Reduced Biquaternion Windowed Linear Canonical Transform: Properties and Applications

Abstract

The quaternion windowed linear canonical transform is a tool for processing multidimensional data and enhancing the quality and efficiency of signal and image processing; however, it has disadvantages due to the noncommutativity of quaternion multiplication. In contrast, reduced biquaternions, as a special case of four-dimensional algebra, possess unique advantages in computation because they satisfy the multiplicative exchange rule. This paper proposes the reduced biquaternion windowed linear canonical transform (RBWLCT) by combining the reduced biquaternion signal and the windowed linear canonical transform that has computational efficiency thanks to the commutative property. Firstly, we introduce the concept of a RBWLCT, which can extract the time local features of an image and has the advantages of both time-frequency analysis and feature extraction; moreover, we also provide some fundamental properties. Secondly, we propose convolution and correlation operations for RBWLCT along with their corresponding generalized convolution, correlation, and product theorems. Thirdly, we present a fast algorithm for RBWLCT and analyze its computational complexity based on two dimensional Fourier transform (2D FTs). Finally, simulations and examples are provided to demonstrate that the proposed transform effectively captures the local RBWLCT-frequency components with enhanced degrees of freedom and exhibits significant concentrations.

1. Introduction

The linear canonical transform (LCT) finds extensive applications across various fields, including applied mathematics, optics, and signal processing [1,2,3,4]. Previous findings underscore the significant role played by LCT in the analysis of chirp signals for parameter estimation and sampling [5,6]. However, due to its global kernel, the LCT is unable to represent the frequency content at the local level. To address this limitation, in references [7,8], researchers have constructed windowed linear canonical transform (WLCT) [9,10,11,12]. The WLCT is a signal processing tool, which conveys both time and LCT frequency information, while originally presenting a local LCT distribution.

The quaternion number system was first described by Hamilton as a generalization of complex numbers [13,14]. In recent years, researchers have extended integral transforms into the quaternion algebra domain, leading to the development of theoretical frameworks such as quaternion Fourier transform (QFT) [15,16,17,18], quaternion fractional Fourier transform (QFRFT) [19], quaternion windowed fractional Fourier transform (QWFRFT) [20,21,22,23,24], quaternion linear canonical transform (QLCT) [25,26], and quaternion offset linear canonical transform [27]. Several important properties of QLCT have been investigated, including linearity, time shift, modulation, reconstruction formula, boundedness, and uncertainty principles in [28,29,30]. Recently, Hu et al. [31] proposed a novel quaternionic convolution operator for QLCT, which shows that the QLCT of the convolution of two quaternion functions can be implemented by the product of their respective QLCTs. The application of QLCT in solving partial differential equations has also been explored based on these established properties. Additionally, Gao and Li extended the WLCT to the quaternion field and defined it as a quaternion windowed linear canonical transform (QWLCT). They further investigated various relevant properties, including several uncertainty principles in this context [32,33].

However, due to the noncommutative nature of quaternion multiplication, certain calculations become intricate, such as the convolution of QFT [34]. To overcome this limitation in the context of digital signal processing, in 1990 Schutte and Wenzel proposed a new number system, reduced biquaternions (RBs), which resemble quaternions but possesses commutative multiplication [35]. RBs have applications in digital signal and image processing [36,37,38], neural networks [39], and electromagnetics [40].

Recently, in [41], the authors combined the RB signal with the LCT, defining the reduced biquaternion linear canonical transform (RBLCT), along with its properties. However, to the best of our knowledge, there has been no literature presenting the reduced biquaternion windowed linear canonical transform (RBWLCT), which overcomes the limitations of the LCT, such as the inability to represent the frequency content at the local level, and also obviates the noncommutativity of quaternion algebra. Therefore, this paper aims to define RBWLCT, establish convolution, correlation, and product theorems for the RBWLCT, and demonstrate the fast algorithm of RBWLCT. Furthermore, it aims to illustrate simulations using RBWLCT for transforming color images.

The structure of the paper is as follows: Section 2 provides an overview of general definitions and basic properties. Section 3 proposes a definition for RBWLCT and derives some fundamental properties; additionally, it presents the convolution and correlation theorems for the RBWLCT. Section 4 develops the fast algorithm for RBWLCT and carries out simulations. Section 5 draws conclusions based on the findings.

2. Preliminaries

In this section, we mainly review some basic facts about the quaternion algebra and RB signal. Additionally, we will provide a brief overview of the QLCT, which will be needed throughout the paper.

2.1. Quaternion Algebra and Reduced Biquaternion Signal

While a complex number consists of two components—the real part and the imaginary part, a quaternion has four parts, with one being the real part and the remaining three being imaginary parts. It was first invented by Hamilton in 1843 [13], is denoted as $\mathbb{H}$, and each of its elements has form given by the following:

$$\mathbb{H}=\left\{q\right|q=q_0+i{q}_1+j{q}_2+k{q}_3,q_0,q_1,q_2,q_3\in \mathbb{R}\},$$

(1)

and $i,j,k$ are imaginary units obeys Hamilton’s multiplication rules [13,14]

$$ij=-ji=k,jk=-kj=i,ki=-ik=j,i^2=j^2=k^2=ijk=-1.$$

(2)

Correspondingly, a quaternionic function $f\left(\mathbf{x}\right)$ with the ${\mathbb{R}}^2$ domain can be expressed as follows:

$$f\left(\mathbf{x}\right)=f_r\left(\mathbf{x}\right)+f_i\left(\mathbf{x}\right)i+f_j\left(\mathbf{x}\right)j+f_k\left(\mathbf{x}\right)k,$$

(3)

where $f_r,f_i,f_j,f_k$ are all real functions; however, the multiplication rule of quaternions is not commutative. In [35], RB was proposed, and commutative multiplication was defined for it. The description of RB algebra is as follows:

$$\tilde{\mathbb{H}}=\left\{q\right|q=q_r+q_{i}i+q_{j}j+q_{k}k\},$$

where $q_r$, $q_i$, $q_j$, $q_k\in \mathbb{R}$, the imaginary units satisfy the following multiplicative rules:

$$ij=ji=k,jk=kj=-i,ki=ik=j,j^2=k^2=-1,i^2=1,$$

(4)

RB can also be represented as $q=q_1+q_{2}j$, where $q_1=q_r+q_{i}i$, $q_2=q_j+q_{k}i$. The norm of a RB $q=q_r+q_{i}i+q_{j}j+q_{k}k$ is as follows [36]:

$$\parallel q\parallel ={[{(q_r^2+q_i^2+q_j^2+q_k^2)}^2-4{(q_r{q}_j+q_i{q}_k)}^2]}^{\frac{1}{4}}.$$

(5)

The RB conjugate $\overline{q}$ is given by the following:

$$\overline{q}={\parallel q\parallel }^2{q}^{-1}={\parallel q\parallel }^2/q.$$

(6)

Correspondingly, a quaternionic function $f\left(\mathbf{x}\right)$ with the ${\mathbb{R}}^2$ domain can be expressed as follows [20]:

$$f\left(\mathbf{x}\right)=f_r\left(\mathbf{x}\right)+f_i\left(\mathbf{x}\right)i+f_j\left(\mathbf{x}\right)j+f_k\left(\mathbf{x}\right)k.$$

(7)

For any two RB signals $f\left(\mathbf{x}\right)$ and $g\left(\mathbf{x}\right)$, according to the multiplication rule of RB, we have the following:

$$\begin{array}{cc}\hfill f\left(\mathbf{x}\right)g\left(\mathbf{x}\right)& =g\left(\mathbf{x}\right)f\left(\mathbf{x}\right).\hfill \end{array}$$

(8)

If $f\left(\mathbf{x}\right)$ and $g\left(\mathbf{x}\right)$ are quaternion signals, Equation (8) will not holds, i.e., $f\left(\mathbf{x}\right)g\left(\mathbf{x}\right)\ne g\left(\mathbf{x}\right)f\left(\mathbf{x}\right)$.

In this paper, for signals $f,g\in L^1({\mathbb{R}}^2,\tilde{\mathbb{H}})$, the convolution operator ∗ and the correlation operator ⊙ is defined as follows [16]:

$$f\left(\mathbf{x}\right)\ast g\left(\mathbf{x}\right)={\int }_{{\mathbb{R}}^2}f\left(\mathbf{\tau }\right)g(x_1-{\tau }_1,x_2-{\tau }_2)d\mathbf{\tau },$$

(9)

and

$$f\left(\mathbf{x}\right)\odot g\left(\mathbf{x}\right)={\int }_{{\mathbb{R}}^2}\overline{f\left(\mathbf{\tau }\right)}g({\tau }_1+x_1,{\tau }_2+x_2)d\mathbf{\tau }.$$

(10)

respectively.

2.2. Reduced Biquaternion Linear Canonical Transform

In this subsection, we shortly review the RBLCT, which is defined in [41].

Definition 1.

Let $f\left(\mathbf{x}\right)$$\in L^2({\mathbb{R}}^2,\tilde{\mathbb{H}})$ be a RB signal, then the RBLCT of $f\left(\mathbf{x}\right)$ with parameters $A_1$ and $A_2$ is defined as

$${\mathcal{L}}_{A_1,A_2}^{i,k}\left\{f\left(\mathbf{x}\right)\right\}\left(\mathbf{u}\right)=\left\{\begin{array}{cc}{\int }_{{\mathbb{R}}^2}f\left(\mathbf{x}\right)K_{A_1}^i(x_1,u_1)K_{A_2}^k(x_2,u_2)d\mathbf{x},\hfill & b_1{b}_2\ne 0,\hfill \\ \sqrt{d_1{d}_2}f(d_1{u}_1,d_2{u}_2)e^{i\left(\frac{c_1{d}_1{u}_1^2}{2}\right)}{e}^{k\left(\frac{c_2{d}_2{u}_2^2}{2}\right)},\hfill & b_1{b}_2=0,\hfill \end{array}\right.$$

(11)

with $i,k\in \tilde{\mathbb{H}}$; such that $i^2=k^2=-1$, including the cases $i=\pm k$ and kernels given as:

$$\begin{array}{cc}\hfill & K_{A_1}^i(x_1,u_1)=\frac{1}{\sqrt{i2\pi b_1}}{e}^{i(\frac{a_1}{2b_1}{x}_1^2-\frac{x_1}{b_1}{u}_1+\frac{d_1}{2b_1}{u}_1^2)},\hfill \\ \hfill & K_{A_2}^k(x_2,u_2)=\frac{1}{\sqrt{k2\pi b_2}}{e}^{k(\frac{a_2}{2b_2}{x}_2^2-\frac{x_2}{b_2}{u}_2+\frac{d_2}{2b_2}{u}_2^2)},\hfill \end{array}$$

(12)

where $A_l=\left(\begin{array}{cc}{a}_l& b_l\\ c_l& d_l\end{array}\right)$, with $a_l,b_l,c_l,d_l\in \mathbb{R}$; such that $a_l{d}_l-b_l{c}_l=1$, for $l=1,2$.

Definition 2.

Let $f\left(\mathbf{x}\right)\in L^2({\mathbb{R}}^2,\tilde{\mathbb{H}})$ be a RB signal, the inverse transform of the RBLCT is given as follows:

$$f\left(\mathbf{x}\right)=\left\{\begin{array}{cc}{\int }_{{\mathbb{R}}^2}{\mathcal{L}}_{A_1,A_2}^{i,k}\left(\mathbf{u}\right)K_{A_1^{-1}}^i(x_1,u_1)K_{A_2^{-1}}^k(x_2,u_2)d\mathbf{x},\hfill & b_1{b}_2\ne 0,\hfill \\ \sqrt{a_1{a}_2}f(a_1{x}_1,a_2{x}_2)e^{-i\left(\frac{a_1{c}_1{x}_1^2}{2}\right)}{e}^{-k\left(\frac{a_2{c}_2{x}_2^2}{2}\right)},\hfill & b_1{b}_2=0,\hfill \end{array}\right.$$

(13)

where $A_l^{-1}=\left(\begin{array}{cc}{d}_l& -b_l\\ -c_l& a_l\end{array}\right)$, for $l=1,2$.

3. Reduced Biquaternion Windowed Linear Canonical Transform

In this section, the RBWLCT is introduced. Subsequently, the convolution and correlation operations for RBWLCT are defined, and the corresponding convolution and correlation theorems are derived. Furthermore, an in–depth investigation into several fundamental properties of RBWLCT is conducted.

3.1. Definition and Properties of RBWLCT

In this subsection, RBWLCT is proposed. And the relationship between RBWLCT and reduced biquaternion windowed Fourier transform (RBWFT) is given.

Definition 3.

Let $A_l=\left(\begin{array}{cc}{a}_l& b_l\\ c_l& d_l\end{array}\right)$ be a matrix parameter such that $a_l,b_l,c_l,d_l\in \mathbb{R}$ and $a_l{d}_l-b_l{c}_l=1$, for $l=1,2$. The RBWLCT of any RB function $f\left(\mathbf{x}\right)\in L^2({\mathbb{R}}^2,\tilde{\mathbb{H}})$ and window function $\varphi \in L^2({\mathbb{R}}^2,\tilde{\mathbb{H}})$ is defined as follows:

$${\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{f\left(\mathbf{x}\right)\right\}(\mathbf{u},\mathbf{t})=\left\{\begin{array}{cc}{\int }_{{\mathbb{R}}^2}f\left(\mathbf{x}\right)\overline{\varphi (\mathbf{x}-\mathbf{t})}{K}_{A_1}^i(x_1,u_1)K_{A_2}^k(x_2,u_2)d\mathbf{x},\hfill & b_1{b}_2\ne 0,\hfill \\ \sqrt{d_1{d}_2}f(d_1{u}_1,d_2{u}_2)e^{i\left(\frac{c_1{d}_1{u}_1^2}{2}\right)}{e}^{k\left(\frac{c_2{d}_2{u}_2^2}{2}\right)},\hfill & b_1{b}_2=0,\hfill \end{array}\right.$$

(14)

where $K_{A_1}^i(x_1,u_1)$, $K_{A_2}^k(x_2,u_2)$ are the same as Equation (12).

In this paper, we deal with the case when $b_1\ne 0$ and $b_2\ne 0$, if $b_1=b_2=0$, then ${\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{f\left(\mathbf{x}\right)\right\}(\mathbf{u},\mathbf{t})$ represents a scaling operation combined with two chirp multiplications, which are not relevant to our research objectives.

Remark 1.

(1) If $A_1=\left(\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right)$, $A_2=\left(\begin{array}{cc}\cos \beta & \sin \beta \\ -\sin \beta & \cos \beta \end{array}\right)$, the RBWLCT boils down to the new transform called reduced biquaternion windowed fractional Fourier transform (RBWFRFT)

$${\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{f\left(\mathbf{x}\right)\right\}(\mathbf{u},\mathbf{t})={\int }_{{\mathbb{R}}^2}f\left(\mathbf{x}\right)\overline{\varphi (\mathbf{x}-\mathbf{t})}{K}_{A_1}^i(x_1,u_1)K_{A_2}^k(x_2,u_2)d\mathbf{x},$$

(15)

where

$$\begin{array}{c}{K}_{A_1}^i(x_1,u_1)=\sqrt{\frac{1-icot\alpha }{2\pi }}{e}^{i((x_1^2+u_1^2)·\frac{cot\alpha }{2}-x_1{u}_{1}csc\alpha )},\hfill \\ K_{A_2}^k(x_2,u_2)=\sqrt{\frac{1-kcot\beta }{2\pi }}{e}^{k((x_2^2+u_2^2)·\frac{cot\beta }{2}-x_2{u}_{2}csc\beta )}.\hfill \end{array}$$

(16)

(2) If $A_1=A_2=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$, the RBWLCT can be reduced to the novel transform called RBWFT as follows:

$${\mathcal{F}}_{\varphi }^{i,k}\left\{f\left(\mathbf{x}\right)\right\}(\mathbf{u},\mathbf{t})={\int }_{{\mathbb{R}}^2}f\left(\mathbf{x}\right)\overline{\varphi (\mathbf{x}-\mathbf{t})}{e}^{-i{x}_1{u}_1}{e}^{-k{x}_2{u}_2}d\mathbf{x}.$$

(17)

(3) Let $h\left(\mathbf{x}\right)=f\left(\mathbf{x}\right)\overline{\varphi (\mathbf{x}-\mathbf{t})}$, then the RBWFT can be reduced to the novel transform called reduced biquaternion Fourier transform (RBFT) as follows:

$${\mathcal{F}}^{ }\left\{h\left(\mathbf{x}\right)\right\}\left(\mathbf{u}\right)={\int }_{{\mathbb{R}}^2}h\left(\mathbf{x}\right)e^{-i{x}_1{u}_1}{e}^{-k{x}_2{u}_2}d\mathbf{x}.$$

(18)

In the subsequent analysis, we will investigate two fundamental characteristics of RBWLCT, namely, its inverse and switching properties, while other properties of RBWLCT are succinctly summarized in

Table 1. Some properties of the RBWLCT.

Property	RBWLCT
Parity	${\mathcal{G}}_{P\varphi }^{A_1,A_2}\left\{Pf\right\}(\mathbf{u},\mathbf{t})={\mathcal{G}}_{\varphi }^{A_1,A_2}f(-\mathbf{u},-\mathbf{t}),Pf=f\left(-\mathbf{x}\right),P\varphi =\varphi \left(-\mathbf{x}\right).$
Conjugation	${\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{\overline{f\left(\mathbf{x}\right)}\right\}(\mathbf{u},\mathbf{t})=\overline{{\mathcal{G}}_{\varphi }^{A_1^{-1},A_2^{-1}}\left\{f\left(\mathbf{x}\right)\right\}(\mathbf{u},\mathbf{t})}.$
Boundedness	$|{\mathcal{G}}_{\varphi }^{A_1,A_2}\left(f\right)(\mathbf{w},\mathbf{u})|\le \frac{1}{\sqrt{2\pi b_1{b}_2}}{\parallel f\parallel }_{L^2\left({\mathbb{R}}^2\right)}{\parallel \varphi \parallel }_{L^2\left({\mathbb{R}}^2\right)}.$
Linearity	${\mathcal{G}}_{\varphi }^{A_1,A_2}\{\alpha \left(\mathbf{x}\right)+\beta \left(\mathbf{x}\right)\}(\mathbf{u},\mathbf{t})=\alpha {\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{f\right\}(\mathbf{u},\mathbf{t})+\beta {\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{g\right\}(\mathbf{u},\mathbf{t}),\alpha ,\beta areconstants.$
Shift	${\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{f(\mathbf{x}-\mathbf{z})\right\}(\mathbf{u},\mathbf{t})={\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{f\right\}(\mathbf{r},\mathbf{v})e^{i{h}_1{u}_1{c}_1}{e}^{k{h}_2{u}_2{c}_2}{e}^{-i\frac{a_1{h}_1^2}{2}}{e}^{-k\frac{a_2{h}_2^2}{2}}.$
Modulation	${\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{M_{\mathbf{s}}f\right\}(\mathbf{u},\mathbf{t})={\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{f\right\}(\mathbf{w},\mathbf{t})e^{i{u}_1{s}_1{d}_1}{e}^{k{u}_2{s}_2{d}_2}{e}^{-i\frac{b_1{d}_1{s}_1^2}{2}}{e}^{-k\frac{b_2{d}_2{s}_2^2}{2}},M_{s}f=e^{i{x}_1{s}_1}f\left(\mathbf{x}\right)e^{k{x}_2{s}_2}.$
Inversion formula	$f\left(\mathbf{x}\right)=\frac{1}{{\Vert \varphi \Vert }^2}{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{\mathcal{G}}_{\varphi }^{A_1,A_2}\left(f\right)(\mathbf{u},\mathbf{t})\overline{K_{A_1}^i(x_1,u_1)}\overline{K_{A_2}^k(x_2,u_2)}\varphi (\mathbf{x}-\mathbf{t})d\mathbf{u}d\mathbf{t}.$
Parseval formula	$f\left(\mathbf{x}\right)\overline{g\left(\mathbf{x}\right)}={∥\frac{-1}{\sqrt{i2\pi b_1}}∥}^2{∥\frac{-1}{\sqrt{k2\pi b_2}}∥}^2\frac{{\Vert \psi \Vert }^2}{{\Vert \varphi \Vert }^2}{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{\mathcal{G}}_{\varphi }^{A_1,A_2}\left(f\right)(\mathbf{u},\mathbf{t})\overline{{\mathcal{G}}_{\psi }^{A_1,A_2}\left(g\right)(\mathbf{u},\mathbf{t})}d\mathbf{u}d\mathbf{t}.$
Switching f with $\varphi$	${\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{f\left(\mathbf{x}\right)\right\}(\mathbf{u},\mathbf{t})=e^{i\frac{a_1}{2b_1}{t}_1^2}·e^{-i\frac{1}{b_1}{t}_1{u}_1}·e^{k\frac{a_2}{2b_2}{t}_2^2}·e^{-k\frac{1}{b_2}{t}_2{u}_2}·\overline{{\mathcal{G}}_f^{A_1^{-1},A_2^{-1}}\varphi (\mathbf{u}-\mathbf{ta},-\mathbf{u})}.$

.

Theorem 1.

(Inversion formula) The inverse RBWLCT is given by the following:

$$f\left(\mathbf{x}\right)=\frac{1}{{\Vert \varphi \Vert }^2}{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{\mathcal{G}}_{\varphi }^{A_1,A_2}\left(f\right)(\mathbf{u},\mathbf{t})\overline{K_{A_1}^i(x_1,u_1)}\overline{K_{A_2}^k(x_2,u_2)}\varphi (\mathbf{x}-\mathbf{t})d\mathbf{u}d\mathbf{t},$$

(19)

with parameters $A_l^{-1}=\left(\begin{array}{cc}{d}_l& -b_l\\ -c_l& a_l\end{array}\right)$, where $a_l,b_l,c_l,d_l\in \mathbb{R}$, for $l=1,2$.

Proof. 

Applying the inversion RBLCT of $f\left(\mathbf{x}\right)$, we have the following:

$$\begin{array}{cc}\hfill & f\left(\mathbf{x}\right)\overline{\varphi (\mathbf{x}-\mathbf{t})}\hfill \\ & ={\mathcal{L}}_{\mathbf{i},\mathbf{j}}^{-A_1,-A_2}\left\{{\mathcal{G}}_{\varphi }^{A_1,A_2}\left(f\right)(\mathbf{u},\mathbf{t})\right\}\hfill \\ & ={\int }_{{\mathbb{R}}^2}{\mathcal{G}}_{\varphi }^{A_1,A_2}\left(f\right)(\mathbf{u},\mathbf{t})\overline{K_{A_1}^i(x_1,u_1)}\overline{K_{A_2}^k(x_2,u_2)}d\mathbf{u}.\hfill \end{array}$$

(20)

Multiplying by $\varphi (\mathbf{x}-\mathbf{t})$ on both sides of Equation (20), and integrating with respect to $d\mathbf{t}$, we have the following:

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^2}f\left(\mathbf{x}\right)\mid \varphi (\mathbf{x}-\mathbf{t}){\mid }^{2}d\mathbf{t}={\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{\mathcal{G}}_{\varphi }^{A_1,A_2}\left(f\right)(\mathbf{u},\mathbf{t})\overline{K_{A_1}^i(x_1,u_1)}& \overline{K_{A_2}^k(x_2,u_2)}\varphi (\mathbf{x}-\mathbf{t})d\mathbf{u}d\mathbf{t}.\hfill \end{array}$$

(21)

Due to ${\int }_{{\mathbb{R}}^2}\mid \varphi (\mathbf{x}-\mathbf{t}){\mid }^{2}d\mathbf{t}={\Vert \varphi \Vert }^2$, we have

$$\begin{array}{cc}\hfill f\left(\mathbf{x}\right)& =\frac{1}{{\Vert \varphi \Vert }^2}{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{\mathcal{G}}_{\varphi }^{A_1,A_2}\left(f\right)(\mathbf{u},\mathbf{t})\overline{K_{A_1}^i(x_1,u_1)}\overline{K_{A_2}^k(x_2,u_2)}\varphi (\mathbf{x}-\mathbf{t})d\mathbf{u}d\mathbf{t},\hfill \end{array}$$

which completes the proof.    □

Theorem 2.

(Switching f with ϕ) Let $f,\varphi \in L^2({\mathbb{R}}^2,\tilde{\mathbb{H}})$, and ϕ be window function, then we obtain

$${\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{f\left(\mathbf{x}\right)\right\}(\mathbf{u},\mathbf{t})=e^{i\frac{a_1}{2b_1}{t}_1^2}·e^{-i\frac{1}{b_1}{t}_1{u}_1}·e^{k\frac{a_2}{2b_2}{t}_2^2}·e^{-k\frac{1}{b_2}{t}_2{u}_2}\overline{{\mathcal{G}}_f^{A_1^{-1},A_2^{-1}}\varphi (\mathbf{u}-\mathbf{ta},-\mathbf{u})}.$$

(22)

Proof. 

It follows from Equation (14) that

$$\begin{array}{cc}\hfill {\mathcal{G}}_{\varphi }^{A_1,A_2}& \left\{f\right(\mathbf{x}\left)\right\}(\mathbf{u},\mathbf{t})\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}f\left(\mathbf{x}\right)\overline{\varphi (\mathbf{x}-\mathbf{t})}\frac{1}{\sqrt{i2\pi b_1}}{e}^{i(\frac{a_1}{2b_1}{x}_1^2-\frac{x_1}{b_1}{u}_1+\frac{d_1}{2b_1}{u}_1^2)}·\frac{1}{\sqrt{k2\pi b_1}}{e}^{k(\frac{a_2}{2b_2}{x}_2^2-\frac{x_2}{b_2}{u}_2+\frac{d_2}{2b_2}{u}_2^2)}d\mathbf{x}\hfill \\ \hfill & =\overline{{\int }_{{\mathbb{R}}^2}\varphi (\mathbf{x}-\mathbf{t})\overline{f\left(\mathbf{x}\right)}\frac{1}{\sqrt{-i2\pi b_1}}{e}^{-i(\frac{a_1}{2b_1}{x}_1^2-\frac{x_1}{b_1}{u}_1+\frac{d_1}{2b_1}{u}_1^2)}\frac{1}{\sqrt{-k2\pi b_1}}{e}^{-k(\frac{a_2}{2b_2}{x}_2^2-\frac{x_2}{b_2}{u}_2+\frac{d_2}{2b_2}{u}_2^2)}}d\mathbf{x}.\hfill \end{array}$$

(23)

Let $\mathbf{y}=\mathbf{x}-\mathbf{t}$, we have the following:

$$\begin{array}{cc}\hfill {\mathcal{G}}_{\varphi }^{A_1,A_2}& \left\{f\right(\mathbf{x}\left)\right\}(\mathbf{u},\mathbf{t})\hfill \\ \hfill =& \overline{{\int }_{{\mathbb{R}}^2}\varphi \left(\mathbf{y}\right)\overline{f(\mathbf{y}+\mathbf{u})}\frac{1}{\sqrt{-i2\pi b_1}}{e}^{-i(\frac{a_1}{2b_1}{y}_1^2-\frac{y_1}{b_1}(u_1-t_1{a}_1)+\frac{d_1}{2b_1}{u}_1^2)}{e}^{-i\frac{a_1}{2b_1}{t}_1^2}·e^{i\frac{1}{b_1}{t}_1{u}_1}}\hfill \\ \hfill & · \overline{\frac{1}{\sqrt{-k2\pi b_1}}{e}^{-k(\frac{a_2}{2b_2}{y}_2^2-\frac{y_2}{b_2}(u_2-t_2{a}_2)+\frac{d_2}{2b_2}{u}_2^2)}{e}^{-k\frac{a_2}{2b_2}{t}_2^2}·e^{k\frac{1}{b_2}{t}_2{u}_2}}d\mathbf{y}\hfill \\ \hfill =& e^{i\frac{a_1}{2b_1}{t}_1^2}·e^{-i\frac{1}{b_1}{t}_1{u}_1}·e^{k\frac{a_2}{2b_2}{t}_2^2}·e^{-k\frac{1}{b_2}{t}_2{u}_2}\overline{{\mathcal{G}}_f^{A_1^{-1},A_2^{-1}}\varphi (\mathbf{u}-\mathbf{ta},-\mathbf{u})},\hfill \end{array}$$

(24)

which completes the proof.    □

From Equations (14) and (17), the RBWLCT can be expressed by RBWFT, as follows:

Theorem 3.

Let $f\in L^2({\mathbb{R}}^2,\tilde{\mathbb{H}})$, the RBWLCT of a signal can be expressed by the ${\mathcal{F}}^{i,k}$ RBWFT as follows:

$${\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{f\left(\mathbf{x}\right)\right\}(\mathbf{u},\mathbf{t})=e^{i\frac{d_1}{2b_1}{u}_1^2}{e}^{k\frac{d_2}{2b_2}{u}_2^2}{\mathcal{F}}_{\varphi }^{i,k}\left\{g\left(\mathbf{x}\right)\right\}(\frac{\mathbf{u}}{\mathbf{b}},\mathbf{t}),$$

(25)

where

$$\begin{array}{cc}\hfill g\left(\mathbf{x}\right)& =f\left(\mathbf{x}\right)\frac{1}{\sqrt{i2\pi b_1}}\frac{1}{\sqrt{k2\pi b_2}}{e}^{i\frac{a_1}{2b_1}{x}_1^2}{e}^{k\frac{a_2}{2b_2}{x}_2^2}.\hfill \end{array}$$

(26)

From Equation (25), we can calculate RBWLCT using classical RBWFT (see Figure 1).

3.2. Convolution of RBWLCT

In this subsection, we present a novel quaternion convolution operation for the RBWLCT, and derive the corresponding convolution theorem of RBWLCT.

Definition 4.

Let $f,g$ ∈ $L^2$ $({\mathbb{R}}^2,\tilde{\mathbb{H}})$, a new convolution operator $\stackrel{A}{\mathsf{\Theta }}$ for the RBWLCT is defined as follows:

$$f\left(\mathbf{x}\right)\stackrel{A}{\mathsf{\Theta }}g\left(\mathbf{x}\right)={\int }_{{\mathbb{R}}^2}{e}^{-i\frac{a_1}{b_1}{x}_1(x_1-{\tau }_1)}{e}^{-k\frac{a_2}{b_2}{x}_2(x_2-{\tau }_2)}f\left(\mathbf{\tau }\right)g(x_1-{\tau }_1,x_2-{\tau }_2)e^{i\frac{a_1}{b_1}{(x_1-{\tau }_1)}^2}{e}^{k\frac{a_2}{b_2}{(x_2-{\tau }_2)}^2}d\mathbf{\tau }.$$

(27)

When $A_1=A_2=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$, then the convolution operator $\stackrel{A}{\mathsf{\Theta }}$ reduces to the classical convolution operator ∗ defined in Equation (9).

Based on the classical convolution operator ∗, the convolution operator for RBWLCT in Equation (27) can be rewritten as follows:

$$h\left(\mathbf{x}\right)=f\left(\mathbf{x}\right)\stackrel{A}{\mathsf{\Theta }}g\left(\mathbf{x}\right)=e^{-i\frac{a_1}{2b_1}{x}_1^2}{e}^{-k\frac{a_2}{2b_2}{x}_2^2}\left[f\left(\mathbf{x}\right)e^{i\frac{a_1}{2b_1}{x}_1^2}{e}^{k\frac{a_2}{2b_2}{x}_2^2}\right]\ast \left[g\left(\mathbf{x}\right)e^{i\frac{a_1}{2b_1}{x}_1^2}{e}^{k\frac{a_2}{2b_2}{x}_2^2}\right].$$

(28)

It means that the convolution operation $\stackrel{A}{\mathsf{\Theta }}$ in Equation (27) can be expressed as classical convolution operation ∗. From Figure 2. we can calculate convolution operation $\stackrel{A}{\mathsf{\Theta }}$ in Equation (27) by using classical convolution operation ∗.

Next, we will derive the convolution and product theorem for the RBWLCT, explore its intricacies and implications in the RB domain.

Theorem 4.

Let $f\left(\mathbf{x}\right)$ and $g\left(\mathbf{x}\right)$ are two RB signals. ${\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{f\left(\mathbf{x}\right)\right\}$ and ${\mathcal{G}}_{\psi }^{A_1,A_2}\left\{g\left(\mathbf{x}\right)\right\}$ are the RBWLCT of $f\left(\mathbf{x}\right)$ and $g\left(\mathbf{x}\right)$, respectively. Then we have the following:

$$\begin{array}{cc}\hfill & {\mathcal{G}}_{\varphi \stackrel{A}{\mathsf{\Theta }}\psi }^{A_1,A_2}\left\{f\left(\mathbf{x}\right)\stackrel{A}{\mathsf{\Theta }}g\left(\mathbf{x}\right)\right\}(\mathbf{u},\mathbf{t})\hfill \\ \hfill =& \sqrt{i2\pi b_1}\sqrt{k2\pi b_2}{e}^{-i\frac{d_1}{2b_1}{u}_{{ }_1}^2}{e}^{-k\frac{d_2}{2b_2}{u}_2^2}{\int }_{{\mathbb{R}}^2}{D}_1{D}_2{\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{f\left(\mathbf{\tau }\right)\right\}(\mathbf{u},\mathbf{s}){\mathcal{G}}_{\psi }^{A_1,A_2}\left\{g\left(\mathbf{y}\right)\right\}(\mathbf{u},\mathbf{t}-\mathbf{s})d\mathbf{s},\hfill \end{array}$$

(29)

where

$$\begin{array}{c}\hfill \begin{array}{c}{D}_1=e^{i\frac{a_1}{b_1}[{[y_1-(t_1-s_1)]}^2-(y_1+{\tau }_1-t_1)(y_1-(t_1-s_1))]},\hfill \\ D_2=e^{k\frac{a_2}{b_2}[{[y_2-(t_2-s_2)]}^2-(y_2+{\tau }_2-t_2)(y_2-(t_2-s_2))]}.\hfill \end{array}\end{array}$$

(30)

Proof. 

Using Equation (14), we obtain the following:

$$\begin{array}{cc}\hfill {\mathcal{G}}_{\varphi \stackrel{A}{\mathsf{\Theta }}\psi }^{A_1,A_2}& \left\{f\left(\mathbf{x}\right)\stackrel{A}{\mathsf{\Theta }}g\left(\mathbf{x}\right)\right\}(\mathbf{u},\mathbf{t})\hfill \\ \hfill =& {\int }_{{\mathbb{R}}^2}\left(f\left(\mathbf{x}\right)\stackrel{A}{\mathsf{\Theta }}g\left(\mathbf{x}\right)\right)\left(\overline{\varphi }\stackrel{A}{\mathsf{\Theta }}\overline{\psi (\mathbf{x}-\mathbf{t})}\right)K_{A_1}^i(x_1, u_1)K_{A_2}^k(x_2, u_2)d\mathbf{x}\hfill \\ \hfill =& {\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}\frac{1}{\sqrt{i2\pi b_1}}\frac{1}{\sqrt{k2\pi b_2}}{e}^{-i\frac{a_1}{b_1}{x}_1(x_1-{\tau }_1)}{e}^{-k\frac{a_2}{b_2}{x}_2(x_2-{\tau }_2)}f({\tau }_1,{\tau }_2)d\mathbf{\tau }\hfill \\ \hfill & ·g(x_1-{\tau }_1,x_2-{\tau }_2)e^{i\frac{a_1}{b_1}{(x_1-{\tau }_1)}^2}{e}^{k\frac{a_2}{b_2}{(x_2-{\tau }_2)}^2}d\mathbf{\tau }\hfill \\ \hfill & ·{\int }_{{\mathbf{R}}^2} e^{-i\frac{a_1}{b_1}(x_1-t_1)(x_1-t_1-r_1)}{e}^{-k\frac{a_2}{b_2}(x_2-t_2)(x_2-t_2-r_2)}\overline{\varphi \left(\mathbf{r}\right)}\hfill \\ \hfill & ·\overline{\psi (\mathbf{x}-\mathbf{t}-\mathbf{r})}{e}^{i\frac{a_1}{b_1}{(x_1-t_1-r_1)}^2}{e}^{k\frac{a_2}{b_2}{(x_2-t_2-r_2)}^2}d\mathbf{r}\hfill \\ \hfill & ·e^{i(\frac{a_1}{2b_1}{x}_1^2-\frac{1}{b_1}{x}_1{u}_1+\frac{d_1}{2b_1}{u}_1^2)}{e}^{k(\frac{a_2}{2b_2}{x}_2^2-\frac{1}{b_2}{x}_2{u}_2+\frac{d_2}{2b_2}{u}_2^2)}d\mathbf{x}.\hfill \end{array}$$

(31)

Let $\mathbf{y}=\mathbf{x}-\mathbf{\tau }$, $\mathbf{r}=\mathbf{\tau }-\mathbf{s}$, we obtain the following:

$$\begin{array}{cc}\hfill {\mathcal{G}}_{\varphi \stackrel{A}{\mathsf{\Theta }}\psi }^{A_1,A_2}& \left\{f\left(\mathbf{x}\right)\stackrel{A}{\mathsf{\Theta }}g\left(\mathbf{x}\right)\right\}(\mathbf{u},\mathbf{t})\hfill \\ \hfill =& {\int }_{{\mathbb{R}}^2}\frac{1}{\sqrt{i2\pi b_1}}\frac{1}{\sqrt{k2\pi b_2}}{e}^{-i\frac{a_1}{b_1}(y_1+{\tau }_1)y_1}{e}^{-k\frac{a_2}{b_2}(y_2+{\tau }_2)y_2}f\left(\mathbf{\tau }\right)\hfill \\ \hfill & ·g(y_1,y_2)e^{i\frac{a_1}{b_1}{y_1}^2}{e}^{k\frac{a_2}{b_2}{y_2}^2}d\tau \hfill \\ \hfill & ·{\int }_{{\mathbb{R}}^2}{e}^{-i\frac{a_1}{b_1}(y_1+{\tau }_1-t_1)(y_1-(t_1-s_1))}·e^{-k\frac{a_2}{b_2}(y_2+{\tau }_2-t_2)(y_2-(t_2-s_2))}\overline{\varphi (\mathbf{\tau }-\mathbf{s})}\hfill \\ \hfill & ·\overline{\psi (\mathbf{y}-(\mathbf{t}-\mathbf{s}\left)\right)}{e}^{i\frac{a_1}{b_1}{(y_1-(t_1-s_1))}^2}{e}^{k\frac{a_2}{b_2}{(y_2-(t_2-s_2))}^2}d\mathbf{s}\hfill \\ \hfill & ·{\int }_{{\mathbb{R}}^2}{e}^{i(\frac{a_1}{2b_1}{(y_1+{\tau }_1)}^2-\frac{1}{b_1}(y_1+{\tau }_1)u_1+\frac{d_1}{2b_1}{u}_1^2)}{e}^{i(\frac{a_2}{2b_2}{(y_2+{\tau }_2)}^2-\frac{1}{b_2}(y_2+{\tau }_2)u_2+\frac{d_2}{2b_2}{u}_2^2)}d\mathbf{y}\hfill \\ \hfill =& \sqrt{i2\pi b_1}\sqrt{k2\pi b_2}{e}^{-i\frac{d_1}{2b_1}{u}_{{ }_1}^2}{e}^{-k\frac{d_2}{2b_2}{u}_2^2}\hfill \\ \hfill & ·{\int }_{{\mathbb{R}}^2}{D}_1·D_2{\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{f\left(\tau \right)\right\}(\mathbf{u},\mathbf{s}){\mathcal{G}}_{\psi }^{A_1,A_2}\left\{g\left(\mathbf{y}\right)\right\}(\mathbf{u},\mathbf{t}-\mathbf{s})d\mathbf{s},\hfill \end{array}$$

(32)

which completes the proof.    □

Theorem 5.

Let $f\left(\mathbf{x}\right)$ and $g\left(\mathbf{x}\right)$ are two RB signals. ${\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{f\left(\mathbf{x}\right)\right\}$ and ${\mathcal{G}}_{\psi }^{A_1,A_2}\left\{g\left(\mathbf{x}\right)\right\}$ are the RBWLCT of $f\left(\mathbf{x}\right)$ and $g\left(\mathbf{x}\right)$, respectively. Then, the product theorem of $f\left(\mathbf{x}\right)$ and $g\left(\mathbf{x}\right)$ for the RBWLCT is as follows:

$$\begin{array}{cc}\hfill {\mathcal{G}}_{\varphi }^{A_1,A_2}& \left\{\sqrt{-i2\pi b_1}\sqrt{-k2\pi b_2}{e}^{i\frac{a_1}{2b_1}{x}_{{ }_1}^2}{e}^{k\frac{a_2}{2b_2}{x}_2^2}f\left(\mathbf{x}\right)g\left(\mathbf{x}\right){\overline{\varphi (\mathbf{x}-\mathbf{u})}}^2\right\}(\mathbf{u},\mathbf{t})\hfill \\ \hfill =& {\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{f\right\}(\mathbf{u},\mathbf{t})\stackrel{\overline{A}}{\mathsf{\Theta }}{\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{g\right\}(\mathbf{u},\mathbf{t}),\hfill \end{array}$$

(33)

where

$$\begin{array}{cc}\hfill & f\left(\mathbf{x}\right)\stackrel{\overline{A}}{\mathsf{\Theta }}g\left(\mathbf{x}\right)\hfill \\ \hfill =& {\int }_{{\mathbb{R}}^2}{e}^{i\frac{d_1}{b_1}{x}_1(x_1-{\tau }_1)}{e}^{k\frac{d_2}{b_2}{x}_2(x_2-{\tau }_2)}f\left(\mathbf{\tau }\right)g(x_1-{\tau }_1,x_2-{\tau }_2)e^{-i\frac{d_1}{b_1}{(x_1-{\tau }_1)}^2}{e}^{-k\frac{d_2}{b_2}{(x_2-{\tau }_2)}^2}d\mathbf{\tau }.\hfill \end{array}$$

Proof. 

From the RBWLCT Definition (14) and inversion Formula (19), we obtain the following:

$$\begin{array}{cc}\hfill {\mathcal{G}}_{\varphi }^{A_1^{-1},A_2^{-1}}& \left\{{\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{f\right\}(\mathbf{u},\mathbf{t})\stackrel{\overline{A}}{\mathsf{\Theta }}{\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{g\right\}(\mathbf{u},\mathbf{t})\right\}(\mathbf{u},\mathbf{t})\hfill \\ \hfill =& {\int }_{{\mathbb{R}}^2}{\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{f\right\}(\mathbf{u},\mathbf{t})\stackrel{\overline{A}}{\mathsf{\Theta }}{\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{g\right\}(\mathbf{u},\mathbf{t})K_{A_1^{-1}}^i(u_1, x_1)K_{A_1^{-1}}^k(u_2, x_2)d\mathbf{u}\hfill \\ \hfill =& {\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}\frac{1}{\sqrt{-i2\pi b_1}}\frac{1}{\sqrt{-k2\pi b_2}}{e}^{i\frac{d_1}{b_1}{u}_1(u_1-{\tau }_1)}{e}^{k\frac{d_2}{b_2}{u}_2(u_2-{\tau }_2)}{\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{f\right\}(\mathbf{\tau },\mathbf{t})\hfill \\ \hfill & ·{\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{g\right\}(\mathbf{u}-\mathbf{\tau },\mathbf{t})e^{-i\frac{d_1}{b_1}{(u_1-{\tau }_1)}^2}{e}^{-k\frac{d_2}{b_2}{(u_2-{\tau }_2)}^2}d\mathbf{\tau }\hfill \\ \hfill & ·e^{-i(\frac{a_1}{2b_1}{x}_1^2-\frac{1}{b_1}{x}_1{u}_1+\frac{d_1}{2b_1}{u}_1^2)}{e}^{-k(\frac{a_2}{2b_2}{x}_2^2-\frac{1}{b_2}{x}_2{u}_2+\frac{d_2}{2b_2}{u}_2^2)}d\mathbf{u}.\hfill \end{array}$$

(34)

Let $\mathbf{u}-\mathbf{\tau }=\mathbf{s}$, we obtain the following:

$$\begin{array}{cc}\hfill {\mathcal{G}}_{\varphi }^{A_1^{-1},A_2^{-1}}& \left\{{\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{f\right\}(\mathbf{u},\mathbf{t})\stackrel{\overline{A}}{\mathsf{\Theta }}{\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{g\right\}(\mathbf{u},\mathbf{t})\right\}(\mathbf{u},\mathbf{t})\hfill \\ \hfill =& {\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}\frac{1}{\sqrt{-i2\pi b_1}}\frac{1}{\sqrt{-k2\pi b_2}}{e}^{i\frac{d_1}{b_1}(s_1+{\tau }_1)s_1}{e}^{k\frac{d_2}{b_2}(s_2+{\tau }_2)s_2}\hfill \\ \hfill & ·{\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{f\right\}(\mathbf{\tau },\mathbf{t}){\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{g\right\}(\mathbf{s},\mathbf{t})e^{-i\frac{d_1}{b_1}{s_1}^2}{e}^{-k\frac{d_2}{b_2}{s_2}^2}d\mathbf{s}\hfill \\ \hfill & ·e^{-i(\frac{a_1}{2b_1}{x}_1^2-\frac{1}{b_1}{x}_1(s_1+{\tau }_1)+\frac{d_1}{2b_1}{(s_1+{\tau }_1)}^2)}{e}^{-k(\frac{a_2}{2b_2}{x}_2^2-\frac{1}{b_2}(s_2+{\tau }_2)x_2+\frac{d_2}{2b_2}{(s_2+{\tau }_2)}^2)}d\mathbf{\tau }\hfill \\ \hfill =& \sqrt{-i2\pi b_1}\sqrt{-k2\pi b_2}{e}^{i\frac{a_1}{2b_1}{x}_1^2}{e}^{k\frac{a_2}{2b_2}{x}_2^2}{\int }_{{\mathbb{R}}^2}{\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{f\right\}(\mathbf{\tau },\mathbf{t})\frac{1}{\sqrt{-i2\pi b_1}}\hfill \\ \hfill & ·\frac{1}{\sqrt{-k2\pi b_2}}·e^{-i(\frac{a_1}{2b_1}{x}_1^2-\frac{1}{b_1}{x}_1{\tau }_1+\frac{d_1}{2b_1}{{\tau }_1}^2)}{e}^{-k(\frac{a_2}{2b_2}{x}_2^2-\frac{1}{b_2}{\tau }_2{x}_2+\frac{d_2}{2b_2}{{\tau }_2}^2)}d\mathbf{\tau }\hfill \\ \hfill & ·{\int }_{{\mathbb{R}}^2}{\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{g\right\}(\mathbf{s},\mathbf{t})\frac{1}{\sqrt{-i2\pi b_1}}\frac{1}{\sqrt{-k2\pi b_2}}{e}^{-i(\frac{a_1}{2b_1}{x}_1^2-\frac{1}{b_1}{x}_1{s}_1+\frac{d_1}{2b_1}{s_1}^2)}\hfill \\ \hfill & ·e^{-k(\frac{a_2}{2b_2}{x}_2^2-\frac{1}{b_2}{s}_2{x}_2+\frac{d_2}{2b_2}{s_2}^2)}d\mathbf{s}\hfill \\ \hfill =& \sqrt{-i2\pi b_1}\sqrt{-k2\pi b_2}{e}^{i\frac{a_1}{2b_1}{x}_{{ }_1}^2}{e}^{k\frac{a_2}{2b_2}{x}_2^2}f\left(\mathbf{x}\right)g\left(\mathbf{x}\right){\overline{\varphi (\mathbf{x}-\mathbf{u})}}^2\hfill \end{array}$$

(35)

which completes the proof    □

The proof of Theorem 5 is completed. The product theorem (Theorem 5) shows that the RBWLCT of the product of f and g multiplied with four different chirp signals is equal to the convolution of the RBWLCT of f and g.

Based on the Definition 4 and Theorem 5, we shall investigate the basic properties of convolution operation $\stackrel{A}{\mathsf{\Theta }}$.

Property 1.

For RB signals f, g, and h, for constants a and b $\in \mathbb{R}$, the RB convolution $\stackrel{A}{\mathsf{\Theta }}$ for RBWLCT satisfied the following linearity property:

$$\left[(af+bg)\stackrel{A}{\mathsf{\Theta }}h\right]\left(\mathbf{x}\right)=a\left[f\stackrel{A}{\mathsf{\Theta }}h\right]\left(\mathbf{x}\right)+b\left[g\stackrel{A}{\mathsf{\Theta }}h\right]\left(\mathbf{x}\right).$$

(36)

Property 2.

(Distributive) For RB signals f, g, and h, we have the following:

$$\left[h\stackrel{A}{\mathsf{\Theta }}(f+g)\right]\left(\mathbf{x}\right)=\left[h\stackrel{A}{\mathsf{\Theta }}f\right]\left(\mathbf{x}\right)+\left[h\stackrel{A}{\mathsf{\Theta }}g\right]\left(\mathbf{x}\right).$$

(37)

Proof. 

The proof of the Property 1 and Property 2 follows from the Definition 4 and Theorem 5.    □

Remark 2.

(1) When $A_1=\left(\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right)$, $A_2=\left(\begin{array}{cc}\cos \beta & \sin \beta \\ -\sin \beta & \cos \beta \end{array}\right)$, we obtain the convolution theorem for the RBWFRFT.

(2) When $A_1=A_2=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$, we obtain the convolution theorem for the RBWFT.

3.3. Correlation of RBWLCT

In this subsection, a correlation operator for the RBWLCT is proposed, and the corresponding correlation theorem is investigated.

Definition 5.

Let $f,g$ ∈ $L^2$ $({\mathbb{R}}^2,\tilde{\mathbb{H}})$, the correlation operator $\stackrel{A}{\oplus }$ for the RBWLCT is defined as follows:

$$\begin{array}{cc}\hfill & f\left(\mathbf{x}\right)\stackrel{A}{\oplus }g\left(\mathbf{x}\right)\hfill \\ \hfill =& {\int }_{{\mathbb{R}}^2}{e}^{-i\frac{a_1}{b_1}{x}_1(x_1+{\tau }_1)}{e}^{-k\frac{a_2}{b_2}{x}_2(x_2+{\tau }_2)}\overline{f\left(\mathbf{\tau }\right)}g(x_1+{\tau }_1,x_2+{\tau }_2)e^{i\frac{a_1}{b_1}{(x_1+{\tau }_1)}^2}{e}^{k\frac{a_2}{b_2}{(x_2+{\tau }_2)}^2}d\mathbf{\tau }.\hfill \end{array}$$

(38)

When $A_1=A_2=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$, then the convolution operator $\stackrel{A}{\oplus }$ can be reduced to the classical correlation operator ⊙ as follows:

$$f\left(\mathbf{x}\right)\odot g\left(\mathbf{x}\right)={\int }_{{\mathbb{R}}^2}\overline{f\left(\tau \right)}g(x_1+{\tau }_1,x_2+{\tau }_2)d\mathbf{\tau }.$$

(39)

Next, we will derive the correlation theorem for the RBWLCT.

Theorem 6.

Let $f\left(\mathbf{x}\right)$ and $g\left(\mathbf{x}\right)$ are two RB signals. ${\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{f\left(\mathbf{x}\right)\right\}$ and ${\mathcal{G}}_{\psi }^{A_1,A_2}\left\{g\left(\mathbf{x}\right)\right\}$ are the RBWLCT of $f\left(\mathbf{x}\right)$ and $g\left(\mathbf{x}\right)$, respectively, then we obtain the following:

$$\begin{array}{cc}\hfill {\mathcal{G}}_{\varphi \stackrel{A}{\oplus }\psi }^{A_1,A_2}& \left\{f\left(\mathbf{x}\right)\stackrel{A}{\oplus }g\left(\mathbf{x}\right)\right\}(\mathbf{u},\mathbf{t})\hfill \\ \hfill =& \sqrt{i2\pi b_1}\sqrt{k2\pi b_2}{\int }_{{\mathbb{R}}^2}{E}_1·E_2{\mathcal{G}}_{\overline{\varphi }}^{A_1,A_2}\left\{\overline{f\left(\mathbf{\tau }\right)}\right\}(\mathbf{u},\mathbf{s}){\mathcal{G}}_{\psi }^{A_1,A_2}\left\{g\left(\mathbf{w}\right)\right\}(\mathbf{u},\mathbf{t}+\mathbf{s})d\mathbf{s},\hfill \end{array}$$

(40)

where

$$\begin{array}{cc}\hfill & E_1=e^{i\frac{a_1}{b_1}[{[w_1-(t_1+s_1)]}^2-(w_1-{\tau }_1-t_1)(w_1-(t_1+s_1))]},\hfill \\ \hfill & E_2=e^{k\frac{a_2}{b_2}[{[w_2-(t_2+s_2)]}^2-(w_2-{\tau }_2-t_2)(w_2-(t_2+s_2))]}.\hfill \end{array}$$

(41)

Proof. 

From the Definitions 3 and the Definitions 5, we have the following:

$$\begin{array}{cc}\hfill {\mathcal{G}}_{\varphi \stackrel{A}{\oplus }\psi }^{A_1,A_2}& \left\{f\left(\mathbf{x}\right)\stackrel{A}{\oplus }g\left(\mathbf{x}\right)\right\}(\mathbf{u},\mathbf{t})\hfill \\ \hfill =& {\int }_{{\mathbb{R}}^2}\left(f\left(\mathbf{x}\right)\stackrel{A}{\oplus }g\left(\mathbf{x}\right)\right)\left(\overline{\varphi }\stackrel{A}{\oplus }\overline{\psi (\mathbf{x}-\mathbf{t})}\right)K_{A_1}^i(x_1, u_1)K_{A_2}^k(x_2, u_2)d\mathbf{x}\hfill \\ \hfill =& {\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}\frac{1}{\sqrt{i2\pi b_1}}\frac{1}{\sqrt{k2\pi b_2}}{e}^{-i\frac{a_1}{b_1}{x}_1(x_1+{\tau }_1)}{e}^{-k\frac{a_2}{b_2}{x}_2(x_2+{\tau }_2)}\hfill \\ \hfill & ·\overline{f\left(\mathbf{\tau }\right)}g(x_1+{\tau }_1,x_2+{\tau }_2)e^{i\frac{a_1}{b_1}{(x_1+{\tau }_1)}^2}{e}^{k\frac{a_2}{b_2}{(x_2+{\tau }_2)}^2}d\mathbf{\tau }\hfill \\ \hfill & ·{\int }_{{\mathbb{R}}^2}{e}^{-i\frac{a_1}{b_1}(x_1-t_1)(x_1-t_1+r_1)}{e}^{-k\frac{a_2}{b_2}(x_2-t_2)(x_2-t_2+r_2)}\overline{\overline{\varphi \left(\mathbf{r}\right)}}\hfill \\ \hfill & ·\overline{\psi (\mathbf{x}-\mathbf{t}+\mathbf{r})}{e}^{i\frac{a_1}{b_1}{(x_1-t_1+r_1)}^2}{e}^{k\frac{a_2}{b_2}{(x_2-t_2+r_2)}^2}d\mathbf{r}\hfill \\ \hfill & ·e^{i(\frac{a_1}{2b_1}{x}_1^2-\frac{1}{b_1}{x}_1{u}_1+\frac{d_1}{2b_1}{u}_1^2)}{e}^{k(\frac{a_2}{2b_2}{x}_2^2-\frac{1}{b_2}{x}_2{u}_2+\frac{d_2}{2b_2}{u}_2^2)}d\mathbf{x}.\hfill \end{array}$$

(42)

Let $\mathbf{w}=\mathbf{x}+\mathbf{\tau }$, $\mathbf{r}=\mathbf{\tau }-\mathbf{s}$, we obtain the following:

$$\begin{array}{cc}\hfill {\mathcal{G}}_{\varphi \stackrel{A}{\oplus }\psi }^{A_1,A_2}& \left\{f\left(\mathbf{x}\right)\stackrel{A}{\oplus }g\left(\mathbf{x}\right)\right\}(\mathbf{u},\mathbf{t})\hfill \\ \hfill =& {\int }_{{\mathbb{R}}^2}\frac{1}{\sqrt{i2\pi b_1}}\frac{1}{\sqrt{k2\pi b_2}}{e}^{-i\frac{a_1}{b_1}(w_1-{\tau }_1)w_1}{e}^{-k\frac{a_2}{b_2}(w_2-{\tau }_2)w_2}\hfill \\ \hfill & ·\overline{f\left(\mathbf{\tau }\right)}g(w_1,w_2)e^{i\frac{a_1}{b_1}{w_1}^2}{e}^{k\frac{a_2}{b_2}{w_2}^2}d\mathbf{\tau }\hfill \\ \hfill & ·{\int }_{{\mathbb{R}}^2}{e}^{-i\frac{a_1}{b_1}(w_1-{\tau }_1-t_1)(w_1-(t_1+s_1))}{e}^{-k\frac{a_2}{b_2}(w_2-{\tau }_2-t_2)(w_2-(t_2+s_2))}\hfill \\ \hfill & ·\varphi (\mathbf{\tau }-\mathbf{s})\overline{\psi (\mathbf{w}-(\mathbf{t}+\mathbf{s}\left)\right)}{e}^{i\frac{a_1}{b_1}{(w_1-(t_1+s_1))}^2}{e}^{k\frac{a_2}{b_2}{(w_2-(t_2+s_2))}^2}d\mathbf{s}\hfill \\ \hfill & ·{\int }_{{\mathbb{R}}^2}{e}^{i(\frac{a_1}{2b_1}{(w_1-{\tau }_1)}^2-\frac{1}{b_1}(w_1-{\tau }_1)u_1+\frac{d_1}{2b_1}{u}_1^2)}{e}^{k(\frac{a_2}{2b_2}{(w_2-{\tau }_2)}^2-\frac{1}{b_2}(w_2-{\tau }_2)u_2+\frac{d_2}{2b_2}{u}_2^2)}d\mathbf{w}\hfill \\ \hfill =& \sqrt{i2\pi b_1}\sqrt{k2\pi b_2}{\int }_{{\mathbb{R}}^2}{E}_1·E_2{\mathcal{G}}_{\overline{\varphi }}^{A_1,A_2}\left\{\overline{f\left(\mathbf{\tau }\right)}\right\}(\mathbf{u},\mathbf{s}){\mathcal{G}}_{\psi }^{A_1,A_2}\left\{g\left(\mathbf{w}\right)\right\}(\mathbf{u},\mathbf{t}+\mathbf{s})d\mathbf{s},\hfill \end{array}$$

(43)

which completes the proof.    □

Remark 3.

(1) If $A_1=\left(\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right)$, $A_2=\left(\begin{array}{cc}\cos \beta & \sin \beta \\ -\sin \beta & \cos \beta \end{array}\right)$, then the Theorem 6 reduces to a version for correlation theorem associated with RBWFRFT.

(2) If $A_1=A_2=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$, then the Theorem 6 reduces to a version for correlation theorem associated with RBWFT.

4. Fast Algorithm of RBWLCT and Simulations

4.1. Fast Algorithm of RBWLCT

In this section, we will discuss the fast algorithm of RBWLCT. For an RB signal $f\left(\mathbf{x}\right)$, based on Equations (14), (17), and (18), we can rewrite the expression for the RBWLCT of $f\left(\mathbf{x}\right)$ as the reduced biquaternion Fourier transform (RBFT) of $h\left(\mathbf{x}\right)$, which is then multiplied by $B_i\left(u_1\right)$ and $B_k\left(u_2\right)$ as follows:

$$\begin{array}{cc}\hfill {\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{f\left(\mathbf{x}\right)\right\}(\mathbf{u},\mathbf{t})=& {\int }_{{\mathbb{R}}^2}f\left(\mathbf{x}\right)\overline{\varphi (\mathbf{x}-\mathbf{t})}{K}_{A_1}^i(x_1,u_1)K_{A_2}^k(x_2,u_2)d\mathbf{x}\hfill \\ \hfill =& B_i\left(u_1\right)B_k\left(u_2\right){\int }_{{\mathbb{R}}^2}h\left(\mathbf{x}\right)e^{-i\frac{x_1{u}_1}{b_1}}{e}^{-k\frac{x_2{u}_2}{b_2}}d\mathbf{x},\hfill \end{array}$$

(44)

where $h\left(\mathbf{x}\right)$ is defined as follows:

$$h\left(\mathbf{x}\right)=f\left(\mathbf{x}\right)\overline{\varphi (\mathbf{x}-\mathbf{u})}{e}^{i\frac{a_1}{2b_1}{x}_1^2}{e}^{k\frac{a_2}{2b_2}{x}_2^2},$$

(45)

and $B_i\left(u_1\right)$, $B_k\left(u_2\right)$ are, respectively, defined as follows:

$$B_i\left(u_1\right)=\frac{1}{\sqrt{i2\pi b_1}}{e}^{i\frac{d_1}{2b_1}{u}_1^2},B_k\left(u_2\right)=\frac{1}{\sqrt{k2\pi b_2}}{e}^{k\frac{d_2}{2b_2}{u}_2^2}.$$

(46)

To implement the RBWLCT, we let

$$W\left(\mathbf{u}\right)={\int }_{{\mathbb{R}}^2}h\left(\mathbf{x}\right)e^{-i\frac{x_1{u}_1}{b_1}}{e}^{-i\frac{x_2{u}_2}{b_2}}d\mathbf{x},$$

(47)

then, we obtain the following:

$$\begin{array}{ccc}\hfill & \frac{W(u_1,u_2)+W(u_1,-u_2)}{2}\hfill & \hfill ={\int }_{{\mathbb{R}}^2}h\left(\mathbf{x}\right)e^{-i\left(\frac{x_1{u}_1}{b_1}\right)}cos\left(\frac{x_2{u}_2}{b_2}\right)d\mathbf{x},\end{array}$$

(48)

$$\begin{array}{ccc}\hfill & \frac{W(u_1,u_2)-W(u_1,-u_2)}{2}\hfill & \hfill ={\int }_{{\mathbb{R}}^2}h\left(\mathbf{x}\right)e^{-i\left(\frac{x_1{u}_1}{b_1}\right)}(-i)sin\left(\frac{x_2{u}_2}{b_2}\right)d\mathbf{x}.\end{array}$$

(49)

Therefore, we have the following:

$$\begin{array}{cc}\hfill \frac{W(u_1,u_2)+W(u_1,-u_2)}{2}+\frac{W(u_1,u_2)-W(u_1,-u_2)}{2}\left(j\right)& ={\int }_{{\mathbb{R}}^2}h\left(\mathbf{x}\right)e^{-i\frac{x_1{u}_1}{b_1}}{e}^{-k\frac{x_2{u}_2}{b_2}}d\mathbf{x}.\hfill \end{array}$$

(50)

From Equations (44), (48)–(50), we can obtain the following:

$$\begin{array}{cc}\hfill {\mathcal{G}}_{\varphi }^{A_1,A_2}& \left\{f\right(\mathbf{x}\left)\right\}(\mathbf{u},\mathbf{t})\hfill \\ \hfill =& \frac{W(u_1,u_2)(1+j)+W(u_1,-u_2)(1-j)}{2}{B}_i\left(u_1\right)B_k\left(u_2\right).\hfill \end{array}$$

(51)

From Equation (51), the major computational load for RBWLCT is the calculation of $W\left(\mathbf{u}\right)$. To implement $W\left(\mathbf{u}\right)$, since $h\left(\mathbf{x}\right)$ is a quaternion function and not a complex function, we cannot use just one complex two-dimensional (2D) Fourier transforms (FT) to implement it. Instead, we can first decompose $h\left(\mathbf{x}\right)$ as follows:

$$\begin{array}{cc}\hfill h\left(\mathbf{x}\right)=& h_1\left(\mathbf{x}\right)+h_2\left(\mathbf{x}\right)j,\hfill \end{array}$$

(52)

where $h_1\left(\mathbf{x}\right)=h_r\left(\mathbf{x}\right)+h_i\left(\mathbf{x}\right)i$ and $h_2\left(\mathbf{x}\right)=h_j\left(\mathbf{x}\right)+h_k\left(\mathbf{x}\right)i$, combining Equations (47) and (52), we obtain the following:

$$\begin{array}{cc}\hfill W\left(\mathbf{u}\right)& ={\int }_{{\mathbb{R}}^2}{h}_1\left(\mathbf{x}\right)e^{-i\frac{x_1{u}_1}{b_1}}{e}^{-i\frac{x_2{u}_2}{b_2}}d\mathbf{x}+\left({\int }_{{\mathbb{R}}^2}{h}_2\left(\mathbf{x}\right)e^{-i\frac{x_1{u}_1}{b_1}}{e}^{-i\frac{x_2{u}_2}{b_2}}d\mathbf{x}\right)j\hfill \\ & =\mathcal{F}\left\{h_1\left(\mathbf{x}\right)\right\}(\frac{u_1}{b_1},\frac{u_2}{b_2})+\mathcal{F}\left\{h_2\left(\mathbf{x}\right)\right\}(\frac{u_1}{b_1},\frac{u_2}{b_2})j.\hfill \end{array}$$

(53)

From Equation (53), the $W\left(\mathbf{u}\right)$ can be implemented by two complex 2D FTs; therefore, the final implementation of RBWLCT also mainly requires two complex 2D FTs. The algorithm and the calculation process for the RBWLCT are summarized in Algorithm 1 and Figure 3, respectively.

Algorithm 1: The algorithm for the RBWLCT

Input: biquaternion signal $f\left(\mathbf{x}\right)$             Step (1): Calculate $h\left(\mathbf{x}\right)$ using Equations (45) and (52).             Step (2): Calculate $W\left(\mathbf{u}\right)$ from $h\left(\mathbf{u}\right)$ using Equations (47) and (53).             Step (3): Calculate $B_i\left(u_1\right)$ and $B_k\left(u_2\right)$ using Equation (46). Output: Calculate ${\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{f\left(\mathbf{x}\right)\right\}(\mathbf{u},\mathbf{t})=\frac{W(u_1,u_2)(1+j)+W(u_1,-u_2)(1-j)}{2}{B}_i\left(u_1\right)B_k\left(u_2\right)$.

Next, we provide the computational complexity analysis of RBWLCT. From Equations (51) and (53), it can be observed that the main computational burden for RBWLCT lies in calculating $W\left(\mathbf{u}\right)$. To implement $W\left(\mathbf{u}\right)$, two complex 2D FTs are required to perform the integration operations mentioned in Equation (47). By employing a discrete algorithm, these complex 2D FTs can be implemented using 2D discrete Fourier transforms (DFTs) to process a 2D discrete signal with dimensions of $M\times N$. The computation of its 2D DFTs necessitates $MN{log}_2\left(MN\right)$ real-number multiplications [42,43]. Therefore, implementing RBWLCT also requires $2MN{log}_2\left(MN\right)$ real-number multiplications. Utilizing Equations (51) and (53), we can determine that the computational complexity of RBWLCT for RB signals $f\left(\mathbf{x}\right)$ is $O\left(2MN{log}_2\left(MN\right)\right)$. Since different types of signals result in different complexities, the corresponding computational complexity of RBWLCT for scalar signal $f\left(\mathbf{x}\right)$ is reduced to $O\left(\frac{1}{2}MN{log}_2\left(MN\right)\right)$, for complex signal it is reduced to $O\left(MN{log}_2\left(MN\right)\right)$, and for vector signal it is reduced to $O\left(\frac{3}{2}MN{log}_2\left(MN\right)\right)$.

On comparing the proposed RBWLCT with the QWLCT, it is pertinent to mention that despite capturing localized time and frequency information comprehensively, the QWLCT’s computational complexity is relatively high due to quaternion arithmetic, making it more intensive than conventional windowed linear canonical transform. In contrast, the proposed RBWLCT reduces computational complexity while preserving essential properties of the QWLCT. Leveraging redundancy in quaternion signal representations, the RBWLCT achieves efficiency by exploiting the symmetries and properties of quaternion exponential functions. Although sacrificing some accuracy compared to the QWLCT, the RBWLCT offers a balanced compromise between accuracy and computational efficiency, making it a valuable alternative in signal processing applications.

4.2. Simulations

In this section, we employ the definition of the RBWLCT for color image processing by encoding the three channel components of the RGB image as imaginary parts of the pure reduced biquaternion. Thus, the pixels at image coordinates $(s,t)$ can be represented as follows:

$$f(s,t)=i{f}_r(s,t)+j{f}_g(s,t)+k{f}_b(s,t),$$

(54)

where $f_r(s,t)$, $f_g(s,t)$, $f_b(s,t)$ represent the red, green, and blue components of the pixels, respectively.

In the experiment, the parameters were set as follows: $A_1=(\sqrt{129},68\pi ,2\pi ,\sqrt{129})$, and $A_2=(10,32\pi ,4\pi ,12.9)$. In Figure 4a, the original color image is shown, which has dimensions of 512 × 512 pixels. The module of the RBWLCT spectrum is presented in Figure 4b. Additionally, we have plotted three imaginary parts of the RBWLCT spectrum in Figure 5.

The image frequency serves as an indicator that quantifies the intensity of gray variations within an image, representing the gradient of gray levels across the spatial plane. However, due to the utilization of a global transform in RBLCT, which averages out frequency properties over time, it fails to capture local time-frequency characteristics. By incorporating a window function $\varphi \left(t\right)$ into the RBWLCT, it acquires localized attributes and becomes both a temporal and spectral function. Thus, for a given time t, ${\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{f\left(\mathbf{x}\right)\right\}(\mathbf{u},\mathbf{t})$ can be interpreted as the spectrum at that specific moment—namely, the local spectrum. Figure 6 and Figure 7 are provided to further explain the different performances in Figure 4a. Specifically, Figure 6 offers a characterization of rectangular window functions in the time domain with a different window step size. Figure 7 shows the spectrum of the different parts of the color image in RBWLCT domains to show the influence of windowed function.

4.3. Applications

For the demonstration of the RBWLCT in (14), we shall present an illustrative example with simulation.

Consider the following 2D biquaternion Gaussian function:

$$\begin{array}{c}\hfill f\left(\mathbf{x}\right)=\alpha e^{-\left(\beta x_1^2+\gamma x_2^2\right)},\beta ,\gamma \in \mathbb{C},\alpha \in L^2({\mathbb{R}}^2,\tilde{\mathbb{H}}).\end{array}$$

(55)

Then, the RBWLCT in (14) of $f\left(\mathbf{x}\right)$ with respect to the uni-modular matrices $A_l=\left(\begin{array}{cc}{a}_l& b_l\\ c_l& d_l\end{array}\right),l=1,2,$ and the following rectangular window function:

$$\begin{array}{c}\hfill \varphi \left(\mathbf{x}\right)=\left\{\begin{array}{cc}1,& \mathrm{if}|x_1|<a,|x_2|<a,a>0,\\ \\ 0,& \mathrm{elsewhere},\end{array}\right.\end{array}$$

(56)

can be computed as follows:

$$\begin{array}{cc}\hfill & {\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{f\right\}(\mathbf{u},\mathbf{t})\hfill \\ \hfill & ={\displaystyle \frac{1}{2\pi \sqrt{b_1{b}_2}}}{\int }_{{\mathbb{R}}^2}\alpha e^{-(\beta x_1^2+\gamma x_2^2)}\overline{\varphi (\mathbf{x}-\mathbf{t})}\hfill \\ \hfill & ·e^{\frac{i}{2b_1}\left[a_1{x}_1^2-2x_1{u}_1+d_1{u}_1^2-\frac{\pi b_1}{2}\right]}{e}^{\frac{k}{2b_2}\left[a_2{x}_2^2-2x_2{u}_2+d_2{u}_2^2-\frac{\pi b_2}{2}\right]}d\mathbf{x}\hfill \\ \hfill & ={\displaystyle \frac{\alpha }{2\pi \sqrt{b_1{b}_2}}}{\int }_{t_1-a}^{t_1+a}{\int }_{t_2-a}^{t_2+a}{e}^{-(\beta x_1^2+\gamma x_2^2)}\hfill \\ \hfill & ·e^{\frac{i}{2b_1}\left[a_1{x}_1^2-2x_1{u}_1+d_1{u}_1^2-\frac{\pi b_1}{2}\right]}{e}^{\frac{k}{2b_2}\left[a_2{x}_2^2-2x_2{u}_2+d_2{u}_2^2-\frac{\pi b_2}{2}\right]}d\mathbf{x}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{\alpha }{2\pi \sqrt{b_1{b}_2}}}{e}^{\frac{i}{2b_1}\left[d_1{u}_1^2-\frac{\pi b_1}{2}\right]}{\int }_{t_1-a}^{t_1+a}{e}^{\frac{i}{2b_1}\left[x_1^2(a_1-2i{b}_1\beta )-2x_1{u}_1\right]}d{x}_1\hfill \\ \hfill & ·{\int }_{t_2-a}^{t_2+a}{e}^{\frac{k}{2b_2}\left[x_2^2(a_2-2k{b}_2\gamma )-2x_2{u}_2\right]}d{x}_2{e}^{\frac{k}{2b_2}\left[d_2{u}_2^2-\frac{\pi b_2}{2}\right]}.\hfill \end{array}$$

For simplicity, we choose $\beta =i{a}_1/\left(2b_1\right)$ and $\gamma =-k{a}_2/\left(2b_2\right),$ to obtain the following:

$$\begin{array}{cc}\hfill & {\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{f\right\}(\mathbf{u},\mathbf{t})\hfill \\ \hfill & ={\displaystyle \frac{\alpha }{2\pi \sqrt{b_1{b}_2}}}{e}^{\frac{i}{2b_1}\left[d_1{u}_1^2-\frac{\pi b_1}{2}\right]}{\int }_{t_1-a}^{t_1+a}{e}^{\frac{-i{x}_1{u}_1}{b_1}}d{x}_1{\int }_{t_2-a}^{t_2+a}{e}^{\frac{-k{x}_2{u}_2}{b_2}}d{x}_2\hfill \\ \hfill & ·e^{\frac{k}{2b_2}\left[d_2{u}_2^2-\frac{\pi b_2}{2}\right]}\hfill \\ \hfill & ={\displaystyle \frac{\alpha }{2\pi \sqrt{b_1{b}_2}}}{e}^{\frac{i}{2b_1}\left[d_1{u}_1^2-\frac{\pi b_1}{2}\right]}{\displaystyle \frac{(-b_1)}{i{u}_1}}\left[e^{\frac{-i{u}_1(t_1+a)}{b_1}}-e^{\frac{-i{u}_1(t_1-a)}{b_1}}\right]\hfill \\ \hfill & ·{\displaystyle \frac{(-b_2)}{k{u}_2}}\left[e^{\frac{-k{u}_2(t_2+a)}{b_1}}-e^{\frac{-k{u}_2(t_2-a)}{b_2}}\right]e^{\frac{k}{2b_2}\left[d_2{u}_2^2-\frac{\pi b_2}{2}\right]}\hfill \\ \hfill & ={\displaystyle \frac{\alpha \sqrt{b_1{b}_2}}{2\pi u_1{u}_2}}{e}^{\frac{i{d}_1{u}_1^2}{2b_1}}{\displaystyle \frac{1}{i\sqrt{i}}}\left[e^{\frac{-i{u}_1(t_1+a)}{b_1}}-e^{\frac{-i{u}_1(t_1-a)}{b_1}}\right]\hfill \\ \hfill & ·{\displaystyle \frac{1}{k\sqrt{k}}}\left[e^{\frac{-k{u}_2(t_2+a)}{b_1}}-e^{\frac{-k{u}_2(t_2-a)}{b_2}}\right]e^{\frac{k{d}_2{u}_2^2}{2b_2}}\hfill \\ \hfill & ={\displaystyle \frac{\alpha \sqrt{b_1{b}_2}}{2\pi u_1{u}_2}}{e}^{\frac{i{d}_1{u}_1^2}{2b_1}}{e}^{\frac{-3\pi i}{4}}\left[e^{\frac{-i{u}_1(t_1+a)}{b_1}}-e^{\frac{-i{u}_1(t_1-a)}{b_1}}\right]\hfill \\ \hfill & ·\left[e^{\frac{-k{u}_2(t_2+a)}{b_1}}-e^{\frac{-k{u}_2(t_2-a)}{b_2}}\right]e^{\frac{k{d}_2{u}_2^2}{2b_2}}{e}^{\frac{-3\pi k}{4}}.\hfill \end{array}$$

(57)

In order to acquire the lucid representation of the transformed signal, we make following choices:

$$A_1=A_2=\left(\begin{array}{cc}1& 1\\ -1& 0\end{array}\right),\alpha =1+2i+3j+4k,a=1,t_1=t_2=1.$$

Therefore, we obtain the following:

$$\begin{array}{cc}\hfill {\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{f\right\}(\mathbf{u},\mathbf{t})& ={\displaystyle \frac{(1+2i+3j+4k)}{2\pi u_1{u}_2}}{e}^{\frac{-3\pi i}{4}}{e}^{-2i{u}_1}{e}^{-2k{u}_2}{e}^{\frac{-3\pi k}{4}}\hfill \\ \hfill & ={\displaystyle \frac{(1+2i+3j+4k)}{2\pi u_1{u}_2}}{e}^{-i(2u_1+3\pi /4)}{e}^{-k(2u_2+3\pi /4)}.\hfill \end{array}$$

(58)

Setting ${\theta }_1=\left(2u_1+3\pi /4\right),$ and ${\theta }_2=\left(2u_2+3\pi /4\right)$, we have the following:

$$\begin{array}{cc}\hfill & {\mathcal{G}}_{\varphi }^{A_1,A_2}\left\{f\right\}(\mathbf{u},\mathbf{t})\hfill \\ \hfill & ={\displaystyle \frac{(1+2i+3j+4k)}{2\pi u_1{u}_2}}{e}^{-i{\theta }_1}{e}^{-k{\theta }_2}\hfill \\ \hfill & ={\displaystyle \frac{(1+2i+3j+4k)}{2\pi u_1{u}_2}}\left(cos{\theta }_1-isin{\theta }_1\right)\left(cos{\theta }_2-ksin{\theta }_2\right)\hfill \\ \hfill & ={\displaystyle \frac{(1+2i+3j+4k)}{2\pi u_1{u}_2}}\left[cos{\theta }_{1}cos{\theta }_2-kcos{\theta }_{1}sin{\theta }_2-isin{\theta }_{1}cos{\theta }_2+i·ksin{\theta }_{1}sin{\theta }_2\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{2\pi u_1{u}_2}}\left[cos{\theta }_{1}cos{\theta }_2-kcos{\theta }_{1}sin{\theta }_2-isin{\theta }_{1}cos{\theta }_2+i·ksin{\theta }_{1}sin{\theta }_2\right]\hfill \\ \hfill & +{\displaystyle \frac{2}{2\pi u_1{u}_2}}\left[icos{\theta }_{1}cos{\theta }_2-i·kcos{\theta }_{1}sin{\theta }_2-i·isin{\theta }_{1}cos{\theta }_2+i·i·ksin{\theta }_{1}sin{\theta }_2\right]\hfill \\ \hfill & +{\displaystyle \frac{3}{2\pi u_1{u}_2}}\left[jcos{\theta }_{1}cos{\theta }_2-j·kcos{\theta }_{1}sin{\theta }_2-j·isin{\theta }_{1}cos{\theta }_2+j·i·ksin{\theta }_{1}sin{\theta }_2\right]\hfill \\ \hfill & +{\displaystyle \frac{4}{2\pi u_1{u}_2}}\left[kcos{\theta }_{1}cos{\theta }_2-k·kcos{\theta }_{1}sin{\theta }_2-k·isin{\theta }_{1}cos{\theta }_2+k·i·ksin{\theta }_{1}sin{\theta }_2\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{2\pi u_1{u}_2}}\left[cos{\theta }_{1}cos{\theta }_2-kcos{\theta }_{1}sin{\theta }_2-isin{\theta }_{1}cos{\theta }_2+jsin{\theta }_{1}sin{\theta }_2\right]\hfill \\ \hfill & +{\displaystyle \frac{2}{2\pi u_1{u}_2}}\left[icos{\theta }_{1}cos{\theta }_2-jcos{\theta }_{1}sin{\theta }_2-sin{\theta }_{1}cos{\theta }_2+ksin{\theta }_{1}sin{\theta }_2\right]\hfill \\ \hfill & +{\displaystyle \frac{3}{2\pi u_1{u}_2}}\left[jcos{\theta }_{1}cos{\theta }_2+icos{\theta }_{1}sin{\theta }_2-ksin{\theta }_{1}cos{\theta }_2-sin{\theta }_{1}sin{\theta }_2\right]\hfill \\ \hfill & +{\displaystyle \frac{4}{2\pi u_1{u}_2}}\left[kcos{\theta }_{1}cos{\theta }_2+cos{\theta }_{1}sin{\theta }_2-jsin{\theta }_{1}cos{\theta }_2-isin{\theta }_{1}sin{\theta }_2\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{2\pi u_1{u}_2}}\left[cos{\theta }_{1}cos{\theta }_2-2sin{\theta }_{1}cos{\theta }_2-3sin{\theta }_{1}sin{\theta }_2+4cos{\theta }_{1}sin{\theta }_2\right]\hfill \\ \hfill & +{\displaystyle \frac{i}{2\pi u_1{u}_2}}\left[-sin{\theta }_{1}cos{\theta }_2+2cos{\theta }_{1}cos{\theta }_2+3cos{\theta }_{1}sin{\theta }_2-4sin{\theta }_{1}sin{\theta }_2\right]\hfill \\ \hfill & +{\displaystyle \frac{j}{2\pi u_1{u}_2}}\left[sin{\theta }_{1}sin{\theta }_2-2cos{\theta }_{1}sin{\theta }_2+3cos{\theta }_{1}cos{\theta }_2-4sin{\theta }_{1}cos{\theta }_2\right]\hfill \\ \hfill & +{\displaystyle \frac{k}{2\pi u_1{u}_2}}\left[-cos{\theta }_{1}sin{\theta }_2+2sin{\theta }_{1}sin{\theta }_2-3sin{\theta }_{1}cos{\theta }_2+4cos{\theta }_{1}cos{\theta }_2\right].\hfill \end{array}$$

The graphical representation of the given biquaternion signal $f\left(\mathbf{x}\right)=\alpha exp\left\{-(\beta x_1^2+\gamma x_2^2)\right\},$ when $\beta =-i/2,\gamma =-j/2\mathrm{and}\alpha =1+2i+3j+4k$ is plotted in Figure 8, whereas its RBWLCT (14) of a signal $f\left(\mathbf{x}\right)$ with respect to the rectangular window $g\left(\mathbf{x}\right)$ and the real part of the transformed signal are depicted in Figure 9, for $a=1,t_1=t_2=1,\alpha =1+2i+3j+4k$ and $A_1=A_2=\left(\begin{array}{cc}1& 1\\ -1& 0\end{array}\right).$ From Figure 9, we see that the proposed transform reveals the local RBWLCT-frequency contents with a higher degree of freedom and enjoys high concentrations.

Moreover, the original image (Figure 4a) can be depicted in its red, green, and blue components as follows (Figure 10):

5. Conclusions

This paper defines RBWLCT, which can be viewed as a generalization of classical LCT and gives its main properties. A typical convolution operator of RBWLCT and an associated correlation operator for RBWLCT are proposed. Furthermore, based on the proposed operators, the corresponding generalized convolution, correlation, and product theorems are derived; then, the RBWLCT fast algorithm was studied. The computational complexity of RBWLCT was calculated by the computational complexity of 2D FT. Finally, the application of RBWLCT to the color images is given. First, to capture the local time-frequency properties, we set different window steps of the rectangular window function and obtained different spectra of the color images. Second, we give a specific 2D biquaternion Gaussian function for RBWLCT as well as its simulations.