Research Article | Open Access
Two-Dimensional Quaternion Linear Canonical Transform: Properties, Convolution, Correlation, and Uncertainty Principle
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.
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 . 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 , 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 ) 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.
The quaternion, which is a type of hypercomplex number, was originally invented by William Hamilton in 1843 . 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 )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  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 bywhere µ is an arbitrarily fixed pure quaternion such that . The inverse transform of the QFT is given byprovided that the integral exists.
Definition 2. The convolution of and , denoted by , is defined byWe have the following important result on the QFT of a convolution of two quaternion functions.
Theorem 1 . 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 bywhere , , 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 .
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 thatthus proving the theorem.
Theorem 2. If and , then the inversion formula of the QLCT is given bywhich is equivalent towhere , .
Proof. A direct calculation givesthus 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 havewhere denotes the k-th derivative of the delta function δ.
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 haveIn 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 byThe modulation operator by is defined byWe 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 givesUsing in Proposition 1, we getAgain taking in Proposition 1 givesSince the delta function is even function and , we haveThen, integration by parts in the sense of distribution implieswhich implies the desired result.
The above properties of the QLCT are summarized in Table 1.
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 byThe above definition implies the following important theorem, which describes how the convolution of two quaternion-valued functions interacts with their QLCTs.
Theorem 8. Letbelong 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 obtainChanging variables in the above expression, we haveApplying the QLCT definition yieldsThe noncommutativity of the quaternion multiplication requires us to decompose into . This givesMultiplying both sides of the above identity by and , we obtain