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A Version of Uncertainty Principle for Quaternion Linear Canonical Transform.

1. Introduction

The quaternion Fourier transform (QFT) is an extension of the classical two-dimensional Fourier transform (FT) [1-4] in the framework of quaternion algebra. It plays an important role in the representation of the two-dimensional quaternion signals. A number of useful properties of the extended transform are generalizations of the corresponding properties of the FT with some modifications (see, e.g., [5-15]). The QFT has found many applications in color image processing and signal analysis; we refer the reader to [16-19] and the references mentioned therein. An extension of the QFT in the framework of the classical linear canonical transform (LCT) (see [20-22]), known as the quaternion linear canonical transform, has received much attention in recent years. Due to the several definitions of the QFT, there are basically three ways of obtaining the quaternion linear canonical transform (QLCT): the right-sided quaternion linear canonical transform, the left-sided quaternion linear canonical transform, and the two-sided quaternion linear canonical transform. The right-sided quaternion linear canonical transform is obtained by substituting the Fourier kernel with the right-sided QFT kernel in the LCT definition, and so on. Recent works related to some important properties of the QLCT such as Parseval's theorem, reconstruction formula, and component-wise uncertainty principles were also published [18, 23-25]. It was found that the properties of the QLCT are extensions of the corresponding version of the QFT with some modifications.

On the other hand, the uncertainty principle plays one important role in signal processing. It describes a function and its FT, which cannot both be simultaneously sharply localized. One example of this is the Heisenberg uncertainty principle concerning position and momentum wave functions in quantum physics. In signal processing, an uncertainty principle states that the product of the variances of the signal in the time and frequency domains has a lower bound. Up till now, several attempts have been made to extend the uncertainty principles associated with the QFT and QLCT domains. The component-wise and directional uncertainty principles associated with the QFT were proposed in [11]. In [26, 27], the authors established a component-wise uncertainty principle for the QLCT and proved that the equality is achieved for optimal quaternion Gaussian function. Recently, the authors [23] proposed the logarithmic uncertainty principle associated with the QLCT which is the generalization of the logarithmic uncertainty principle for the QFT.

Therefore, the main objective of the present paper is to establish full uncertainty principle for the two-sided QLCT, which is a new general form of component-wise uncertainty principle for the two-sided QLCT. This uncertainty principle is derived using the connection between the QFT and QLCT. We also obtain full uncertainty principle for the right-sided QLCT using a relation between the right-sided QLCT and two-sided QLCT. We also derive the Hausdorff-Young inequality associated with the two-sided QLCT.

This paper is organized as follows. In Section 2, we briefly review the basic knowledge of quaternion and its split used in the next section. In Section 3, we introduce the definition of the QFT and the QLCT. The relationship between the QFT and the QLCT is also discussed in this section. In Section 4, we derive a new version uncertainty principle associated with the QLCT, which shows that the spread of a quaternion-valued function and its QLCT are inversely proportional. Section 5 concludes this paper.

2. Preliminaries

In this section, let us briefly recall some basic definitions and properties of the quaternions (for more details, see [28]).

2.1. Quaternions. The quaternions, a generalization of complex numbers, are members of a noncommutative division algebra. The set of quaternions is denoted by H. Every element of H can be written in the following form:

[mathematical expression not reproducible], (1)

with the units i, j, k, which obey the following:

[mathematical expression not reproducible]. (2)

For a quaternion q = [q.sub.0] + i[q.sub.1] + j[q.sub.2] + k[q.sub.3] [member of] H, [q.sub.0] is simply called the scalar part of q denoted by Sc(q) and q = i[q.sub.1] + j[q.sub.2] + k[q.sub.3] is called the vector part of q denoted by Vec(q).

Let p,q [member of] H and p, q be their vector parts, respectively. Equation (2) yields the quaternionic multiplication qp as

qp = [q.sub.0[p.sub.0] + q x p + [q.sub.0]p + [p.sub.0]q + q x p, (3)

where

[mathematical expression not reproducible]. (4)

Analogously to the complex case, a quaternionic conjugation [bar.q] is given by

[bar.q] = [q.sub.0] - j[q.sub.2] - k[q.sub.3], (5)

which leads to the anti-involution; that is,

[bar.pq] = [bar.p][bar.q]. (6)

With the help of (5), we get the norm or modulus of q [member of] H as

[mathematical expression not reproducible]. (7)

One can easily verify that

[mathematical expression not reproducible]. (8)

Using conjugate (5) and the modulus of q, we can define the inverse of q [member of] H\{0} as

[q.sup.-1] = [bar.q]/[[absolute value of q].sup.2], (9)

which shows that H is a normed division algebra.

In quaternionic notation, we may define an inner product for quaternion-valued functions f,g: [R.sup.2] [right arrow] H as follows:

[mathematical expression not reproducible] (10)

with symmetric real scalar part

[mathematical expression not reproducible]. (11)

In particular, for f = g, we obtain the [L.sup.2]([R.sup.2]; H)-norm

[mathematical expression not reproducible]. (12)

A quaternion module [L.sup.2]([R.sup.2]; H) is then defined as

[mathematical expression not reproducible]. (13)

2.2. Split Quaternion and Properties. In this section, we study some of the basic formulas of split quaternion (see [9]), which will be used to prove the fundamental results in the sequel.

Definition 1. For two quaternion square roots [mu], v such that [[mu].sup.2] = [v.sup.2] = -1, one may express a quaternion q as

q = [q.sub.-] + [q.sub.+], [q.sub.[+ or -]] = 1/2 ([q.sub.[+ or -]] [mu]qv). (14)

For the special case of [mu] = v, any quaternion q may be split up into commuting and anticommuting parts with respect to [mu]; that is,

[mu] [q.sub.-] = [q.sub.-] [mu] [mu] [q.sub.+] = [q.sub.+] [mu] (15)

It easily seems that the commuting and anticommuting parts satisfy the interesting properties:

[[mu].sup.2] = [[mu].sup.2.sub.-] + [[mu].sup.2.sub.+] = -1 [[mu].sub.+] [[mu].sub.-] + [[mu].sub.-] [[mu].sub.+]] = 0. (16)

We learn from the above equation that

[mathematical expression not reproducible], (17)

where

[mathematical expression not reproducible]. (18)

In particular, taking [mu] = i and v = j, (14) becomes

q = [q.sub.+] + [q.sub.-], [q.sub.[+ or -]] = 1/2 (q [+ or -] iqj). (19)

The above gives

[mathematical expression not reproducible]. (20)

This leads to the following modulus identity:

[[absolute value of q].sup.2] = [[absolute value of [q.sub.-].sup.2] + [[absolute value of [q.sub.+].sup.2]. (21)

Furthermore, one can obtain

Sc ([p.sub.+] [[bar.q].sub.-]) = 0. (22)

3. Relationship between Quaternion Fourier Transform (QFT) and Quaternion Linear Canonical Transform (QLCT)

In this section, we introduce the QFT and its relationship to the QLCT. We begin by introducing different types of the QFT.

Definition 2 (two-sided, right-sided, and left-sided QFTs). The two-sided, right-sided, and left-sided quaternion Fourier transforms (QFTs) of f [member of] [L.sup.1]([R.sup.2]; H) are given by, respectively,

[mathematical expression not reproducible] (23)

[mathematical expression not reproducible] (24)

[mathematical expression not reproducible], (25)

where x = [x.sub.1][e.sub.1] + [x.sub.2][e.sub.2], [omega] = [[omega].sub.1][e.sub.1] + [[omega].sub.2] [e.sub.2], and the quaternion exponential product [mathematical expression not reproducible] is the quaternion Fourier kernel. From (24), we get the partial right-sided QFT:

[mathematical expression not reproducible]. (26)

It is not difficult to check that the relationship between the two-sided QFT and the right-sided QFT takes the form

[mathematical expression not reproducible]. (27)

Definition 3 (left-sided, right-sided, and two-sided QLCTs). Suppose that [A.sub.1] = ([a.sub.1], [b.sub.1], [c.sub.1], [d.sub.1]) and [A.sub.2] = ([a.sub.2], [b.sub.2], [c.sub.2], [d.sub.2]) are real matrix parameters satisfying det([A.sub.1]) = det([A.sub.2]) = 1. The left-sided, right-sided, and two-sided QLCTs of a quaternion signal f [member of] [L.sup.1] ([R.sup.2]; H) are defined by, respectively,

[mathematical expression not reproducible] (28)

[mathematical expression not reproducible], (29)

[mathematical expression not reproducible], (30)

where the kernel functions of the QLCT above are given by

[mathematical expression not reproducible]. (31)

From the definition of the QLCT, we can easily see that when [b.sub.1][b.sub.2] = 0 and [b.sub.1] = [b.sub.2] = 0, the QLCT of a signal is essentially a quaternion chirp multiplication. Therefore, in this work, we always assume [b.sub.1][b.sub.2] [not equal to] 0. As a special case, when [A.sub.1] = [A.sub.2] = ([a.sub.i], [b.sub.i], [c.sub.i], [d.sub.i]) = (0,1,-1,0), for i = 1,2, the QLCT definition (30) will lead to the QFT definition; that is,

[mathematical expression not reproducible]. (32)

The following lemma describes the general relationship between the two-sided QFT and the two-sided QLCT of 2D quaternion-valued signals.

Lemma 4. The QLCT of a signal f with matrix parameters [A.sub.1] = ([a.sub.1],[b.sub.1],[c.sub.1],[d.sub.1]) and [A.sub.2] = ([a.sub.2],[b.sub.2],[c.sub.2],[d.sub.2]) can be seen as the QFT of the signal f in the following form:

[mathematical expression not reproducible]. (33)

We introduce

[mathematical expression not reproducible]. (34)

This implies that (33) can be rewritten in the form

[mathematical expression not reproducible]. (35)

Further, we have the following lemma which describes a relation between the right-sided QLCT and the two-sided QLCT of 2D quaternion-valued signals.

Lemma 5. For f [member of] [L.sup.1] ([R.sup.2]; H), one has

[mathematical expression not reproducible], (36)

[mathematical expression not reproducible], (37)

where the matrix parameters [A.sup.*.sub.1] = ([a.sub.1], -[b.sub.1], [c.sub.1], [d.sub.1]) and [A.sub.2] = ([a.sub.2], -[b.sub.2], [c.sub.2], [d.sub.2]).

The following lemma allows us to represent the right-sided QLCT to the single right-sided QFT of 2D quaternion-valued signals.

Lemma 6. For f [member of] [L.sup.1] ([R.sup.2]; H), one has

[mathematical expression not reproducible]. (38)

Proof. Proofs of Lemmas 4, 5, and 6 are straightforward and are therefore omitted for brevity.

4. A Version of Uncertainty Principle Associated with QLCT

The classical uncertainty principle of harmonic analysis describes that a nontrivial function and its Fourier transform cannot be sharply localized simultaneously. 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). Let us now establish a version of the uncertainty principle associated with the QLCT. However, before proceeding the statement of this main result, we need to introduce a modified uncertainty principle for the QFT as follows (see [29] for more details).

Theorem 7 (the two-sided QFT uncertainty principle). Let S([R.sup.2]; H) be the quaternion Schwartz space. If the quaternion-valued function f [member of] S([R.sup.2]; H), then the following inequality holds:

[mathematical expression not reproducible]. (39)

It is shown that the quaternionic Gabor filters minimize the above uncertainty. Analogously, we get the uncertainty principle associated with the right-sided QFT in the following lemma.

Lemma 8 (the right-sided QFT uncertainty principle). Under the hypotheses of Theorem 7, one has the following inequality:

[mathematical expression not reproducible]. (40)

Theorem 7 has been recently generalized in the context of the QLCT by the authors of [30]. Our generalization is given by the following theorem.

Theorem 9 (the two-sided QLCT uncertainty principle). Under the assumptions of Theorem 7, one has

[mathematical expression not reproducible], (41)

where [??](x) is defined by

[mathematical expression not reproducible]. (42)

It is worth noting here that if [mathematical expression not reproducible] can be reduced to

[mathematical expression not reproducible]. (43)

Proof. By replacing f by h defined by (34) on both sides of (39), we immediately get

[mathematical expression not reproducible]. (44)

Let [omega] = [omega]/b; then, (44) becomes

[mathematical expression not reproducible]. (45)

In view of (34), we obtain

[mathematical expression not reproducible]. (46)

This leads to

[mathematical expression not reproducible]. (47)

Inserting (35) into (47), we easily obtain

[mathematical expression not reproducible]. (48)

The above identity can be further simplified to

[mathematical expression not reproducible], (49)

where [??](x) is defined in (42). Therefore, the proof is complete.

The above theorem is also valid for the right-sided QLCT as described in the following lemma.

Lemma 10 (the right-sided QLCT uncertainty principle). Under the assumptions of Theorem 7, one has

[mathematical expression not reproducible]. (50)

Proof. The proof follows directly from (37).

Now, we define a module of [F.sub.q]{f}([omega]) as

[mathematical expression not reproducible], (51)

where

[mathematical expression not reproducible]. (52)

Furthermore, we obtain the [L.sub.p]([R.sup.2]; H)-norm

[mathematical expression not reproducible]. (53)

It should be noticed that if [F.sub.q]{[f.sub.i]}, i = 0,2,3, is real-valued function, then (53) reduces to

[mathematical expression not reproducible], (54)

where

[mathematical expression not reproducible]. (55)

By Riesz's interpolation theorem, one can get the Hausdorff-Young inequality related to the QFT (see [10]):

[mathematical expression not reproducible] (56)

holds for 1 [less than or equal to] p [less than or equal to] 2 with 1/p + 1/p'. This gives the following important theorem.

Theorem 11 (Hausdorff-Young inequality). If 1 [less than or equal to] p [less than or equal to] 2 and letting p' be such that 1/p + 1/p' = 1, then, for all f [member of] LP([R.sup.2]; H), it holds that

[mathematical expression not reproducible], (57)

where

[mathematical expression not reproducible]. (58)

Proof. From the Hausdorff-Young inequality for the QFT, we have

[mathematical expression not reproducible]. (59)

Based on the arguments used in the proof of Theorem 9, we immediately get

[mathematical expression not reproducible]. (60)

Hence,

[mathematical expression not reproducible]. (61)

Or, equivalently,

[mathematical expression not reproducible]. (62)

Thus, the theorem is completely proved.

5. Conclusion

In this paper, we derived a version of the uncertainty principle for the QLCT using a relation between the QFT and the QLCT. We presented Hausdorff-Young inequality associated with the QLCT. This inequality is very useful for establishing a variation on Heisenberg's uncertainty principle related to the QLCT.

https://doi.org/10.1155/2018/8732457

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work is supported by Research Collaboration Between Universities (PEKERTI) (no. 622.48/UN28.2/PL/2017) of Indonesia.

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Mawardi Bahri (iD), (1) Resnawati, (2) and Selvy Musdalifah (2)

(1) Department of Mathematics, Hasanuddin University, Makassar 90245, Indonesia

(2) Department of Mathematics, Tadulako University, Palu 94148, Indonesia

Correspondence should be addressed to Mawardi Bahri; mawardibahri@gmail.com

Received 2 October 2017; Revised 18 March 2018; Accepted 2 April 2018; Published 7 May 2018

Academic Editor: Feyzi Basar
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