# Parameter selection and solution algorithm for TGV-based image restoration model

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## Abstract

In this paper, image restoration problem is formulated to solve a total generalized variation (TGV)-based minimization problem. The minimization problem includes an unknown regularization parameter. A Morozov’s discrepancy principle-based method is used to choose a suitable regularization parameter. Computationally, by introducing two dual variables, the TGV-based image restoration problem is reformulated as a convex-concave saddle-point problem. Meanwhile, the Chambolle–Pock’s first-order primal–dual algorithm is transformed into a different equivalent form which can be seen as a proximal-based primal–dual algorithm. Then, the different equivalent form is used to solve the saddle-point problem. At last, compared with several existing state-of-the-art methods, experimental results demonstrate the performance of our proposed method.

## Keywords

Image restoration Total generalized variation (TGV) Morozov’s discrepancy principle Primal–dual algorithm## 1 Introduction

*H*is a linear blurring operator, and

*n*1 represents the Gaussian white noise with variance \(\sigma ^{2}\). Our goal is to obtain the original image

*u*from the blurred image

*g*. The problem of finding

*u*from (1) is a discrete linear inverse problem. The TV-based model [2] has been proposed to solve this inverse problem. The TV-based image restoration model can be written as

*u*, and \(\lambda >0\) is a regularization parameter.

*w*represents an approximation of the first-order gradient \(\nabla u\), \(\alpha _{1}\) and \(\alpha _{0}\) are the constants, \(\varepsilon (w)=\frac{1}{2}(\nabla w+\nabla w^{T})\) is a symmetrized derivative.

The regularization parameter plays an important role in the TGV-based image restoration problem. If the regularization parameter is too large, the regularized solution is under-smoothed, whereas if regularization parameter is too small, the regularized solution does not fit the given data properly. Recently, Bioucas-Dias et al. [14] proposed the majorization–minimization (MM) method to estimate the regularization parameter. Liao et al. [15] utilized the generalized cross-validation (GCV) method to estimate the regularization parameter. Babacan et al. [16] proposed the parameter estimation using the variational distribution approximations, which considered the Gamma distribution for the prior. Meanwhile, the discrepancy principle has been used to the problem of TV-based regularization parameter selection [17, 18, 19, 20].

Our main contributions in this paper can be summarized into three aspects listed as follows. First, we consider the choice of regularization parameter in the TGV-based image restoration problem. Secondly, the Morozov’s discrepancy principle is used to choose a suitable regularization parameter. Thirdly, a proximal-based primal–dual algorithm is used to solve the TGV-based image restoration problem.

The outline of this paper is organized as follows. In Sect. 2, the TGV-based image restoration model is introduced. In Sect. 3, when the regularization parameter is fixed, the numerical Algorithm 1 is given, and when the regularization parameter is chosen by the Morozov’s discrepancy principle, the numerical Algorithm 2 is given. In Sect. 4, experimental results are given to show that Morozov’s discrepancy principle-based parameter selection method is effective in improving the restoration effect. Finally, some discussions are given in Sect. 5.

## 2 The TGV-based image restoration model

*p*and

*q*are the introduced dual variables. The indicator function \(\delta _{P}(p)\) and \(\delta _{Q}(q)\) are defined as

*T*represents the transpose operation. For \(u,v\in {\mathbb {R}}^{m\times n}\), the inner product and induced norm are defined as

## 3 Two numerical algorithms

In this section, the Chambolle–Pock’s first-order primal–dual algorithm is firstly transformed into a different equivalent form which can be seen as a proximal-based primal–dual algorithm. Then, two proximal-based primal–dual algorithms are proposed. On the one hand, when the regularization parameter \(\lambda\) is fixed, Algorithm 1 is given. On the other hand, when the regularization parameter \(\lambda\) is chosen by the Morozov’s discrepancy principle, Algorithm 2 is given.

### 3.1 The transformed Chambolle–Pock’s first-order primal–dual algorithm

*x*is primal variable,

*y*is dual variable,

*F*and

*G*are proper, convex, lower-semicontinuous functions, \(F^{*}\) is the convex conjugate of

*F*, and \(<.,.>\) represents the inner product,

*K*is a linear operator. Here, the Chambolle–Pock’s first-order primal–dual algorithm is transformed into a different equivalent form which can be written as

*O*(1 /

*n*) when \(\theta =1\) and the step sizes satisfy \(s~t<1/\left\| K\right\| _{2}^{2}\).

### 3.2 The first numerical algorithm

*I*is the identity matrix. It is easy to check that (6) completely fits into the framework of (15).

When the regularization parameter \(\lambda\) is fixed, the resulting proximal-based primal–dual algorithm is summarized into Algorithm 1.

*u*; the subproblem for

*u*can be written as

### 3.3 The Morozov’s discrepancy principle

Here, the basic theory of choosing the regularization parameter by the Morozov’s discrepancy principle is firstly introduced. Then, when the regularization parameter is chosen by the Morozov’s discrepancy principle, the numerical Algorithm 2 is given.

*u*is always in a feasible region

*D*. The feasible region

*D*can be written as

### 3.4 The second numerical algorithm

When the regularization parameter \(\lambda\) is chosen by the Morozov’s discrepancy principle, the resulting proximal-based primal–dual algorithm is summarized into Algorithm 2.

*u*can be written as

*w*can be written as

*p*can be written as

*P*, which can be computed by

*q*, we have

### 3.5 Convergence analysis

*O*(1 /

*n*) when \(\theta =1\) and the step sizes satisfy \(s~t\left\| K\right\| _{2}^{2}<1\). Thus, an estimate on \(\left\| K\right\| _{2}^{2}\) is needed. According to the definition of the divergence operator \(\mathrm{div}\), it is easy to show that

## 4 Experimental results

*u*denotes the original clean image, \(\widehat{u}\) is the restored image. Generally speaking, the larger the SNR, PSNR and ISNR, and the lower the MSE, the better the performance.

Figure 1 consists of the original Lena, Barbara and Cameraman images, and the blurred Lena, Barbara and Cameraman images. In Fig. 1, the first row are the original three images, the second row are the blurred three images under the uniform blur of size \(9\times 9\), and the third row are the blurred three images under the Gaussian blur of size \(9\times 9\) with variance 9.

The restored Lena images obtained by four different methods under the uniform blur of size \(9\times 9\) are provided in Fig. 2. Table 1 is the restoration effect comparison of Lena image under the uniform blur of size \(9\times 9\) and the Gaussian blur of size \(9\times 9\) with variance 9. By combining with Fig. 2 and Table 1, it is clear to see that our proposed method is effective in improving the restoration effect. And the SNR and PSNR values of our proposed method are 3–4 dB higher than the MM, TV-LR and TGV-fixed methods.

In order to show that the TGV-based image restoration model and our method can effectively eliminate the staircase effect and preserve the edge of image, the samples of the red bounding box in Fig. 3 are taken for experiments. The locally enlarged restored Barbara images using four different methods under the uniform blur of size \(9\times 9\) are provided in Fig. 5. From Fig. 5a, b, it is clear to see that the MM and TV-LR methods lead to very blocky images in the smooth regions. From Fig. 5c, it is clear to see that the TGV-fixed method has the advantage for eliminating the staircase effect, but it causes the unexpected edge blurring. Figure 5d is the locally enlarged restored Barbara image obtained by our method. It is clear to see that our method can effectively restore the patterned fabric. In a word, our method can effectively eliminate the staircase effect, and it is able to overcome the blocky images while preserving edge details.

The restoration effect comparison of Lena image under the uniform blur of size \(9\times 9\) (BSNR\(=\)40) and the Gaussian blur of size \(9\times 9\) with variance 9 (BSNR\(=\)40)

The restoration effect comparison of Barbara image under the uniform blur of size \(9\times 9\) (BSNR\(=\)40) and the Gaussian blur of size \(9\times 9\) with variance 9 (BSNR\(=\)40)

The restoration effect comparison of Cameraman image under the uniform blur of size \(9\times 9\) (BSNR\(=\)40) and the Gaussian blur of size \(9\times 9\) with variance 9 (BSNR\(=\)40)

## 5 Conclusion

In this paper, the choice of regularization parameter in the TGV-based image restoration model is considered, and the regularization parameter is chosen by the Morozov’s discrepancy principle. The TGV-based image restoration problem is reformulated as a saddle-point problem. When the regularization parameter is chosen by our proposed method, the proximal-based primal–dual algorithm is applied to solve the saddle-point problem. For the subproblems of the primal variables and dual variables, each subproblem has a closed-form solution. Therefore, the regularization parameter associated with the solution of the restored image is chosen by the Morozov’s discrepancy principle in the each iteration. Experimental results show that the Morozov’s discrepancy principle-based regularization parameter selection method can effectively improve the effect of TGV-based image restoration in terms of SNR, PSNR and MSE quality, compared with several existing state-of-the-art methods.

## Notes

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest, whether financial or non-financial.

### Human participants and animals

This research did not involve human participants and animals.

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