Abstract

Total variation regularization is well-known for recovering sharp edges; however, it usually produces staircase artifacts. In this paper, in order to overcome the shortcoming of total variation regularization, we propose a new variational model combining high-order total variation regularization and regularization. The new model has separable structure which enables us to solve the involved subproblems more efficiently. We propose a fast alternating method by employing the fast iterative shrinkage-thresholding algorithm (FISTA) and the alternating direction method of multipliers (ADMM). Compared with some current state-of-the-art methods, numerical experiments show that our proposed model can significantly improve the quality of restored images and obtain higher SNR and SSIM values.

1. Introduction

Image restoration is an important and fundamental problem in the literature of image processing. It is widely used in many fields, such as machine identification, biomedicine, astronomy, and medical imaging [14]. The purpose of image restoration is to get a better visual image from the degraded image. There are many factors that cause image degradation, for instance, atmospheric turbulence, camera shake, and the relative motion between an imaging device and an object [5].

The typical representation of image degradation is the blurring and additional noise of image. The original model of image degradation can generally be defined aswhere is an original image, is a blur operator, is an additive noise, and is a degraded image. Our aim is to recover from , which is known as deconvolution or deblurring.

Recovering from is a typically ill-posed problem. So we cannot solve it simply by using the normal least-squares method. In order to overcome the ill-condition of the problem, adding some regularization terms to the energy functional is usually adopted. One of the commendable regularization methods is Tikhonov regularization [6]. However, this method often tends to make images overly smooth and fails to preserve image details. In order to overcome this shortcoming, in 1992, replacing -norm of with the TV-norm of , Rudin, Osher, and Fatemi in [7] introduced the following famous Rudin-Osher-Fatemi (ROF) model:where denotes the Euclidean norm, the first term of (2) is called fidelity term, and is the total variation regularization term. is a positive regularization parameter which controls the trade-off between the first and second terms. The isotropic discrete total variation regularization term can be defined as where denotes the discrete gradient operator which is defined as follows: with for . Here refers to the entry of the vector . It is the pixel location of the image; see [8].

As well known, TV function is nondifferentiable and nonlinear; it is difficult to solve the ROF model (2). A number of efficient and robust methods are proposed to solve the ROF model. In [7], by seeking the steady-state solution of a parabolic PDE, Rudin et al. proposed a time marching scheme to solve the associated Euler-Lagrange equation of (2). In [8, 9], a lagged diffusivity fixed point iteration method was proposed to solving the same problem. Later, Newton’s method [1015] and Chambolle’s projection algorithm [1618] were proposed. Another class of algorithms for TV problems are the iterative shrinkage/thresholding (IST) algorithms, which are independently proposed and analyzed by several authors in different fields [1924]. Recently, some first-order optimization methods were proposed, such as fast TV deconvolution algorithm (FTVd) [25], augmented Lagrangian method (ALM), and split Bregman method [2629]. Alternating direction method (ADM) [3032], which is a variant of the ALM, goes back to the work of Gabay and Mercier [33]. The ADM was first studied extensively in variational optimization problems and then introduced to the fields of image processing [30, 31]. There are also many other methods for image restoration, such as partial differential equation (PDE) based methods [3440] and the neural network methods [41, 42].

Recently, to further study the solving methods for ROF model, Huang et al. [46] introduced a new auxiliary variable and proposed a fast total variation minimization method as follows:where , are positive regularization parameters. The main purpose of this model (6) is to realize the separation of denoising and deblurring steps. The authors solved system (6) by an alternating minimization algorithm. The experimental results show the effectiveness of their method. In the process of image processing, there are some abnormal values, such as the sudden jump of pixels in a certain area. However, regularization is less sensitive to abnormal values and promotes sparse solutions. In [44], an unconstrained regularized minimization problem was proposed:where , , are positive regularization parameters. Due to and norm used in model (7), the experimental results show a good recovery effect.

It has been shown in [47, 48] that total variation methods can realize significantly sharper edges and overall more visually pleasing images. However, on the other hand, they tend to create piecewise-constant images even in regions with smooth transitions of grey or color values in the original image. The undesired artifact is usually called staircase effect. Staircase solution fails to satisfy the evaluation of visual quality and develops false edges that do not exist in the true image.

To overcome the adverse effects in the application of image processing, a popular method is to replace the TV norm by a higher-order TV norm [45, 4952]. Lysaker et al. first proposed second-order total variation regularizer in [53] for additive noise removal in the process of medical image processing. Their method shows robustness in the protection of the edges. Later, a second-order model to substantially reduce the staircase effect was proposed by Chan et al. in [48]. In particular, the second-order TV regularization schemes are widely studied to overcome the staircase effects while preserving the edges well in the restored image.

Recently, Lv et al. in [45] proposed a high-order total variation minimization model:where , are positive regularization parameters and denotes higher-order total variation. The minimization model can reduce the staircase effect in the process of image restoration. The authors adopted an alternating minimization algorithm to solve the high-order minimization problem (8). Their experimental results showed that the proposed method gave the restored images with higher quality than some existing first-order total variation restoration methods.

Motivated by the works of [4446], we propose a new unconstrained minimization model:where , , and are positive regularization parameters. The definition of second-order total variation is similar to the definition of the first-order one: where denote the second-order discrete gradient of the entry of the vector . For more details about the second-order difference, refer to [54]. The second-order TV regularization and regularization are used; the edges in the restored image can be preserved quite well and the staircase effect is reduced simultaneously. An alternating minimization method is used to solve the new unconstrained minimization model (9). By comparing with the other existing methods about total variation, our numerical experiments show the effectiveness of the minimization model (9).

The rest of this paper is organized as follows. In Section 2, we propose our alternating minimization algorithm to solve model (9), which is based on the fast iterative shrinkage-thresholding algorithm (FISTA) and the alternating direction method of multipliers (ADMM). In Section 3, we give some numerical results to demonstrate the effectiveness of the proposed algorithm. Finally, concluding remarks of this paper are given in Section 4.

2. Alternating Minimization Method

In this section, we apply the ADMM idea to derive an algorithm for solving problem (9). We first introduce an auxiliary variable to transform problem (9) into an equivalent constrained optimization problem,and then minimize the augmented Lagrangian function of problem (11) by using the fast iterative shrinkage-thresholding algorithm and the alternating direction multiplier method.

For the above constrained optimization problem (11), its augmented Lagrange function is defined byThen, using the alternating minimization method to solve problem (12) can be expressed as follows:

Fixing , , the subproblem can be obtained by solving the following minimization problem. Recently, various acceleration techniques of iterative algorithms are proposed [5563]. For this subproblem, in order to accelerate the convergence of the above iteration, we adopt the acceleration technique in [55] to solve it.where is a Lipschitz constant [55], and . So, the minimizer of is given by the one-dimensional shrinkage:

Fixing , , the subproblem for is equivalent to According to [64], the solution of can be obtained by the two-dimensional shrinkage:where the convention is followed.

Fixing , the subproblem for can be written as follows:We can get the exact minimizer solution from the optimal conditions of (19) as follows:where refers to the first-order difference operator and is the adjoint of . represents the identity matrix. Assuming that the problem satisfies the periodic boundary condition, then is block circulant matrix [65, 66], which can be diagonalized by the two-dimensional discrete Fourier transform. So, by using the convolution theorem, we getwhere “” denotes complex conjugacy, “” denotes component-wise multiplication, , and is the inverse Fourier transform.

Finally, we update Lagrange multiplier :Our method in this paper can be described as shown in Algorithm 1.

Input: , , , , , and
Initialization:, ,
While   and , Do
Compute from (14).
   ,
Compute from (17).
   ,
Compute via solving (20).
,
Update from (22).
  End Do

3. Numerical Experiments

In this section, we present some numerical examples of image restoration to illustrate the effectiveness of our proposed approach. In our experiments, we compare the proposed algorithm with the other three state-of-the-art methods, namely, FCP [44], high-order TV [45], and SALSA [43], under three different blurring operators. All experiments are performed under Windows 10 and MATLAB 2012a running on a desktop with an Intel® Core i5 Duo central processing unit at 2.50 GHz and 4 GB memory. The signal-to-noise ratio (SNR), the structural similarity index measurement (SSIM) [67], and the blurred signal to noise ratio (BSNR) are employed to measure the quality of the restored images by different algorithms. They are defined as follows: where , , and are the original image, the restored image, and the noise vector added in the test, respectively. is the mean intensity value of , and are the mean value of and , respectively. and are the variance of and , respectively. And is covariance of and . and are constants. The SSIM is an index used to measure the similarity between restored image and ideal image. The closer the SSIM value to 1, the better the restored image.

Three classical grayscale images, named, “Lena” (256 256), “Man” (512 512), and “Cameraman” (256 256), are used as the test images and shown in Figure 1. To make it easier to compare different methods, we set as the stopping criterion in all experiments.

3.1. Parameters Settings

In this subsection, we discuss the selection of the three regularization parameters , , and in our method. In order to explain how to choose parameters in our algorithm, we recover the “Lena” image, which is blurred by Gaussian blur with size “9 9” and added the Gaussian white noise with BSNR=35. The influence of different , , and on the restoration image is explained in Figure 2. As shown in Figure 2, if is too small, the restored image will be overly smoothed with smeared image details. In contrast, the restored image will be dominated by noise component with larger values. We can see from Figures 2(e) and 2(f) that when the parameter changes within a certain range, there is little effect on image restoration. The parameter has the same effect as the one of has. The restored image will be overly smoothed with small values and dominated by noise component with larger values. The parameters , , and in our method are manually adjusted to the optimal value. Therefore, the parameters used in the tests will be illustrated in the following experiments.

3.2. Gaussian Blur

In this example, we consider the Gaussian blur. The “Man” image shown in Figure 3(b) is degraded by Gaussian blur. The function has two parameters: bandwidth and standard deviation . In the test, we choose . We added Gaussian noise with BSNR = 35 to the test image. The parameters and are selected in SALSA. The parameters , , and are selected in our method. The parameters , , and are used in FCP method. For the high-order TV method, the parameters and are selected. The restored images obtained by SALSA [43], FCP [44], high-order TV [45], and our method are shown in Figures 3(c)3(f). The changing curve of SNR of the reconstructed image during the experiment is shown in Figure 4. It can be seen from the changing graph that our method can get higher SNR than the other three methods. In Table 1, we list the restoration results of the four different methods in SNRs and SSIMs. From Figures 3(c)3(f), we can see that the SALSA method can make restoration image smooth, the FCP method causes staircase effect, and our method can induce sparse structure of the image compared with the high-order TV method. The visual quality of the restored image by our method is better than the other three methods. More precisely, the same regions of image are zoomed and processed in the same way; they are presented in Figure 5. It is not difficult to see that our method has some advantages over the other three methods in eliminating the staircase effect.

3.3. Average Blur

In this example, we choose “Cameraman” image with size as an object of study. The image is degraded by average kernel with window size 11 and contaminated by the Gaussian noise with BSNR = 40, which is shown in Figure 6(b). In this example, we choose the parameters and for SALSA method. The parameters ,, and are used in our method. The parameters , , and are selected in FCP method. For high-order TV, the parameters and are used. Figures 6(c)6(f) represent the restored images by SALSA, FCP, high-order TV, and our method, respectively. The change of SNR of the reconstructed image during the experiment is shown in Figure 7. The experimental results on SNR and SSIM are shown in Table 1. We see that both the SNR and SSIM values of the restored image by our method are higher than those provided by the SALSA [43], FCP [44], and high-order TV [45] method. The quality of restored images by our method is better than the other three methods.

Our method has some advantages in detail processing. In Figure 6, we have enlarged some details of image. The recovery results are presented in Figures 8(b)8(e). Obviously, our method can well overcome the staircase effect and retain more image details.

3.4. Disk Blur

For this experiment, we consider the well-known “Lena” as the test image. The image is corrupted by the Gaussian noise with BSNR = 35. The PSF for the linear disk blur represented a line segment of radius . In this experiment, we select . The original image and the observed image are shown in Figures 9(a) and 9(b). The parameters , , and are selected in our method. In parameters, we set and for SALSA method. The parameters , , and are used in FCP method. For the high-order TV method, the parameters and are selected. The restored images by the SALSA, FCP, high-order TV, and our method are shown in Figures 9(c)9(f). In Figure 10, we plot the changes of SNR value versus iteration number for the four methods. Figure 11 shows the enlarged results of a part of Figure 9. The values of SNR and SSIM are listed in Table 1. We can see that our method can reduce the staircase effect effectively while preserving the edges.

We can see from the above three experiments that the images restored by our method have better qualities with less staircase effect and more detail textures than the other three methods. In order to further verify the effectiveness of our method, for the same experiment, we tested all the images which are listed in Figure 1, and the experimental results are shown in Table 1. We can find from the experimental results that the SNR and SSIM values obtained by our method are higher than the other three methods.

4. Conclusion

In this paper, by combining high-order total variation regularization and regularization, we propose a new variational model. Numerical experiments show that our method can overcome the staircase effects while preserving edges of the restored image. We also show that our proposed method can obtain better results than the three state-of-the-art methods with higher SSIM and SNR values.

Data Availability

The test images in the article can be freely downloaded from this site “http://sipi.usc.edu/database/”.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Training Program of the Major Research Plan of National Science Foundation of China under Grant 91746104, by National Science Foundation of China under Grant 61101208 and Grant 11326186, by Qingdao Postdoctoral Science Foundation, China (2016114), by a project of Shandong Province Higher Educational Science and Technology Program, China (J17KA166), and by Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources, Shandong Province of China, and SDUST Research Fund (2014TDJH102).