A Variation on Uncertainty Principle and Logarithmic Uncertainty Principle for Continuous Quaternion Wavelet Transforms

and Applied Analysis 3 Definition 1. A couple α = (α1, α2) of nonnegative integers is called a multi-index. One denotes |α| = α1 + α2, α! = α1!α2!, (19) and, for x ∈ R, xα = xα1 1 xα2 2 . (20) Derivatives are conveniently expressed by multi-indices: ∂α = ∂|α| ∂xα1 1 ∂xα2 2 . (21) Next, we obtain the Schwartz space as (compared to [21]) S (R2;H) = {f ∈ C∞ (R2,H) : sup x∈R2 (1 + |x|k) 󵄨󵄨󵄨󵄨∂αf (x)󵄨󵄨󵄨󵄨 < ∞} , (22) whereC∞(R2,H) is the set of smooth functions fromR toH. Elements in the dual space S󸀠(R2;H) of S(R2;H) are called tempered distribution. 3. Quaternion Fourier Transform and Its Heisenberg Uncertainty Principle 3.1. QFT and Properties. In the following, we introduce the (right-sided) QFT and some of its fundamental properties such as Riemann-Lebesgue lemma and continuity. Definition 2 (right-sided QFT). The (right-sided) quaternion Fourier transform (QFT) of f ∈ L1(R2;H) is the transform FQ{f}: R → H given by FQ {f} (ω) = ?̂? (ω) = ∫ R f (x) e−iω1x1e−jω2x2dx, dx = dx1dx2, (23) where x = x1e1 + x2e2,ω = ω1e1 + ω2e2, and the quaternion exponential product e−iω1x1e−jω2x2 is the quaternion Fourier kernel. Theorem 3 (inverse QFT). Suppose that f ∈ L1(R2;H) and FQ{f} ∈ L1(R2;H). Then, the QFT of f is an invertible transform and its inverse is given by F −1 Q [FQ {f}] (x) = f (x) = 1 (2π)2 ∫R2FQ {f} (ω) ejω2x2eiω1x1dω, (24) where the quaternion exponential product ejω2x2eiω1x1 is called the inverse (right-sided) quaternion Fourier kernel. Since S(R2;H) ⊂ L1(R2;H), the definition of QFT (23) may be extended to the Schwartz space. It is important to note that FQ{f} is not necessary in L1(R2;H) even if f is in L1(R2;H), so in generalFQ{f}might not be well defined. However, the QFT of a Schwartz quaternion function is also in the Schwartz space. Applying (23), we have FQ {f} (ω) = ∫ R (f0 (x) + if1 (x) + jf2 (x) + kf3 (x)) ⋅ e−iω1x1e−jω2x2dx = FQ {f0} (ω) + iFQ {f1} (ω) + jFQ {f2} (ω) + kFQ {f3} (ω) . (25) Now, we define a module ofFQ{f}(ω) as 󵄨󵄨󵄨󵄨FQ {f} (ω)󵄨󵄨󵄨󵄨Q = (󵄨󵄨󵄨󵄨FQ {f0} (ω)󵄨󵄨󵄨󵄨2 + 󵄨󵄨󵄨󵄨FQ {f1} (ω)󵄨󵄨󵄨󵄨2 + 󵄨󵄨󵄨󵄨FQ {f2} (ω)󵄨󵄨󵄨󵄨2 + 󵄨󵄨󵄨󵄨FQ {f3} (ω)󵄨󵄨󵄨󵄨2)1/p . (26) Furthermore, we obtain the Lp(R2;H)-norm 󵄩󵄩󵄩󵄩FQ {f}󵄩󵄩󵄩󵄩Q,p = (∫ R 󵄨󵄨󵄨󵄨FQ {f} (ω)󵄨󵄨󵄨󵄨pQ dω) 1/p . (27) Remark 4. It is worth noting here that ifFQ{fi}, i = 0, 2, 3, is real-valued, (26) can be written in the form 󵄩󵄩󵄩󵄩FQ {f}󵄩󵄩󵄩󵄩Q,p = 󵄩󵄩󵄩󵄩FQ {f} (ω)󵄩󵄩󵄩󵄩p , (28) where 󵄨󵄨󵄨󵄨FQ {f} (ω)󵄨󵄨󵄨󵄨Q = ((FQ {f0} (ω))2 + (FQ {f1} (ω))2 + (FQ {f2} (ω))2 + (FQ {f3} (ω))2) . (29) Some important properties of the QFT are stated in the following lemmas. Lemma 5 (QFT Plancherel). If f ∈ L1(R2;H) ∩ L2(R2;H), then 1 (2π)2 ∫R2 󵄨󵄨󵄨󵄨FQ {f} (ω)󵄨󵄨󵄨󵄨2 dω = ∫R2 󵄨󵄨󵄨󵄨f (x)󵄨󵄨󵄨󵄨2 dx. (30) Moreover, 1 (2π)2 ∫R2 󵄨󵄨󵄨󵄨FQ {f} (ω)󵄨󵄨󵄨󵄨2Q dω = ∫R2 󵄨󵄨󵄨󵄨f (x)󵄨󵄨󵄨󵄨2 dx. (31) 4 Abstract and Applied Analysis Proof. We prove expression (31) of Lemma 5. Using (26), we immediately get 1 (2π)2 ∫R2 󵄨󵄨󵄨󵄨FQ {f} (ω)󵄨󵄨󵄨󵄨2Q dω = 1 (2π)2 ⋅ ∫ R (󵄨󵄨󵄨󵄨FQ {f0} (ω)󵄨󵄨󵄨󵄨2 + 󵄨󵄨󵄨󵄨FQ {f1} (ω)󵄨󵄨󵄨󵄨2 + 󵄨󵄨󵄨󵄨FQ {f2} (ω)󵄨󵄨󵄨󵄨2 + 󵄨󵄨󵄨󵄨FQ {f3} (ω)󵄨󵄨󵄨󵄨2) dω = 1 (2π)2 (∫R2 󵄨󵄨󵄨󵄨FQ {f0} (ω)󵄨󵄨󵄨󵄨2 dω + ∫ R 󵄨󵄨󵄨󵄨FQ {f1} (ω)󵄨󵄨󵄨󵄨2 dω + ∫ R 󵄨󵄨󵄨󵄨FQ {f2} (ω)󵄨󵄨󵄨󵄨2 dω + ∫ R 󵄨󵄨󵄨󵄨FQ {f3} (ω)󵄨󵄨󵄨󵄨2 dω) . (32) Applying (30) into the right-hand side of the above identity gives 1 (2π)2 ∫R2 󵄨󵄨󵄨󵄨FQ {f} (ω)󵄨󵄨󵄨󵄨2Q dω = ∫ R 󵄨󵄨󵄨󵄨f0 (x)󵄨󵄨󵄨󵄨2 dx + ∫ R 󵄨󵄨󵄨󵄨f1 (x)󵄨󵄨󵄨󵄨2 dx + ∫ R 󵄨󵄨󵄨󵄨f2 (x)󵄨󵄨󵄨󵄨2 dx + ∫ R 󵄨󵄨󵄨󵄨f3 (x)󵄨󵄨󵄨󵄨2 dx. (33) Since fi(x), i = 0, 1, 2, 3, is real-valued, the above equation can be written in the form 1 (2π)2 ∫R2 󵄨󵄨󵄨󵄨FQ {f} (ω)󵄨󵄨󵄨󵄨2Q dω = ∫ R (f2 0 (x) + f2 1 (x) + f2 2 (x) + f2 3 (x)) dx, (34) which completes the proof of the theorem. Remark 6. Equation (30) shows that the QFT is a bounded linear operator on L1(R2;H) ∩ L2(R2;H). Hence, using standard density arguments, one may extend the QFT in a unique way to the whole of L2(R2;H). Lemma7 (see [14]). Iff ∈ L1(R2;H)∩L2(R2;H) and (∂/xk)f exists and is also in L2(R2;H), then one has for every n ∈ N FQ { ∂n ∂xn 1 f} (ω) = ω n 1FQ {fi} (ω) , (35) FQ { ∂n ∂xn 2 f} (ω) = FQ {f} (ω) (jω2) n . (36) By Riesz’s interpolation theorem, we get that the Hausdorff-Young inequality (see [15]) 󵄩󵄩󵄩󵄩FQ {f}󵄩󵄩󵄩󵄩Q,p󸀠 ≤ 󵄩󵄩󵄩󵄩f󵄩󵄩󵄩󵄩p (37) holds for 1 ≤ p ≤ 2 with 1/p + 1/p󸀠 = 1. Using inversion formula of the QFT, (37) can be rewritten in the form 󵄩󵄩󵄩󵄩f󵄩󵄩󵄩󵄩p󸀠 ≤ 󵄩󵄩󵄩󵄩FQ {f}󵄩󵄩󵄩󵄩Q,p . (38) The following theorem is an extension of the RiemannLebesgue lemma in the QFT domain. Theorem 8 (Riemann-Lebesgue lemma of QFT). For a function in f ∈ L1(R2;H), one has that lim |ω1|→∞ 󵄨󵄨󵄨󵄨FQ {f} (ω)󵄨󵄨󵄨󵄨 = 0, lim |ω2|→∞ 󵄨󵄨󵄨󵄨FQ {f} (ω)󵄨󵄨󵄨󵄨 = 0. (39) Proof. Notice first that e−iω1x1 = −e−iω1(x1+π/ω1), e−jω2x2 = −e−jω2(x2+π/ω2). (40)


Introduction
As it is known, the classical wavelet transform (WT) is a very useful mathematical tool.It has been discussed extensively in the literature and has been proven to be powerful and useful in the communication theory, quantum mechanics, and many other fields [1][2][3][4].Of great interest is the study of the quaternion wavelet transform, which can be considered as a generalization of the WT using quaternion algebra.Some research papers on the continuous and discrete quaternion wavelet transforms have been published.In [5][6][7], the authors constructed the continuous quaternion wavelet transform (CQWT) using the quaternionic affine group and similitude group, respectively.Several fundamental properties of this extended wavelet transform, which correspond to classical continuous wavelet transform properties, were also investigated.Further, in regard to a numerical concept of the quaternion wavelet transforms, Bayro-Corrochano [8] developed the discrete quaternion wavelet transform and applied it for optical flow estimation.In [9,10], the authors studied the discrete reduced biquaternion wavelet transform and applied it to multiscale texture classification.Another approach of the CQWT based on a natural convolution of quaternion-valued functions was recently proposed by Akila and Roopkumar [11,12].The essential part in the study of the quaternion wavelet transform, as usual, is to establish its Heisenberg type uncertainty principle.It plays an important role in quaternionic signal processing.Based on the Heisenberg type uncertainty principle for the quaternion Fourier transform (QFT) [13,14], the authors in [7] proposed a component-wise uncertainty principle associated with the CQWT.
Motivated by the authors in [15][16][17], in the present paper, we propose the directional uncertainty principle related to the CQWT and then apply this uncertainty to obtain a variation on the Heisenberg type uncertainty principle and the logarithmic uncertainty principle in the context of the CQWT.The uncertainty principle describes the relation between the QFT of a quaternion function and its CQWT.These principles are more general forms of Heisenberg's uncertainty principle related to the CQWT [7].To achieve the results, our first step is to derive the directional uncertainty principle for the QFT using the component-wise QFT uncertainty principle.Due to this principle, we can easily derive directional QFT uncertainty principle using a representation of polar coordinate from the ones proposed in [18].We then study an important theorem which describes interactions between the CQWT and QFT in frequency domain.Applying the cyclic multiplication of quaternion, we obtain some useful properties of the CQWT.Based on the relationship between the extended Heisenberg uncertainty principle and properties related to the CQWT, we finally establish the logarithmic uncertainty principles associated with the CQWT.

Preliminaries
The concept of the quaternion algebra [19,20] was introduced by Sir Hamilton in 1842 and is denoted by H in his honor.It is an extension of the complex numbers to a four-dimensional (4-D) algebra.Every element of H is a linear combination of a real scalar and three imaginary units i, j, and k with real coefficients, which obey Hamilton's multiplication rules (2) For a quaternion  =  0 + i 1 + j 2 + k 3 ∈ H,  0 is called the scalar (or real) part of  denoted by Sc() and i 1 + j 2 + k 3 is called the vector (or pure) part of .The vector part of  is conventionally denoted by q or Vec() = i 1 + j 2 + k 3 .
Let ,  ∈ H and p, q be their vector parts, respectively.Equation (2) yields the quaternionic multiplication  as where The conjugate  of the quaternion  is the quaternion given by It is an anti-involution; that is, From ( 5), we obtain the norm or modulus of  ∈ H defined as Using conjugate (5) and the modulus of q, we can define the inverse of  ∈ H \ {0} as which shows that H is a normed division algebra.Now we notice that These will lead to the cyclic multiplication; that is, Any quaternion  may be split up into The above gives This leads to the following modulus identity: It is convenient to introduce an inner product for two quaternion functions ,  : R 2 → H as follows: where the overline indicates the quaternion conjugation of the function.In particular, for  = , we obtain the   (R 2 ; H)-norm where As a consequence of the inner product (15), we obtain the quaternion Cauchy-Schwarz inequality Abstract and Applied Analysis 3 Definition 1.A couple  = ( 1 ,  2 ) of nonnegative integers is called a multi-index.One denotes and, for x ∈ R 2 , Derivatives are conveniently expressed by multi-indices: Next, we obtain the Schwartz space as (compared to [21]) where  ∞ (R 2 , H) is the set of smooth functions from R 2 to H. Elements in the dual space S  (R 2 ; H) of S(R 2 ; H) are called tempered distribution.

Quaternion Fourier Transform and Its Heisenberg Uncertainty Principle
3.1.QFT and Properties.In the following, we introduce the (right-sided) QFT and some of its fundamental properties such as Riemann-Lebesgue lemma and continuity.
Theorem 3 (inverse QFT).Suppose that  ∈  1 (R 2 ; H) and F  {} ∈  1 (R 2 ; H).Then, the QFT of  is an invertible transform and its inverse is given by where the quaternion exponential product  j 2  2  i 1  1 is called the inverse (right-sided) quaternion Fourier kernel.
Since S(R 2 ; H) ⊂  1 (R 2 ; H), the definition of QFT (23) may be extended to the Schwartz space.It is important to note that F  {} is not necessary in  1 (R 2 ; H) even if  is in  1 (R 2 ; H), so in general F  {} might not be well defined.However, the QFT of a Schwartz quaternion function is also in the Schwartz space.
Applying (23), we have where Some important properties of the QFT are stated in the following lemmas. Moreover, Proof.We prove expression (31) of Lemma 5. Using (26), we immediately get Applying (30) into the right-hand side of the above identity gives Since   (x),  = 0, 1, 2, 3, is real-valued, the above equation can be written in the form which completes the proof of the theorem.
Remark 6. Equation (30) shows that the QFT is a bounded linear operator on  1 (R 2 ; H) ∩  2 (R 2 ; H).Hence, using standard density arguments, one may extend the QFT in a unique way to the whole of  2 (R 2 ; H).
Theorem 8 (Riemann-Lebesgue lemma of QFT).For a function in  ∈  1 (R 2 ; H), one has that Proof.Notice first that Applying (40) gives Representing F  {} = (1/2)[F  {}+F  {}] and changing variable  1 + / 1 =  1 in the above identity, we immediately obtain This means that lim Analogously, it can be shown that lim The proof is complete.where (R 2 ; H) is the space of continuous quaternion functions from R 2 to H.
Proof.From the QFT definition (23), we readily see that Using the triangle inequality for quaternions, we easily get This means that we have The quaternion function (x) is integrable and the Lebesgue dominated convergence theorem with (46) then gives This proves that F  {}() is continuous on R 2 .Again, since (48) is independent of , F  {}() is, in fact, uniformly continuous on R 2 .

Uncertainty Principle for QFT.
In what follows, we investigate the uncertainty principles associated with the QFT.These results will be needed in the next section.
Theorem 10 (QFT component-wise uncertainty principle [14]).If  ∈  1 (R 2 ; H) ∩  2 (R 2 ; H) and (/  ) exists and is also in Remark 11.An alternative form of Theorem 10 is Notice that for 1 ≤  ≤ 2 we can replace the  2 norms to   norms on the left-hand side of (50) and obtain the following theorem.
Theorem 12.Under the assumptions of Theorem 10, one has (x)      x) (x)     2 x. (52) For  = 2, we can take similar steps as above using (36) and get This concludes the proof of the theorem.
A generalized version of Theorem 10 is directional uncertainty principle for the QFT given by the following.
Remark 14.A different proof of Theorem 13 using the logarithmic uncertainty principle for the QFT can be found in [15].
Using the polar coordinate form of quaternion function , Yang et al. [18] obtained an alternative form of the directional uncertainty principle for the QFT as follows.
Applying (51), we can easily generalize the uncertainty principle (61) to the directional QFT uncertainty principle; that is, It is obvious that the result is the same as Theorem 4.3 in [18].

Continuous Quaternion Wavelet Transform
This section briefly introduces the continuous quaternion wavelet transform (CQWT).We shall derive two theorems of the CQWT which will be used in proving the main theorem.A more complete and detailed discussion of the properties of the CQWT can be found in [5,[7][8][9].
Definition 16.A quaternion-valued function is admissible if and only if it satisfies the following admissibility condition: Here,   is a real positive constant independent of  satisfying || = 1.
Let  ∈  1 (R 2 ; H) be a quaternion mother wavelet.We consider the family of the wavelets  ,b defined by where  b (x) = (x − b) and   (x) = (1/)(x/).Here,  is a dilation parameter and b is a translation vector parameter.The relationship between (64) and its QFT is given in the following lemma.

Lemma 17. Let 𝜓 be an admissible quaternion function. The family of the wavelets (64) can be written in terms of the QFT as
If we assume that F  {  }(),  = 0, 1, 2, 3, is real-valued (in the next section, we will always assume that the QFT of quaternion mother wavelet, i.e., F  {}() = ψ(), is realvalued), (65) can be rewritten in the form Definition 18 (CQWT).The CQWT of a quaternion function  ∈  2 (R 2 ; H) with respect to the quaternion mother wavelet  is defined by As an easy consequence of the above definition, we further obtain the following useful theorem.Theorem 19.Let  ∈  2 (R 2 ; H) be a quaternion admissible wavelet; then, CQWT (67) can be expressed as We need the following two important results, which will be useful in proving the logarithmic uncertainty principle for the CQWT.
Theorem 20.Let  ∈  2 (R 2 ; H) be a quaternion admissible wavelet; then, CQWT (67) has a quaternion Fourier representation of the form Proof.From the definition of QFT (23), it follows that Using the assumption that the QFT of quaternion mother wavelet is real-valued and then applying Fubini's theorem, we obtain where ).The proof is complete.
Theorem 21.Let  ∈  2 (R 2 ; H) be a quaternion admissible wavelet which satisfies the admissibility condition defined by (63).Then, for every  ∈  2 (R 2 ; H), one has Proof.Applying Plancherel's theorem for QFT (30) to the bintegral into the left-hand side of (72) yields Taking into consideration Fubini's theorem about the inversion of order of integration and applying Plancherel's theorem (30), we get which gives the desired result.

Logarithmic Uncertainty Principle for CQWT
The simplest formulation of the uncertainty principle in harmonic analysis is Heisenberg-Weyl inequality, which gives us the information that a nontrivial function and its Fourier transform cannot both be simultaneously sharply localized [1,22].In this section, we first derive a variation on uncertainty principle associated with the CQWT.From this, we establish the logarithmic uncertainty principle which is valid for the QFT [14] to the setting of the CQWT.Due to the uncertainty principle for QFT (58), we have the logarithmic uncertainty principle for the QFT [15] as follows.
Theorem 22 (QFT logarithmic uncertainty principle).For  ∈ S(R It is proved that, for every ,  ∈ S(R 2 ; H), the Heisenberg type uncertainty principle for the CQWT is given [7]: A generalization of the above uncertainty principle is given in the following theorem.
Theorem 23.Let ,  ∈ S(R 2 ; H) be a quaternion admissible wavelet.Let   (, b) be the CQWT of .If 1 ≤  ≤ 2, then For the proof of Theorem 23, we use the following lemma.
Lemma 24.Suppose that ,  ∈ S(R Now, integrating both sides of (81) with respect to the Haar measure / 3 , we obtain Then, inserting Lemma 24 into the second term of (82), we easily obtain Substituting (72) into the right-hand side of (83) and simplifying it, we finally get which concludes the proof of Theorem 23.
Let us derive a logarithmic uncertainty principle associated with the continuous quaternion wavelet transform (CQWT).
Theorem 25 (CQWT logarithmic uncertainty principle).Let  ∈ S(R 2 ; H) be a quaternion admissible wavelet.Let   (, b) be the CQWT of  ∈ S(R 2 ; H).Then, the following inequality is satisfied: Next, we need the following lemma to assist the proof of the above theorem.

Lemma 26. Under the same conditions as in Theorem 25, one has
which was to be proved.
Remark 27.It is worth nothing that, following the steps of the proof of the above theorem, we can also obtain Theorem 25 using (76).

Conclusion
Based on the logarithmic uncertainty principle in the quaternion Fourier domain, we have established a logarithmic uncertainty principle related to the CQWT.It is a more general form of component-wise uncertainty principle associated with the CQWT, which describes the relationship between the QFT of a quaternion function and its CQWT.We also presented a variation on uncertainty principle related to the QFT and then found the variation on uncertainty principle related to the CQWT.