Journal of Mathematics
Volume 2019 (2019), Article ID 1062979, 13 pages
https://doi.org/10.1155/2019/1062979

Research Article

Two-Dimensional Quaternion Linear Canonical Transform: Properties, Convolution, Correlation, and Uncertainty Principle

Mawardi Bahri¹ and Ryuichi Ashino²

¹Department of Mathematics, Universitas Hasanuddin, Tamalanrea, Makassar, Indonesia
²Mathematics and Informatics, Osaka Kyoiku University, Osaka 582-8582, Japan

Correspondence should be addressed to Ryuichi Ashino; ashino@cc.osaka-kyoiku.ac.jp

Received 21 March 2019; Revised 6 July 2019; Accepted 4 August 2019; Published 9 September 2019

Academic Editor: Viliam Makis

Copyright © 2019 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.

A definition of the two-dimensional quaternion linear canonical transform (QLCT) is proposed. The transform is constructed by substituting the Fourier transform kernel with the quaternion Fourier transform (QFT) kernel in the definition of the classical linear canonical transform (LCT). Several useful properties of the QLCT are obtained from the properties of the QLCT kernel. Based on the convolutions and correlations of the LCT and QFT, convolution and correlation theorems associated with the QLCT are studied. An uncertainty principle for the QLCT is established. It is shown that the localization of a quaternion-valued function and the localization of the QLCT are inversely proportional and that only modulated and shifted two-dimensional Gaussian functions minimize the uncertainty.

1. Introduction

The quaternion Fourier transform (QFT) is a nontrivial generalization of the classical Fourier transform (FT) using the quaternion algebra. The QFT has been shown to relate to the other quaternion signal analysis tools, such as quaternion wavelet transform [1–3], fractional quaternion Fourier transform [4, 5], quaternionic windowed Fourier transform [6–9], and quaternion Wigner transform [10]. Because of the noncommutative property of quaternion multiplication, we obtain at least three different kinds of two-dimensional QFTs as follows (see [11–15]):

where and are any two unit pure quaternions () that are orthogonal to each other. These three QFTs are so-called left-sided, right-sided, and double-sided QFTs or type I, II, and III QFTs, respectively. As is well known, the linear canonical transform (LCT) is a general form of the FT, and the quaternion linear canonical transform (QLCT) is a generalization of the QFT in the LCT domain.

In the recent years, the LCT and QLCT have received much attention. Kou et al. [16, 17], for instance, constructed the windowed LCT, with a local window function. It can reveal the local LCT-frequency contents and enjoys high concentrations and eliminates the cross term. It is used to study the generalized prolate spheroidal wave functions and the connection with energy concentration problems. In [18–20], Kou et al. also proposed the (right-sided) quaternion linear canonical transform (QLCT) which is a generalization of the quaternion Fourier transform (QFT) in the LCT domain. This generalization is obtained by replacing the Fourier kernel with the right-sided QFT (type III QFT) kernel in the LCT definition. Some important properties such as Parseval’s theorem, reconstruction formula, uncertainty principles, and asymptotic behaviour are discussed. Like the uncertainty principle for the QFT [3], they also showed that only a two-dimensional Gaussian signal minimizes the uncertainty. However, the convolution theorem is an important result of the QLCT which does not hold using this construction because of the noncommutative property of the right-sided quaternion Fourier kernel.

Our attention in this article is to introduce a definition of the QLCT (compared to [21–23]). This definition is obtained by substituting the Fourier kernel with the type II QFT kernel (see [24]) in the LCT definition, which is essentially different from the kernel of the type III QFT. We investigate some important properties such as linearity, shift, and modulation and Plancherel’s and Parseval’s theorems. We study the convolution theorems associated with the QLCT, which can be useful in digital signal and image processing. Based on the convolution definitions of the LCT [25–27] and QFT, we also propose a new correlation definition for the QLCT and obtain its correlation theorems. We finally establish an uncertainty principle for the QLCT and show that only modulated and shifted two-dimensional Gaussian functions minimize the uncertainty.

We display here the organization of the paper: Section 2 describes the preliminaries about the quaternion Fourier transform and its basic properties, which will be used in the next section. The construction of the QLCT is presented in Section 3; its important properties are also discussed in this section. The definition of the convolution in the QLCT domain is introduced and the theorem on the QLCT of the convolution of two quaternion-valued functions is constructed in Section 4. Section 5 provides the definition of the correlation in the QLCT domain and discusses the theorem on the QLCT of a correlation of two quaternion functions. Section 6 establishes an uncertainty principle for the QLCT, which shows that the spread of a quaternion-valued function and its QLCT are inversely proportional. It is shown that only modulated and shifted two-dimensional Gaussian functions minimize the uncertainty.

Throughout this paper, is used to denote a set of real numbers.

2. Preliminaries

2.1. Quaternion

The quaternion, which is a type of hypercomplex number, was originally invented by William Hamilton in 1843 [28]. It is a generalization of a complex number to a 4D algebra and is denoted by . Every element of can be written in the hypercomplex form as follows:

Here, the three different imaginary parts satisfy the following multiplication rules:

For a quaternion , is called the scalar part of q denoted by and is called the pure part of q denoted by . A quaternionic conjugation is given by

Any quaternion q can be represented in the polar form as (see [29])

where , , is the eigen angle or phase of q and μ is an arbitrarily fixed unit quaternion such that . When , q is a unit quaternion. Note that Euler’s formula holds for quaternions; that is, . We also have . Hereinafter, besides the quaternion units , , and and the vector part of a quaternion , we shall use the following real vector notation:

and so on when there is no confusion. We may define the left quaternion inner product for two functions f and by

We see that, for , we obtain the norm induced by the above inner product as

We find it convenient to introduce the space as follows:

2.2. Convolution Associated with QFT

The quaternion Fourier transform (QFT) was originally proposed by Ell [11] and applied in quaternion color image processing [12, 30]. Some results related to the fundamental properties of the QFT can be found in [31–33]. We introduce a definition of the QFT as follows.

Definition 1. The QFT of is the function given by

where µ is an arbitrarily fixed pure quaternion such that . The inverse transform of the QFT is given by

provided that the integral exists.

Definition 2. The convolution of and , denoted by , is defined by

We have the following important result on the QFT of a convolution of two quaternion functions.

Theorem 1 [34]. Let f, have the following representations:

Then, we have

It is easily seen that the convolution theorem in the FT domain is a special case of Theorem 1.

3. Quaternion Linear Canonical Transform

The linear canonical transform (LCT) is a linear integral transformation with three free parameters which has widely been used in various fields such as spectral analysis, image processing, and optical system analysis [35, 36]. Several famous transforms such as the Fourier transform, the fractional Fourier transform, and the other transformations are special cases. This section will consider generalizing the LCT using the quaternion algebra. This extension is then called the quaternion linear canonical transform (QLCT).

3.1. Definition of QLCT

Based on the definition of the type II QFT, we obtain a definition of the QLCT by replacing the kernel of the FT with the kernel of the type II QFT in the classical LCT definition. The special linear group of degree 2 over , that is, the group of all real matrices with determinant one, is denoted by . Let

When , we define the kernels , , of the QLCT by

Definition 3. The QLCT of is defined by

where , , are called chirp signals in signal processing. Here after, we will deal with the case when because is trivial for . When , the QLCT reduces to the QFT, that is,

The polar form of the QLCT (17) can be found in [37].
The following describes the general relationship between the QLCT and the type II QFT of a signal f.

Lemma 1. The QLCT of a signal f with , , can be seen as the QFT of a signal f in the form

Proof. By a straightforward computation, it follows from Definition 3 that

thus proving the theorem.

Theorem 2. If and , then the inversion formula of the QLCT is given by

which is equivalent to

where , .

Proof. A direct calculation gives

thus proving the theorem.
The following proposition will be very useful when proving the uncertainty principle for the QLCT.

Proposition 1. Let the kernel functions , , be defined by (16). Then, we have

where denotes the k-th derivative of the delta function δ.

Proof. For , we have

Since the distributional support of as a distribution of is , we have

in the distributional sense. With (25) and (26), we have (24).

3.2. Useful Properties of the QLCT

Parseval’s theorem for Fourier transform can be generalized to the QLCT as well. Let us now formulate Parseval’s theorem in the QLCT domain.

Theorem 3 (QLCT Parseval). Let f, . Then, we have

Proof. For f, , we have

In the second equality in Theorem 3, we have replaced the quaternion function f with its inverse QLCT expression in Theorem 2. In the third equality, we have interchanged the order of integration. In the fourth equality, we have applied the quaternion conjugation rule for p, , thus proving the theorem.
Due to the noncommutativity of the kernel of the QLCT, we only have a left linearity property with quaternion constants; that is,

In the following, we will summarize the important properties of the QLCT. We will see that the results are generalizations of the basic properties of the LCT. Let . The translation operator by is defined by

The modulation operator by is defined by

We now begin with the shift property of the QLCT.

Theorem 4 (shift property). For , we have

Next, we are concerned about the behavior of the QLCT under modulation.

Theorem 5 (modulation property). For , we have

Theorem 6 (time-frequency shift). For , we have

We present the following result, which is needed for deriving the uncertainty principle related to the QLCT.

Theorem 7. Let and for . Assume that , that , and that , . Then, we have

Proof. Recall the quaternion conjugation rule for p, . The distributional calculation gives

Using in Proposition 1, we get

Again taking in Proposition 1 gives

Since the delta function is even function and , we have

Then, integration by parts in the sense of distribution implies

which implies the desired result.

The above properties of the QLCT are summarized in Table 1.

Table 1: Useful properties of the QLCT.

Property
Quaternion func.
QLCT

Left linearity

Shift

Modulation

Scaling

Time-frequency shift

Formula

Parseval’s theorem

Plancherel’s theorem

Reconstruction

4. Convolution Associated with QLCT

As we know, convolution is one of the fundamental results in the Fourier transform and LCT. Because the QLCT is a generalization of the LCT using the quaternion algebra, it makes possible to build the convolution theorem in the QLCT domain. For this reason, we introduce the following definition.

Definition 4. For f, , we define the convolution operator of the QLCT by

The above definition implies the following important theorem, which describes how the convolution of two quaternion-valued functions interacts with their QLCTs.

Theorem 8. Let

belong to . Then, the QLCTs of the convolution of f and are given by

Proof. Let and denote the QLCTs of f and , respectively. Expanding the QLCT of the left-hand side of the above identity, we obtain

Changing variables in the above expression, we have

Applying the QLCT definition yields

The noncommutativity of the quaternion multiplication requires us to decompose into . This gives

Multiplying both sides of the above identity by and , we obtain

thus proving the theorem.

5. Correlation Theorem for QLCT

In this section, we define the correlation of the QLCT. Then, we investigate the relationship between the QLCT and the correlation of two quaternion functions.

Definition 5. For f, , we define the correlation operator of the QLCT by

Then, we reap a consequence of the definition above.

Theorem 9. For f, , we have the QLCTs of correlation of f and by

When , the above expression reduces to

Proof. From the definitions of the QLCT (17) and the convolution, we get

Setting , we get

Using the QLCT definition, we obtain

Substituting , we have

Multiplying both sides of the above identity by and , we obtain

We finally obtain

thus proving the theorem.

6. Uncertainty Principle for QLCT

As in the case of the LCT [38, 39], the uncertainty principle is an important result of the QLCT. In [18, 19, 21], the authors have established the uncertainty principle for the two-sided QLCT. Based on the definition of our proposed QLCT in this article, we investigate an uncertainty principle related to the QLCT. The proof of the uncertainty differs from ones mentioned above. This uncertainty principle describes that the spread of a quaternion-valued function and its QLCT are inversely proportional. It is found that modulated and shifted two-dimensional Gaussian functions minimize the uncertainty.

Theorem 10. Let , , and , , , and . Then, the following inequalities hold:

Moreover, we have equality in the above expression for if and only if f is a modulated and shifted two-dimensional Gaussian function; that is, it has the form

where is a quaternion constant and , are positive real constants.

Proof. We will prove this theorem in two parts. Firstly, we set . By inserting Theorem 7 into the left-hand side of (58), we obtain

where the third equality follows from the quaternion Cauchy–Schwarz inequality.
Secondly, for , we can take similar steps as above and have

Now notice that, for , using integration by parts, we have

Note that the equality in equation (58) only holds for a two-dimensional Gaussian function:

It is known that modulation and shift properties are valid for the type II QFT (compare to [40]). Moreover, equality in (58) holds if and only if f is a modulated and shifted two-dimensional Gaussian function, thus proving the theorem.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was partially supported by JSPS KAKENHI under Grant nos. JP16K05216, JP17K05298, and JP17K05363 and by Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University.

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