Applied Mathematics and Computation

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1 Applied Mathematics and Computation 8 () Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: Two-dimensional quaternion wavelet transform Mawardi Bahri a, yuichi Ashino b,, émi Vaillancourt c a Department of Mathematics, Hasanuddin University, KM Tamalanrea, Makassar, Indonesia b Division of Mathematical Sciences, Osaka Kyoiku University, Osaka , Japan c Department of Mathematics and Statistics, University of Ottawa, 585 Kind Edward Ave., Ottawa ON, Canada KIN 6N5 article info abstract Keywords: Quaternion Fourier transform Admissible quaternion wavelets In this paper we introduce the continuous quaternion wavelet transform (CQWT). We express the admissibility condition in terms of the (right-sided) quaternion Fourier transform. We show that its fundamental properties, such as inner product, norm relation, and inversion formula, can be established whenever the quaternion wavelets satisfy a particular admissibility condition. We present several examples of the CQWT. As an application we derive a Heisenberg type uncertainty principle for these extended wavelets. Ó Elsevier Inc. All rights reserved.. Introduction The quaternion Fourier transform (QFT), which is a nontrivial generalization of the real and complex Fourier transform (FT) using quaternion algebra has been of interest to researchers for some years (see e.g. [,,7,6]). It was found that many FT properties still hold but others have to be modified. Based on the (right-sided) QFT, one can extend the classical wavelet transform (WT) to quaternion algebra while enjoying the same properties as in the classical case. He [] and hao and Peng [3] constructed the continuous quaternion wavelet transform of quaternion-valued functions. They also demonstrated a number of properties of these extended wavelets using the classical Fourier transform (FT). In [6], using the (two-sided) QFT Traversoni proposed a discrete quaternion wavelet transform which was applied by Bayro-Corrochano [] and hou et al. [3]. ecently, in [8,9], we introduced an extension of the WT to Clifford algebra by means of the kernel of the Clifford Fourier transform [8]. The purpose of this paper is to construct the -D continuous quaternion wavelet transform (CQWT) based on quaternion algebra. We emphasize that our approach is significantly different from previous work in the definition of the exponential kernel. Our construction uses the kernel of the (right-sided) QFT which in general does not commute with quaternions. The previous papers considered the kernel of the FT which commutes with the quaternions so that the properties of the extension of the WT to quaternion algebra is a quite similar to the classical wavelets. In the present paper we use the (right-sided) QFT to investigate some important properties of the CQWT. Special attention is devoted to inner product, norm relation, and inversion formula. We show that these fundamental properties can be established whenever the admissible quaternion wavelets satisfy a particular admissibility condition. Using the properties of the CQWT and the uncertainty principle for the (right-sided) QFT [6] we establish an uncertainty principle for the CQWT.. Basics For convenience of further discussions, we briefly review some basic ideas on quaternions, the (right-sided) QFT and the similitude group SIM(). The quaternion algebra over, denoted by H, is an associative non-commutative four-dimensional algebra, Corresponding author. addresses: mawardibahri@gmail.com (M. Bahri), ashino@cc.osaka-kyoiku.ac.jp (. Ashino), remi@uottawa.ca (. Vaillancourt) /$ - see front matter Ó Elsevier Inc. All rights reserved. doi:.6/j.amc..5.3

2 M. Bahri et al. / Applied Mathematics and Computation 8 () H ¼fq ¼ q þ iq þ jq þ kq 3 jq ; q ; q ; q 3 g; which obey Hamilton s multiplication rules ðþ ij ¼ ji ¼ k; jk ¼ kj ¼ i; ki ¼ ik ¼ j; i ¼ j ¼ k ¼ ijk ¼ : ðþ The quaternion conjugate of a quaternion q is given by q ¼ q iq jq kq 3 ; q ; q ; q ; q 3 ð3þ and it is an anti-involution, i.e. qp ¼ pq: From (3) we obtain the norm of q H defined as p jqj ¼ ffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qq ¼ q þ q þ q þ q 3: ð5þ It is not difficult to see that jqpj ¼ jqjjpj; 8p; q H: ð6þ It is convenient to introduce the inner product of two quaternion functions, f ; g :! H, as follows: ðf ; gþ ¼ L ð ;HÞ f ðxþgðxþd x: In particular, if f = g, then we obtain the associated norm = kf k L ð ;HÞ ¼ðf ; f Þ= ¼ jf ðxþj d x : ð8þ L ð ;HÞ As a consequence of the inner product (7) we obtain the quaternion Cauchy Schwarz inequality = = f gd x 6 jf j d x jgj d x ; 8f ; g L ð ; HÞ: ð9þ Based on quaternions we can define the (right-sided) QFT. Definition. The QFT of f L ð ; HÞ is the function F q ff g :! H given by F q ff gðxþ ¼^f ðxþ ¼ f ðxþe ix x e jx x d x; where x = x e + x e, x = x e + x e, and the quaternion exponential product e ix x e jx x is called the quaternion Fourier kernel. ðþ ð7þ ðþ Theorem. Suppose that f L ð ; HÞ and F q ff gl ð ; HÞ. Then the QFT of f is an invertible transform and its inverse is given by F q ½F qff gšðxþ ¼f ðxþ ¼ F ðpþ q ff gðxþe jx x e ix x d x: ðþ As in the classical case, we obtain Plancherel s formula, specific to the (right-sided) QFT [,7,6], ðf ; gþ L ð ;HÞ ¼ ðpþ ðf qff g; F q fggþ L ð ;HÞ : ðþ In particular, if f = g we get Parseval s formula, kf k L ð ;HÞ ¼ ðpþ kf qff gk L ð ;HÞ : Table presents some useful properties of the (right-sided) QFT. Detailed information about the QFT and its properties can be found in [,7,6]. Following Antoine et al. [3,], we consider the similitude group SIM() denoted by G on associated with wavelets as follows. G¼ þ SOðÞ ¼fða; r h ; bþja þ ; r h SOðÞ; b g; where SO() is the special orthogonal group of. The multiplication defined by () follows the group law ð3þ ðþ

3 M. Bahri et al. / Applied Mathematics and Computation 8 () Table Properties of the (right-sided) QFT of f ; g L ð ; HÞ, where a; b H; a nfgare constants, x ¼ x e þ y e and n N. Property Quaternion function Quaternionic Fourier transform Left linearity af(x)+bg(x) af qff gðxþþbf qfggðxþ Scaling f(ax) F jaj qff gð x a Þ Shift f(x x ) F qffe ixx gðxþe jxy otation f ðr h ðxþþ F qff gðr h ðxþþ Part. deriv. ðxþi x n F qff gðxþ; f L ð ; HÞ ðxþ ðix Þ n F qff gðxþ f ¼ f þ ðxþ F qff gðxþðjx Þ n ; f L ð ; Formula Plancherel s formula ðf ; f Þ L ð ;HÞ ¼ ðf ðpþ qff g; F qff gþ L ð ;HÞ Parseval s formula kf k L ð ;HÞ ¼ p kf qff gk L ð ;HÞ fa; b; r h gfa ; b ; r h g¼faa ; b þ ar h b ; r hþh g: The rotation r h SO() acts on x as usual, r h ðxþ ¼ðx cos h x sin h; x sin h þ x cos hþ; 6 h < p: ð6þ The left Haar measure on G is given by dkða; h; bþ ¼dlða; hþd b; where dlða; hþ ¼ dadh and dh is the Haar measure on SO(). For the sake of simplicity, we write dl = dl(a,h) and dk = dk(a,h,b). a 3 3. Construction of -D quaternion wavelets Based on quaternions and the (right-sided) QFT, one can extend the real (or complex) wavelet transform to a quaternion wavelet transform. This section constructs the -D CQWT from a group theoretical point of view. We shall characterize the admissibility condition in terms of the (right-sided) QFT and define the CQWT in terms of an admissible quaternion wavelet. 3.. Admissible quaternion wavelet ð5þ Definition (Admissible quaternion wavelet). Let AQW denote the class of admissible quaternion wavelets w L ð ; HÞ which satisfy the following admissibility condition, i.e. j^wðar h ðxþþj dadh ð7þ SOðÞ þ a is a real positive constant independent of x satisfying jxj =. Denote by C w, the real positive constant. emark. If w AWQ, then (7) is a real positive constant independent of x for x. Let us show this fact. Denote x = jxjx, where jx j =. Since r h is linear and da/a is the Haar measure of the multiplicative group þ, j^wðar h ðxþþj dadh ¼ j^wðar h ðjxjx ÞÞj dadh ¼ j^wðajxjr h ðx ÞÞj dadh SOðÞ þ a SOðÞ þ a SOðÞ þ a ¼ j^wðar h ðx ÞÞj dadh SOðÞ þ a : emark. Assume that w L ð ; HÞ is radially symmetric, that is, rotation invariant, j^wj is continuous at x =, and j^wðþj ¼. Then w AWQ. Let us show this fact. Denote e = (,). For x satisfyingjxj =, there exists g [,p) such that x = r g (e). Since w is radially symmetric, ^w is also radially symmetric. Then, we have ^wðar h ðxþþ ¼ ^wðar hþg ðeþþ ¼ ^wðaeþ; which implies SOðÞ j^wðar h ðxþþj dadh þ a ¼ dh j^wðaeþj da SOðÞ þ a : The condition j^wðþj ¼ ensures the integrability of the right-hand side of (8). ð8þ

4 M. Bahri et al. / Applied Mathematics and Computation 8 () 3 Notice that according to (5) C w is an invertible real constant. Using () we may decompose w AQW into the following form wðxþ ¼w ðxþþiw ðxþþjw ðxþþkw 3 ðxþ; where w s L ð ; Þ for s =,,,3. Using () and the linearity of the (right-sided) QFT we may write (9) in the quaternionic frequency domain in the form F q fwgðxþ ¼ ðw ðxþþiw ðxþþjw ðxþþkw 3 ðxþþe ix x e jx x d x ¼F q fw gðxþþif q fw gðxþþjf q fw gðxþþkf q fw 3 gðxþ; ðþ where we assume that F q fw s gl ð ; Þ, for s =,,,3. Like for classical wavelets [5,], the zeroth moment of w AQW vanishes, wðxþd x ¼ ðw ðxþþiw ðxþþjw ðxþþkw 3 ðxþþd x ¼ : ðþ It means that the integral of every component w s of the quaternion mother wavelet is zero w s d x ¼ ; s ¼ ; ; ; 3: ðþ Definition 3. For w L ð ; HÞ; a þ ; b, and r h SO(), we define the unitary linear operator U a;h;b : L ð ; HÞ!L ðg; HÞ; as ðu a;h;b ðwþþ ¼ w a;h;b ðxþ ¼ a w r x b h : ð3þ a The family of wavelets w a,h,b are called daughter quaternion wavelets where a is a dilation parameter, b a translation vector parameter, and h an SO() rotation parameter. By straightforward calculations we obtain the following lemma. Lemma. Let w be an admissible quaternion function. Daughter quaternion wavelets (3) can be written in terms of the (rightsided) QFT as n F q fw a;h;b gðxþ ¼ae ix b w cðar h ðxþþ þ iw c o n ðar h ðxþþ e jx b þ ae ix b jw c ðar h ðxþþ þ kw c o 3 ðar h ðxþþ e jx b : ðþ ð9þ Proof. Definition gives F q fw a;h;b gðxþ ¼ a w r x b h a e ix x e jx x d x: Performing the change of variables x b ¼ y into the above expression, we immediately obtain a F q fw a;h;b gðxþ ¼ a wðr hyþe ix ðb þay Þ e jx ðb þay Þ a d y ¼ a wðr h ðyþþe ix b e iax y e jax y d ye jx b : e ix b fw ðr ðyþþ þ iw ðr ðyþþge iax y e jax y d ye jx b b Observe, first, that w =(w + iw )+(jw + kw 3 ) and use the fact that jw e ix b = w e ix b j and kw 3 e ix b = w 3 e ix b k. The above identity leads to F q fw a;h;b gðxþ ¼a fw ðr h ðyþþ þ iw ðr h ðyþþ þ jw ðr h ðyþþ þ kw 3 ðr h ðyþþg e ix b e iax y e jax y d ye jx b ¼ a e ix b fw ðr h ðyþþ þ iw ðr h ðyþþg þ e ix b fjw ðr h ðyþþ þ kw 3 ðr h ðyþþg e iax y e jax y d ye jx b n o ¼ a h h n o þ a e ix b fjw ðr h ðyþþ þ kw 3 ðr h ðyþþge iax y e jax y d ye jx ¼ ae ix b w ðr h ðyþþe iax y e jax y d ye jx b þ ae ix b iw ðr h ðyþþe iax y e jax y d ye jx b þ ae ix b jw ðr h ðyþþe iax y e jax y d ye jx b þ ae ix b kw 3 ðr h ðyþþge iax y e jax y d ye jx b ¼ ae ix b c wl ðar h ðxþþe jx b þ ae ix b c wl ðar h ðxþþe jx b ; ð5þ ð6þ ð7þ

5 M. Bahri et al. / Applied Mathematics and Computation 8 () where we write cw l ðar h ðxþþ ¼ w c ðar h ðxþþ þ iw c ðar h ðxþþ; cw l ðar h ðxþþ ¼ jw c ðar h ðxþþ þ kw c 3 ðar h ðxþþ: ð8þ This proves the lemma. h emark 3. Notice that if we assume that iw = wi, i.e. w ¼ w þ iw ; w ; w : ð9þ Then Lemma takes the following form F q fw a;h;b gðxþ ¼ae ix b b wðar h ðxþþe jx b : ð3þ 3.. -D continuous quaternion wavelet transform (CQWT) Definition (CQWT). The CQWT of a quaternion-valued function f L ð ; HÞ with respect to w AQW in dimensions is defined by T w : L ð ; HÞ!L ð ; HÞ f # T w f ða; h; bþ ¼ðf ; w a;h;b Þ ¼ L ð ;HÞ f ðxþ a w r x b h d x: a ð3þ It must be remarked that the order of the terms in (3) is fixed because of the non-commutativity of the product of quaternions. Changing the order yields another quaternion valued function which differs by the signs of the terms. Eq. (3) clearly shows that the CQWT can be regarded as the inner product of a quaternion-valued signal f with daughter quaternion wavelets. Lemma. Suppose that w AQW. If w L ð ; HÞ, then the CQWT (3) has a quaternion Fourier representation of the form T w f ða; h; bþ ¼ a h ^f ðxþe jb x wl c ðar ðpþ h ðxþþe ib x þ c i w l ðar h ðxþþe ib x d x; ð3þ where c w l ðar h ðxþþ and c w l ðar h ðxþþ are defined in (8). Proof. We have T w f ða; h; bþ ¼ðf ; w a;h;b Þ ðþ ¼ L ð ;HÞ ðpþ ð^f ; d wa;h;b Þ L ð ;HÞ ¼ ^f ðxþwa;h;b d ðxþd x ð3þ ¼ ðpþ ðpþ h a^f ðxþ e ix b wl c ðar h ðxþþe jx b þ e ix b wl c ðar h ðxþþe jx b ¼ h i a^f ðxþ e ix b wl c ðar ðpþ h ðxþþe jx b d x þ ðpþ h i a^f ðxþ e ix b wl c ðar h ðxþþe jx b d x ðþ ¼ ðpþ a^f ðxþe jb x wl c ðar h ðxþþe ib x d x: This proves (3). h i d x a^f ðxþe jb x c wl ðar h ðxþþe ib x d x þ ðpþ ð33þ Lemma 3. Let w L ð ; HÞ be a quaternion valued wavelet. If F q fwg ¼F q fw gþkf q fw 3 g, then Eq. (3) can be expressed as T w f ða; h; bþ ¼F q a b f ðþw c ðar h ðþþ ðbþþf q a^f ðþkw c 3 ðar h ðþþ ð bþ: ð3þ Proof. For F q fwg ¼F q fw gþkf q fw 3 g we have F q fw a;h;b gðxþ ¼ae ix b c w ðar h ðxþþe jx b þ ae ix b k c w 3 ðar h ðxþþe jx b : ð35þ

6 M. Bahri et al. / Applied Mathematics and Computation 8 () 5 In view of (35), Eq. (3) takes the following form T w f ða; h; bþ ¼ a ^f ðxþe jb x w cðar ðpþ h ðxþþe ib x d x þ a ^f ðxþe jb x kw c ðpþ 3 ðar h ðxþþe ib x d x ¼ a ^f ðxþw cðar ðpþ h ðxþþe jb x e ib x d x þ a ^f ðxþkw3 cðar ðpþ h ðxþþe jb x e ib x d x ; ð36þ where the second equality we have used the fact that k c w 3 ðar h ðxþþe jb x ¼ e jb x k c w 3 ðar h ðxþþ: ð37þ Next, applying the inverse of the (right-sided) QFT () yields T w f ða; h; bþ ¼F q a b f ðþw c ðar h ðþþ ðbþþf q a b f ðþkw c 3 ðar h ðþþ ð bþ: ð38þ emark. It is easy to see that for F q fwg Eq. (35) reduces to F q fw a;h;b gðxþ ¼ae ix b ^wðar h ðxþþe jx b ; ð39þ and for F q fwg ¼kF q fw 3 g Eq. (35) takes the form F q fw a;h;b gðxþ ¼ae ix b b wðar h ðxþþe jx b : The following proposition is a particular case of the lemma proved above. ðþ Proposition. Let w L ð ; HÞ be a quaternion valued wavelet. (i) If F q fwg, then Eq. (3) has the form T w f ða; h; bþ ¼ a ^f ðxþ^wðar h ðxþþe jb x e ib x d x: ðpþ ðþ Or, equivalently, F q ðt w f ða; h;:þþðxþ ¼a^f ðxþ^wðar h ðxþþ: (ii) If F q fwg ¼kF q fw 3 g, then we may rewrite Eq. (3) in the form T w f ða; h; bþ ¼ a ^f ðxþwðar h b ðxþþe jb x e ib x d x: ðpþ Or, equivalently, T w f ða; h; bþ ¼F q a^f ðþwðar b h ðþþ ð bþ: ðþ ð3þ ðþ 3.3. Examples of -D quaternion wavelets As examples of AQW we first take the difference of Gaussian (DOG) wavelet which the mother wavelet w obtained by subtracting a wide Gaussian from a narrow Gaussian. Example. Consider the two-dimensional DOG wavelets or difference-of-gaussian wavelets (see [3]): wðxþ ¼ c e ðx þx Þ=c e ðx þx Þ= ; < c < : ð5þ The DOG wavelet for c = 7/5 is illustrated in Fig.. Notice that F q fwg. When h =, the representation (39) implies F q fw a;;b gðxþ ¼ae ix b ^wðar ðxþþe jx b ¼ ae ix b pe ðacþ ðx þx Þ= pe a ðx þx Þ= e jx b ¼ ae ix b e jx b pe ðacþ ðx þx Þ= pe a ðx þx Þ= ; ð6þ where we used the fact that the (right-sided) QFT of the Gaussian function is another Gaussian function (see [6]). The quaternion Fourier transform F q fw a;h;b g of the DOG wavelet is illustrated in Fig. for c = /, h =,b = b = and a =.

7 6 M. Bahri et al. / Applied Mathematics and Computation 8 () x x.8 Fig.. The DOG wavelet w for c = 7/5. Now, we take ( f ðxþ ¼ e ðx þx Þ ; if x > and x > ; ; otherwise: ð7þ It is known (see [7]) that the (right-sided) QFT of f is given by F q ff gðxþ ¼ ix jx kx x ðpþ ð þ x þ x þ x x Þ : ð8þ The CQWT with respect to the DOG wavelets (5) are obtained as follows: T w f ða; h; bþ ¼F q ðt w f ða; h; ÞÞðxÞ ¼aF q ff gðxþf q fw a;h;b gðxþ ¼ a ð ix jx kx x Þðpe ðacþ ðx þx Þ= e jx b Þ ðpþ ð þ x þ x þ x x Þ a ð ix jx kx x Þðpe a ðx þx Þ= e ix b Þ ðpþ ð þ x þ x þ x x Þ : ð9þ Example. The two-dimensional quaternionic Hermite wavelets (compare to [9,]) are defined by w l ðxþ ¼e ðx þx Þ= H l ðxþ ¼ð Þ l ðe ðx þx Þ= Þ; l ¼ ; ; ð5þ where the two-dimensional quaternionic Hermite polynomials H n and Dirac are given by, respectively, H l ðxþ ¼ð Þ n e ðx þx l e ðx þx Þ= þ j It is easy to see that Eq. (5) are alternatively real or quaternion-valued. In the following we show that in terms of the QFT them are real-valued. Notice that for l = we have F q fw gðxþ ¼ e ðx þx þ j e ðx þx Þ= Þe ix x e jx x @ ¼ i e ðx þx Þ= e ix x e jx x x j e ðx þx Þ= e ix x e jx x ¼ i x e x = e ix x dx e x = e jx x dx þ j x e x = e ix x dx e x = e jx x dx ¼ pðx þ x Þe x þ x : ð5þ For l = we first observe ¼ ¼ Using the properties of the (right-sided) QFT in Table we

8 M. Bahri et al. / Applied Mathematics and Computation 8 () ω - - ω ω - - ω ω ω ω - ω Fig.. The quaternion Fourier transform F qfw a;h;b g of the DOG wavelet: the real part and imaginary part i (top row), j, and k (bottom row) of(6), for the scale parameter values c = /, h =,b = b = and a =. F q fw gðxþ @ e ðx þx Þ= e þx!e Þ= ix x e jx x d x e þx Þ= e ix x e jx x d x þ ¼ pðix Þ e x þ x þ pðjx x þ x e ðx þx Þ= Þe ix x e jx x d x ¼ pðx þ x Þe x þ x : ð53þ 3.. Basic properties Some basic properties of the CQWT are summarized in the following proposition. The properties correspond to classical wavelet transform properties. Their proofs are verified by straightforward calculations and can be found in [5,8,]. Proposition. Suppose that w, / AQW. If w = w + iw + jw + kw 3 and / = / + i/ + j/ + k/ 3 and if f, gare two quaternion functions belonging to L ð ; HÞ, then for every ða; bþ þ we have the following properties. (i) (Left linearity) ½T w ðaf þ bgþšða; h; bþ ¼aT w f ða; h; bþþbt w gða; h; bþ; where a and b are quaternion constants in H. estricting the constants to a; b we get right linearity of the CQWT.

9 8 M. Bahri et al. / Applied Mathematics and Computation 8 () (ii) (Translation covariance) ½T w f ð x ÞŠða; h; bþ ¼T w f ða; h; b x Þ for any constant x. (iii) (Dilation covariance) ½T w f ðcþšða; h; bþ ¼ c T wf ðac; h; bcþ; where c is a real positive constant. (iv) (otation covariance) ½T w f ðr h ÞŠða; h; bþ ¼T w f ða; h ; r h bþ with r h ¼ r h r h. (v) (Parity) ½T Pw Pf Šða; h; bþ ¼T w f ða; h; bþ; where P is the parity operator defined by Pf(x) = f( x). (vi) (Antilinearity) ½T awþb/ f Šða; h; bþ ¼T w f ða; h; bþa þ T / f ða; h; bþ b; for any quaternion constants a, b in H. (vii) If we introduce the translation operator M x wðxþ ¼wðx x Þ, then ½T Mx wf Šða; h; bþ ¼T w f ða; h; b þ x aþ: (viii) Consider the dilation operator D c wðxþ ¼ c w x c ½T D c wf Šða; h; bþ ¼ c T wf ðac; h; bþ:,c>. Then we have 3.5. eproducing formula In this section we show that the quaternion function f can be recovered from its CQWT whenever the quaternion wavelets satisfy the following admissibility condition. Theorem (Inner product relation). Suppose that w = w + iw + jw + kw 3 AQW be a quaternion admissible wavelet which defines the CQWT (3). If F q fwg L ð ; Þ satisfies the admissibility condition defined by (7). Then for every f ; g L ð ; HÞ\L ð ; HÞ we have SOðÞ þ T w f ða; h; bþt w gða; h; bþd dadh b ¼ C w ðf ; gþ a : L ð ;HÞ ð5þ Proof. Applying Placherel s formula for the (right-sided) QFT () to the b-integral into the left side of (5) yields (compare to Gr ochenig [5]) T w f ða; h; bþt w gða; h; bþd b dl ¼ F SOðÞ þ ðpþ q ðt w f ða; h;:þþðxþf q ðt w gða; h; ÞÞðxÞd x dl SOðÞ þ ðþ ¼ a ^f ðxþ^wðar ðpþ h ðxþþ^wðar h ðxþþ^gðxþd x dl SOðÞ þ ¼! ^f ðxþ j^wðar ðpþ h ðxþþj dadh ^gðxþd x SOðÞ þ a In the third equality we applied Fubini s theorem to reverse the integration order. In particular, if f = g in (5) we have fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} C w is a real constant ¼ C w ^f ðxþ^gðxþd x ðþ ¼ C ðpþ w f ðxþgðxþd x ¼ C w ðf ; gþ : L ð ;HÞ h ð55þ kt w f k L ðg;hþ ¼ C wkf k L ð ;HÞ : ð56þ This shows that, except for the factor C w, the CQWT is an isometry from L ð ; HÞ to L ðg; HÞ.

10 M. Bahri et al. / Applied Mathematics and Computation 8 () 9 Theorem 3 (Inversion formula). Under the assumptions of Theorem, any quaternion function f L ð ; HÞ can be decomposed as f ðxþ ¼ T w f ða; b; hþw C a;h;b dk; ð57þ w G where the integral converges in the weak sense. Proof. An application of Theorem gives for every g L ð ; HÞ C w ðf ; gþ ¼ L ð ;HÞ T w f ða; h; bþt w gða; h; bþd b dl ¼ T w f ða; h; bþt w gða; h; bþdk SOðÞ þ G ¼ T w f ða; h; bþw a;h;b ðxþgðxþd xdk ¼ T w f ða; h; bþw a;h;b ðxþgðxþdkd x G G ¼ T w f ða; h; bþw a;h;b dk; g : ð58þ G L ð ;HÞ Because the inner product identity holds for every g L ð ; HÞ we conclude that C w f ðxþ ¼ T w f ða; b; hþw a;b;h ðxþdk; which completes the proof. G h ð59þ Theorem (eproducing kernel). Suppose that w AQW. If K w ða; h; b; a ; h ; b Þ¼C w ðw a;h;b; w a ;h ;b Þ L ð ;HÞ ; ð6þ then K w (a,h,b;a,h,b ) is a reproducing kernel in L ðg; dkþ, i.e., T w f ða ; h ; b Þ¼ T w f ða; h; bþk w ða; h; b; a ; h ; b Þdk: G ð6þ Proof. By inserting (57) into the definition of the CQWT (3) we have T w f ða ; h ; b Þ¼ f ðxþw a ;h ;b ðxþd x ¼ C w T w f ða; h; bþw a ;bðxþdk w h a ;h ;b ðxþd x G ¼ T w f ða; h; bþ C w w a;h;b ðxþw a ;h ;b ðxþd x dk ¼ T w f ða; b; hþk w ða; h; b; a ; h ; b Þdk: G G The proof is complete. h ð6þ. Uncertainty principle for the CQWT The classical uncertainty principle of harmonic analysis states that a non-trivial function and its FT cannot both be simultaneously sharply localized []. In quantum mechanics the uncertainty principle asserts that one cannot at the same time be certain of the position and of the velocity of an electron (or any particle). That is, increasing the knowledge of the position decreases the knowledge of the velocity or momentum of an electron. This section extends the uncertainty principle which is valid for the QFT to the setting of the CQWT. Let us now formulate an uncertainty principle for the CQWT. This principle describes how the CQWT relates to the (rightsided) QFT of a quaternion function. Theorem 5. Let w L ð ; HÞ be an admissible quaternion wavelet that satisfies the admissibility condition (7). If w = w + iw + jw + kw 3 and assume that F q fwg, then for every f L ð ; HÞ we have the inequality (no summation over k) qffiffiffiffiffi C w kf k L ð ;HÞ; k ¼ ; : ð63þ kb k T w f ða; h; bþk L ðg;hþ kx k^f k L ð ;HÞ P In order to prove this theorem, we need to introduce the following lemma. Lemma. SOðÞ kx k FfT w f ða; h; Þgk dl ¼ C þ L ð ;HÞ wkx k^f k L ð ;HÞ; k ¼ ; : ð6þ

11 M. Bahri et al. / Applied Mathematics and Computation 8 () Proof. We observe that SOðÞ kx k FfT w f ða; h; Þgk dl ¼ þ L ð ;HÞ x k FfT w f gfft w f gx k dld x ðþ ¼ SOðÞ þ SOðÞ a x k^f ðxþ^wðar h ðxþþ^wðar h ðxþþ b dadh f ðxþx k d x ð7þ ¼ þ a 3 x k^f ðxþj^wðar h ðxþþj dadh b f ðxþd x þ a We begin with the proof of Theorem 5. ¼ C w kx k^f k L ð ;HÞ : ð65þ SOðÞ Proof. Using the uncertainty principle for the (right-sided) QFT (see [6] for more details), we get h i = h i = kb k T w f ða; h; Þk L ð ;HÞ kxk FfT w f ða; h; Þgk L ð ;HÞ P kt wf ða; h; Þk : ð66þ L ð ;HÞ Now integrating both sides of (66) with respect to the Haar measure dl, we obtain h i = h i = kb k T w f ða; h; Þk L ð ;HÞ kxk FfT w f ða; h; Þgk L ð ;HÞ dl P SOðÞ þ By applying the quaternion Cauchy Schwartz inequality (9) on the left-hand side of (67), we see that! =! = SOðÞ P þ kb k T w f ða; h; Þk L ð ;HÞ ; dl SOðÞ þ kt w f ða; h; Þk L ð ;HÞ dl: SOðÞ Then, inserting (6) into the second term of (68), we easily obtain! = kb k T w f ða; h; Þk dl = SOðÞ þ L ð ;HÞ C w kx k^f k L ð ;HÞ P SOðÞ kt w f ða; h; Þk dl: ð67þ þ L ð ;HÞ kx k ; FfT w f ða; h; Þgk dl þ L ð ;HÞ ð68þ SOðÞ kt w f ða; h; Þk dl: ð69þ þ L ð ;HÞ We recognize that the first and third terms of (69) are L ðg; HÞ-norms. This implies that qffiffiffiffiffi kb k T w f ða; h; bþk L ðg;hþ C w kx k^f kl P ð ;HÞ kt wf k : L ðg;hþ Substituting (56) into the right-hand side of (7) and simplifying it we finally get qffiffiffiffiffi kf k ; L ð ;HÞ kb k T w f ða; h; bþk L ðg;hþ kx k^f k L ð ;HÞ P C w ð7þ ð7þ which concludes the proof of Theorem 5. h eferences [] T.A Ell, Quaternionic-Fourier transforms for analysis of two-dimensional linear time-invariant partial differential systems, in: Proceedings of 3nd IEEE Conference on Decision and Control, pp. 8 8, San Antonio, TX, 993. [] S.C. Pei, J.J. Ding, J.H. Chang, Efficient implementation of quaternion Fourier transform, convolution, and correlation by -D complex FFT, IEEE Trans. Signal Process 9 () () [3] J.P. Antoine,. Murenzi, Two-dimensional directional wavelet and the scale-angle representation, Signal Process. 5 (3) (996) [] J.P. Antoine, P. Vandergheynst, Two-dimensional directional wavelet in imaging processing, Int. J. Imag. Syst. Technol. 7 (3) (996) [5] K. Gröchenig, Foundation of Time-Frequency Analysis, Birkhäuser, Boston,. [6] L. Traversoni, Imaging analysis using quaternion wavelet, in geometric algebra with applications, in: E.B. Corrochano, G. Sobczyk (Eds.), Science and Engineering, Birkhäuser, Boston,. [7] E. Hitzer, Quaternion Fourier transform on quaternion fields and generalizations, Adv. Appl. Clifford Algebr. 7 (3) (7) [8] E. Hitzer, B. Mawardi, Clifford Fourier transform on multivector fields and uncertainty principle for dimensions n = (mod ) and n = 3 (mod ), Adv. Appl. Clifford Algebr. 8 (3 ) (8) [9] F. Brackx,. Delange, F. Sommen, Clifford Hermite wavelets in Euclidean space, J. Fourier Anal. Appl. 8 (3) () [] F. Brackx, F. Sommen, Benchmarking of three-dimensional Clifford wavelet functions, Complex Variables: Theory and Applications 7 (7) () [] E. Bayro-Corrochano, The theory and use of the quaternion wavelet transform, J. Math. Imag. Vision () (6) [3] J. hou, Y. Xu, X. Yang, Quaternion wavelet phase based stereo matching for uncalibrated images, Pattern ecogn. Lett. 8 () (7) [5] S. Mallat, A Wavelet Tour of Signal Processing, second ed., Academic Press, San Diego, CA, 999. [6] B. Mawardi, E. Hitzer, A. Hayashi,. Ashino, An uncertainty principle for quaternion Fourier transform, Comput. Math. Appl. 56 (9) (8) 7. [7] B. Mawardi, E. Hitzer,. Ashino,. Vaillancourt, Windowed Fourier transform of two-dimensional quaternionic signals, App. Math. Comput. 6 (8) ()

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