© Wuhan University 2025

0 Introduction

Image deblurring, a key challenge in image processing, aims to accurately restore potentially sharp images from degraded image observations. The blur phenomenon is usually caused by the convolution of the point spread function (PSF) of the image capture device with the real information of the scene. This process can be mathematically expressed as:

$\mathit{f}=\mathit{K}\mathit{u}+\mathit{n}.$(1)

where $\mathit{u}$ represents a clear image, K represents the blur kernel, $\mathit{f}$ is a known blurred image, and n is the noise. The non-blind image deblurring refers to the restoration of a clear image, knowing the blurred image and the blur kernel.

Image deblurring is a challenging inverse problem, the complexity of which lies in the fact that the solution may neither exist nor be unique. In the reconstruction process, small perturbations in the data may cause relatively obvious errors. Hence, regularization technique is very important to transform the inverse problem into well-posed problem in order to solve the ill-conditioned problem. Over time, people have introduced many methods of image deblurring in various applications^[1-4]. The prior information of images is typically incorporated into the deblurring process through a regularization technology, which involves minimizing the objective function:

$\underset{\mathit{u}}{\mathrm{m}\mathrm{i}\mathrm{n}}\left\{\frac{\mathrm{1}}{\mathrm{2}}{\Vert \mathit{K}\mathit{u}-\mathit{f}\Vert }_{\mathrm{2}}^{\mathrm{2}}+\lambda R(\mathit{u})\right\}.$(2)

In model (2), the first term is referred to as data fidelity term, which ensures that the model accurately fits the data. The second term $R(\mathit{u})$, acts as a regularization functional to prevent overfitting and improve the performance of the model. And $\lambda >\mathrm{0}$, as a regularization parameter, effectively adjusts the trade-off between fidelity and regularization terms, ensuring that the model finds the optimal solution between fitting the data and avoiding excessive complexity.

The key to enhancing image deblurring is to design effective regularization term $R(\mathit{u})$ and enforce the prior knowledge of the image. Classical regularization models, including quadratic Tikhonov regularization^[5], Mumford-Shah (MS) model^[6] and total variational (TV) regularization^[7], are widely used because of their good use of local structural patterns to preserve image edges and restore smooth regions. However, these models often fall short in preserving complex image details and tend to result in excessive smoothing effects. This is largely due to their reliance on piece-wise constant assumptions, which can oversimplify the complexity of the image. To solve this problem, advanced regularization strategies should be explored to more accurately simulate the statistical features of natural images, thereby enhancing detail retention while minimizing excessive smoothing. In our previous work^[8], we extended the TV norm to the shear total variation (STV) norm by adding two shear gradient (SG) terms, and then proposed a shear total variation deblurring model.

In fact, apart from the extensively employed $L_{\mathrm{1}}$ norm, various regularization terms exhibit distinct advantages in diverse scenarios. For example, the nuclear norm minimization (NNM) algorithm proposed by Ji et al^[9] established a model by approximating the rank function with the nuclear norm and effectively restored the low-rank matrix to achieve image denoising. However, the fixed threshold processing method overlooks the disparity among singular values, potentially introducing bias in the data. To address this limitation, Gu et al^[10] proposed the weighted nuclear norm minimization (WNNM) algorithm, which significantly improves the denoising effect by dynamically adjusting the weight of singular values. At the same time, Bai et al^[11] innovatively introduced the adaptive correction term, effectively alleviating the excessive constraint of $L_{\mathrm{1}}$ norm and further optimizing the performance of the model.

In the pursuit of more efficient regularization methods, researchers have also proposed non-convex regularization terms, including $L_{\mathrm{1}}/L_{\mathrm{2}}$ regularization^[12], $L_{\mathrm{1}/\mathrm{2}}$ regularization^[13], $L_p$ regularization^[14], and Logarithmic regularization term^[15]. These methods, while maintaining sparsity, avoid some inherent defects of $L_{\mathrm{1}}$ norm and show superiority in some specific problems. These regularization strategies not only enrich the toolbox of image processing, but also promote the continuous optimization and innovation of algorithms, providing more possibilities for solving complex problems.

In order to make more efficient use of the sparsity of image gradient, many researchers have studied the regularization terms to significantly enhance the sparse representation of image gradient, thus optimizing the image processing effect. As Huo et al^[16] showed, the $L_{\mathrm{1}}-\beta L_q$ minimization provides an effective way to address the q-ratio sparsity minimization model. This approach takes into account a new minimization method and applies it to signal recovery and image reconstruction. It maintains image detail during processing while reducing noise and distortion. Li et al^[8] proposed a new SG operator, which has been experimentally verified to detect richer edge information than the gradient operator. Additionally, the alternating direction method of multipliers (ADMM) algorithm is used to efficiently solve the proposed model. The SG operator has significant advantages in image deblurring. It can capture gradient information in multiple directions and perform fine analysis on local areas, thereby precisely locating blurred regions and addressing them specifically. Compared with the gradient operator, it places greater emphasis on suppressing noise, reducing the interference of noise on gradient information, and providing more accurate gradient information, thus significantly enhancing the performance of deblurring. The method of Joshi et al^[17] focused on edge detection and extracted sharp edges by identifying areas with significant gradient changes in blur images. Then, optimized adjustments are made along the detected edges to obtain a more accurate image reconstruction.

In this paper, we propose a novel approach that combines STV norm and $L_{\mathrm{1}}/L_{\mathrm{2}}$regularization to enhance gradient information in multiple directions of the image and mitigate the excessive penalty on large attitude image gradients caused by $L_{\mathrm{1}}$ regularization, thereby improving image deblurring performance. The contributions of this paper are summarized as follows:

First, a brief introduction is given to the definitions of shear operators and SG operators. In comparison to Ref. [8], we have conducted a more comprehensive study on some important properties of the shear operator and the SG operator, and applies these properties to the deblurring model.

Second, a novel image deblurring model is proposed by integrating the STV norm with $L_{\mathrm{1}}/L_{\mathrm{2}}$ regularization to improve the efficacy of image deblurring. The proposed model is successfully solved by the ADMM algorithm.

Finally, we compare our algorithm with some existing algorithms in image deblurring experiments, and find that the deblurring effect of the proposed algorithm is significantly improved.

The structure of this paper is summarized as follows: Section 1 provides comprehensive definitions of the SG operator and the STV norm, and briefly describes the shear operator and Block-Circulant-with-Circulant-Blocks (BCCB) matrix, in order to lay a solid theoretical foundation for the subsequent content. In Section 2, the concrete matrix forms and some properties of the shear operators and the SG operators are described. In Section 3, we propose the $L_{\mathrm{1}}/L_{\mathrm{2}}-\mathrm{S}\mathrm{T}\mathrm{V}$ deblurring model to deal with the image deblurring problem, and employ the ADMM to solve the non-convex optimization problem. In Section 4, we verify the validity of the model through a series of numerical experiments. These experimental results demonstrate the practicability of the model and further highlight the advantages of the model. Finally, Section 5 is the conclusion.

1 Preliminaries

1.1 Shear Operator

Let ${\mathcal{S}}_{{\theta }_d}$ be a shear operator with the shear angle ${\theta }_d\in [\mathrm{0},\mathrm{\pi }]$ (see Ref. [18]), and $d\in \{\mathrm{t},\mathrm{b},\mathrm{l},\mathrm{r}\}$,

${\mathcal{S}}_{{\theta }_d}:{\mathbb{R}}^{n\times n}\mapsto {\mathbb{R}}^{n\times n}:\mathit{X}\mapsto {\mathit{X}}^{\mathrm{s}\mathrm{h}\mathrm{r}},$(3)

where the $(\widehat{i},\widehat{j})$-th entry of ${\mathit{X}}^{\mathrm{s}\mathrm{h}\mathrm{r}}$ is given, for $\widehat{i}=\mathrm{1},\cdots ,m$ and$\widehat{j}=\mathrm{1},\cdots ,n$, by ${\mathit{X}}_{\widehat{i},\widehat{j}}^{\mathrm{s}\mathrm{h}\mathrm{r}}:={\mathit{X}}_{\overline{i},\overline{j}}$ with

$\overline{i}:=\{\begin{array}{c}\left[\left(\widehat{i}-\mathrm{1}+[\frac{\mathrm{4}}{\mathrm{\pi }}\theta \widehat{j}]\right)\mathrm{m}\mathrm{o}\mathrm{d}n\right]+\mathrm{1},\mathrm{i}\mathrm{f}d=\mathrm{t},\\ \left[\left(\widehat{i}-\mathrm{1}+[\frac{\mathrm{4}}{\mathrm{\pi }}\theta \left(n-\widehat{j}\right)]\right)\mathrm{m}\mathrm{o}\mathrm{d}n\right]+\mathrm{1},\mathrm{i}\mathrm{f}d=\mathrm{b},\\ \widehat{i},\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e},\end{array}$

$\overline{j}:=\{\begin{array}{c}\left[\left(\widehat{j}-\mathrm{1}+[\frac{\mathrm{4}}{\mathrm{\pi }}\theta \widehat{i}]\right)\mathrm{m}\mathrm{o}\mathrm{d}n\right]+\mathrm{1},\mathrm{i}\mathrm{f}d=\mathrm{r},\\ \left[\left(\widehat{j}-\mathrm{1}+[\frac{\mathrm{4}}{\mathrm{\pi }}\theta \left(n-\widehat{i}\right)]\right)\mathrm{m}\mathrm{o}\mathrm{d}n\right]+\mathrm{1},\mathrm{i}\mathrm{f}d=\mathrm{l},\\ \widehat{j},\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e},\end{array}$

where $\left[\cdot \right]$ represents the procedure rounding the real number to the nearest integer in the zero direction. Parameter $d$ identifies the direction of the shear axis, which can be represented in four directions: top (t), bottom (b), left (l) and right (r). ${\theta }_d$ represents the strength of the shear, that is, the degree to which the image is deformed in a particular direction. In the case of ${\mathcal{S}}_{{\theta }_{\mathrm{t}}}\mathit{X}$, this expression means that the image $\mathit{X}$ is sheared at the ${\theta }_d$ angle through the top axis. It should be noted that when ${\theta }_d=\mathrm{0}$, the operator ${\mathcal{S}}_{{\theta }_d}$ degenerates into a unit transformation, meaning that the image does not undergo any shear deformation.

1.2 Shear Gradient Operator and Shear TV Norm

Total variation was first proposed by Rudin, Osher and Fatemi^[19] to optimize image denoising and restoration problems. Given a grayscale image $\mathit{u}\in {\mathbb{R}}^{n\times n}$, its discrete TV norm is

$\mathrm{T}\mathrm{V}(\mathit{u})={\Vert {\mathit{D}}^i\mathit{u}\Vert }_{\mathrm{1}}.$(4)

where ${\mathcal{D}}^{\mathrm{1}}={\mathcal{D}}_x$ and ${\mathcal{D}}^{\mathrm{2}}={\mathcal{D}}_y$ represent linear transformation operators for first-order differences in the horizontal and vertical directions, respectively. In (4), using only finite differences in these two directions simplifies the calculation, but it has limitations. Specifically, focusing only on the finite differences in horizontal and vertical directions will overlook details in the image, resulting in loss of information in other directions in the recovered or reconstructed image.

To address this issue, more sophisticated directional difference operators are employed to comprehensively capture the intricate details in the image, such as the SG operator^[8], which takes the following form:

$\mathcal{D}\triangleq ({\mathcal{D}}^{\mathrm{3}},{\mathcal{D}}^{\mathrm{4}})=({\mathcal{S}}_{{\theta }_d}^{\mathrm{T}}{\mathcal{D}}_x{\mathcal{S}}_{{\theta }_d},{\mathcal{S}}_{{\theta }_d}^{\mathrm{T}}{\mathcal{D}}_y{\mathcal{S}}_{{\theta }_d}),$(5)

where the superscript $\mathrm{T}$ means transpose. Then, for the grayscale image $u$, its discrete STV norm is defined as

$\mathrm{S}\mathrm{T}\mathrm{V}(\mathit{u})={\Vert {\mathit{D}}^i\mathit{u}\Vert }_{\mathrm{1}}.$(6)

Therefore, the SG operator is a generalization of the gradient operator and is used to capture directional features in image.

1.3 The Advantages of SG Operator in Image Processing

The gradient operator of the image can help us better understand and analyze the edge and detail information of the image. Gradient operator is used in many ways in image processing tasks, including Tikhonov regularization^[20] and TV regularization^[7].

The comparison of the effect between the gradient operator and the SG operator in processing the Barbara image is visually demonstrated in Fig. 1. From the comparison of Fig. 1 (a) and (d), we found that the gradient operator is insufficient in capturing details, failing to effectively preserve the subtle features of image, which leads to unsatisfactory results in image processing. However, when we turned to images processed with the SG operator, as shown in Fig. 1 (b)-(c) and (e)-(f), the SG operator accurately captures and enhances the texture of the image, which significantly enhances the sense of layer and detail expression of the image. The operator not only improves the visual impact of the image, but also improves the effect of image restoration by mining more image information, which plays a key role in image processing tasks such as image restoration.

Fig. 1 The gradient operator and the shear gradient operator from different angles are applied to the image u-'Barbara' (a) ${\mathit{D}}_{\mathit{x}}\mathit{u}$; (b) ${\mathit{S}}_{{\mathit{\theta }}_{\mathbf{b}}}^{\mathbf{T}}{\mathit{D}}_{\mathit{x}}{\mathit{S}}_{{\mathit{\theta }}_{\mathbf{b}}}\mathit{u}$ with ${\mathit{\theta }}_{\mathbf{b}}=\mathrm{45}°$; (c) ${\mathit{S}}_{{\mathit{\theta }}_{\mathbf{b}}}^{\mathbf{T}}{\mathit{D}}_{\mathit{x}}{\mathit{S}}_{{\mathit{\theta }}_{\mathbf{b}}}\mathit{u}$ with ${\mathit{\theta }}_{\mathbf{b}}=\mathrm{90}°$; (d) ${\mathit{D}}_{\mathit{y}}\mathit{u}$; (e) ${\mathit{S}}_{{\mathit{\theta }}_{\mathbf{l}}}^{\mathbf{T}}{\mathit{D}}_{\mathit{y}}{\mathit{S}}_{{\mathit{\theta }}_{\mathbf{l}}}\mathit{u}$with ${\mathit{\theta }}_{\mathbf{l}}=\mathrm{45}°$; (f) ${\mathit{S}}_{{\mathit{\theta }}_{\mathbf{l}}}^{\mathbf{T}}{\mathit{D}}_{\mathit{y}}{\mathit{S}}_{{\mathit{\theta }}_{\mathbf{l}}}\mathit{u}$with ${\mathit{\theta }}_{\mathbf{l}}=\mathrm{90}°$

1.4 Definition and Diagonalizations of BCCB Matrix

Under the periodic boundary conditions, both the blur matrix and gradient matrix exhibit BCCB matrix. This special structure means that the matrix can be diagonalized by two-dimensional discrete Fourier transform (2D DFT).

Definition 1   (BCCB matrix) Let C be a block circulant matrix of the following form

$\mathit{C}=\left[\begin{array}{ccccc}{\mathit{C}}_{\mathrm{0}}& {\mathit{C}}_{n-\mathrm{1}}& {\mathit{C}}_{n-\mathrm{2}}& \cdots & {\mathit{C}}_{\mathrm{1}}\\ {\mathit{C}}_{\mathrm{1}}& {\mathit{C}}_{\mathrm{0}}& {\mathit{C}}_{n-\mathrm{1}}& \cdots & {\mathit{C}}_{\mathrm{2}}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ {\mathit{C}}_{n-\mathrm{2}}& {\mathit{C}}_{n-\mathrm{3}}& {\mathit{C}}_{n-\mathrm{4}}& \dots & {\mathit{C}}_{n-\mathrm{1}}\\ {\mathit{C}}_{n-\mathrm{1}}& {\mathit{C}}_{n-\mathrm{2}}& {\mathit{C}}_{n-\mathrm{3}}& \dots & {\mathit{C}}_{\mathrm{0}}\end{array}\right],$(7)

where ${\mathit{C}}_i\in {\mathbb{R}}^{n\times n}$ is a cyclic matrix. In this case, we call C the BCCB matrix.

It is rather straightforward to demonstrate that every BCCB matrix is a unitary matrix; that is, ${\mathit{C}}^{*}\mathit{C}=\mathit{C}{\mathit{C}}^{*}.$Consequently, $\mathit{C}$ possesses a unitary spectral decomposition.

$\mathit{C}={\mathit{F}}^{\mathbf{*}}\mathit{\Lambda }\mathit{F},{\mathit{F}}^{\mathbf{*}}\mathit{F}=N\mathit{I}$(8)

where $\mathit{F}$ is the two-dimensional unitary DFT matrix, $\mathit{\Lambda }$ is a diagonal matrix consisting of all the eigenvalues of $\mathit{C}$, $N=n^{\mathrm{4}}$, $\mathit{I}$ represents the identity matrix.

2 The Matrix Form and Properties of the Shear Operator and the SG Operator

In this section, we first discuss the matrix representation of the shear operator, and find that these matrices can be expressed in the form of special matrices. Next, we analyze the SG operator and also obtain its matrix form, which can also be represented by special matrices. Finally, we discuss the matrix forms and properties of shear operator and SG operator in different cases by changing the angle and direction.

2.1 The Matrix Form and Properties of the Shear Operator

Let ${\mathit{E}}_{ij}$ be a special matrix in which the elements in position $(i,j)$ are 1 and all other positions are 0. The set of matrices defined as $\{{\mathit{E}}_{ij}|i,j=\mathrm{1,2},\cdots ,n\}$ forms an orthogonal basis of the linear space ${\mathbb{R}}^{n\times n}$. Further, if we define ${\mathit{S}}_{{\mathit{\theta }}_{\mathit{d}}}$ as the shear operator matrix represented under the above basis$\{{\mathit{E}}_{ij}|i,j=\mathrm{1,2},\cdots ,n\}$, i.e.

${\mathcal{S}}_{{\mathit{\theta }}_{\mathit{d}}}\left({\mathit{E}}_{\mathrm{11}},{\mathit{E}}_{\mathrm{21}},\cdots ,{\mathit{E}}_{nn}\right)=\left({\mathit{E}}_{\mathrm{11}},{\mathit{E}}_{\mathrm{21}},\cdots ,{\mathit{E}}_{nn}\right){\mathit{S}}_{{\theta }_d}$(9)

where ${\theta }_d=\mathrm{45}°$. By Eq. (9), we can obtain that ${\mathit{S}}_{{\theta }_d}$ is a matrix with the following forms

${\mathit{S}}_{{\theta }_{\mathrm{l}}}=\left[\begin{array}{ccccc}{\mathit{E}}_{\mathrm{11}}& {\mathit{E}}_{nn}& {\mathit{E}}_{(n-\mathrm{1})(n-\mathrm{1})}& \cdots & {\mathit{E}}_{\mathrm{22}}\\ {\mathit{E}}_{\mathrm{22}}& {\mathit{E}}_{\mathrm{11}}& {\mathit{E}}_{nn}& \cdots & {\mathit{E}}_{(n-\mathrm{1})(n-\mathrm{1})}\\ ⋮& ⋮& ⋮& \cdots & ⋮\\ {\mathit{E}}_{nn}& {\mathit{E}}_{(n-\mathrm{1})(n-\mathrm{1})}& {\mathit{E}}_{(n-\mathrm{2})(n-\mathrm{2})}& \cdots & {\mathit{E}}_{\mathrm{11}}\end{array}\right],$

${\mathit{S}}_{{\theta }_{\mathrm{r}}}=\left[\begin{array}{ccccc}{\mathit{E}}_{\mathrm{11}}& {\mathit{E}}_{\mathrm{22}}& {\mathit{E}}_{\mathrm{33}}& \cdots & {\mathit{E}}_{nn}\\ {\mathit{E}}_{nn}& {\mathit{E}}_{\mathrm{11}}& {\mathit{E}}_{\mathrm{22}}& \cdots & {\mathit{E}}_{(n-\mathrm{1})(n-\mathrm{1})}\\ ⋮& ⋮& ⋮& \cdots & ⋮\\ {\mathit{E}}_{\mathrm{22}}& {\mathit{E}}_{\mathrm{33}}& {\mathit{E}}_{\mathrm{44}}& \cdots & {\mathit{E}}_{\mathrm{11}}\end{array}\right]$

${\mathit{S}}_{{\theta }_{\mathrm{t}}}=\left[\begin{array}{ccccc}\mathit{E}& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}\\ \mathrm{0}& {\mathit{E}}_{n-\mathrm{1}}^c& \mathrm{0}& \cdots & \mathrm{0}\\ ⋮& ⋮& ⋮& \cdots & ⋮\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \cdots & {\mathit{E}}_{\mathrm{1}}^c\end{array}\right],$

${\mathit{S}}_{{\theta }_{\mathrm{b}}}=\left[\begin{array}{ccccc}\mathit{E}& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}\\ \mathrm{0}& {\mathit{E}}_{\mathrm{1}}^c& \mathrm{0}& \cdots & \mathrm{0}\\ ⋮& ⋮& ⋮& \cdots & ⋮\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \cdots & {\mathit{E}}_{n-\mathrm{1}}^c\end{array}\right]$(10)

where E is the identity matrix of order n, ${\mathit{E}}_i^c$ is circulant matrix that circularly shifts the elements in the arry $\mathit{E}$ along columns by i units. The following describes its properties:

1) ${\mathit{E}}_{ii}{\mathit{E}}_{jj}=\{\begin{array}{c}\mathrm{0},\mathrm{i}\mathrm{f}ii\ne jj\\ {\mathit{E}}_{ii},\mathrm{i}\mathrm{f}ii=jj\end{array}$; ${\displaystyle \sum _{i=\mathrm{1}}^n}{\mathit{E}}_{ii}=\mathit{E}.$

2) ${\mathit{E}}_i^c{\mathit{E}}_j^c={\mathit{E}}_{\left(i+j\right)\mathrm{m}\mathrm{o}\mathrm{d}n}^c$; ${\mathit{E}}_{\mathrm{0}}^c=\mathit{E}$. $i,j=\mathrm{0,1},\mathrm{2},\cdots ,n-\mathrm{1}.$

3) ${\left({\mathit{E}}_i^c\right)}^{\mathrm{T}}={\mathit{E}}_{n-i}^c$, $i=\mathrm{1,2},\cdots ,n-\mathrm{1}.$

Similarly, when ${\theta }_d=\mathrm{90}°$, the shear matrix ${\mathit{S}}_{{\theta }_d}$ has the following forms

${\mathit{S}}_{{\theta }_{\mathrm{l}}}=\left[\begin{array}{ccccc}{\mathit{E}}_{\mathrm{11}}& {\mathit{E}}_{\mathrm{22}}& {\mathit{E}}_{\mathrm{33}}& \cdots & {\mathit{E}}_{nn}\\ {\mathit{E}}_{nn}& {\mathit{E}}_{\mathrm{11}}& {\mathit{E}}_{\mathrm{22}}& \cdots & {\mathit{E}}_{(n-\mathrm{1})(n-\mathrm{1})}\\ ⋮& ⋮& ⋮& \cdots & ⋮\\ {\mathit{E}}_{\mathrm{22}}& {\mathit{E}}_{\mathrm{33}}& {\mathit{E}}_{\mathrm{44}}& \cdots & {\mathit{E}}_{\mathrm{11}}\end{array}\right],$

${\mathit{S}}_{{\theta }_{\mathrm{r}}}=\left[\begin{array}{ccccc}{\mathit{E}}_{\mathrm{11}}& {\mathit{E}}_{nn}& {\mathit{E}}_{(n-\mathrm{1})(n-\mathrm{1})}& \cdots & {\mathit{E}}_{\mathrm{22}}\\ {\mathit{E}}_{\mathrm{22}}& {\mathit{E}}_{\mathrm{11}}& {\mathit{E}}_{nn}& \cdots & {\mathit{E}}_{(n-\mathrm{1})(n-\mathrm{1})}\\ ⋮& ⋮& ⋮& \cdots & ⋮\\ {\mathit{E}}_{nn}& {\mathit{E}}_{(n-\mathrm{1})(n-\mathrm{1})}& {\mathit{E}}_{(n-\mathrm{2})(n-\mathrm{2})}& \cdots & {\mathit{E}}_{\mathrm{11}}\end{array}\right],$

${\mathit{S}}_{{\theta }_{\mathrm{t}}}=\left[\begin{array}{ccccc}\mathit{E}& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}\\ \mathrm{0}& {\mathit{E}}_{\mathrm{1}}^c& \mathrm{0}& \cdots & \mathrm{0}\\ ⋮& ⋮& ⋮& \cdots & ⋮\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \cdots & {\mathit{E}}_{n-\mathrm{1}}^c\end{array}\right],$

${\mathit{S}}_{{\theta }_{\mathrm{b}}}=\left[\begin{array}{ccccc}\mathit{E}& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}\\ \mathrm{0}& {\mathit{E}}_{n-\mathrm{1}}^c& \mathrm{0}& \cdots & \mathrm{0}\\ ⋮& ⋮& ⋮& \cdots & ⋮\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \cdots & {\mathit{E}}_{\mathrm{1}}^c\end{array}\right].$(11)

From the shear matrix form obtained above, we can obtain some properties of the shear operator:

Property 1 (Matrix form of ${\mathcal{S}}_{{\theta }_d}$ ) The operator ${\mathcal{S}}_{{\theta }_d}$ is linear, so it can be represented as a matrix denoted by ${\mathit{S}}_{{\theta }_d}$. Moreover, since ${\mathit{S}}_{{\theta }_{\mathit{d}}}$ is a permutation matrix, it is orthonormal.

Property 2 No matter what value ${\theta }_d$ takes, ${\mathit{S}}_{{\theta }_{\mathrm{b}}}={\mathit{S}}_{{\theta }_{\mathrm{t}}}^{\mathrm{T}}$ and ${\mathit{S}}_{{\theta }_{\mathrm{l}}}={\mathit{S}}_{{\theta }_{\mathrm{r}}}^{\mathrm{T}}$ holds.

Property 3 Regardless of d taken, the shear operator with $\theta =\mathrm{45}°$ is equal to the transpose of the shear operator with $\theta =\mathrm{90}°$, i.e. ${\mathcal{S}}_{{\theta }_d}{|}_{\theta =\mathrm{45}°}={\mathcal{S}}_{{\theta }_d}^{\mathrm{T}}{|}_{\theta =\mathrm{90}°}$.

Property 4 When both $\theta$ and $d$ change, the shear operator holds the following equation,

${\mathcal{S}}_{{\theta }_{\mathrm{t}}}{|}_{\theta =\mathrm{45}°}={\mathcal{S}}_{{\theta }_{\mathrm{b}}}{|}_{\theta =\mathrm{90}°},{\mathcal{S}}_{{\theta }_{\mathrm{t}}}{|}_{\theta =\mathrm{90}°}={\mathcal{S}}_{{\theta }_{\mathrm{b}}}{|}_{\theta =\mathrm{45}°},{\mathcal{S}}_{{\theta }_{\mathrm{r}}}{|}_{\theta =\mathrm{45}°}={\mathcal{S}}_{{\theta }_{\mathrm{l}}}{|}_{\theta =\mathrm{90}°},{\mathcal{S}}_{{\theta }_{\mathrm{r}}}{|}_{\theta =\mathrm{90}°}={\mathcal{S}}_{{\theta }_{\mathrm{l}}}{|}_{\theta =\mathrm{45}°}.$(12)

2.2 The Matrix Form and Properties of the SG Operator

Given that the gradient matrix exhibits the BCCB structure under periodic boundary conditions, it can be seen that the matrices of operators ${\mathcal{D}}^{\mathrm{3}}$ and ${\mathcal{D}}^{\mathrm{4}}$ in equation (5) also possess the BCCB structure. The matrix representation of the SG operator ${\mathcal{D}}^{\mathrm{3}}={\mathcal{S}}_{{\theta }_d}^{\mathrm{T}}{\mathcal{D}}_x{\mathcal{S}}_{{\theta }_d}$ with $\theta =\mathrm{45}°$ has the following forms:

${\mathit{D}}^{\mathrm{3}}={\mathit{S}}_{{\theta }_{\mathrm{l}}}^{\mathrm{T}}{\mathit{D}}_x{\mathit{S}}_{{\theta }_{\mathrm{l}}}={\mathit{S}}_{{\theta }_{\mathrm{r}}}^{\mathrm{T}}{\mathit{D}}_x{\mathit{S}}_{{\theta }_{\mathrm{r}}}=\left[\begin{array}{ccccccc}-\mathit{E}& \mathit{E}& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}& \mathrm{0}\\ \mathrm{0}& -\mathit{E}& \mathit{E}& \mathrm{0}& \cdots & \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& -\mathit{E}& \mathit{E}& \cdots & \mathrm{0}& \mathrm{0}\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ \mathit{E}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}& -\mathit{E}\end{array}\right]={\mathit{D}}_x,$

${\mathit{D}}^{\mathrm{3}}={\mathit{S}}_{{\theta }_{\mathrm{t}}}^{\mathrm{T}}{\mathit{D}}_x{\mathit{S}}_{{\theta }_{\mathrm{t}}}=\left[\begin{array}{ccccccc}-\mathit{E}& {\mathit{E}}_{n-\mathrm{1}}^c& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}& \mathrm{0}\\ \mathrm{0}& -\mathit{E}& {\mathit{E}}_{n-\mathrm{1}}^c& \mathrm{0}& \cdots & \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& -\mathit{E}& {\mathit{E}}_{n-\mathrm{1}}^c& \cdots & \mathrm{0}& \mathrm{0}\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ {\mathit{E}}_{n-\mathrm{1}}^c& \mathrm{0}& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}& -\mathit{E}\end{array}\right],{\mathit{D}}^{\mathrm{3}}={\mathit{S}}_{{\theta }_{\mathrm{b}}}^{\mathrm{T}}{\mathit{D}}_x{\mathit{S}}_{{\theta }_{\mathrm{b}}}=\left[\begin{array}{ccccccc}-\mathit{E}& {\mathit{E}}_{\mathrm{1}}^c& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}& \mathrm{0}\\ \mathrm{0}& -\mathit{E}& {\mathit{E}}_{\mathrm{1}}^c& \mathrm{0}& \cdots & \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& -\mathit{E}& {\mathit{E}}_{\mathrm{1}}^c& \cdots & \mathrm{0}& \mathrm{0}\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ {\mathit{E}}_{\mathrm{1}}^c& \mathrm{0}& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}& -\mathit{E}\end{array}\right]$(13)

Similarly, the matrix representation of ${\mathcal{D}}^{\mathrm{4}}={\mathcal{S}}_{{\theta }_d}^{\mathrm{T}}{\mathcal{D}}_y{\mathcal{S}}_{{\theta }_d}$ with $\theta =\mathrm{45}°$ has the following forms:

${\mathit{D}}^{\mathrm{4}}={\mathit{S}}_{{\theta }_{\mathrm{l}}}^{\mathrm{T}}{\mathit{D}}_y{\mathit{S}}_{{\theta }_{\mathrm{l}}}=\left[\begin{array}{ccccccc}-\mathit{E}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}& {\mathit{E}}_{n-\mathrm{1}}^c\\ {\mathit{E}}_{n-\mathrm{1}}^c& -\mathit{E}& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}& \mathrm{0}\\ \mathrm{0}& {\mathit{E}}_{n-\mathrm{1}}^c& -\mathit{E}& \mathrm{0}& \cdots & \mathrm{0}& \mathrm{0}\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \cdots & {\mathit{E}}_{n-\mathrm{1}}^c& -\mathit{E}\end{array}\right],{\mathit{D}}^{\mathrm{4}}={\mathit{S}}_{{\theta }_{\mathrm{r}}}^{\mathrm{T}}{\mathit{D}}_y{\mathit{S}}_{{\theta }_{\mathrm{r}}}=\left[\begin{array}{ccccccc}-\mathit{E}& {\mathit{E}}_{n-\mathrm{1}}^c& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}& \mathrm{0}\\ \mathrm{0}& -\mathit{E}& {\mathit{E}}_{n-\mathrm{1}}^c& \mathrm{0}& \cdots & \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& -\mathit{E}& {\mathit{E}}_{n-\mathrm{1}}^c& \cdots & \mathrm{0}& \mathrm{0}\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ {\mathit{E}}_{n-\mathrm{1}}^c& \mathrm{0}& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}& -\mathit{E}\end{array}\right].$

${\mathit{D}}^{\mathrm{4}}={\mathit{S}}_{{\theta }_{\mathrm{t}}}^{\mathrm{T}}{\mathit{D}}_y{\mathit{S}}_{{\theta }_{\mathrm{t}}}={\mathit{S}}_{{\theta }_{\mathrm{b}}}^{\mathrm{T}}{\mathit{D}}_y{\mathit{S}}_{{\theta }_{\mathrm{b}}}=\left[\begin{array}{ccccccc}-\mathit{E}+{\mathit{E}}_{n-\mathrm{1}}^c& \mathrm{0}& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}& \mathrm{0}\\ \mathrm{0}& -\mathit{E}+{\mathit{E}}_{n-\mathrm{1}}^c& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& -\mathit{E}+{\mathit{E}}_{n-\mathrm{1}}^c& \mathrm{0}& \cdots & \mathrm{0}& \mathrm{0}\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}& -\mathit{E}+{\mathit{E}}_{n-\mathrm{1}}^c\end{array}\right]={\mathit{D}}_y.$(14)

When $\theta =\mathrm{90}°$, ${\mathcal{D}}^{\mathrm{3}}$and ${\mathcal{D}}^{\mathrm{4}}$ also has the following forms:

${\mathit{D}}^{\mathrm{3}}={\mathit{S}}_{{\theta }_{\mathrm{l}}}^{\mathrm{T}}{\mathit{D}}_x{\mathit{S}}_{{\theta }_{\mathrm{l}}}={\mathit{S}}_{{\theta }_{\mathrm{r}}}^{\mathrm{T}}{\mathit{D}}_x{\mathit{S}}_{{\theta }_{\mathrm{r}}}={\mathit{D}}_x,$

${\mathit{D}}^{\mathrm{3}}={\mathit{S}}_{{\theta }_{\mathrm{t}}}^{\mathrm{T}}{\mathit{D}}_x{\mathit{S}}_{{\theta }_{\mathrm{t}}}=\left[\begin{array}{ccccccc}-\mathit{E}& {\mathit{E}}_{\mathrm{1}}^c& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}& \mathrm{0}\\ \mathrm{0}& -\mathit{E}& {\mathit{E}}_{\mathrm{1}}^c& \mathrm{0}& \cdots & \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& -\mathit{E}& {\mathit{E}}_{\mathrm{1}}^c& \cdots & \mathrm{0}& \mathrm{0}\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ {\mathit{E}}_{\mathrm{1}}^c& \mathrm{0}& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}& -\mathit{E}\end{array}\right],{\mathit{D}}^{\mathrm{4}}={\mathit{S}}_{{\theta }_{\mathrm{b}}}^{\mathrm{T}}{\mathit{D}}_x{\mathit{S}}_{{\theta }_{\mathrm{b}}}=\left[\begin{array}{ccccccc}-\mathit{E}& {\mathit{E}}_{n-\mathrm{1}}^{\mathit{c}}& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}& \mathrm{0}\\ \mathrm{0}& -\mathit{E}& {\mathit{E}}_{n-\mathrm{1}}^{\mathit{c}}& \mathrm{0}& \cdots & \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& -\mathit{E}& {\mathit{E}}_{n-\mathrm{1}}^{\mathit{c}}& \cdots & \mathrm{0}& \mathrm{0}\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ {\mathit{E}}_{n-\mathrm{1}}^{\mathit{c}}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}& -\mathit{E}\end{array}\right],$

${\mathit{D}}^{\mathrm{4}}={\mathit{S}}_{{\theta }_{\mathrm{b}}}^{\mathrm{T}}{\mathit{D}}_y{\mathit{S}}_{{\theta }_{\mathrm{b}}}$=${\mathit{S}}_{{\theta }_{\mathrm{t}}}^{\mathrm{T}}{\mathit{D}}_y{\mathit{S}}_{{\theta }_{\mathrm{t}}}={\mathit{D}}_y$

${\mathit{D}}^{\mathrm{4}}={\mathit{S}}_{{\theta }_{\mathrm{l}}}^{\mathrm{T}}{\mathit{D}}_y{\mathit{S}}_{{\theta }_{\mathrm{l}}}=\left[\begin{array}{ccccccc}-\mathit{E}& {\mathit{E}}_{n-\mathrm{1}}^c& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}& \mathrm{0}\\ \mathrm{0}& -\mathit{E}& {\mathit{E}}_{n-\mathrm{1}}^c& \mathrm{0}& \cdots & \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& -\mathit{E}& {\mathit{E}}_{n-\mathrm{1}}^c& \cdots & \mathrm{0}& \mathrm{0}\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ {\mathit{E}}_{n-\mathrm{1}}^c& \mathrm{0}& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}& -\mathit{E}\end{array}\right],{\mathit{D}}^{\mathrm{4}}={\mathit{S}}_{{\theta }_{\mathrm{r}}}^{\mathrm{T}}{\mathit{D}}_y{\mathit{S}}_{{\theta }_{\mathrm{r}}}=\left[\begin{array}{ccccccc}-\mathit{E}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}& {\mathit{E}}_{n-\mathrm{1}}^c\\ {\mathit{E}}_{n-\mathrm{1}}^c& -\mathit{E}& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}& \mathrm{0}\\ \mathrm{0}& {\mathit{E}}_{n-\mathrm{1}}^c& -\mathit{E}& \mathrm{0}& \cdots & \mathrm{0}& \mathrm{0}\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \cdots & {\mathit{E}}_{n-\mathrm{1}}^c& -\mathit{E}\end{array}\right].$(15)

In the same way, we can also deduce the properties of the SG operator from its matrix form mentioned above.

Property 5 The shear operator${\mathcal{S}}_{{\theta }_d}$ and the horizontal gradient operator ${\mathcal{D}}_x$ are commutative when $d="\mathrm{l}"\mathrm{o}\mathrm{r}"\mathrm{r}"$; Similarly, ${\mathcal{S}}_{{\theta }_d}$ and the vertical operator ${\mathcal{D}}_y$ are commutative when d= "t" or "b". i.e.,

${\mathcal{S}}_{{\theta }_d}^{\mathrm{T}}{\mathcal{D}}_x{\mathcal{S}}_{{\theta }_d}={\mathcal{D}}_x\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n}d="\mathrm{l}"\mathrm{o}\mathrm{r}"\mathrm{r}"$(16)

${\mathcal{S}}_{{\theta }_d}^{\mathrm{T}}{\mathcal{D}}_y{\mathcal{S}}_{{\theta }_d}={\mathcal{D}}_y\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n}d="\mathrm{t}"\mathrm{o}\mathrm{r}"\mathrm{b}"$(17)

Proof   When $\theta =\mathrm{45}°$ and $d="\mathrm{l}"\mathrm{o}\mathrm{r}"\mathrm{t}"$, the desired results can be obtained by multiplying the three matrices, i.e., ${\mathit{S}}_{{\theta }_{\mathrm{l}}}^{\mathrm{T}}{\mathit{D}}_x{\mathit{S}}_{{\theta }_{\mathrm{l}}}$and ${\mathit{S}}_{{\theta }_{\mathrm{t}}}^{\mathrm{T}}{\mathit{D}}_y{\mathit{S}}_{{\theta }_{\mathrm{t}}}$:

${\mathit{S}}_{{\mathit{\theta }}_{\mathrm{l}}}^{\mathrm{T}}{\mathit{D}}_{\mathit{x}}{\mathit{S}}_{{\mathit{\theta }}_{\mathrm{l}}}=\left[\begin{array}{ccccc}{\mathit{E}}_{\mathrm{11}}& {\mathit{E}}_{\mathrm{22}}& {\mathit{E}}_{\mathrm{33}}& \cdots & {\mathit{E}}_{nn}\\ {\mathit{E}}_{nn}& {\mathit{E}}_{\mathrm{11}}& {\mathit{E}}_{\mathrm{22}}& \cdots & {\mathit{E}}_{(n-\mathrm{1})(n-\mathrm{1})}\\ ⋮& ⋮& ⋮& \cdots & ⋮\\ {\mathit{E}}_{\mathrm{22}}& {\mathit{E}}_{\mathrm{33}}& {\mathit{E}}_{\mathrm{44}}& \cdots & {\mathit{E}}_{\mathrm{11}}\end{array}\right]\left[\begin{array}{ccccccc}-\mathit{E}& \mathit{E}& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}& \mathrm{0}\\ \mathrm{0}& -\mathit{E}& \mathit{E}& \mathrm{0}& \cdots & \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& -\mathit{E}& \mathit{E}& \cdots & \mathrm{0}& \mathrm{0}\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ \mathit{E}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}& -\mathit{E}\end{array}\right]\left[\begin{array}{ccccc}{\mathit{E}}_{\mathrm{11}}& {\mathit{E}}_{nn}& {\mathit{E}}_{(n-\mathrm{1})(n-\mathrm{1})}& \cdots & {\mathit{E}}_{\mathrm{22}}\\ {\mathit{E}}_{\mathrm{22}}& {\mathit{E}}_{\mathrm{11}}& {\mathit{E}}_{nn}& \cdots & {\mathit{E}}_{(n-\mathrm{1})(n-\mathrm{1})}\\ ⋮& ⋮& ⋮& \cdots & ⋮\\ {\mathit{E}}_{nn}& {\mathit{E}}_{(n-\mathrm{1})(n-\mathrm{1})}& {\mathit{E}}_{(n-\mathrm{2})(n-\mathrm{2})}& \cdots & {\mathit{E}}_{\mathrm{11}}\end{array}\right]={\mathit{D}}_x,$

${\mathit{S}}_{{\theta }_{\mathrm{t}}}^{\mathrm{T}}{\mathit{D}}_y{\mathit{S}}_{{\theta }_{\mathrm{t}}}=\left[\begin{array}{ccccc}\mathit{E}& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}\\ \mathrm{0}& {\mathit{E}}_{\mathrm{1}}^c& \mathrm{0}& \cdots & \mathrm{0}\\ ⋮& ⋮& ⋮& \cdots & ⋮\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \cdots & {\mathit{E}}_{n-\mathrm{1}}^c\end{array}\right]\left[\begin{array}{ccccccc}-\mathit{E}+{\mathit{E}}_{n-\mathrm{1}}^c& \mathrm{0}& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}& \mathrm{0}\\ \mathrm{0}& -\mathit{E}+{\mathit{E}}_{n-\mathrm{1}}^c& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& -\mathit{E}+{\mathit{E}}_{n-\mathrm{1}}^c& \mathrm{0}& \cdots & \mathrm{0}& \mathrm{0}\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}& -\mathit{E}+{\mathit{E}}_{n-\mathrm{1}}^c\end{array}\right]\left[\begin{array}{ccccc}\mathit{E}& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}\\ \mathrm{0}& {\mathit{E}}_{n-\mathrm{1}}^c& \mathrm{0}& \cdots & \mathrm{0}\\ ⋮& ⋮& ⋮& \cdots & ⋮\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \cdots & {\mathit{E}}_{\mathrm{1}}^c\end{array}\right]={\mathit{D}}_y$(18)

Similarly, the same result can be obtained when the angle and d are taken in other cases.

Property 6 When $\theta$ is the same, there are some equality relationships between the SG operator ${\mathcal{D}}^{\mathrm{3}}$ and ${\mathcal{D}}^{\mathrm{4}}$, i.e.

${\mathcal{S}}_{{\theta }_{\mathrm{t}}}^{\mathrm{T}}{\mathcal{D}}_x{\mathcal{S}}_{{\theta }_{\mathrm{t}}}={\mathcal{S}}_{{\theta }_{\mathrm{r}}}^{\mathrm{T}}{\mathcal{D}}_y{\mathcal{S}}_{{\theta }_{\mathrm{r}}}\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n}\theta =\mathrm{45}°$(19)

${\mathcal{S}}_{{\theta }_{\mathrm{b}}}^{\mathrm{T}}{\mathcal{D}}_x{\mathcal{S}}_{{\theta }_{\mathrm{b}}}={\mathcal{S}}_{{\theta }_{\mathrm{l}}}^{\mathrm{T}}{\mathcal{D}}_y{\mathcal{S}}_{{\theta }_{\mathrm{l}}}\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n}\theta =\mathrm{90}°$(20)

Property 7 When the direction d of two shear operators is same and the angle difference between them is $\frac{\mathrm{3}\mathrm{\pi }}{\mathrm{4}}$, the two shear operators are equal $\left({\theta }_d=k\frac{\mathrm{\pi }}{\mathrm{4}}\left(k=\mathrm{0,1},\cdots ,n\right)\right)$. The same is true of the SG operator. That is, the shear operator and the SG operator is periodic, i.e

$\mathrm{1}){\mathcal{S}}_{{\theta }_d}{|}_{\theta =\mathrm{0}°}={\mathcal{S}}_{{\theta }_d}{|}_{\theta =\mathrm{135}°}=\mathit{E},{\mathcal{S}}_{{\theta }_d}{|}_{\theta =\mathrm{45}°}={\mathcal{S}}_{{\theta }_d}{|}_{\theta =\mathrm{180}°},{\mathcal{S}}_{{\theta }_d}{|}_{\theta =\mathrm{90}°}={\mathcal{S}}_{{\theta }_d}{|}_{\theta =\mathrm{225}°}.$

$\mathrm{2}){\mathcal{S}}_{{\theta }_d}^{\mathrm{T}}{\mathcal{D}}_x{\mathcal{S}}_{{\theta }_d}{|}_{\theta =\mathrm{0}°}={\mathcal{S}}_{{\theta }_d}^{\mathrm{T}}{\mathcal{D}}_x{\mathcal{S}}_{{\theta }_d}{|}_{\theta =\mathrm{135}°}={\mathcal{D}}_x,{\mathcal{S}}_{{\theta }_d}^{\mathrm{T}}{\mathcal{D}}_x{\mathcal{S}}_{{\theta }_d}{|}_{\theta =\mathrm{45}°}={\mathcal{S}}_{{\theta }_d}^{\mathrm{T}}{\mathcal{D}}_x{\mathcal{S}}_{{\theta }_d}{|}_{\theta =\mathrm{180}°},{\mathcal{S}}_{{\theta }_d}^{\mathrm{T}}{\mathcal{D}}_x{\mathcal{S}}_{{\theta }_d}{|}_{\theta =\mathrm{90}°}={\mathcal{S}}_{{\theta }_d}^{\mathrm{T}}{\mathcal{D}}_x{\mathcal{S}}_{{\theta }_d}{|}_{\theta =\mathrm{225}°}.$

$\mathrm{3}){\mathcal{S}}_{{\theta }_d}^{\mathrm{T}}{\mathcal{D}}_y{\mathcal{S}}_{{\theta }_d}{|}_{\theta =\mathrm{0}°}={\mathcal{S}}_{{\theta }_d}^{\mathrm{T}}{\mathcal{D}}_y{\mathcal{S}}_{{\theta }_d}{|}_{\theta =\mathrm{135}°}={\mathcal{D}}_y,{\mathcal{S}}_{{\theta }_d}^{\mathrm{T}}{\mathcal{D}}_y{\mathcal{S}}_{{\theta }_d}{|}_{\theta =\mathrm{45}°}={\mathcal{S}}_{{\theta }_d}^{\mathrm{T}}{\mathcal{D}}_y{\mathcal{S}}_{{\theta }_d}{|}_{\theta =\mathrm{180}°},{\mathcal{S}}_{{\theta }_d}^{\mathrm{T}}{\mathcal{D}}_y{\mathcal{S}}_{{\theta }_d}{|}_{\theta =\mathrm{90}°}={\mathcal{S}}_{{\theta }_d}^{\mathrm{T}}{\mathcal{D}}_y{\mathcal{S}}_{{\theta }_d}{|}_{\theta =\mathrm{225}°}.$(21)

3 ${\mathit{L}}_{\mathrm{1}}/{\mathit{L}}_{\mathrm{2}}-\mathbf{S}\mathbf{T}\mathbf{V}$ Model and Algorithm

In this section, we will propose a deblurring model that combines STV norm with $L_{\mathrm{1}}/L_{\mathrm{2}}$ minimization. Considering the advantages of STV norm and the SG operator, we propose the following optimization model:

$\underset{\mathit{u}}{\mathrm{m}\mathrm{i}\mathrm{n}}{\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{4}}}\frac{{\Vert {\mathit{D}}^i\mathit{u}\Vert }_{\mathrm{1}}}{{\Vert {\mathit{D}}^i\mathit{u}\Vert }_{\mathrm{2}}}\mathrm{s}.\mathrm{t}\mathit{K}\mathit{u}=\mathit{f}$(22)

where $\frac{{\Vert \cdot \Vert }_{\mathrm{1}}}{{\Vert \cdot \Vert }_{\mathrm{2}}}$ represents the ratio of the $L_{\mathrm{1}}$ and $L_{\mathrm{2}}$ norm.

Two significant advantages of $L_{\mathrm{1}}/L_{\mathrm{2}}$ minimization in optimization problem (22) are the inherent scale invariance and the unconstrained parameters. However, the non-convex nonlinearity of the $L_{\mathrm{1}}$ to $L_{\mathrm{2}}$ ratio introduces complexity to the model solution and the theoretical analysis of convergence. To solve this problem and ensure the robustness of the optimization process, we introduce the ADMM framework. ADMM significantly reduces the complexity of problems by decomposing them into multiple simple sub problems and solving them alternately, making it particularly suitable for tasks such as image deblurring and denoising. In addition, ADMM can effectively handle non-convex optimization problems, enabling it to perform excellently in image processing. Specifically, the original problem is neatly transformed into a constrained optimization problem:

$\underset{\mathit{u}}{\mathrm{m}\mathrm{i}\mathrm{n}}{\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{4}}}\frac{{\Vert {\mathit{P}}_i\Vert }_{\mathrm{1}}}{{\Vert {\mathit{Q}}_i\Vert }_{\mathrm{2}}}\mathrm{s}.\mathrm{t}\mathit{K}\mathit{u}=\mathit{f},{\mathit{P}}_i={\mathit{D}}^i\mathit{u},{\mathit{Q}}_i={\mathit{D}}^i\mathit{u}$(23)

The corresponding augmented Lagrange function of (23) can be written as

$\begin{array}{l}{\mathrm{ℒ}}_{{\rho }_{\mathrm{1}},{\rho }_{\mathrm{2}},{\rho }_{\mathrm{3}}}\left(\mathit{u},{\mathit{P}}_i,{\mathit{Q}}_i,{\mathit{\mu }}_i,{\mathit{\nu }}_i,\mathit{\lambda }\right)\\ ={\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{4}}}\frac{{\Vert {\mathit{P}}_i\Vert }_{\mathrm{1}}}{{\Vert {\mathit{Q}}_i\Vert }_{\mathrm{2}}}+〈\mathit{\lambda },\mathit{K}\mathit{u}-\mathit{f}〉+\frac{{\rho }_{\mathrm{1}}}{\mathrm{2}}{\Vert \mathit{K}\mathit{u}-\mathit{f}\Vert }_F^{\mathrm{2}}+{\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{4}}}\left\{〈{\mathit{\mu }}_i,{\mathit{D}}^i\mathit{u}-{\mathit{P}}_i〉+{\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{4}}}〈{\mathit{\nu }}_i,{\mathit{D}}^i\mathit{u}-{\mathit{Q}}_i〉+\frac{{\rho }_{\mathrm{2}}}{\mathrm{2}}{\Vert {\mathit{D}}^i\mathit{u}-{\mathit{P}}_i\Vert }_F^{\mathrm{2}}+\frac{{\rho }_{\mathrm{3}}}{\mathrm{2}}{\Vert {\mathit{D}}^i\mathit{u}-{\mathit{Q}}_i\Vert }_F^{\mathrm{2}}\right\}\end{array}$

$={\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{4}}}\frac{{\Vert {\mathit{P}}_i\Vert }_{\mathrm{1}}}{{\Vert {\mathit{Q}}_i\Vert }_{\mathrm{2}}}+\frac{{\rho }_{\mathrm{1}}}{\mathrm{2}}{\Vert \mathit{K}\mathit{u}-\mathit{f}+\frac{\mathit{\lambda }}{{\rho }_{\mathrm{1}}}\Vert }_F^{\mathrm{2}}+{\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{4}}}\left\{\frac{{\rho }_{\mathrm{2}}}{\mathrm{2}}{\Vert {\mathit{D}}^i\mathit{u}-{\mathit{P}}_i+\frac{{\mathit{\mu }}_i}{{\rho }_{\mathrm{2}}}\Vert }_F^{\mathrm{2}}+\frac{{\rho }_{\mathrm{3}}}{\mathrm{2}}{\Vert {\mathit{D}}^i\mathit{u}-{\mathit{Q}}_i+\frac{{\mathit{\nu }}_i}{{\rho }_{\mathrm{3}}}\Vert }_F^{\mathrm{2}}\right\},$(24)

where ${\rho }_{\mathrm{1}},{\rho }_{\mathrm{2}}$ and ${\rho }_{\mathrm{3}}$ are three penalty parameters, Variables $\mathit{\lambda },{\mathit{\mu }}_i$ and ${\mathit{\nu }}_i$ are the Lagrange multipliers. For the optimization problem (24), our strategy is to delicately decompose it into a series of independent sub-problems, thus minimizing the separation of each sub-problem. Based on this, we employ the ADMM algorithm as follows:

$\begin{array}{l}{\mathit{u}}^{k+\mathrm{1}}=\underset{\mathit{u}}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}\frac{{\rho }_{\mathrm{1}}}{\mathrm{2}}{\Vert \mathit{K}\mathit{u}-\mathit{f}+\frac{{\mathit{\lambda }}^k}{{\rho }_{\mathrm{1}}}\Vert }_F^{\mathrm{2}}\\ +{\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{4}}}\left\{\frac{{\rho }_{\mathrm{2}}}{\mathrm{2}}{\Vert {\mathit{D}}^i\mathit{u}-{\mathit{P}}_i^k+\frac{{\mathit{\mu }}_i^k}{{\rho }_{\mathrm{2}}}\Vert }_F^{\mathrm{2}}+\frac{{\rho }_{\mathrm{3}}}{\mathrm{2}}{\Vert {\mathit{D}}^i\mathit{u}-{\mathit{Q}}_i^k+\frac{{\mathit{\nu }}_i^k}{{\rho }_{\mathrm{3}}}\Vert }_F^{\mathrm{2}}\right\}\end{array}$(25)