seen that in the specific case when 0 A A, 0 equation 8) can be reduced to L H A,A ω k k ij e 4k + ω 4k ), which shows that the above identity is a Gaussian quaternion function 3 Concusion Due to the non-commutative property of quaternion mutipication, there are three different types of two-dimensiona QFTs These three QFTs are so-caed a eft-sided QFT, a rightsided QFT, and a two-sided QFT, respectivey In this work, we have proposed an extension of the two-side QFT to the LCT domain and so-caed the quaternion inear canonica transform QLCT) We show that the QLCT can be reduced to the quaternion Fourier transform QFT) and its inverse can be obtained using the properties of the QFT Acknowedgments The first author is supported by Hibah Peneitian Kompetisi Interna tahun 03 sesuai surat perjanjian nomor: 0/UN4-4/LK6/SP-UH/03 from the Hasanuddin University, Indonesia The second author is partiay supported by JSP- SKAKENHIC)54000 of Japan eferences [] M Bahri, Ashino and Vaiancourt, Convoution theorems for quaternion Fourier transform: properties and appications, Abstr App Ana, 03), Art ID 6769, 0 pp [] M Bahri, E Hitzer, A Hayashi, and Ashino, An uncertainty principe for quaternion Fourier transform, Comput Math App, 56 9) 008), 4 47 [3] K I Kou, J Y Ou and J Morais, On uncertainty principe for quaternionic inear canonica transform, Abstr App Ana, 03), Art ID 7595, 4 pp 0

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