International Journal of Wavelets, Multiresolution and Information Processing c World Scientific Publishing Company

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1 October 6, 014 1:7 WSPC/WS-IJWMIP QTF-ijwmip revf International Journal of Wavelets, Multiresolution and Information Processing c World Scientific Publishing Company CONTINUOUS QUATENION FOUIE AND WAVELET TANSFOMS MAWADI BAHI Department of Mathematics, Universitas Hasanuddin, Makassar 9045, Indonesia mawardibahri@gmail.com YUICHI ASHINO Division of Mathematical Sciences, Osaka Kyoiku University, Osaka , Japan ashino@cc.osaka-kyoiku.ac.jp ÉMI VAILLANCOUT Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario, Canada K1N 6N5. remi@uottawa.ca eceived (Day Month Year) evised (Day Month Year) Communicated by (xxxxxxxxxx) A two-dimensional quaternion Fourier transform (QFT) defined with the kernel ω x is proposed. Some fundamental properties, such as convolution theorem, Plancherel theorem, and vector differential, are established. The heat equation in quaternion algebra is presented as an example of the application of the QFT to partial differential equations. The wavelet transform is extended to quaternion algebra using the kernel of the QFT. e i+j+k Keywords: quaternion-valued function; quaternion algebra; quaternion Fourier transform. AMS Subject Classification: 15A66, 4B10, 0G5 1. Introduction It is well-known that every two dimensional rotation around the origin in the plane can be represented by the multiplication of the complex number e iθ = cos θ + i sin θ, 0 θ < π. Similarly, every three dimensional rotation in the space can be represented by the multiplications of the quaternion q from the left-hand side and its conjugate q from the right-hand side, where q = cos(θ/) + α sin(θ/) with a unit quaternion α representing the axis of rotation and the angle θ of rotation, 0 θ < π. By this reason, quaternions are commonly used in threedimensional computer graphics and computer vision. ecently it has become popular to generalize the classical Fourier transform (FT) to quaternion algebra and 1

2 October 6, 014 1:7 WSPC/WS-IJWMIP QTF-ijwmip revf Mawardi Bahri, yuichi Ashino, and émi Vaillancourt used in image analysis. For example, in order to compute the Fourier transform of a color image without separating a color image into three gray-scale images, a single and holistic Fourier transform, which treats a color image as a vector field, was proposed. 6 In this framework, quaternions represent color image pixels. These extensions are broadly called the quaternion Fourier transform (QFT). Due to the non-commutative property of quaternion multiplication, there are at least three different types of two-dimensional QFTs as follows (see Ell 5, Hitzer 10, Hitzer 11, Mawardi, Hitzer, Hayashi, and Ashino 19, Pei, Ding, and Chan, Said, Bihan, and Sangwine ), Fq I {f}(ω) = e µ1ω x f(x) d x, ω x = ω 1 x 1 + ω x, (1.1) Fq II {f}(ω) = f(x) e µ1ω x d x, ω x = ω 1 x 1 + ω x, (1.) Fq III {f}(ω) = e µ1ω1x1 f(x) e µωx d x, (1.) where µ 1 and µ are any two unit pure quaternions (µ 1 = µ = 1) that are orthogonal to each other. These three QFTs are so-called left-side, right-side, and double-side, or type I, II, and III, respectively. Assefa, Mansinha, Tiampo, asmussen, and Abdell 1 used QFTs of type II to establish the two-dimensional quaternion Stockwell (QS) transform and then apply it for the analysis of local color image spectra. ecently, Guo and Zhu 8 introduced the quaternion Fourier-Mellin moments as a generalization of traditional Fourier- Mellin moments to quaternion algebra. Several properties of this generalization are investigated using QFTs of type II. On the other hand, a number of attempts have been previously made to generalize the classical wavelet transform (WT) to quaternion algebra. He 9, Zhao and Peng 5 constructed the continuous quaternion wavelet transform of quaternionvalued functions. They also demonstrated a number of properties of these extended wavelets using the classical FT. Using the (two-sided) QFT, Traversoni 4 proposed a discrete quaternion wavelet transform (QWT) which was applied by Bayro and Zhou, Xu, and Yang 6. Olhede and Metikas 1 have proposed the monogenic wavelet transform based on 1-D analytic WT. ecently, Mawardi, Adji, and Zhao 14, and Mawardi and Hitzer 18 introduced an extension of the WT to Clifford algebra by means of the kernel of the Clifford Fourier transform 1. It was found that many WT properties still hold but others have to be modified. In this paper, we concentrate on QFTs of type II with kernel µ 1 = i+j+k. We derive the shift, modulation, and convolution properties and also establish the Plancherel and vector differential theorems. Then, we define a continuous quaternion wavelet transform using the kernel of QFTs of type II, which is essentially different from the kernel of QFTs of type III studied in Mawardi, Ashino, and Vaillancourt 15 and Mawardi 17 ).

3 October 6, 014 1:7 WSPC/WS-IJWMIP QTF-ijwmip revf Continuous Quaternion Fourier and Wavelet Transforms. Basics Let us present the notation which will be used in this paper. The quaternion algebra over, denoted by H, is an associative non-commutative four-dimensional algebra, H = {q = q 0 + iq 1 + jq + kq q 0, q 1, q, q }, (.1) which obey Hamilton s multiplication rules: ij = ji = k, jk = kj = i; ki = ik = j, i = j = k = ijk = 1. (.) For a quaternion q = q 0 + iq 1 + jq + kq H, q 0 is called the scalar (or real) part of q, denoted by Sc(q), and iq 1 + jq + kq is called the vector (or pure) part of q. The vector part of q is conventionally denoted by q or Vec(q). Let p, q H and p, q be their vector parts, respectively. Equation (.) yields the quaternionic product qp as where we define the scalar product: qp = q 0 p 0 + q p + q 0 p + p 0 q + q p, (.) q p = (q 1 p 1 + q p + q p ), and the antisymmetric cross type product: q p = i(q p q p ) + j(q p 1 q 1 p ) + k(q 1 p q p 1 ). The quaternion conjugate of a quaternion q is given by and it is an anti-involution, that is, q = q 0 iq 1 jq kq, q 0, q 1, q, q, (.4) qp = p q. (.5) From (.4), we obtain the norm or modulus of q H defined as q = q q = q0 + q 1 + q + q. (.6) It is easy to see that qp = q p, p, q H. (.7) Using the conjugate (.4) and the modulus of q, we can define the reciprocal of q H \ {0} by q 1 = q q, (.8) which shows that H is a normed division algebra. Any quaternion q can be written as q = q e i+j+k θ, (.9)

4 October 6, 014 1:7 WSPC/WS-IJWMIP QTF-ijwmip revf 4 Mawardi Bahri, yuichi Ashino, and émi Vaillancourt where θ = arctan Sc(q) /Vec(q), 0 θ π is the eigen angle or phase of q. When q = 1, q is a unit quaternion. Euler s and De Moivre s formulas still hold in quaternion space, that is, for a pure unit quaternion the following holds: e i+j+k θ = cos θ + i + j + k ( e i+j+k n θ = cos θ + i + j + k sin θ ) n sin θ = cos nθ + i + j + k sin nθ. As in the algebra of complex numbers, we can define three nontrivial algebra involutions for quaternions α(q) = iqi = i(q 0 + iq 1 + jq + kq )i = q 0 + iq 1 jq kq, β(q) = jqj = j(q 0 + iq 1 + jq + kq )j = q 0 iq 1 + jq kq, γ(q) = kqk= k(q 0 + iq 1 + jq + kq )k = q 0 iq 1 jq + kq. (.10) It is convenient to introduce the inner product of two quaternion-valued functions, f, g : H, as follows: (f, g) L ( ;H) = f(x)g(x) d x. (.11) In particular, if f = g, then we obtain the associated norm: f L ( ;H) = f(x) d x. (.1). Basic Properties of QFT Before giving the fundamental properties of two-dimensional QFTs, we list their basic properties and detailed proofs. Most of them are essentially straightforward extensions of the -D FT properties. Denote by {e 1, e } the standard basis of. Definition.1. Let f L 1 ( ; H). The QFT of f is the function F q {f} : H defined by F q {f}(ω) = ˆf(ω) = f(x) e i+j+k ω x d x, (.1) where x = x 1 e 1 + x e, ω = ω 1 e 1 + ω e, and e i+j+k ω x is called the quaternion Fourier kernel. Theorem.1. Suppose that f L ( ; H) and F q {f} L 1 ( ; H). Then the QFT is invertible with inverse Fq 1 [F q {f}](x) = f(x) = 1 (π) F q {f}(ω) e i+j+k ω x d ω. (.)

5 October 6, 014 1:7 WSPC/WS-IJWMIP QTF-ijwmip revf Continuous Quaternion Fourier and Wavelet Transforms 5.1. Left Linearity It is easy to show the following lemma. Lemma.1. Let f 1, f L ( ; H). The QF T is left linear, that is, F q {αf 1 + βf }(ω) = αf q {f 1 }(ω) + βf q {f }(ω), α, β H. (.) Note that right linearity is not valid for the QFT... Shift Property Lemma.. The QFT of a shifted function is given by F q {f(x b)}(ω) = F q {f}(ω) e i+j+k ω b. (.4) Proof. Equation (.1) gives F q {f(x b)}(ω) = f(x b)e i+j+k ω x d x. Substituting t = x b in the above expression, we have x = t + b and d x= d t. Hence, F q {f(x b)}(ω) = f(t)e i+j+k ω (t+b) d t = f(t)e i+j+k ω t d t e i+j+k ω b This proves (.4). = F q {f}(ω) e i+j+k ω b... Scaling Property Lemma.. Let a \ {0}. The QFT of the scaled function f a (x) = f(ax) is given by F q {f a }(ω) = 1 ( ω ) a F q{f}. (.5) a Proof. We first assume that a > 0. Definition.1 gives F q {f a }(ω) = f(ax)e i+j+k ω x d x. Substituting u for ax, we have F q {f a }(ω) = 1 a = 1 a F q{f} f(u)e ( ω a ). i+j+k ω a u d u

6 October 6, 014 1:7 WSPC/WS-IJWMIP QTF-ijwmip revf 6 Mawardi Bahri, yuichi Ashino, and émi Vaillancourt For a < 0, we have which completes the proof. F q {f a }(ω) = 1 ( ω ) ( a) F q{f}, a.4. Modulation Property Lemma.4. Let ω 0 and F 0 (x) = f(x) e i+j+k ω 0 x. Then, we have F q {F 0 }(ω) = F q {f}(ω ω 0 ). (.6) Proof. Using Definition.1 and simplifying it, we obtain F q {F 0 }(ω) = f(x)e i+j+k ω 0 x e i+j+k ω x d x = f(x)e i+j+k (ω ω 0) x d x which was to be proved. = F q {f}(ω ω 0 ), Notation.1. We use the conventional notation f(x), which means that the function f(x) is real-valued. emark.1. Note that this property is different from the usual modulation property of the two-dimensional FT. If the modulation term is multiplied from the left, that is, F 0 (x) = e i+j+k then the modulation property holds only for ω 0 x f(x), f(x) = f 0 (x) + (i + j + k)f 1 (x), f 0 (x), f 1 (x). 4. Major Properties of the QFT This section describes important properties of the QFT, such as the Plancherel and convolution theorems. Firstly, let us establish the Plancherel Theorem. Theorem 4.1 (QFT Plancherel). Let f, g L ( ; H). Then, we have (f, g) L ( ;H) = 1 (π) (F q{f}, F q {g}) L ( ;H). (4.1) In particular, with f = g, we have the Parseval Theorem: f L ( ;H) = 1 π F q{f} L ( ;H). (4.)

7 October 6, 014 1:7 WSPC/WS-IJWMIP QTF-ijwmip revf Continuous Quaternion Fourier and Wavelet Transforms 7 Proof. Equation (4.1) follows from (f, g) L ( ;H) = f(x)g(x) d x [ (.) 1 = (π) F q {f}(ω) e i+j+k (.5) = = 1 (π) [ 1 (π) F q {f}(ω) g(x) e which completes the proof of Theorem 4.1. ] ω x d ω g(x) d x i+j+k ω x d x ] d ω F q {f}(ω)f q {g}(ω) d ω, (4.) The most important property of the QFT for applications in signal processing is the convolution theorem. Due to the non-commutativity of quaternion multiplication, we have the following definition. Definition 4.1. Let f, g L ( ; H). The convolution f g is defined by (f g)(x) = f(y)g(x y) d y. (4.4) emark 4.1. In general, f g g f because the quaternion multiplication is not commutative: f(y)g(x y) g(x y)f(y). Theorem 4.. Let f, g L ( ; H) be two quaternion-valued functions. If f has the following representation f(x) = f 0 (x) + if 1 (x) + jf (x) + kf (x), then, the QFT of the convolution f g is given by F q {f g}(ω) = F q {g}(ω)f q {f 0 }(ω) + if q {g}(ω)f q {f 1 }(ω) + jf q {g}(ω)f q {f }(ω) + kf q {g}(ω)f q {f }(ω) = F q {g}(ω)f q {f 0 }(ω) + α(f q {g}(ω))i F q {f 1 }(ω) + β(f q {g}(ω))j F q {f }(ω) + γ(f q {g}(ω))k F q {f }(ω), (4.5) where α, β and γ denote the three nontrivial automorphisms of the quaternion algebra defined in (.10). Proof. Applying the definition of the QFT on the left-hand side of (4.4), we have [ ] F q {f g}(ω) = f(y)g(x y)d y e i+j+k ω x d x [ ] = f(y) g(x y)e i+j+k ω x d x d y. (4.6)

8 October 6, 014 1:7 WSPC/WS-IJWMIP QTF-ijwmip revf 8 Mawardi Bahri, yuichi Ashino, and émi Vaillancourt Put z = x y and applying again the definition of the QFT, (4.6) can be rewritten as F q {f g}(ω) [ ] = f(y) g(z) e i+j+k [ω (y+z)] d z d y [ ] = f(y) g(z) e i+j+k ω z e i+j+k ω y d z d y = (f 0 (y) + if 1 (y) + jf (y) + kf (y))[f q {g}(ω)]e i+j+k ω y d y ( = Fq {g}(ω)f 0 (y) + if q {g}(ω)f 1 (y) + jf q {g}(ω)f (y) + kf q {g}(ω)f (y) ) e i+j+k ω y d y = F q {g}(ω)f q {f 0 }(ω) + if q {g}(ω)f q {f 1 }(ω) + jf q {g}(ω)f q {f }(ω) + kf q {g}(ω)f q {f }(ω), (4.7) which finishes the proof by (.10). As a special case of Theorem 4., we have the following lemma. Lemma 4.1. (i) If F q {g}(ω), then F q {f g}(ω) = F q {f}(ω)f q {g}(ω). (4.8) (ii) If f(x), then F q {f g}(ω) = F q {g}(ω)f q {f}(ω). (4.9) Note that the convolution of two Gaussian functions is again a Gaussian function by Lemma 4.1. Applying the inverse QFT to the left-hand side of (4.8), we have the following corollary, which is important for solving partial differential equations in quaternion algebra. Corollary 4.1. Assume that F q {g}(ω). Then, we have F 1 q [F q {f}f q {g}](x) = (f g)(x). (4.10)

9 October 6, 014 1:7 WSPC/WS-IJWMIP QTF-ijwmip revf Continuous Quaternion Fourier and Wavelet Transforms 9 Proof. By the QFT inversion, we have Fq 1 [F q {f}f q {g}](x) (.) 1 = (π) f(y) e i+j+k ω y d y F q {g}(ω) e i+j+k ω x d ω [ ] 1 = f(y) (π) F q {g}(ω) e i+j+k ω (x y) d ω d y = f(y) g(x y) d y = (f g)(x). (4.11) In the second equality of (4.11), we used the assumption to interchange and combine the quaternion Fourier kernel. This completes the proof of (4.10). 5. Differentiation of QFT Differentiation properties of the two-dimensional FT can be generalized to the proposed QFT without changing their invariant (independent of coordinates) expressions in terms of vector differentials and vector derivatives. Firstly, we define the vector differential (compare to Hitzer and Mawardi 1, Mawardi and Hitzer 16 ). Let ( n, Q) = V be a real vector space with basis {ẽ 1,..., ẽ n } and nondegenerate quadratic form Q : V : Q(x) = x 1 + x + + x p x p+1 x p+q, p + q = n, p, q Z +, where x = n i=1 x iẽ i represents an arbitrary element of ( n, Q) and Z + = {n Z n > 0}. It is required that the following basic multiplication rules hold: ẽ i = 1, 1 i p, ẽ i = 1, p + 1 i n, ẽ i ẽ j + ẽ j ẽ i = 0, i j, 1 i n, 1 j n. Sylvester s theorem guarantees that p and q do not depend on the choice of the basis {ẽ 1,..., ẽ n }. We use the abbreviation p,q for ( n, Q). The pair p, q is c ed the signature of p,q. The spaces n,0 are called Euclidean spaces while all spaces of type 0,n are called anti-euclidean spaces. Definition 5.1. Let a 0,. The vector differential a along the direction a is defined by where = ẽ ẽ, a k = a ẽ k, and k = Applying the vector derivative twice, we have ( ) ( ) = = ẽ 1 + ẽ ẽ 1 + ẽ x 1 x x 1 x a = a a, (5.1) x k, k = 1,. ( ) = x + 1 x, (5.)

10 October 6, 014 1:7 WSPC/WS-IJWMIP QTF-ijwmip revf 10 Mawardi Bahri, yuichi Ashino, and émi Vaillancourt where we used the fact that ẽ 1 = ẽ = 1 and ẽ 1 ẽ = ẽ ẽ 1. Theorem 5.1 (Vector differential). The QFT of the vector differential of f L ( 0, ; H) is given by F q {a f}(ω) = F q {f}(ω) i + j + k a ω. (5.) Proof. Applying the vector differential a to the inversion formula (.), we have a f(x) = a 1 (π) F q {f}(ω) e i+j+k ω x d ω = 1 (π) F q {f}(ω) (a ) e i+j+k ω x d ω = 1 ( i + j + k ) (π) F q {f}(ω) a ω e i+j+k ω x d ω [ = Fq 1 F q {f}(ω) i + j + k ] a ω (x), which implies (5.). Example 5.1. Setting a = ẽ k, k = 1,, we have the QFT of partial derivatives of f(x), F q { k f}(ω) = F q {f}(ω) i + j + k ω k, k = 1,. (5.4) A similar calculation gives the following theorem. Theorem 5.. Let a, b 0,. Then, we have F q {(a ) (b )f}(ω) = a ω b ω F q {f}(ω). (5.5) Example 5.. For a = ẽ k, b = ẽ l, we have F q { k l }(ω) = ω k ω l F q {f}(ω), k, l = 1,. (5.6) Theorem 5.. Let x 0,. Then, the QFT of the m th vector moment is given by ( ) m i + j + k F q {x m f(x)}(ω) = m ω F q {f}(ω), m N, (5.7) where ω is the vector derivative with respect to the vector variable index ω, i.e., ω = ẽ 1 ω1 + ẽ ω.

11 October 6, 014 1:7 WSPC/WS-IJWMIP QTF-ijwmip revf Continuous Quaternion Fourier and Wavelet Transforms 11 Proof. A direct calculation of the first vector moment (m = 1) gives F q {xf(x)}(ω) = xf(x) e i+j+k ω x d x = ω f(x) i + j + k e i+j+k ω x d x = ω f(x) i + j + k e i+j+k ω x d x = ω f(x) e i+j+k ω x d x i + j + k = ω F q {f}(ω) i + j + k. In the fourth equality, we used the fact that e i+j+k ω x i + j + k = i + j + k e i+j+k ω x. (5.8) epeating this procedure m 1 times, we have which completes the proof. F q {x m f(x)}(ω) = m ω F q {f}(ω) ( i + j + k ) m, (5.9) Theorem 5.4. The QFT of the m th vector derivative is given by ( m i + j + k F q { m f}(ω) = F q {f}(ω) ω), m N. (5.10) In particular, the case of the Laplacian = is F q { f}(ω) = ω F q {f}(ω). (5.11) Proof. A simple computation gives f(x) = 1 (π) F q {f}(ω) e i+j+k ω x d ω = 1 (π) F q {f}(ω) i + j + k ω e i+j+k ω x d ω [ = Fq 1 F q {f}(ω) i + j + k ] ω (x). (5.1) Therefore, we have F q { f}(ω) = F q {f}(ω) i + j + k ω.

12 October 6, 014 1:7 WSPC/WS-IJWMIP QTF-ijwmip revf 1 Mawardi Bahri, yuichi Ashino, and émi Vaillancourt Applying the vector differential,, to equation (5.1) once more, we have F q { f} = F q { ( f)} = F q { f}(ω) i + j + k ω ( ) i + j + k = F q {f}(ω) ω = ω F q {f}(ω). (5.1) Then, mathematical induction implies ( m i + j + k F q { m f} = F q {f}(ω) ω), m N. (5.14) emark 5.1. Notice that ω = ω ω + ω ω = ω ω. This means that ω m is a scalar if m is even and ω m is a vector if m is odd. 6. Application of the QFT In this section, we present apply the QFT to partial differential equations in quaternion algebra (compare to Obolashvili 0 ). Let us consider the following initial value problem: and u t u = 0, on 0, (0, ), (6.1) u(x, 0) = f(x), f S( 0, ; H), (6.) where S( 0, ; H) is the quaternion Schwartz space, that is, the set of rapidly decreasing functions from 0, to H. Applying the definition of the QFT to both sides of (6.1) with respect to x and using (5.11), we have F q {u t }(ω) = F q {u}(ω) ( i + j + k ω 1 ) + F q {u}(ω) ( ) i + j + k ω = (ω 1 + ω )F q {u}(ω). (6.) Assume that u(x, t) is sufficiently nice to ensure the interchange of differentiation with respect to t and the QFT, that is, F q { u t } = t F q{u}. Then, the general solution of (6.) is given by F q {u}(ω, t) = C e (ω 1 +ω )t, (6.4) where C is a quaternion constant. We impose the initial condition F q {u}(ω, 0) = F q {f}(ω) to obtain F q {u}(ω, t) = e (ω 1 +ω )t F q {f}(ω). (6.5)

13 October 6, 014 1:7 WSPC/WS-IJWMIP QTF-ijwmip revf Continuous Quaternion Fourier and Wavelet Transforms 1 Note that the QFT of a Gaussian quaternion function is also a Gaussian quaternion function (compare to Mawardi, Hitzer, Hayashi, and Ashino 19 ). More precisely, we have 1 { 4πt F q e (x 1 +x)/(4t)} = e (ω 1 +ω )t. (6.6) Applying the inverse QFT, we have [ ] u(x, t) = Fq 1 e (ω 1 +ω )t F q {f} (x) [ 1 { = Fq 1 4πt F q e (x 1 +x)/(4t)} ] F q {f} (x). (6.7) Since F q {e (x 1 +x )/(4t)} (ω) = 4πte (ω 1 +ω )t, then we can apply the convolution theorem (4.8) to have u(x, t) = K t (x) f, (6.8) where K t (x) = 1 4πt e (x 1 +x )/(4t). If we decompose f = f 0 + if 1 + jf + kf, then equation (6.8) reduces to u(x, t) = K t (x) f 0 + ik t (x) f 1 + jk t (x) f + kk t (x) f, (6.9) where f i, i = 0, 1,,. By Definition(4.1) of the convolution, (4.4) gives u(x, t) = 1 e (x y) /(4t) f 0 (y) d y + i e (x y) /(4t) f 1 (y) d y 4πt 4πt 0, 0, + j /(4t) f (y) d y + k /(4t) f (y) d y. (6.10) 4πt 4πt 0, e (x y) Thus, we have the following theorem. 0, e (x y) Theorem 6.1. Let f i, i = 0, 1,,, belong to L p ( 0, ; ), 1 p. Put u i (x, t) = K t (x) f i. Then, each u i (x, t), i = 0, 1,,, is a solution of and u i t u i = 0 on 0, (0, ) is a solution of (6.1). u(x, t) = u 0 (x, t) + iu 1 (x, t) + ju (x, t) + ku (x, t) Proof. It is well-known that each u i (x, t) = K t (x) f i satisfies the heat equation 7. Using the superposition principle, we have that K t (x) f 0 + ik t (x) f 1 + jk t (x) f + kk t (x) f (6.11) is a solution of the quaternion heat equation (6.1).

14 October 6, 014 1:7 WSPC/WS-IJWMIP QTF-ijwmip revf 14 Mawardi Bahri, yuichi Ashino, and émi Vaillancourt A similar argument gives the following theorem. Theorem 6.. Let f i, i = 0, 1,,, be bounded and continuous. Then, we have the following: (i) Each u i (x, t) = K t (x) f i, i = 0, 1,,, is continuous on 0, (0, ) and satisfies u i (x, 0) = f i (x). (ii) The solution u is continuous on 0, (0, ) and satisfies u(x, 0) = f(x). 7. Construction of -D QWT To extend the classical wavelet theory to quaternion algebra, we use the kernel of the QFT in Definition.1 (compare to Mawardi, Ashino, and Vaillancourt 15 ). This section constructs the two-dimensional continuous quaternion wavelet transform (CQWT) Admissible Quaternion Wavelet Denote by SO(n), n N, the special orthogonal group, that is, the rotation group in n. Definition 7.1 (Admissible quaternion wavelet). A pair {ψ, φ} of functions in L ( ; H) is said to be an admissible quaternion wavelet (AQW) pair if ψ and φ satisfy the following admissibility condition: C {ψ,φ} = SO() + ˆψ(ar θ (ω)) ˆφ(ar θ (ω)) da dθ a (7.1) is a nonzero quaternion constant independent of ω satisfying ω = 1. Denote by AQW, the class of admissible quaternion wavelet pairs {ψ, φ}, ψ, φ L ( ; H). A function ψ L ( ; H) is said to be an admissible quaternion wavelet if ψ satisfies the following admissibility condition: C ψ = ˆψ(ar da dθ θ (ω)) (7.) SO() a + is a real positive constant independent of ω satisfying ω = 1. As a matter of convenience, we denote ψ AQW when ψ is an admissible quaternion wavelet. A ψ AW Q is sometimes called a quaternion mother wavelet and ψ a,θ,b (x) are called daughter quaternion wavelets, where a is a dilation parameter, b is a translation vector parameter, and θ is an SO() rotation parameter. emark 7.1. If ˆψ(ξ) is continuous near ξ = 0, then the existence of integral (7.) guarantees that ˆψ(0) = 0. Since the QFT ˆψ of each quaternion mother wavelet

15 October 6, 014 1:7 WSPC/WS-IJWMIP QTF-ijwmip revf Continuous Quaternion Fourier and Wavelet Transforms 15 ψ L 1 L ( ; H) is bounded and continuous, we have ˆψ(0) = (ψ 0 (x) + iψ 1 (x) + jψ (x) + kψ (x)) e i+j+k 0 x d x = ψ 0 (x) d x + i ψ 1 (x) d x + j ψ (x) d x + k ψ (x) d x = 0. This means that the integral of every component ψ s of the quaternion mother wavelet is zero: ψ s (x) d x = 0, s = 0, 1,,, (7.) where ψ s L 1 L ( ; ), s = 0, 1,,. Definition 7.. For ψ L ( ; H), a + = {a a > 0}, b, and r θ SO(), we define the unitary linear operator U a,θ,b : L ( ; H) L (G; H), by (U a,θ,b (ψ)) = ψ a,θ,b (x) = 1 ( ( )) x b a ψ r θ. (7.4) a Here we denote by G, the similarity group SIM(), which is defined by SIM() = + SO() = {(a, r θ, b) a +, r θ SO(), b }. (7.5) For an admissible pair {ψ, ψ} AW Q, ψ is called a quaternion mother wavelet and ψ a,θ,b (x) are called daughter quaternion wavelets, where a is a dilation parameter, b is a translation vector parameter, and θ is an SO() rotation parameter. Straightforward calculations give the following lemma. Lemma 7.1. Let ψ be a quaternion mother wavelet. The daughter quaternion wavelets (7.4) can be represented in terms of the QFT as F q {ψ a,θ,b }(ω) = a ψ(ar θ ω) e i+j+k ω b. (7.6) Proof. By definition, we have F q {ψ a,θ,b }(ω) = 1 a ψ ( ( )) x b r θ e i+j+k ω x d x. (7.7) a

16 October 6, 014 1:7 WSPC/WS-IJWMIP QTF-ijwmip revf 16 Mawardi Bahri, yuichi Ashino, and émi Vaillancourt Performing the change of variables x b a 1 F q {ψ a,θ,b }(ω) = a ψ(r θy) e = a ψ(r θ y) e = a ψ(r θ y) e Putting z = r θ y, we easily obtain F q {ψ a,θ,b }(ω) = a ψ(z) e = a ψ(z) e which was to be proved. = y into the above expression, we have i+j+k i+j+k i+j+k ω (b+ay) a d y i+j+k ω b e i+j+k aω y d y i+j+k aω y d y e i+j+k ω b. aω (r θ z) det(r θ )d z e i+j+k ω b (aωr θ ) z d z e i+j+k ω b = a ψ(ar θ ω) e i+j+k ω b, (7.8) 7.. -D Continuous Quaternion Wavelet Transform (CQWT) Definition 7. (CQWT). The two-dimensional continuous quaternion wavelet transform (CQWT), T ψ : L ( ; H) L ( ; H) of a quaternion-valued function f L ( ; H) with respect to a quaternion mother wavelet ψ is defined by f T ψ f(a, θ, b) = (f, ψ a,θ,b ) L ( ;H) = f(x) 1 a ψ ( r θ ( x b a )) d x. (7.9) It must be remarked that the order of the terms in (7.9) 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. Equation (7.9) clearly shows that the CQWT can be regarded as the inner product of a quaternionvalued signal f with a daughter quaternion wavelet. Lemma 7.. Let ψ be a quaternion mother wavelet. Then, the CQWT of ψ has a quaternion Fourier representation of the form T ψ f(a, θ, b) = 1 (π) a ˆf(ω) e i+j+k ω b ψ(ar θ ω) d ω. (7.10)

17 October 6, 014 1:7 WSPC/WS-IJWMIP QTF-ijwmip revf Continuous Quaternion Fourier and Wavelet Transforms 17 Proof. Simple computations yield This proves (7.10). T ψ f(a, θ, b) = (f, ψ a,θ,b ) L ( ;H) (4.1) 1 = (π) ( f, ψ a,θ,b ) L ( ;H) = 1 ˆf(ω) ψ (π) a,θ,b (ω) d ω { } (7.6) 1 = (π) a ˆf(ω) ψ(ar θ ω) e i+j+k ω b d ω (.5) 1 = (π) a ˆf(ω) e i+j+k ω b ψ(ar θ ω) d ω. (7.11) We now deduce the following proposition from Lemma 7.. Proposition 7.1. Let ψ be a quaternion mother wavelet. If we assume that then equation (7.10) has the form T ψ f(a, θ, b) = a (π) Or, equivalently, e i+j+k ω b ψ(ar θ ω) = ψ(ar θ ω) e i+j+k ω b, = F 1 q ˆf(ω) ψ(ar θ (ω)) e i+j+k ω b d ω { a ˆf(ω) ψ(ar θ (ω)) } (b). (7.1) F q (T ψ f(a, θ, ))(ω) = a f(ω) ψ(ar θ (ω)). (7.1) emark 7.. Let ω 0 = ue 1 + ve be an arbitrary frequency vector, and ξ(x) = e 1 ω 0 e 1 x a correction term in order for equation (7.) to be satisfied. Then, the quaternionic Gabor wavelet is given by (compare to Mawardi and Hitzer 18 ) ψ q (x) = (e i+j+k ω 0 x e 1 x) ξ(x). (7.14) Using the modulation property of the QFT, we can represent the quaternionic Gabor wavelet (7.14) in terms of the QFT as F q {ψ q }(ω) = e 1 (ω ω0) e 1 (ω +ω 0). (7.15) It is easy to show that the quaternionic Gabor wavelet satisfies the assumption of Preposition 7.1. emark 7. leads to the following remark. emark 7.. It is not difficult to check that Proposition 7.1 holds if the quaternion mother wavelet is of the form F q {ψ}(ω) = F q {ψ 0 }(ω) + (i + j + k) F q {ψ 1 }(ω), F q {ψ 0 }, F q {ψ 1 }. (7.16)

18 October 6, 014 1:7 WSPC/WS-IJWMIP QTF-ijwmip revf 18 Mawardi Bahri, yuichi Ashino, and émi Vaillancourt 7.. eproducing Formula In this section, we show that a quaternion function, f, can be recovered from its CQWT whenever the quaternion wavelets satisfy the following admissibility condition. Theorem 7.1 (Inner product relation). Let {ψ, φ} AQW, where ψ = ψ 0 + iψ 1 + jψ + kψ, φ = φ 0 + iφ 1 + jφ + kφ. Assume that both quaternion mother wavelets ψ and φ satisfy the assumption of Preposition 7.1. Then, for every f, g L 1 L ( ; H), we have SO() + ( ) da dθ T ψ f(a, θ, b) T φ g(a, θ, b) d b a = (fc {ψ,φ}, g) L ( ;H). (7.17) Proof. Applying the QFT Plancherel Theorem 4.1 to the integration with respect to b on the left-hand side of (7.17), we have ( ) da dθ T ψ f(a, θ, b) T φ g(a, θ, b) d b + a = 1 ( ) da dθ (π) F q (T ψ f(a, θ, ))(ω) F q (T φ g(a, θ, ))(ω) d ω SO() + a ( ) (7.1) 1 = (π) a ˆf(ω) ˆψ(ar θ (ω)) ˆφ(ar da dθ θ (ω)) ĝ(ω) d ω SO() + a = 1 ( ) ˆf(ω) (π) ˆψ(ar θ (ω)) ˆφ(ar da dθ θ (ω)) ĝ(ω) d ω SO() a } + {{} C {ψ,φ} = 1 (π) (F q{f}c {ψ,φ}, F q {g}) L ( ;H) SO() (4.1) = (fc {ψ,φ}, g) L ( ;H), (7.18) where, in the third equality, we applied Fubini s Theorem to exchange the order of integrations. Denote C {ψ,φ} = C (0) {ψ,φ} + ic(1) {ψ,φ} + jc() {ψ,φ} + kc() {ψ,φ}, C(s) {ψ,φ}, s = 0, 1,,.

19 October 6, 014 1:7 WSPC/WS-IJWMIP QTF-ijwmip revf Continuous Quaternion Fourier and Wavelet Transforms 19 By (.10), we can rewrite equation (7.17) in the form: ( ) da dθ T ψ f(a, θ, b) T φ g(a, θ, b) d b SO() + a = C (0) {ψ,φ} (f, g) L ( ;H) + C (1) {ψ,φ} (fi, g) L ( ;H) + C () {ψ,φ} (fj, g) L ( ;H) + C () {ψ,φ} (fk, g) L ( ;H) = C (0) {ψ,φ} (f, g) L ( ;H) + ic (1) {ψ,φ} (α(f), g) L ( ;H) + jc () {ψ,φ} (β(f), g) L ( ;H) + kc () {ψ,φ} (γ(f), g) L ( ;H). (7.19) By Theorem 7.1, we have the following corollary. Corollary 7.1. Let (ψ, ψ) AQW. Then we have (T ψ f, T ψ g) L (G;H) = C ψ (f, g) L ( ;H). (7.0) emark 7.4. Note that (7.0) is of the same form as the inner product relation of the classical wavelet transform (see Debnath 4 and Mallat 1 ). In particular, if f = g in Corollary 7.1 we have the CQWT Plancherel Formula: T ψ f L (G;H) = C ψ f L ( ;H). (7.1) Theorem 7. (Inversion formula). Let {ψ, φ} AQW. Then, f L ( ; H) can be reconstructed from T ψ f(a, b, θ) by 1 da dθ f(x) = C {ψ,φ} T ψ f(a, b, θ) φ a,θ,b C {ψ,φ} a d b, (7.) where the integral converges in the weak sense. G Proof. Applying Theorem 7.1 to g L ( ; H), we have ( ) da dθ (fc {ψ,φ}, g) L ( ;H) = T ψ f(a, θ, b) T φ g(a, θ, b) d b SO() + a da dθ = T ψ f(a, θ, b)t φ g(a, θ, b) G a d b = T ψ f(a, θ, b)φ a,θ,b (x)g(x) d da dθ x G a d b da dθ = T ψ f(a, θ, b)φ a,θ,b (x)g(x) G a d b d x ( ) da dθ = T ψ f(a, θ, b)φ a,θ,b G a d b, g. (7.) L ( ;H) Since the inner product identity holds for every g L ( ; H), we conclude that da dθ f(x) C {ψ,φ} = T ψ f(a, b, θ)φ a,b,θ (x) G a d b, f(x) = T ψ f(a, b, θ)φ a,b,θ (x) C 1 da dθ {ψ,φ} a d b, (7.4) G

20 October 6, 014 1:7 WSPC/WS-IJWMIP QTF-ijwmip revf 0 Mawardi Bahri, yuichi Ashino, and émi Vaillancourt which implies (7.) by (.8). 8. Conclusion Using basic quaternion properties, we introduced the quaternion Fourier transform (QFT). Because of the non-commutativity of the multiplication in the quaternion space H, several important properties of the classical FT, such as modulation and convolution, must be modified. We also studied a simple application of the QFT to partial differential equations in quaternion algebra. Using the kernel of the QFT, we proposed a two-dimensional continuous quaternion wavelet transform (CQWT). We extended to the CQWT some fundamental properties of the continuous wavelet transform, such as the inner product and the inversion formula. This generalization also enables us to establish the quaternionic Gabor wavelets, which have potential abilities to go beyond the application area of the two-dimensional complex Gabor wavelets. Acknowledgment This work was partially supported by Bantuan Seminar Luar Negeri oleh DPM DIKTI (S-44/PB/011) Indonesia, JSPS.KAKENHI (C)54010 of Japan, and NSEC of Canada. eferences 1. D. Assefa, L. Mansinha, K. F. Tiampo, H. asmussen, and K. Abdella, Local quaternion Fourier transform and color images texture analysis, Signal Process., 90(6), 010, T. Bülow, Hypercomplex Spectral Signal epresentations for the Processing and Analysis of Images, Ph.D. thesis, University of Kiel, Germany, E. B. Corrochano, The theory and use of the quaternion wavelet transform, J. Math. Imaging and Vision 4(1), 006, L. Debnath, Wavelet Transforms and Their Applications, Birkhäuser, Boston, T. A. Ell, Quaternion-Fourier transforms for analysis of two-dimensional linear timeinvariant partial differential systems, Proceeding of the nd Conference on Decision and Control, IEEE 199, T. A. Ell and S. J. Sangwine, Hypercomplex Fourier transform of color images, IEEE Trans. Image Process., 16(1), 007, G. B. Folland, Introduction to Partial Differential Equations, second edition, Princeton University Press, New Jersey, L. O. Guo and M. Zhu, Quaternion Fourier-Mellin moments for color images, Pattern ecognition, 48, 011, J. X. He, Continuous wavelet transform on the space L (, H; dx), Appl. Math. Lett. 17(1), 001, E. Hitzer, Quaternion Fourier transform on quaternion fields and generalizations, Adv. Appl. Clifford Algebr., 17(), 007, E. Hitzer, Directional uncertainty principle for quaternion Fourier transform, Adv. Appl. Clifford Algebr., 0(), 010,

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