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Abstract and Applied Analysis
Volume 2017, Article ID 3795120, 11 pages
https://doi.org/10.1155/2017/3795120
Research Article

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

1Department of Mathematics, Hasanuddin University, Makassar 90245, Indonesia
2Division of Mathematical Sciences, Osaka Kyoiku University, Osaka 582-8582, Japan

Correspondence should be addressed to Ryuichi Ashino; pj.ca.ukioyk-akaso.cc@onihsa

Received 23 October 2016; Accepted 5 January 2017; Published 30 January 2017

Academic Editor: Lucas Jodar

Copyright © 2017 Mawardi Bahri and Ryuichi Ashino. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The continuous quaternion wavelet transform (CQWT) is a generalization of the classical continuous wavelet transform within the context of quaternion algebra. First of all, we show that the directional quaternion Fourier transform (QFT) uncertainty principle can be obtained using the component-wise QFT uncertainty principle. Based on this method, the directional QFT uncertainty principle using representation of polar coordinate form is easily derived. We derive a variation on uncertainty principle related to the QFT. We state that the CQWT of a quaternion function can be written in terms of the QFT and obtain a variation on uncertainty principle related to the CQWT. Finally, we apply the extended uncertainty principles and properties of the CQWT to establish logarithmic uncertainty principles related to generalized transform.

1. 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 [14]. 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 [57], 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 [1517], 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.

2. Preliminaries

The concept of the quaternion algebra [19, 20] was introduced by Sir Hamilton in 1842 and is denoted by in his honor. It is an extension of the complex numbers to a four-dimensional (4-D) algebra. Every element of is a linear combination of a real scalar and three imaginary units ,  , and with real coefficients,which obey Hamilton’s multiplication rulesFor a quaternion , is called the scalar (or real) part of denoted by and is called the vector (or pure) part of . The vector part of is conventionally denoted by or .

Let , and , be their vector parts, respectively. Equation (2) yields the quaternionic multiplication aswhere The conjugate of the quaternion is the quaternion given byIt is an anti-involution; that is, From (5), we obtain the norm or modulus of defined asIt is not difficult to see thatUsing conjugate (5) and the modulus of q, we can define the inverse of as which shows that is a normed division algebra.

Now we notice thatThese will lead to the cyclic multiplication; that is,

Any quaternion may be split up intoThe above givesThis leads to the following modulus identity:

It is convenient to introduce an inner product for two quaternion functions as follows:where the overline indicates the quaternion conjugation of the function. In particular, for , we obtain the -normwhereAs a consequence of the inner product (15), we obtain the quaternion Cauchy-Schwarz inequality

Definition 1. A couple of nonnegative integers is called a multi-index. One denotesand, for ,Derivatives are conveniently expressed by multi-indices:Next, we obtain the Schwartz space as (compared to [21])where is the set of smooth functions from to . Elements in the dual space of 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 is the transform : given bywhere , and the quaternion exponential product is the quaternion Fourier kernel.

Theorem 3 (inverse QFT). Suppose that and . Then, the QFT of is an invertible transform and its inverse is given bywhere the quaternion exponential product is called the inverse (right-sided) quaternion Fourier kernel.

Since , the definition of QFT (23) may be extended to the Schwartz space. It is important to note that is not necessary in even if is in , so in general might not be well defined. However, the QFT of a Schwartz quaternion function is also in the Schwartz space.

Applying (23), we haveNow, we define a module of asFurthermore, we obtain the -norm

Remark 4. It is worth noting here that if , , is real-valued, (26) can be written in the formwhere

Some important properties of the QFT are stated in the following lemmas.

Lemma 5 (QFT Plancherel). If , thenMoreover,

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 , , 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 . Hence, using standard density arguments, one may extend the QFT in a unique way to the whole of .

Lemma 7 (see [14]). If and exists and is also in , then one has for every

By Riesz’s interpolation theorem, we get that the Hausdorff-Young inequality (see [15])holds for with . Using inversion formula of the QFT, (37) can be rewritten in the form

The following theorem is an extension of the Riemann-Lebesgue lemma in the QFT domain.

Theorem 8 (Riemann-Lebesgue lemma of QFT). For a function in , one has that

Proof. Notice first thatApplying (40) gives Representing and changing variable in the above identity, we immediately obtainThis means that Analogously, it can be shown that The proof is complete.

Theorem 9 (continuity). If , then the quaternion Fourier transform is continuous on . Moreover,where is the space of continuous quaternion functions from to .

Proof. From the QFT definition (23), we readily see thatUsing the triangle inequality for quaternions, we easily get This means that we haveThe quaternion function is integrable and the Lebesgue dominated convergence theorem with (46) then givesThis proves that is continuous on . Again, since (48) is independent of is, in fact, uniformly continuous on .

3.2. 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 and exists and is also in , then

Remark 11. An alternative form of Theorem 10 is

Notice that for we can replace the 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

Proof. It is not difficult to check that Using Holder’s inequality, we further get By application of the Hausdorff-Young inequality (38) and then integration by parts, we obtain We set . Using (35) gives For , we can take similar steps as above using (36) and getThis concludes the proof of the theorem.

A generalized version of Theorem 10 is directional uncertainty principle for the QFT given by the following.

Theorem 13 (QFT directional uncertainty principle). If and exists and is also in , then

Proof. We haveUsing (51) givesThe result follows.

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.

Theorem 15. If and exists and is also in , thenwhere is the absolute covariance.

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 in [18].

4. 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, 79].

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 .

Let be a quaternion mother wavelet. We consider the family of the wavelets defined bywhere and . Here, is a dilation parameter and is a translation vector parameter.

The relationship between (64) and its QFT is given in the following lemma.

Lemma 17. Let be an admissible quaternion function. The family of the wavelets (64) can be written in terms of the QFT as

If we assume that , , is real-valued (in the next section, we will always assume that the QFT of quaternion mother wavelet, i.e., , is real-valued), (65) can be rewritten in the form

Definition 18 (CQWT). The CQWT of a quaternion function 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 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 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 obtainwhere . The proof is complete.

Theorem 21. Let be a quaternion admissible wavelet which satisfies the admissibility condition defined by (63). Then, for every , one has

Proof. Applying Plancherel’s theorem for QFT (30) to the -integral 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 getwhich gives the desired result.

5. 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 ,where and is Gamma function. Here, denotes the Schwartz class on quaternion function.

Applying Plancherel’s theorem for QFT (30) to the right-hand side of (75), we easily obtain

It is proved that, for every , 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 be a quaternion admissible wavelet. Let be the CQWT of . If , then

For the proof of Theorem 23, we use the following lemma.

Lemma 24. Suppose that . Then,

Proof. A straightforward computation yields The proof is complete.

Proof. Using the uncertainty principle in Theorem 12, we immediately obtainNow, integrating both sides of (81) with respect to the Haar measure , we obtainThen, inserting Lemma 24 into the second term of (82), we easily obtainSubstituting (72) into the right-hand side of (83) and simplifying it, we finally getwhich 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 be a quaternion admissible wavelet. Let be the CQWT of . 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

Proof. A simple calculation revealsThe proof is complete.

Proof. It is known thatNotice that . This implies that . Therefore, we may replace by on both sides of (88) and get Integrating both sides of this equation with respect to yieldsInserting Lemma 26 into the first term on the left-hand side of (90), we obtain