Research on Two-Dimensional Linear Canonical Transformation Series and Its Applications

Abstract

In light of the computational efficiency bottleneck and inadequate regional feature representation in traditional global data approximation methods, this paper introduces the concept of non-uniform partition to transform global continuous approximation into multi-region piecewise approximation. Additionally, we propose an image representation algorithm based on linear canonical transformation and non-uniform partitioning, which enables the regional representation of sub-signal features while reducing computational complexity. The algorithm first demonstrates that the two-dimensional linear canonical transformation series has a least squares solution within each region. Then, it adopts the maximum likelihood estimation method and the scale transformation characteristics to achieve conversion between the nonlinear and linear expressions of the two-dimensional linear canonical transformation series. It then uses the least squares method and the recursive method to convert the image information into mathematical expressions, realize image vectorization, and solve the approximation coefficients in each region more quickly. The proposed algorithm better represents complex image texture areas while reducing image quality loss, effectively retains high-frequency details, and improves the quality of reconstructed images.

1. Introduction

Theoretical tools such as transformations play a crucial role in engineering applications, particularly in signal processing. Key transformations include Fourier transform, fractional Fourier transform, and linear canonical transform [1,2]. These tools are widely utilized in signal processing and are essential in various fields, including optics, image processing, and pattern recognition.However, with the continuous improvement and development of technology and theory, the analysis and processing of two-dimensional signals have gradually increased. Simple one-dimensional signal processing tools are not fully applicable in some cases. For instance, in image processing, the two-dimensional signal must be decomposed into multiple independent, one-dimensional components for processing, resulting in spatial correlation destruction and feature information loss. In addition, the freedom of the one-dimensional linear canonical transformation parameter is insufficient. One-dimensional linear canonical transformation contains only four free parameters, which makes it difficult to characterize the nonlinear modulation characteristics of complex multidimensional signals. Consequently, more and more generalized forms of these two-dimensional transformations have emerged [3,4]. The two-dimensional linear canonical transform (2D LCT) is a generalized linear canonical transform with more flexible six degrees of freedom, and it has become one of the core tools of modern signal processing.

A theoretical framework of two-dimensional linear regular transformation has been developed in recent years. High-order transformation forms have emerged by integrating two-dimensional linear regular transformations with quaternions, octonions, and hyperbolic geometric structures, and these have been applied to image processing, multidimensional signal analysis, and other fields, demonstrating powerful effects and potential. The two-dimensional linear canonical transform has garnered significant scholarly attention as a more generalized form of the two-dimensional Fourier transform and the two-dimensional fractional Fourier transform. In terms of the theoretical extension and mathematical structure of LCT, in LCT under a high-order algebraic structure, the quaternion linear canonical transform (QLCT) [5] and the octonion linear canonical transform (OLCT) expand the application of LCT in the processing of multidimensional signals (such as color images and three-dimensional signals) by introducing hypercomplex structures. For example, Jiang [6] derived the differential properties and convolution theorem of the left-hand octonion linear canonical transform (LOCLCT) and used the properties and the corresponding convolution theorem to discuss and analyze three-dimensional linear time-invariant systems, highlighting its enhanced flexibility and multi-scale analysis capabilities. Dar et al. [7] proposed Wigner distribution (WDOL) in the OLCT domain, realized the efficient characterization of a high-dimensional signal phase space by combining octonion algebra, and formulated the generalized form of the Heisenberg uncertainty principle. The QLCT performs well in quaternion signal processing. Prasad et al. [8] studied the characterization range, reproducing kernel, one-to-one mapping, Donoho–Stark inequality, and Pitt inequality based on quaternion, windowed, linear canonical transformation. Some useful uncertainty principles were discussed, such as the Heisenberg, Riebre, and local uncertainty principles. Hu et al. [9] implemented the QLCT of the convolution of two quaternion functions and derived related operators and theorems, which were used in the design of multiplication filters through the product of their QLCTs, or the sum of the products of their QLCTs, and solved the problem of the Fredholm integral equation with special kernels. Kit Ian KOU [10] studied the principles of hypercomplex signals in the linear canonical transform domain and proposed the Plancherel theorem of quaternion Hilbert transform related to linear canonical transform. To detect single-component and two-component linear frequency modulation signals, Bhat et al. [11] introduced scaled Wigner distribution on a biased linear canonical domain (SWDOLC), which expanded the application range of Wigner distribution. Introducing the offset linear canonical transform (OLCT) and the scaling factor k, it has greater flexibility on the frequency axis and can effectively reduce the cross terms. This provides a new tool for the detection of linear frequency modulation signals and improves the accuracy and reliability of signal detection. For graph signal processing and data compression, Chen et al. [12] proposed the graph linear canonical transform (GLCT), which overcomes the limitation of capturing local information through parameter decomposition of fractional Fourier transform, scale transform, and chirp modulation and verifies the effectiveness of its filter design in image classification tasks. Ravi [13] implemented nonlinear dual image encryption using inseparable, two-dimensional linear canonical transformation. Li [14] proposed a graph linear canonical transform (GLCT) based on linear frequency modulation multiplication–linear frequency modulation convolution–linear frequency modulation multiplication decomposition (CM-CC-CM-GLCT), which provides a new tool for the field of graph signal processing. Through CM-CC-CM-GLCT decomposition, it achieves lower computational complexity, similar additivity, and better reversibility. The effectiveness and superiority of CM-CC-CM-GLCT in data compression were verified through theoretical analysis and simulation. In order to improve the performance of graph signal processing, Zhang et al. [15] proposed an uncertainty principle based on GLCT and used this principle to establish the conditions for recovering GLCT band-limited signals from sample subsets. Subsequently, they constructed the sampling theory of GLCT and explored the relationship between the uncertainty principle and sampling. Li [16] introduced a two-dimensional quaternion linear canonical series to describe color images and analyzed the ability of two-dimensional quaternion linear canonical sequences in image representation and reconstruction. Furthermore, this method studied the relationship between the two-dimensional quaternion linear canonical transformation series and the two-dimensional linear canonical transformation series in detail. Pankaj Rakheja et al. [17] proposed a dual image encryption mechanism that combines a three-dimensional Lorentz chaotic system and QR decomposition in a two-dimensional inseparable linear canonical transform domain. The simulation results show that the scheme has strong robustness against occlusion and special attacks. Wang et al. [18] proposed an innovative zero-watermarking method specifically designed for stereo images, utilizing an accurate ternary polar linear canonical transform. It introduced a precise polar linear canonical transform to solve the numerical integration issues associated with the polar linear canonical transform. Experiments show that this method offers improved performance and greater robustness. Liu et al. [19] proposed the discrete quaternion offset linear canonical transform using quaternion algebra and presented a new image encryption scheme that combines double random phase encoding with the generalized Arnold transform.

From the existing literature, the two-dimensional linear canonical transform and its extended forms integrate high-order algebra, a geometric structure, and modern signal processing requirements, which continuously drive technological innovation in time–frequency analysis, image processing, and graph signal processing and have important academic value and application prospects. However, the development of two-dimensional linear canonical transformation theory is facing two dilemmas. One aspect is the obstacle of computational complexity. Existing linear canonical transform algorithms reconstruct image pixels by global approximation, which requires an iterative solution, and the data and computational time increase superlinearly. On the other hand, fundamental frameworks, such as the two-dimensional linear canonical transformation series theory, have not been systematically established, and there is a gap in the theoretical system. To solve the double dilemma of two-dimensional linear canonical transform, fill in the structural gap in multidimensional transformation theory, construct a theoretical system of the two-dimensional LCT series, develop a new fast algorithm framework, and solve the mutually exclusive problem between computational complexity and accuracy, this paper introduces the concept of non-uniform partition to convert global data approximation into approximation operations under different partitions. First, we conduct mathematical derivation and logical reasoning on the two-dimensional linear canonical transformation series, along with proof by contradiction to demonstrate that it has a unique least squares solution within the signal sub-region. To realize the conversion between the nonlinear and linear expressions of the two-dimensional linear canonical transformation series, we utilize mathematical methods, such as maximum likelihood estimation and the properties of scale transformation. Furthermore, a fast linear canonical transformation algorithm is constructed to improve the computational efficiency. Second, for complex texture areas and edge features of images, the non-uniform partition method and two-dimensional linear canonical transformation series are used to represent the image, and the least squares method is employed to convert the image information into a mathematical expression, achieving image vectorization. Finally, an adaptive two-dimensional linear canonical transform partition algorithm is constructed to represent the complex texture areas of the image more effectively, retain the texture details, and improve the quality of the reconstructed image. The proposed algorithm can enhance the quality of the non-uniform partition while reducing the program execution time and image quality loss.

In summary, the contributions of this paper are highlighted as follows:

• To solve the computational efficiency bottleneck and insufficient regional feature representation in traditional global data approximation methods, this paper proposes an adaptive non-uniform partition algorithm based on the two-dimensional linear canonical transformation series. Its mathematical essence can be extended to the approximation problem of linear canonical transformation series on partition sub-regions. Under this framework, the existence and uniqueness of the least squares solution for the linear canonical transformation series in each sub-region is proved, which is not only the theoretical basis for ensuring the mathematical completeness of the image representation model but also the core scientific proposition for achieving stable convergence of the algorithm.
• This paper introduces the concept of non-uniform partition to convert global data approximation into an approximation operation under different partitions. In order to speed up its operation, different transformation coefficients are employed to represent the sub-signals in different regions. Concurrently, the least-squares and maximum likelihood estimation methods are applied to swiftly determine the approximation coefficients for each area. The proposed algorithm aims to improve the quality of reconstructed images while minimizing program execution time and image quality loss.

The rest of this paper is organized as follows: Section 2 reviews some preliminaries of the 2D linear canonical transform and adaptive non-uniform image partition. Section 3 formulates an adaptive non-uniform partition algorithm based on 2D-LCT and presents some examples to illustrate how to partition the image flexibly using this algorithm. Section 4 demonstrates the effectiveness of the proposed 2D-LCT-based partition algorithm for image representation through simulation experiments. Finally, conclusions are presented in Section 5.

2. Preliminaries

In this section, we briefly review some of the knowledge acquired in the remaining parts of this paper.

2.1. The Mathematics Form for 2D LCT

First, we introduce the Theorem of the 2-D linear canonical transform. The two-dimensional linear canonical transform with parameter matrices has eight parameters. Since the two-parameter matrices are orthogonal, the two-dimensional linear canonical transform has six free parameters. The two-dimensional Fourier transform, the two-dimensional fractional Fourier transform, and the two-dimensional Fresnel transform are all special forms of the two-dimensional linear canonical transform. When $A=B=\left(\begin{array}{cc}cos\theta & sin\theta \\ -sin\theta & cos\theta \end{array}\right)$, the two-dimensional fractional Fourier transform can be derived from the two-dimensional linear canonical transform. And when $\theta =pi/2$, the two-dimensional Fourier transform can be reduced from the two-dimensional fractional Fourier transform.

Theorem 1.

Assume that the parameter matrix $A=\left(\begin{array}{cc}{\mathrm{a}}_1& b_1\\ c_1& d_1\end{array}\right)$, $a_1,b_1,c_1,d_1\in R,b_1\ne 0,a_1{d}_1-b_1{c}_1=1$. $B=\left(\begin{array}{cc}{\mathrm{a}}_2& b_2\\ c_2& d_2\end{array}\right)$, $a_2,b_2,c_2,d_2\in R,b_2\ne 0,a_2{d}_2-b_2{c}_2=1$. Then, the linear canonical transform of a image or function $f\left(t\right)$ with parameter matrix A is defined as

$$\begin{array}{l}{F}^{A,B}\left[f(x,y)\right](u,v)=F_f^{A,B}(u,v)\\ =\left\{\begin{array}{c}{\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }f(x,y)K_{A,B}(u,v,x,y)dxdy,b_1{b}_2\ne 0\hfill \\ \\ \sqrt{d_1{d}_2}{e}^{i\left(c_1{d}_1\right)u^2+c_2{d}_2)v^2/2}f(d_{1}u,d_{2}v),b_1^2+b_2^2=0\hfill \end{array}\right.,\end{array}$$

(1)

where

$$\begin{array}{c}\hfill K_{A,B}(u,v,x,y)=K_A(u,x)K_B(v,y)\\ \hfill K_A(u,x)=C_{A}exp(i{\displaystyle \frac{a_1{x}^2}{2b_1}}-i{\displaystyle \frac{ux}{b_1}}+i{\displaystyle \frac{d_1{u}^2}{2b_1}}),C_A={\displaystyle \frac{1}{\sqrt{i2\pi b_1}}}.\\ \hfill K_B(v,y)=C_{B}exp(i{\displaystyle \frac{a_2{y}^2}{2b_2}}-i{\displaystyle \frac{vy}{b_2}}+i{\displaystyle \frac{d_2{v}^2}{2b_2}}),C_B={\displaystyle \frac{1}{\sqrt{i2\pi b_2}}}.\end{array}$$

When the parameter matrix $A=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$ and $B=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$, the above equation is reduced to a two-dimensional Fourier transform as follows:

$$F(u,v)={\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }f(x,y)exp(-i2\pi (ux+vy))dxdy.$$

(2)

When the parameter matrix $A=\left(\begin{array}{cc}cos\alpha & sin\alpha \\ -sin\alpha & cos\alpha \end{array}\right)$ and $B=\left(\begin{array}{cc}cos\beta & sin\beta \\ -sin\beta & cos\beta \end{array}\right)$, Equation (1) is transformed into a two-dimensional fractional Fourier transform as follows:

$$\begin{array}{cc}\hfill & F^{A,B}\left[f(x,y)\right](u,v)=F_f^{A,B}(u,v)={\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }f(x,y)K_{A,B}(u,v,x,y)dxdy,\hfill \end{array}$$

(3)

with

$$\begin{array}{c}\hfill K_{A,B}(u,v,x,y)=\sqrt{(1-icot\alpha )(1-icot\beta )}/2\pi \\ \hfill \ast exp[i(x^2+u^2)/2tan\alpha -ixu/sin\alpha ]\\ \hfill \ast exp[i(y^2+v^2)/2tan\beta -iyv/sin\beta ].\end{array}$$

where $\alpha ,\beta$ are expressed as the rotation angles of the two-dimensional fractional Fourier transform.

2.2. The Property for 2D LCT

The application of two-dimensional linear canonical transform in image processing requires corresponding research on some vital properties of two-dimensional linear canonical transform, such as time-shift modulation, scale feature, superposition, reversibility, etc. In this subsection, we derive and prove the scale feature and time–frequency shift properties of the two-dimensional linear canonical transform based on the inferential proofs related to the one-dimensional linear canonical transform.

Property 1

(Time-shift modulation). Let $g(x,y)=f(x-{\tau }_1,y-{\tau }_2),{\tau }_1,{\tau }_2\in R$. Then, $F_g^{A,B}(U,V)=e^{(u{c}_1{\tau }_1+v{c}_2{\tau }_2-(i/2)(a_1{c}_1{\tau }_1^2+a_2{c}_2{\tau }_2^2)}{F}^{A,B}(u-a_1{\tau }_1,v-a_2{\tau }_2)$.

Proof of Property 1.

According to (1),

$$\begin{array}{cc}\hfill F_g^{A,B}(U,V)& ={\displaystyle \frac{1}{\sqrt{i2\pi b_1}}}{\displaystyle \frac{1}{\sqrt{i2\pi b_2}}}{\int }_{-\infty }^{+\infty }{\int }_{\infty }^{+\infty }f(x-{\tau }_1,y-{\tau }_2)\hfill \\ & \ast exp(i{\displaystyle \frac{a_1{(x-{\tau }_1)}^2}{2b_1}}-i{\displaystyle \frac{u(x-{\tau }_1)}{b_1}}+i{\displaystyle \frac{d_1{u}^2}{2b_1}})\hfill \\ & \ast exp(i{\displaystyle \frac{a_2{(y-{\tau }_2)}^2}{2b_2}}-i{\displaystyle \frac{v(y-{\tau }_2)}{b_2}}+i{\displaystyle \frac{d_2{v}^2}{2b_2}})d(x-{\tau }_1)d(y-{\tau }_2)\hfill \\ \hfill & =exp(u{c}_1{\tau }_1-i{\displaystyle \frac{a_1{c}_1{\tau }_1^2}{2}})\ast exp(v{c}_2{\tau }_2-i{\displaystyle \frac{a_2{c}_2{\tau }_2^2}{2}})\hfill \\ & {\displaystyle \frac{1}{\sqrt{i2\pi b_1}}}{\displaystyle \frac{1}{\sqrt{i2\pi b_2}}}{\int }_{-\infty }^{+\infty }{\int }_{\infty }^{+\infty }f(x-{\tau }_1,y-{\tau }_2)\hfill \\ & \ast exp(i{\displaystyle \frac{a_1{x}^2}{2b_1}}-i{\displaystyle \frac{(u-a_1{\tau }_1)x}{b_1}}+i{\displaystyle \frac{d_1{(u-a_1{\tau }_1)}^2}{2b_1}})\hfill \\ & \ast exp(i{\displaystyle \frac{a_2{y}^2}{2b_2}}-i{\displaystyle \frac{(v-a_2{\tau }_2)y}{b_2}}+i{\displaystyle \frac{d_2{(v-a_2{\tau }_2)}^2}{2b_2}})d(u-a_1{\tau }_1)d(v-a_2{\tau }_2)\hfill \\ \hfill & =exp(u{c}_1{\tau }_1+v{c}_2{\tau }_2-(i/2)(a_1{c}_1{\tau }_1^2+a_2{c}_2{\tau }_2^2)F^{A,B}(u-a_1{\tau }_1,v-a_2{\tau }_2).\hfill \end{array}$$

□

Property 2

(Time–frequency shift). Let $g(x,y)=e^{i(x{u}_0+y{v}_0)}f(x-{\tau }_1,y-{\tau }_1),{\tau }_1,{\tau }_2,u_0,v_0\in R$, then $F_g^{A,B}(u,v)=e^{i(c_1{\tau }_1+d_1{u}_0)u+i(c_2{\tau }_2+d_2{u}_0)v-i(b_1{c}_1{\tau }_1{u}_0+b_2{c}_2{\tau }_2{u}_0)}{e}^{-(i/2)(a_1{c}_1{\tau }_1^2+a_2{c}_2{\tau }_2^2+b_1{d}_1{u}_0^2+b_2{d}_2{v}_0^2)}{F}^{A,B}(u-u_0{b}_1-a_1{\tau }_1,v-v_0{b}_2-a_2{\tau }_2)$.

Property 2 can be obtained by the same method as above, and will not be proved here.

Property 3

(Scale feature). Let $g(x,y)=f(\alpha x,\beta y)$, then $F_g^{A,B}(U,V)(u,v)=\alpha \beta F^{A^{\prime }{B}^{\prime }}\left(f(x,y)\right)$$(u,v),\alpha ,\beta \in R$, where $A=(\alpha a_1,b_1/\alpha ,\alpha c_1,d_1/\alpha ),B=\left((\alpha a_2,b_2/\alpha ,\alpha c_2,d_2/\alpha )\right)$.

Proof of Property 3.

We substitute $g(x,y)=f(\alpha x,\beta y)$ into the Theorem of the two-dimensional linear canonical transform and obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill F_g^{A,B}(U,V)& ={\displaystyle \frac{1}{\sqrt{i2\pi b_1}}}{\displaystyle \frac{1}{\sqrt{i2\pi b_2}}}{\int }_{-\infty }^{+\infty }{\int }_{\infty }^{+\infty }f(\alpha x,\beta y)\hfill \\ & \ast exp(i{\displaystyle \frac{a_1{\left(\alpha x\right)}^2}{2b_1}}-i{\displaystyle \frac{u\left(\alpha x\right)}{b_1}}+i{\displaystyle \frac{d_1{u}^2}{2b_1}})\hfill \\ & \ast exp(i{\displaystyle \frac{a_2{\left(\beta y\right)}^2}{2b_2}}-i{\displaystyle \frac{v\left(\beta y\right)}{b_2}}+i{\displaystyle \frac{d_2{v}^2}{2b_2}})d\left(\alpha x\right)d\left(\beta y\right)\hfill \\ & =\alpha \beta {\displaystyle \frac{1}{\sqrt{i2\pi b_1/\alpha }}}{\displaystyle \frac{1}{\sqrt{i2\pi b_2/\beta }}}{\int }_{-\infty }^{+\infty }{\int }_{\infty }^{+\infty }f(x,y)\hfill \\ & \ast exp(i{\displaystyle \frac{\alpha a_1{x}^2}{2b_1/\alpha }}-i{\displaystyle \frac{ux}{b_1/\alpha }}+i{\displaystyle \frac{(d_1/\alpha )u^2}{2b_1/\alpha }})\hfill \\ & \ast exp(i{\displaystyle \frac{\beta a_2{y}^2}{2b_2/\beta }}-i{\displaystyle \frac{vy}{b_2/\beta }}+i{\displaystyle \frac{(d_2/\beta )v^2}{2b_2/\beta }})dxdy\hfill \\ & =\alpha \beta F^{A^{\prime }{B}^{\prime }}\left(f(x,y)\right)(u,v).\hfill \end{array}\end{array}$$

□

2.3. 2D LCT Series

The Fourier series is the basis of modern Fourier analysis and transformation. It is also one of the core concepts in modern signal processing theory. In this subsection, the concept of a one-dimensional linear canonical transformation series based on the characteristics of the Fourier series and the fractional Fourier sequence is put forward.

Theorem 2.

Assume that the parameter matrix $A=\left(\begin{array}{cc}\mathrm{a}& b\\ c& d\end{array}\right)$, $a,b,c,d\in R,b\ne 0$, $ad-bc=1$. If the signal $f\left(t\right),t\in [-T/2,T/2]$ is a finite-length signal, then the signal $f\left(t\right)$ can be expanded into the linear canonical transformation series below:

$$\begin{array}{c}\hfill f\left(t\right)=\sum _{n=-\infty }^{\infty }{C}_{A,n}\sqrt{{\displaystyle \frac{i}{T}}}exp(-{\displaystyle \frac{id}{2b}}{\left(n{\displaystyle \frac{2\pi b}{T}}\right)}^2+{\displaystyle \frac{it}{b}}\left(n{\displaystyle \frac{2\pi b}{T}}\right)-{\displaystyle \frac{ia}{2b}}{t}^2),\end{array}$$

(4)

where $C_{A,n}$ is the expansion coefficent of the linear canonical series

$$\begin{array}{c}\hfill C_{A,n}={\int }_{-T/2}^{T/2}f\left(t\right)\sqrt{{\displaystyle \frac{-i}{T}}}exp\left({\displaystyle \frac{id}{2b}}{\left(n{\displaystyle \frac{2\pi b}{T}}\right)}^2-{\displaystyle \frac{it}{b}}\left(n{\displaystyle \frac{2\pi b}{T}}\right)+{\displaystyle \frac{ia}{2b}}{t}^2\right)dt.\end{array}$$

This paper starts with the Fourier transform and the fractional Fourier transform, proposes the Theorem of two-dimensional linear canonical transform series in combination with the Theorem of one-dimensional linear canonical transform series, and the two-dimensional linear canonical transform is equally orthogonal to the one-dimensional linear canonical transform. Next, we introduce the concept of the 2-D linear canonical transform series.

Theorem 3.

Assume that the parameter matrix $A=\left(\begin{array}{cc}{\mathrm{a}}_1& b_1\\ c_1& d_1\end{array}\right)$, $a_1,b_1,c_1,d_1\in R,b_1\ne 0,a_1{d}_1-b_1{c}_1=1$. $B=\left(\begin{array}{cc}{\mathrm{a}}_2& b_2\\ c_2& d_2\end{array}\right)$, $a_2,b_2,c_2,d_2\in R,b_2\ne 0,a_2{d}_2-b_2{c}_2=1$. Then, the linear canonical transformation series is expressed as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill f(x,y)=& \sum _{m,n}{C}_{m,n}^{A,B}\sqrt{{\displaystyle \frac{-1}{T_1{T}_2}}}\hfill \\ \hfill & \ast exp\left({\displaystyle \frac{-i}{2b_1}}\left[d_1{\left(m{\displaystyle \frac{2\pi b_1}{T_1}}\right)}^2-2x\left(m{\displaystyle \frac{2\pi b_1}{T_1}}\right)+a_1{x}^2\right]\right)\hfill \\ \hfill & \ast exp\left({\displaystyle \frac{-i}{2b_2}}\left[d_2{\left(n{\displaystyle \frac{2\pi b_2}{T_2}}\right)}^2-2y\left(n{\displaystyle \frac{2\pi b_2}{T_2}}\right)+a_2{y}^2\right]\right),\hfill \end{array}\end{array}$$

(5)

where $C_{m,n}^{A,B}$ is the expansion coefficients of the linear canonical series

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_{m,n}^{A,B}=& {\int }_{-{\displaystyle \frac{T_1}{2}}}^{{\displaystyle \frac{T_1}{2}}}{\int }_{-{\displaystyle \frac{T_2}{2}}}^{{\displaystyle \frac{T_2}{2}}}f(x,y)\left(-\sqrt{{\displaystyle \frac{-1}{T_1{T}_2}}}\right)\hfill \\ \hfill & \ast exp\left({\displaystyle \frac{i}{2b_1}}\left[d_1{\left(m{\displaystyle \frac{2\pi b_1}{T_1}}\right)}^2-2x\left(m{\displaystyle \frac{2\pi b_1}{T_1}}\right)+a_1{x}^2\right]\right)\hfill \\ \hfill & \ast exp\left({\displaystyle \frac{i}{2b_2}}\left[d_2{\left(n{\displaystyle \frac{2\pi b_2}{T_2}}\right)}^2-2y\left(n{\displaystyle \frac{2\pi b_2}{T_2}}\right)+a_2{y}^2\right]\right)dxdy.\hfill \end{array}\end{array}$$

2.4. Adaptive Non-Uniform Partition Algorithm of Image

The adaptive non-uniform partition of the image is performed by least-squares fitting based on the image’s gray values. The non-uniform partition algorithm mainly includes the rectangular partition [20] and the triangular partition [21]. Both the non-uniform rectangular partition algorithm and the non-uniform triangular partition algorithm are classic methods. The adaptive triangular partition algorithm [22] is an improvement on the non-uniform triangular partition. This algorithm establishes a separation partition model and uses a fitting function to solve it. Additionally, it can reduce redundant calculations and preserve image quality by removing shared edges and vertices between adjacent triangles in the partition grid. However, compared with the rectangular partition, the rectangular partition has certain advantages in running time and the number of sub-regions.

For a given image, the process of the non-uniform partition is performed as follows: First, we set the sub-region, denoted as $G_m$, starting from the initial partition. $G_m$ contains a larger number of pixels of known data, labeled $Z_i$ (where $i\in \left[0,n-1\right]$), n is the number of pixels on $G_m$, p is the set of undetermined coefficients in bi-variate polynomial $f_m\left(P\right)$, and $Q_i$ is the set of determined coefficients after applying least squares approximation [23]. $f_m\left(Q_i\right)$ is the reconstructed pixel set in the current regions.

$$e=\sum _{i=0}^{n-1}{\left(f_m\left(Q_i\right)-z_i\right)}^2<\epsilon .$$

(6)

The partition process stops when (6) occurs; this means that the reconstruction error e is less than the set control error $\epsilon$. Otherwise, the $G_m$ is unqualified, and the $G_{\mathrm{mn}}\left(n=0,1,2,3\right)$ will be checked one by one in the same way. The qualified $f_{mn}\left(P\right)$ are recorded when $e<\epsilon$ until the whole process finishes. The nonuniform partition of images is one kind of image vectorization [24] that employs fractional polynomials to approximate the pixel values in subregions. Different partition accuracies have a direct impact on the quality of reconstruction. Figure 1 indicates the reconstruction effect of the image after nonuniform partition under different error accuracies.

Most non-uniform partition algorithms currently rely on bivariate polynomials to approximate the pixels in a region using least-squares fitting. However, the original non-uniform partition algorithm uses linear bivariate polynomials, which may not effectively approximate complex textures.

3. Method and Algorithm Analysis

In this section, we first prove that the two-dimensional linear canonical transform has a unique least squares solution in the selected region to find the best matching function of the data by minimizing the sum of squared errors. Next, we use the least squares method to find the functional relationship between the independent variable and the dependent variable, also aiming to minimize the sum of squared errors. Finally, a fast two-dimensional linear canonical transform algorithm is constructed to reconstruct the image.

3.1. Theory and Design

Based on the theory of two-dimensional linear canonical transform series, this paper investigates fast algorithms of two-dimensional linear canonical transform and image representation, reduces the computational complexity of image pixel approximation and reconstruction by linear canonical transform, and improves the adaptability of image representation in complex texture regions. As an extension of the two-dimensional Fourier transform, the two-dimensional linear canonical transform also shows many advantages in image processing due to its six free parameters. Inspired by the signal reconstruction method in [25], this paper uses two-dimensional LCT series as nonuniform decomposition polynomials to reconstruct the image.

Theorem 4.

For any finite function $f(x,y)$, its 2-D LCTS expansion with parameters $A=\left(\begin{array}{cc}{\mathrm{a}}_1& b_1\\ c_1& d_1\end{array}\right)$ and $B=\left(\begin{array}{cc}{\mathrm{a}}_2& b_2\\ c_2& d_2\end{array}\right)$ is expressed as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill f(x,y)=& \sum _{m,n}{C}_{m,n}^{A,B}\sqrt{{\displaystyle \frac{-1}{T_1{T}_2}}}\hfill \\ \hfill & \ast exp\left({\displaystyle \frac{-i}{2b_1}}\left[d_1{\left(m{\displaystyle \frac{2\pi b_1}{T_1}}\right)}^2-2x\left(m{\displaystyle \frac{2\pi b_1}{T_1}}\right)+a_1{x}^2\right]\right)\hfill \\ \hfill & \ast exp\left({\displaystyle \frac{-i}{2b_2}}\left[d_2{\left(n{\displaystyle \frac{2\pi b_2}{T_2}}\right)}^2-2y\left(n{\displaystyle \frac{2\pi b_2}{T_2}}\right)+a_2{y}^2\right]\right),\hfill \end{array}\end{array}$$

(7)

with the 2-D LCT expansion coefficients $C_{m,n}^{A,B}$ defined as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_{m,n}^{A,B}=& {\int }_{-{\displaystyle \frac{T_1}{2}}}^{{\displaystyle \frac{T_1}{2}}}{\int }_{-{\displaystyle \frac{T_2}{2}}}^{{\displaystyle \frac{T_2}{2}}}f(x,y)\left(-\sqrt{{\displaystyle \frac{-1}{T_1{T}_2}}}\right)\hfill \\ \hfill & \ast exp\left({\displaystyle \frac{i}{2b_1}}\left[d_1{\left(m{\displaystyle \frac{2\pi b_1}{T_1}}\right)}^2-2x\left(m{\displaystyle \frac{2\pi b_1}{T_1}}\right)+a_1{x}^2\right]\right)\hfill \\ \hfill & \ast exp\left({\displaystyle \frac{i}{2b_2}}\left[d_2{\left(n{\displaystyle \frac{2\pi b_2}{T_2}}\right)}^2-2y\left(n{\displaystyle \frac{2\pi b_2}{T_2}}\right)+a_2{y}^2\right]\right)dxdy.\hfill \end{array}\end{array}$$

where $(x,y)\in [-{\displaystyle \frac{T_1}{2}},{\displaystyle \frac{T_1}{2}}]\ast [-{\displaystyle \frac{T_2}{2}},{\displaystyle \frac{T_2}{2}}]$.

Proof of Theorem 4.

The orthogonality of the 2D-LCT series has been proven in the literature [26], and then we solve the expression of $C_{A,n}$.

Multiplying both sides of the equation by $-\sqrt{{\displaystyle \frac{-1}{T_1{T}_2}}}\ast$$exp\left({\displaystyle \frac{i}{2b_1}}\left[d_1{\left(m^{\prime }{\displaystyle \frac{2\pi b_1}{T_1}}\right)}^2-2x\left(m^{\prime }{\displaystyle \frac{2\pi b_1}{T_1}}\right)+a_1{x}^2\right]\right)\ast exp\left({\displaystyle \frac{i}{2b_2}}\left[d_2{\left(n^{\prime }{\displaystyle \frac{2\pi b_2}{T_2}}\right)}^2-2y\left(n^{\prime }{\displaystyle \frac{2\pi b_2}{T_2}}\right)+a_2{y}^2\right]\right)$ and taking the integral, we have

$$\begin{array}{c}\hfill \begin{array}{cc}& {\int }_{-{\displaystyle \frac{T_1}{2}}}^{{\displaystyle \frac{T_1}{2}}}{\int }_{-{\displaystyle \frac{T_2}{2}}}^{{\displaystyle \frac{T_2}{2}}}f(x,y)(-\sqrt{{\displaystyle \frac{-1}{T_1{T}_2}}})\hfill \\ & \ast exp\left({\displaystyle \frac{i}{2b_1}}\left[d_1{\left(m{\displaystyle \frac{2\pi b_1}{T_1}}\right)}^2-2x\left(m{\displaystyle \frac{2\pi b_1}{T_1}}\right)+a_1{x}^2\right]\right)\hfill \\ & \ast exp\left({\displaystyle \frac{i}{2b_2}}\left[d_2{\left(n{\displaystyle \frac{2\pi b_2}{T_2}}\right)}^2-2y\left(n{\displaystyle \frac{2\pi b_2}{T_2}}\right)+a_2{y}^2\right]\right)dxdy=\hfill \\ & {\int }_{-{\displaystyle \frac{T_1}{2}}}^{{\displaystyle \frac{T_1}{2}}}{\int }_{-{\displaystyle \frac{T_2}{2}}}^{{\displaystyle \frac{T_2}{2}}}\sum _{m,n}{C}_{m,n}^{A,B}\sqrt{{\displaystyle \frac{-1}{T_1{T}_2}}}\ast (-\sqrt{{\displaystyle \frac{-1}{T_1{T}_2}}})\hfill \\ & \ast exp\left({\displaystyle \frac{-i}{2b_1}}\left[d_1{\left(m{\displaystyle \frac{2\pi b_1}{T_1}}\right)}^2-2x\left(m{\displaystyle \frac{2\pi b_1}{T_1}}\right)+a_1{x}^2\right]\right)\hfill \\ & \ast exp\left({\displaystyle \frac{-i}{2b_2}}\left[d_2{\left(n{\displaystyle \frac{2\pi b_2}{T_2}}\right)}^2-2y\left(n{\displaystyle \frac{2\pi b_2}{T_2}}\right)+a_2{y}^2\right]\right)\hfill \\ & \ast exp\left({\displaystyle \frac{i}{2b_1}}\left[d_1{\left(m^{\prime }{\displaystyle \frac{2\pi b_1}{T_1}}\right)}^2-2x\left(m^{\prime }{\displaystyle \frac{2\pi b_1}{T_1}}\right)+a_1{x}^2\right]\right)\hfill \\ & \ast exp\left({\displaystyle \frac{i}{2b_2}}\left[d_2{\left(n^{\prime }{\displaystyle \frac{2\pi b_2}{T_2}}\right)}^2-2y\left(n^{\prime }{\displaystyle \frac{2\pi b_2}{T_2}}\right)+a_2{y}^2\right]\right)dxdy.\hfill \end{array}\end{array}$$

□

When $m=m^{\prime },n=n^{\prime }$, after expanding the formula on the right side of the equation, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& -\sqrt{{\displaystyle \frac{-1}{T_1{T}_2}}}{\int }_{-{\displaystyle \frac{T_1}{2}}}^{{\displaystyle \frac{T_1}{2}}}{\int }_{-{\displaystyle \frac{T_2}{2}}}^{{\displaystyle \frac{T_2}{2}}}f(x,y)exp\left({\displaystyle \frac{i}{2b_1}}\left[d_1{\left(m{\displaystyle \frac{2\pi b_1}{T_1}}\right)}^2-2x\left(m{\displaystyle \frac{2\pi b_1}{T_1}}\right)+a_1{x}^2\right]\right)\hfill \\ & \ast exp\left({\displaystyle \frac{i}{2b_2}}\left[d_2{\left(n{\displaystyle \frac{2\pi b_2}{T_2}}\right)}^2-2y\left(n{\displaystyle \frac{2\pi b_2}{T_2}}\right)+a_2{y}^2\right]\right)dxdy={\int }_{-{\displaystyle \frac{T_1}{2}}}^{{\displaystyle \frac{T_1}{2}}}{\int }_{-{\displaystyle \frac{T_2}{2}}}^{{\displaystyle \frac{T_2}{2}}}\sum _{m,n}{C}_{m,n}^{A,B}dxdy,\hfill \end{array}\end{array}$$

(8)

from (8); the solution is

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_{m,n}^{A,B}=& {\int }_{-{\displaystyle \frac{T_1}{2}}}^{{\displaystyle \frac{T_1}{2}}}{\int }_{-{\displaystyle \frac{T_2}{2}}}^{{\displaystyle \frac{T_2}{2}}}f(x,y)\left(-\sqrt{{\displaystyle \frac{-1}{T_1{T}_2}}}\right)\hfill \\ \hfill & \ast exp\left({\displaystyle \frac{i}{2b_1}}\left[d_1{\left(m{\displaystyle \frac{2\pi b_1}{T_1}}\right)}^2-2x\left(m{\displaystyle \frac{2\pi b_1}{T_1}}\right)+a_1{x}^2\right]\right)\hfill \\ \hfill & \ast exp\left({\displaystyle \frac{i}{2b_2}}\left[d_2{\left(n{\displaystyle \frac{2\pi b_2}{T_2}}\right)}^2-2y\left(n{\displaystyle \frac{2\pi b_2}{T_2}}\right)+a_2{y}^2\right]\right)dxdy.\hfill \end{array}\end{array}$$

(9)

When $m\ne m^{\prime },n\ne n^{\prime }$, after expanding the formula on the right side of the equation, we have

$$\begin{array}{c}\hfill \begin{array}{cc}& {\int }_{-{\displaystyle \frac{T_1}{2}}}^{{\displaystyle \frac{T_1}{2}}}{\int }_{-{\displaystyle \frac{T_2}{2}}}^{{\displaystyle \frac{T_2}{2}}}\sum _{m,n}{C}_{m,n}^{A,B}{\displaystyle \frac{1}{T_1{T}_2}}\hfill \\ & \ast exp\left({\displaystyle \frac{-i}{2b_1}}\left[d_1{\left(m{\displaystyle \frac{2\pi b_1}{T_1}}\right)}^2-2x\left(m{\displaystyle \frac{2\pi b_1}{T_1}}\right)+a_1{x}^2\right]\right)\hfill \\ & \ast exp\left({\displaystyle \frac{-i}{2b_2}}\left[d_2{\left(n{\displaystyle \frac{2\pi b_2}{T_2}}\right)}^2-2y\left(n{\displaystyle \frac{2\pi b_2}{T_2}}\right)+a_2{y}^2\right]\right)\hfill \\ & \ast exp\left({\displaystyle \frac{i}{2b_1}}\left[d_1{\left(m^{\prime }{\displaystyle \frac{2\pi b_1}{T_1}}\right)}^2-2x\left(m^{\prime }{\displaystyle \frac{2\pi b_1}{T_1}}\right)+a_1{x}^2\right]\right)\hfill \\ & \ast exp\left({\displaystyle \frac{i}{2b_2}}\left[d_2{\left(n^{\prime }{\displaystyle \frac{2\pi b_2}{T_2}}\right)}^2-2y\left(n^{\prime }{\displaystyle \frac{2\pi b_2}{T_2}}\right)+a_2{y}^2\right]\right)dxdy=0.\hfill \end{array}\end{array}$$

(10)

In the one-dimensional linear canonical transform, we have demonstrated that approximation to the regional signal with the 1-D LCT series has a unique least squares solution. Similarly, it can be inferred that the approximation to regional image pixels with the 2-D LCT series also has a unique least squares solution.

Likewise, suppose $L,P$ is the number of terms in the 2-D LCT series expansion of the image $f(x,y)$, and the expansion is defined as the finite length LCT series of the image, expressed as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill f_{L,P}(x,y)=& \sum _{m=1,n=1}^{L,P}{C}_{m,n}^{A,B}\sqrt{{\displaystyle \frac{-1}{T_1{T}_2}}}\hfill \\ \hfill & \ast exp\left({\displaystyle \frac{-i}{2b_1}}\left[d_1{\left(m{\displaystyle \frac{2\pi b_1}{T_1}}\right)}^2-2x\left(m{\displaystyle \frac{2\pi b_1}{T_1}}\right)+a_1{x}^2\right]\right)\hfill \\ \hfill & \ast exp\left({\displaystyle \frac{-i}{2b_2}}\left[d_2{\left(n{\displaystyle \frac{2\pi b_2}{T_2}}\right)}^2-2y\left(n{\displaystyle \frac{2\pi b_2}{T_2}}\right)+a_2{y}^2\right]\right).\hfill \end{array}\end{array}$$

(11)

Assuming that the value of the function of the image at $(x_i,y_j)$ is the same as f, it is transformed to

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill f_{L,P}(x,y)=& \sum _{m=1,n=1}^{L,P}{C}_{m,n}^{A,B}\sqrt{{\displaystyle \frac{-1}{T_1{T}_2}}}\hfill \\ \hfill & \ast exp\left({\displaystyle \frac{-i}{2b_1}}\left[d_1{\left(m{\displaystyle \frac{2\pi b_1}{T_1}}\right)}^2-2x_i\left(m{\displaystyle \frac{2\pi b_1}{T_1}}\right)+a_1{x}_i^2\right]\right)\hfill \\ \hfill & \ast exp\left({\displaystyle \frac{-j}{2b_2}}\left[d_2{\left(n{\displaystyle \frac{2\pi b_2}{T_2}}\right)}^2-2y_j\left(n{\displaystyle \frac{2\pi b_2}{T_2}}\right)+a_2{y}_j^2\right]\right),\hfill \end{array}\end{array}$$

(12)

where $i=1,2,\cdots ,M,j=1,2,\cdots ,N$. Decompose $f(x,y)$ into the product of three independent parts and define the matrices U, V, and O. U is a matrix $M\times L$, where the elements are

$$U=exp\left({\displaystyle \frac{-i}{2b_1}}\left[-2x_i\left(m{\displaystyle \frac{2\pi b_1}{T_1}}\right)+a_1{x}_i^2\right]\right),$$

(13)

V is a $N\times P$ matrix, where the elements are

$$V=exp\left({\displaystyle \frac{-i}{2b_2}}\left[-2y_j\left(n{\displaystyle \frac{2\pi b_2}{T_2}}\right)+a_2{y}_j^2\right]\right),$$

(14)

$O\in R^{L\times P}$ and O is

$$O=C_{m,n}^{A,B}\sqrt{{\displaystyle \frac{-1}{T_1{T}_2}}}exp\left({\displaystyle \frac{-i{d}_1}{2b_1}}{\left(m{\displaystyle \frac{2\pi b_1}{T_1}}\right)}^2\right)exp\left({\displaystyle \frac{-i{d}_2}{2b_2}}{\left(n{\displaystyle \frac{2\pi b_2}{T_2}}\right)}^2\right).$$

(15)

Perform Kronecker product operation on U and V, and the final four-dimensional tensor $f(x,y)\in R^{M\times N\times L\times P}$ is combined by outer product and element-by-element multiplication to obtain the final matrix expression as follows:

$$f(x,y)=(U\otimes V)\odot (1_{M\times N}\otimes O).$$

(16)

Here, ⊗ represents the Kronecker product, which results in a $(M·N)(L·P)$ matrix with elements

$$U\otimes V=exp({\displaystyle \frac{-i}{2b_1}}\left[-2x_i\left(m{\displaystyle \frac{2\pi b_1}{T_1}}\right)+a_1{x}_i^2\right])\otimes exp({\displaystyle \frac{-i}{2b_2}}\left[-2y_j\left(n{\displaystyle \frac{2\pi b_2}{T_2}}\right)+a_2{y}_j^2\right]),$$

(17)

and ⊙ represents the multiplication of elements by element; $1_{M\times N}$ is a full 1-matrix, which is used to match dimensions.

The matrix representation of $f(x,y)$ is

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill f(x,y)=& exp({\displaystyle \frac{-i}{2b_1}}\left[-2x_i\left(m{\displaystyle \frac{2\pi b_1}{T_1}}\right)+a_1{x}_i^2\right])\otimes exp({\displaystyle \frac{-i}{2b_2}}\left[-2y_j\left(n{\displaystyle \frac{2\pi b_2}{T_2}}\right)+a_2{y}_j^2\right])\hfill \\ \hfill & \odot C_{m,n}^{A,B}\sqrt{{\displaystyle \frac{-1}{T_1{T}_2}}}exp\left({\displaystyle \frac{-i{d}_1}{2b_1}}{\left(m{\displaystyle \frac{2\pi b_1}{T_1}}\right)}^2\right)exp\left({\displaystyle \frac{-i{d}_2}{2b_2}}{\left(n{\displaystyle \frac{2\pi b_2}{T_2}}\right)}^2\right).\hfill \end{array}\end{array}$$

(18)

Rewrite (18) as the form

$$\begin{array}{l}\left(\begin{array}{cccc}{X}_1& 0& \cdots & 0\\ 0& X_2& \cdots & 0\\ ⋮& ⋮& & ⋮\\ 0& 0& \cdots & X_M\end{array}\right)\left(\begin{array}{ccc}{Y}_1^1& \cdots & Y_1^L\\ ⋮& & ⋮\\ Y_M^1& \cdots & Y_M^L\end{array}\right)\\ \otimes \left(\begin{array}{cccc}{X}_1^{\prime }& 0& \cdots & 0\\ 0& X_2^{\prime }& \cdots & 0\\ ⋮& ⋮& & ⋮\\ 0& 0& \cdots & X_N^{\prime }\end{array}\right)\left(\begin{array}{ccc}{Y}_1^{\prime 1}& \cdots & Y_1^{\prime L}\\ ⋮& & ⋮\\ Y_N^{\prime 1}& \cdots & Y_N^{\prime L}\end{array}\right)\\ \odot C_{m,n}^{A,B}{Z}_{M,N}=f_{M.N},\end{array}$$

(19)

with

$\begin{array}{l}{X}_i=exp\left(-i{\displaystyle \frac{a_1}{2b_1}}{x}_i^2\right),Y_{\mathrm{i}}^m=exp\left({\displaystyle \frac{2m\pi i{\mathrm{x}}_i}{T_1}}\right),X_j^{\prime }=exp\left(-i{\displaystyle \frac{a_2}{2b_2}}{y}_j^2\right),Y_{\mathrm{j}}^{\prime n}=exp\left({\displaystyle \frac{2n\pi j{\mathrm{y}}_j}{T_2}}\right),\\ O_n=C_{m,n}^{A,B}{Z}_{M,N}=C_{m,n}^{A,B}\sqrt{{\displaystyle \frac{-1}{T_1{T}_2}}}exp({\displaystyle \frac{-2d_1{b}_{1}i{m}^2{\pi }^2}{T_1^2}}+{\displaystyle \frac{-2d_2{b}_{2}i{n}^2{\pi }^2}{T_2^2}}).\end{array}$

The above matrix form can be abbreviated as

$$\begin{array}{c}\hfill XY\otimes X^{\prime }{Y}^{\prime }\odot O=f.\end{array}$$

(20)

Then, the problem is transformed into the following optimization model by using the following idea of the least squares method:

$$\begin{array}{c}\hfill min{∥f-U\otimes V\odot O∥}^2.\end{array}$$

(21)

Theorem 5.

When $(M\times N)\ge (L\times P)$, the column vectors of matrix B are linearly independent.

Proof of Theorem 5.

The matrix of B has the form

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & B=XY\otimes X^{\prime }{Y}^{\prime }=\hfill \\ \hfill & \left(\begin{array}{cccccccc}{X}_1{Y}_1^1{X}_1^{\prime }{Y}_1^{\prime 1}& \cdots & X_1{Y}_1^1{X}_1^{\prime }{Y}_1^{\prime P}& \cdots & \cdots & X_1{Y}_1^L{X}_1^{\prime }{Y}_1^{\prime 1}& \cdots & X_1{Y}_1^L{X}_1^{\prime }{Y}_1^{\prime P}\\ X_1{Y}_1^1{X}_2^{\prime }{Y}_2^{\prime 1}& \cdots & X_1{Y}_1^1{X}_2^{\prime }{Y}_2^{\prime P}& \cdots & \cdots & X_1{Y}_1^L{X}_2^{\prime }{Y}_2^{\prime 2}& \cdots & X_1{Y}_1^L{X}_2^{\prime }{Y}_2^{\prime P}\\ ⋮& \ddots & ⋮& & & ⋮& \ddots & ⋮\\ X_1{Y}_1^1{X}_N^{\prime }{Y}_N^{\prime 1}& \cdots & X_1{Y}_1^1{X}_N^{\prime }{Y}_N^{\prime P}& \cdots & \cdots & X_1{Y}_1^L{X}_N^{\prime }{Y}_N^{\prime 1}& \cdots & X_1{Y}_1^L{X}_N^{\prime }{Y}_N^{\prime P}\\ ⋮& & ⋮& \ddots & & ⋮& & ⋮\\ ⋮& & ⋮& & \ddots & ⋮& & ⋮\\ X_M{Y}_M^1{X}_1^{\prime }{Y}_1^{\prime 1}& \cdots & X_M{Y}_M^1{X}_1^{\prime }{Y}_1^{\prime P}& \cdots & \cdots & X_M{Y}_M^L{X}_1^{\prime }{Y}_1^{\prime 1}& \cdots & X_M{Y}_M^L{X}_1^{\prime }{Y}_1^{\prime P}\\ X_M{Y}_M^1{X}_2^{\prime }{Y}_2^{\prime 1}& \cdots & X_M{Y}_M^1{X}_2^{\prime }{Y}_2^{\prime P}& \cdots & \cdots & X_M{Y}_M^L{X}_2^{\prime }{Y}_2^{\prime 1}& \cdots & X_M{Y}_M^L{X}_2^{\prime }{Y}_2^{\prime P}\\ ⋮& \ddots & ⋮& & & ⋮& \ddots & ⋮\\ X_M{Y}_M^1{X}_N^{\prime }{Y}_N^{\prime 1}& \cdots & X_M{Y}_M^1{X}_N^{\prime }{Y}_N^{\prime P}& \cdots & \cdots & X_M{Y}_M^L{X}_N^{\prime }{Y}_N^{\prime 1}& \cdots & X_M{Y}_M^L{X}_N^{\prime }{Y}_N^{\prime P}\end{array}\right),\hfill \end{array}\end{array}$$

(22)

according to (22); the matrix of B is transformed to

$$\begin{array}{c}\hfill {\beta }_0=\left(\begin{array}{c}{X}_1{Y}_1^1{X}_1^{\prime }{Y}_1^{\prime 1}\\ X_1{Y}_1^1{X}_2^{\prime }{Y}_2^{\prime 1}\\ ⋮\\ X_1{Y}_1^1{X}_N^{\prime }{Y}_N^{\prime 1}\\ ⋮\\ ⋮\\ X_M{Y}_M^1{X}_1^{\prime }{Y}_1^{\prime 1}\\ X_M{Y}_M^1{X}_2^{\prime }{Y}_2^{\prime 1}\\ ⋮\\ X_M{Y}_M^1{X}_N^{\prime }{Y}_N^{\prime 1}\end{array}\right),{\beta }_1=\left(\begin{array}{c}{X}_1{Y}_1^1{X}_1^{\prime }{Y}_1^{\prime 2}\\ X_1{Y}_1^1{X}_2^{\prime }{Y}_2^{\prime 2}\\ ⋮\\ X_1{Y}_1^1{X}_N^{\prime }{Y}_N^{\prime 2}\\ ⋮\\ ⋮\\ X_M{Y}_M^1{X}_1^{\prime }{Y}_1^{\prime 2}\\ X_M{Y}_M^1{X}_2^{\prime }{Y}_2^{\prime 2}\\ ⋮\\ X_M{Y}_M^1{X}_N^{\prime }{Y}_N^{\prime 2}\end{array}\right),\cdots ,{\beta }_{L\times P}=\left(\begin{array}{c}{X}_1{Y}_1^L{X}_1^{\prime }{Y}_1^{\prime P}\\ X_1{Y}_1^L{X}_2^{\prime }{Y}_2^{\prime P}\\ ⋮\\ X_1{Y}_1^L{X}_N^{\prime }{Y}_N^{\prime P}\\ ⋮\\ ⋮\\ X_M{Y}_M^L{X}_1^{\prime }{Y}_1^{\prime P}\\ X_M{Y}_M^L{X}_2^{\prime }{Y}_2^{\prime P}\\ ⋮\\ X_M{Y}_M^L{X}_N^{\prime }{Y}_N^{\prime P}\end{array}\right).\end{array}$$

assuming that ${\beta }_0,{\beta }_1,\cdots {\beta }_{L\times P}$ are linear correlation vectors, then

$$\begin{array}{c}\hfill s_0{\beta }_0+s_1{\beta }_1+\cdots +s_{L\times P}{\beta }_{L\times P},\end{array}$$

(23)

and suppose that the first non-zero value among $k_0,k_1,\cdots k_{L\times P}$ is $k_q$, that is $k_0=k_1=k_2=\cdots =k_{q-1}=0,k_q\ne 0,q\ge 1$. Equation (23) is transformed to

$$\begin{array}{cc}\hfill B& =s_0{\beta }_0+s_1{\beta }_1+\cdots +s_q{\beta }_q+\cdots +s_{L\times P}{\beta }_{L\times P}+k_q{\beta }_q+k_{q+1}{\beta }_{q+1}+\cdots +k_{L\times P}{\beta }_{L\times P}\hfill \\ \hfill & =s_0{\beta }_0+s_1{\beta }_1+\cdots +\left(s_q+k_q\right){\beta }_q+\left(s_{q+1}+k_{q+1}\right){\beta }_{q+1}\cdots +\left(s_{L\times P}+k_{L\times P}\right){\beta }_{L\times P}.\hfill \end{array}$$

(24)

□

According to (24), due to $s_q+k_q\ne s_q$, this contradicts the notion that B can be unique linear representation by the vectors ${\beta }_0,{\beta }_1,\cdots {\beta }_{L\times P}$. Therefore, we have contradicted our assumption that ${\beta }_0,{\beta }_1,\cdots {\beta }_{L\times P}$ are linear correlation vectors. This indicates that the column vectors of the matrix B are irrelevant.

If the columns of ${\displaystyle B}$ are linearly independent, when $M+N\ge L+P$, the equations of $XY\otimes X^{\prime }{Y}^{\prime }\odot O=f$ is a over-determined system, and the matrix $B=XY\otimes X^{\prime }{Y}^{\prime }$ has a unique Moore–Penrose [27] generalized inverse matrix $B^{+}$, so that the equation has a unique least squares solution, as follows:

$$\begin{array}{c}\hfill O=B^{+}f,\end{array}$$

(25)

where $B^{+}={\left(B^{H}B\right)}^{-1}{B}^H$ is the Moore–Penrose generalized inverse matrix of B, and $B^H$ is the complex conjugate transpose matrix of B.

Next, the identity deformation of the approximation function is obtained by the maximum likelihood estimation method. Since the logarithmic function is monotonic, the logarithmic transformation does not affect the extreme points; it does not change the positions of the extreme points. The logarithmic transformation can also reduce or eliminate the influence of extreme values in the data and improve the numerical stability of the calculation. We usually choose the logarithmic likelihood function in the maximum likelihood estimation (MLE) [28]. Therefore, the logarithm operation is adopted here for the two-dimensional linear canonical transformation series.

When $L=1,P=1$, the 2-D LCT-based adaptive non-uniform algorithm has a unique least squares solution in the selected region. Taking logarithms on both sides of Equation (11), then (11) is simplified as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill lnf(x,y)=& ln\sqrt{{\displaystyle \frac{-1}{T_1{T}_2}}}{C}_{1,1}^{A,B}\hfill \\ & +\left({\displaystyle \frac{-i}{2b_1}}\left[d_1{\left({\displaystyle \frac{2\pi b_1}{T_1}}\right)}^2-2x\left({\displaystyle \frac{2\pi b_1}{T_1}}\right)+a_1{x}^2\right]\right)\hfill \\ & +\left({\displaystyle \frac{-i}{2b_2}}\left[d_2{\left({\displaystyle \frac{2\pi b_2}{T_2}}\right)}^2-2y\left({\displaystyle \frac{2\pi b_2}{T_2}}\right)+a_2{y}^2\right]\right).\hfill \end{array}\end{array}$$

(26)

According to the scaling invariance of 2-D LCT [29], we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill lnf=& ln\sqrt{{\displaystyle \frac{-1}{T_1{T}_2}}}{C}_{1,1}^{A,B}\hfill \\ & +\left({\displaystyle \frac{-i2{\pi }^2{b}_1{d}_1}{{T_1}^2}}-{\displaystyle \frac{-i2\pi k}{T_1}}x+{\displaystyle \frac{-i{a}_1{k}^2}{2b_1}}{x}^2\right)\hfill \\ & +\left({\displaystyle \frac{-i2{\pi }^2{b}_2{d}_2}{{T_2}^2}}-{\displaystyle \frac{-i2\pi q}{T_2}}y+{\displaystyle \frac{-i{a}_2{q}^2}{2b_2}}{y}^2\right),\hfill \end{array}\end{array}$$

(27)

Equation (27) is converted into

$$\begin{array}{c}\hfill lnf=A{x}^2+B{y}^2+Cx+Dy+E,\end{array}$$

(28)

with

$$\begin{array}{c}\hfill \left\{\begin{array}{c}A={\displaystyle \frac{-i{a}_1{k}^2}{2b_1}}\hfill \\ B={\displaystyle \frac{-i{a}_2{q}^2}{2b_2}}\hfill \\ C=-{\displaystyle \frac{-i2\pi k}{T_1}}\hfill \\ D=-{\displaystyle \frac{-i2\pi q}{T_2}}\hfill \\ E=ln\sqrt{{\displaystyle \frac{-1}{T_1,T_2}}}{C}_{1,1}^{A,B}+{\displaystyle \frac{-i2{\pi }^2{b}_1{d}_1}{{T_1}^2}}+{\displaystyle \frac{-i2{\pi }^2{b}_2{d}_2}{{T_2}^2}}\hfill \end{array}\right.\end{array}$$

(29)

in this paper. S is the least square solution to minimize the sum of squared residuals.

$$\begin{array}{c}\hfill S=\min \sum {(A{x}^2+B{y}^2+Cx+Dy+E-lnf)}^2.\end{array}$$

(30)

After solving the partial derivative for each variable, and setting it to zero, the system of equations can be represented as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}2\sum \left(A{x}^2+B{y}^2+Cx+Dy+E-lnf)\right)\sum x^2=0,\hfill \\ 2\sum \left(A{x}^2+B{y}^2+Cx+Dy+E-lnf)\right)\sum y^2=0,\hfill \\ 2\sum \left(A{x}^2+B{y}^2+Cx+Dy+E-lnf)\right)\sum x=0,\hfill \\ 2\sum \left(A{x}^2+B{y}^2+Cx+Dy+E-lnf)\right)\sum y=0,\hfill \\ 2\sum \left(A{x}^2+B{y}^2+Cx+Dy+E-lnf)\right)1=0.\hfill \end{array}\right.\end{array}$$

(31)

To solve (31), the system of linear equations is obtained as follows:

$$A=\left\{\begin{array}{c}\sum x^4\sum x^2{\mathrm{y}}^2\sum x^{3}y\sum x^3\sum x^{2}y\sum x^2\hfill \\ \sum x^2{\mathrm{y}}^2\sum {\mathrm{y}}^4\sum x{\mathrm{y}}^3\sum x{y}^2\sum {\mathrm{y}}^3\sum {\mathrm{y}}^2\hfill \\ \sum x^{2}y\sum {\mathrm{y}}^3\sum \mathrm{x}{\mathrm{y}}^2\sum xy\sum {\mathrm{y}}^2\sum \mathrm{y}\hfill \\ \sum x^2\sum {\mathrm{y}}^2\sum xy\sum x\sum \mathrm{y}\sum 1\hfill \end{array}\right.$$

(32)

$$B=\left\{\begin{array}{c}\sum x^{2}lnf\hfill \\ \sum {\mathrm{y}}^{2}lnf\hfill \\ \sum xlnf\hfill \\ \sum ylnf\hfill \\ \sum lnf\hfill \end{array}\right.C=\left\{\begin{array}{c}a\\ b\\ c\\ d\\ e\end{array}.\right.$$

(33)

The parameters of C can be obtained by solving the linear equation.

The 2-D LCT series is used to approximate pixels according to the least squares method. Supposing $Z_i(i\in \left[0,n-1\right])$ is used to represent the gray value of the pixel in the image, n is the number of all pixels in the region. Equation (21) can be rewritten as

$$\begin{array}{c}\hfill e=\sum _{i=0}^{n-1}{\left[U_i\otimes V_i\odot O_i-Z_i\right]}^2.\end{array}$$

(34)

Assuming that the global control error is set as $\epsilon$, $Z_i^{\prime }$ is $U_i\otimes V_i\odot O_i$, (34) is transformed as follows:

$$\begin{array}{c}\hfill e=\sum _{i=0}^{n-1}{\left[Z_i^{\prime }-Z_i\right]}^2<\epsilon .\end{array}$$

(35)

3.2. Algorithm Description

In this section, we present the detailed steps of the 2-D LCT-based adaptive non-uniform algorithm and perform a recursive partitioning of the initial region based on a predetermined condition ($e<\epsilon$). Initially, we take an image set, denoted as G, and divide it into four small rectangular sub-regions. For each rectangular sub-region, if it meets the preset control error, the partition will be stopped. If the rectangular sub-region does not meet the control error condition, it will be further divided into four smaller rectangular sub-regions using the self-similarity rule. This process continues until the reconstructed grayscale value in the current area reaches the termination condition; i.e., when the sum of the squared residual of current region calculated by the least-square fitting satisfies $e={\displaystyle \sum _{i=0}^{n-1}}{\left(Z_i^{\prime }-Z_i\right)}^2<\epsilon$, this region stops dividing further and saves encoded data after a finite number of steps.

To explain its process, the algorithm is described in detail below.

1

For the initial image, G is the rectangular region, and $G_m$ is the sub-region. Regarding G as the initial region, it will be divided into four sub-rectangular regions—$G_0,G_1,G_2,G_3$. The region pixels of an image can be represented by the two-dimensional LCT series $Z=f(x,y),(x,y)\in G$. x and y are the rows and columns of the image pixel matrix. Z is the grayscale value set of the regional pixels. The pixel contained in $G_m$ is data known as $Z_i$ (where $i\in \left[0,n-1\right]$), n is the number of pixels on $G_m$, and $Y_i$ is the set of determined coefficients after applying least squares approximation. $f_m\left(X_i,Y_i\right)$ is the reconstructed pixels set in the current region.

2

If the error accuracy of the reconstructed region meets the preset error accuracy, i.e., the reconstructed region approximates the original region under the control error, then the partition stops. For $m=0,1,2,3$, use the least squares method to obtain and record m and $f_m\left(P\right)$ of $G_{\mathrm{m}}$, satisfying $\mathrm{e}={\displaystyle \sum _{i=0}^{n-1}}{\left(f_m\left(X_i,Y_i\right)-Z_i\right)}^2<\epsilon$.

3

For the rectangular sub-region $G_{\mathrm{m}}$ that does not satisfy the condition of $e<\epsilon$, the current region should be further divided into four smaller subregions in a similar manner, and as this process continues, $G_{\mathrm{m}\mathrm{n}}\left(n=0,1,2,3\right)$ should be examined one by one, recording $\mathrm{m},n$, and $f_{mn}\left(P\right)$ that satisfy the condition $e<\epsilon$, and so on.

Although LCT shows many advantages in the field of image processing, whether the image reconstruction problem explored by combining the LCT series and the adaptive non-uniform partition algorithm is superior to the traditional Fourier transform is a problem we need to examine.

4. Simulation Results and Performance Analyses

In this section, we will evaluate our method based on two objective evaluation indicators: Peak Signal-to-Noise Ratio (PSNR) [30] and Structural Similarity Index (SSIM) [31]. We will analyze the proposed method and other methods to determine these under the same $ϵ$. Furthermore, all simulation experiments were conducted on a 64-bit Windows 11 PC with a 2.10 GHz Intel Core i7 CPU and 32 GB of RAM. The simulation software is MATLAB R2022b.

4.1. Objective Evaluation Standard

The objective evaluation of images is based on the characteristics of human visual perception. It employs mathematical models to quantify the differences between a reference image and the image being tested, and the quality of the image is assessed based on the calculated numerical results. This method has become the standard technology for image quality assessment due to its automated processing abilities and scoring consistency, which is unaffected by individual differences among observers. Typical objective evaluation indicators include the peak signal-to-noise ratio and the structural similarity index, among others.

PSNR is a full-reference image quality evaluation metric based on pixel-level error. It is suitable for rapid assessment of pixel-level distortion. The unit of measurement is the decibel (dB). Generally, a higher PSNR value indicates better image quality, with PSNR values greater than 30 dB considered acceptable and those above 40 dB indicating high quality. However, PSNR is based solely on pixel-level errors and cannot reflect structural information. When it comes to tasks involving human eye perception optimization, it is advisable to combine more advanced metrics such as SSIM and Learned Perceptual Image Patch Similarity (LPIPS).

$$PSNR=10\ast lo{g}_{10}\left({\displaystyle \frac{MA{X}_I^2}{MSE}}\right),$$

(36)

where $MSE={\displaystyle \frac{1}{MN}}{\sum }_{i=1}^M{\sum }_{j=1}^N{\left(I\left(i,j\right)-K\left(i,j\right)\right)}^2$, and $I\left(i,j\right)$ is the value of the reference image at pixel $(i,j)$, $K\left(i,j\right)$ is the value of the distorted image at pixel $(i,j)$. $MA{X}_I$ represents the maximum value of the pixels.

SSIM is a full-reference image quality assessment method based on human visual characteristics. It measures the similarity between the reference image and the distorted image from three dimensions: Luminance, Contrast, and Structure. The SSIM values range from 0 to 1, where a higher value indicates less image distortion and greater similarity between the two images.

$$SSIM(x,y)={\displaystyle \frac{(2{\mu }_x{\mu }_y+C_1)(2{\sigma }_x{\sigma }_y+C_2)}{({\mu }_x^2+{\mu }_y^2+C_1)({\sigma }_x^2+{\sigma }_y^2+C_2)}},$$

(37)

where $\mu$ is the average value, $\sigma$ is the standard deviation, and ${\sigma }_{xy}$ is the covariance of x and y. $C_1$ and $C_2$ are constants.

4.2. Objective Quantitative Analysis

In this subsection, a series of experiments are conducted to verify the feasibility of the algorithm. We use PSNR and SSIM as indicators to evaluate image quality. We set $\epsilon =\mathrm{10,100,300}$. In terms of images, we use six 512 × 512 8-bit grayscale images from Set 14. The number of image partitions reflects time complexity to a certain extent. That is to say, the fewer the number of image partitions, the higher the efficiency of image reconstruction. Therefore, we still need to analyze the number of image partitions to evaluate the efficiency of image reconstruction.

We will now conduct some experiments to compare the advantages of the proposed method. First, we analyze the signal-to-noise ratio, SSIM, and the number of partitions of the proposed algorithm under different values of $\epsilon$. This will help us evaluate the visual quality of the image reconstruction. Figure 2 indicates that when $\epsilon =\mathrm{10,100,300}$, from left to right, the resulting images are the original images—the reconstructed image when $\epsilon =10$, the reconstructed image when $\epsilon =100$, and the reconstructed image when $\epsilon =300$, respectively.

Table 1. The PSNRs, SSIMs, and TPcnt values of the image under different $ϵ$ conditions.

Image	$\mathit{ϵ}$	PSNRs (dB)	SSIMs	TPcnt
coastguard	10	42.29	0.9914	27,874
	100	31.08	0.8158	5837
	300	26.43	0.5638	2021
barbara	10	37.34	0.9925	36,074
	100	32.63	0.9526	19,280
	300	26.67	0.8232	7421
zebra	10	39.91	0.9971	44,083
	100	31.27	0.8841	16,004
	300	26.69	0.7142	9161
foreman	10	41.62	0.9809	13,318
	100	33.17	0.9063	4730
	300	28.23	0.8156	2453
flowers	10	40.99	0.9929	30,790
	100	32.37	0.9266	10,238
	300	26.45	0.7857	4226
comic	10	41.69	0.9965	36,691
	100	32.63	0.9549	14,210
	300	27.28	0.8654	6122

presents the signal-to-noise ratio, SSIM index, and the number of image partitions (TPcnt) under varying error accuracy thresholds.

It can be seen from the experimental data in Figure 2 and Table 1 that when $ϵ$ gradually decreases, both the number of partitions generated by the proposed algorithm and the corresponding PSNR and SSIM values illustrates a monotonically increasing trend. It is particularly noteworthy that when $ϵ=10$, the objective evaluation indicators of most reconstructed images approach the high-quality standard ($PSNR>$ 40 dB, $SSIM>0.99$). These images also exhibit clear edge details and smooth transitions in texture, as confirmed by the subjective visual evaluation. This sensitivity to the parameter indicates that the refined regional partition strategy effectively captures high-frequency components in the image, especially along contour boundaries and complex texture areas.

Next, under the same conditions, we compare and analyze the partition numbers, PSNR, and SSIM of the Fourier transform algorithm, the adaptive non-uniform rectangular partition algorithm, and the proposed algorithm to demonstrate the superiority of the proposed algorithm. Figure 3 illustrates the original image, the reconstructed image by the proposed algorithm, the reconstructed image by the Fourier algorithm, and the reconstructed image by the adaptive non-uniform rectangular partition algorithm when the error accuracy is 100.

Table 2. PSNR (dB), TPcnt, and SSIM comparison between different methods.

Image	Coastguard			Barbara			Comic
	PSNRs	TPcnt	SSIMs	PSNRs	TPcnt	SSIMs	PSNRs	TPcnt	SSIMs
Our proposed	31.08	5837	0.8158	32.63	19,280	0.9526	32.63	14,210	0.9549
Fourier	31.20	9146	0.7818	32.87	24,089	0.8986	32.90	20,057	0.9107
RectPartition	31.14	8390	0.8175	32.86	23,813	0.9511	32.91	19,601	0.9576

and

Table 3. PSNR (dB), TPcnt, and SSIM comparison between different methods.

Image	Zebra			Foreman			Flowers
	PSNRs	TPcnt	SSIMs	PSNRs	TPcnt	SSIMs	PSNRs	TPcnt	SSIMs
Our proposed	31.27	16,004	0.8841	33.17	4730	0.9063	32.37	10,238	0.9266
Fourier	31.38	20,357	0.8220	33.39	6491	0.8986	32.41	13,916	0.8729
RectPartition	31.47	18,656	0.8865	33.21	5576	0.9072	32.36	13,541	0.9229

show the corresponding PSNR, SSIM, and the number of image partitions.

It can be seen from Figure 3 and Table 2 and Table 3 that when the error accuracy is 100, the PSNR of the proposed algorithm is almost the same as that of the Fourier algorithm and the adaptive non-uniform rectangular partition algorithm in the same image. Nevertheless, the proposed algorithm requires significantly fewer partition numbers than the other two methods, which indicates that the proposed algorithm can achieve the same level of reconstructed image quality with lower computational complexity. In addition, the comparison of SSIM values also shows that the proposed algorithm performs better in subjective vision. These advantage originates from the dynamic partitioning strategy and adaptive sampling mechanism, which massively reduces the consumption of computing resource while ensuring accuracy, thereby demonstrating the algorithm’s efficiency and practicality.

5. Conclusions

This paper proposes a fast adaptive non-uniform two-dimensional linear canonical transformation partition algorithm based on the theory of two-dimensional linear canonical transformation series. The goal is to solve the issue of repeated superposition when the original linear canonical transformation approximates the image pixel globally and to apply this algorithm in image representation research. The experimental results indicate that the proposed algorithm can effectively reduce the computational complexity of linear canonical transformation when approximating and reconstructing image pixels. Additionally, it enhances the adaptability of image representation in complex image texture areas. The research results will provide theoretical support for time–frequency joint processing in fields such as remote sensing, image encryption, and medical tomography reconstruction, while also promoting the evolution of high-dimensional time–frequency analysis technology towards efficiency and intelligence.