Adaptive Image Restoration via a Relaxed Regularization of Mean Curvature

Ma, Mingxi; Zhang, Jun; Deng, Chengzhi; Liu, Zhaoyang; Wang, Yuanyun

doi:https://doi.org/10.1155/2020/3416907

Mathematical Problems in Engineering

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Advances in Numerical Optimisation: Theory, Models, and Applications

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Research Article | Open Access

Volume 2020 | Article ID 3416907 | https://doi.org/10.1155/2020/3416907

Adaptive Image Restoration via a Relaxed Regularization of Mean Curvature

Mingxi Ma,¹Jun Zhang,^1,2Chengzhi Deng,¹Zhaoyang Liu,¹and Yuanyun Wang¹

Academic Editor: S. A. Edalatpanah

Received22 Jul 2020

Revised29 Oct 2020

Accepted03 Nov 2020

Published01 Dec 2020

Abstract

In this paper, a new relaxation model based on mean curvature for adaptive image restoration is proposed. To solve the problem efficiently, an alternating direction method of multipliers (ADMMs) is proposed. Furthermore, a rigorous convergence theory of the proposed algorithm is established. We also give the complexity analysis of our proposed method. Experimental results are provided to demonstrate the effectiveness and efficiency of the proposed method over a state-of-the-art method on synthetic and natural images.

1. Introduction

Image restoration has always been a research hotspot in the field of image processing. To solve this inverse problem, one of the most popular research directions is applying variational regularization methods. It mainly includes total variation (TV) regularization [1–3], nonlocal regularization [4], sparse regularization [5], higher-order regularization based on higher-order derivatives [6–9], and fractional-order regularization based on fractional-order derivatives [10, 11].

As a classic regularization method, TV regularization has been successfully applied to image restoration, image segmentation, image reconstruction, image decomposition, and phase recovery. When using it for image denoising, the edge-preserving effect is better. For the TV-based image denoising problem, researchers have proposed a variety of fast and efficient numerical algorithms, such as dual algorithm [12], primal-dual algorithm [13], split Bregman method [14], augmented Lagrangian method [15, 16], alternating direction method of multipliers (ADMMs) [17], proximity algorithm [18], domain decomposition method [19], and nonlinear multigrid method [3]. However, after denoising with TV regularization, the image often exhibits staircase effects in the flat area. Second, this type of regularization model will cause image contrast loss.

To reduce the staircase effects, a number of higher-order models have been developed [20, 21]. Among them, let us specially note the mean curvature model [9, 22] and Euler’s elastica model [23–27], which are nonconvex so that the numerical algorithms converge to the local optimal solution. The mean curvature-based image denoising model (called the MC model) can be formulated as follows:where and represent the restored image and the noisy image, is the gradient operator, the weighting parameter controls the amount of denoising, and is the mean curvature of the image surface . As mentioned in [28], the MC model preserves image contrast better for cleaned images than the TV model. In addition, this model handles corners better for synthetic images than the TV and Euler’s elastica models. In view of a number of valuable properties of the MC model, designing fast iterative algorithms for its solution has important theoretical and practical significance. There are some different augmented Lagrangian methods for solving the MC model. In [28], original unconstrained problem (1) was transformed into a constrained optimization problem and the saddle-points of the corresponding augmented Lagrangian functional are obtained from solving several subproblems alternatively. Some subproblems had closed form solutions while others were solved by fast Fourier transform (FFT). Subsequently, in [29], the solutions of two subproblems were obtained by a Gauss–Seidel method. In addition to the reduction of computational cost, this approach can be applied to computational domains with nonperiodic boundaries. Besides, a fast linearized augmented Lagrangian method was proposed to further improve the convergence rates [30]. In this method, all subproblems have closed form solutions. However, due to the nonconvexity of the MC model, it is still very hard to find suitable model parameter and penalization parameters.

By choosing the regularization parameter based on the inverse gradient, in [31], Thanh et al. proposed some adaptive image restoration methods. Besides, in [32, 33], the authors proposed the adaptive regularization and the adaptive weighted regularization for image denoising, respectively. In [34], a novel adaptive image denoising method was proposed, which combines a Tchebichef moment-based sparse regularizer with an adaptive steerable total variation regularizer.

By minimizing the norm of the gradient of the unit normal, Duan et al. proposed a novel variational model as follows [8]:and proved this model with the ability to preserve image contrast and sharp edges during denoising. Due to the high nonlinearity and nonconvexity of its regularization term, the relaxation form of model (2) was studied. Namely, they proposed a spatially adaptive hybrid regularization-based minimization problem, which can be given aswhere operator represents the second-order gradient, and . For solving this relaxation model, the ADMM-based numerical algorithm was presented and its convergence was proved [8, 35]. ADMM is a fast method for solving many regularization problems. It has been widely applied in many fields, such as hyperspectral image unmixing [36], classification [37], and fusion [38].

Inspired by the successful application of the relaxation technique, in this paper, we present a new form for the MC model (1) and propose its relaxation model. Moreover, we propose an efficient ADMM with guaranteed convergence. Compared with the ADMM-based algorithm for solving model (3), our method is faster.

The organization of the paper is as follows: In Section 2, the new form of the MC model (1) is presented, and then its geometric properties and its relaxation form together with connections to the existing models are introduced. Section 3 details an efficient ADMM for solving the proposed relaxation model. The convergence and complexity of the proposed algorithm are also studied in this section. Experimental results on synthetic and natural images are presented in Section 4 to show the effectiveness and efficiency of our proposed method. Finally, Section 5 concludes the paper.

2. Adaptive TV and Laplacian Regularization Model

In this section, we first give a new form for the MC model (1). Then, we introduce its geometric properties for image denoising. Finally, we propose a relaxation version using a spatially adaptive TV and Laplacian regularization and analyze the connections to the existing models.

2.1. New Form for the MC Model

Let us first review the MC model.

For a given image where is the image domain, the corresponding image surface is or . The aim of denoising is to find a piecewise smooth surface to approximate it. Meanwhile, we would like to preserve the geometric features, such as edges of objects and corners. We also expect to preserve image contrasts. To this end, the norm of mean curvature of the image surface was used in [9]. In the following, to introduce the mean curvature of surface , a level set function is considered. Obviously, its zero level set corresponds to the surface. Then, the mean curvature of this surface can be defined as

Using the norm of mean curvature as a regularization term, the MC model was proposed as follows:

Sincewe can obtain a new form of the MC model which is

Because this model is nonconvex, high order, and highly nonlinear, its complete study is nontrivial. However, one can get some insight into its features.

2.2. Features of Model (7)

In this subsection, the features of model (7) will be introduced. One can refer to [9] for more details.

If and for each point in a region, the evolution equation corresponding to the gradient descent flow of model (7) behaves like the biharmonic heat equation in that region. Owing to the property that the biharmonic heat equation can damp high frequency signals quickly, model (7) can remove noise efficiently.

In order to illustrate why model (7) can preserve image contrasts, one may consider a simple image defined on with being the characteristic function, being an open disk in centered at the origin with radius , and . Since is radial symmetric, and one can approximate it by a sequence of smooth radial symmetric function . Referring to [9], the properties of preserving image contrasts and sharp edges about model (7) are guaranteed by the following lemma and theorem:

Lemma 1. Assuming that is an image defined as above, we havewhere .

The integral indicates that the regularizer of model (7) does not rely on heights of signals. With the conclusion of this lemma, one can further obtain the following theorem:

Theorem 1. Let be an image defined as above. Then, there exists a constant such that if , is a minimizer of model (7).

This theorem demonstrates that model (7) is able to preserve image contrasts, once is chosen to be small enough. In contrast, a loss of image contrast may appear while using the ROF model, for whatever small . Besides, this theorem illustrates that the model can keep sharp edges as the ROF model.

To illustrate why the model can preserve object corners, one needs to consider a different image defined on with [9]. Similar results are presented in the following lemma and theorem:

Lemma 2. Let be an image defined on a rectangle with . Then,

Theorem 2. Let be defined as the above lemma, then there exists a constant such that if , is a minimizer of model (7).

Theorem 2 indicates another key feature of model (7) on preserving object edges once is small enough.

2.3. Proposed Relaxation Model and Connections to the Previous Models

Although the new form of the MC model has good properties in preserving image contrasts, sharp edges, and object corners, it is still nonconvex and highly nonlinear. Motivated by the success of relaxation in [8], by relaxing the regularization term, we reformulate (7) into a spatially adaptive TV and Laplacian regularization model, which can be expressed aswhere

It is straightforward to see that model (10) is just the relaxation version of the MC model (1). In [39], an effective model which combines the total variation with the Laplacian regularizer (TVL) was introduced:where and represent the blurring operator and the Abel transform, respectively. and are two constants. The regularization of our model (10) can be seen as an adaptive form of this TVL regularization. Replacing in (10) by , it becomes the spatially adaptive first- and second-order regularization model (3) proposed in [8]. Compared with the isotropic model, the corresponding anisotropic model is faster when it achieves almost the same denoising effect. Therefore, in order to improve the computational complexity, it is reasonable to use the Laplacian operator to replace the isotropic second-order gradient operator .

3. The Proposed Algorithm

In this section, we propose an efficient numerical algorithm based on the popular ADMM for solving the proposed model (10). Besides, we analyze its convergence and complexity.

3.1. Algorithm Description

Firstly, we introduce two auxiliary variables and to reformulate (10) as follows:

The augmented Lagrangian functional corresponding to (13) iswhere are Lagrangian multipliers and are positive penalty parameters. By applying the ADMM, one needs to minimize three subproblems with other variables fixed and update two Lagrangian multipliers at each iteration which iswhere are three penalty parameters and can be explicitly updated as

To solve the -subproblem in (15), we consider its Euler–Lagrange equation, which has the formnamely,

In order to efficiently solve this nonlinear equation, we use the fast Fourier transform (FFT) with periodic boundary conditions. More concretely, is given bywhere is the FFT.

Both and subproblems have closed form solutions. Specifically, the solutions of -subproblem and -subproblem are given by the shrinkage operator:andwhere . Here, denotes the component-wise multiplication.

We summarize the overall algorithm for solving the spatially adaptive relaxation model (10) in Algorithm 1.

	Step 1. Input . Initialization: , . Let .
	Step 2. Compute by (19).
	Step 3. Update using according to (16).
	Step 4. Compute by (20).
	Step 5. Compute by (21).
	Step 6. Update by the last two equations of (15).
	Step 7. Stop if the stopping criterion is satisfied. Otherwise, let , and go to Step 2.

3.2. Convergence Analysis

In this subsection, we give a convergence theory for the proposed algorithm (Algorithm 1) under the assumption that a solution of the proposed model (10) exists.

Theorem 3. Assume that is a sequence generated by Algorithm 1 and are the points satisfying the following first-order optimality conditions:

Let be the error sequences. If for any , and ,are nonnegative. Then, the generated sequence converges to a limit point which satisfies the first-order optimality conditions (22).

Our proposed model (10) can be seen as a variation of the previous model in [8]. Specifically, the operator in (3) is changed to . Furthermore, an ADMM-based algorithm same to [8] is applied to solve the proposed model (10). Therefore, the rigorous convergence proof of Algorithm 1 can refer to the proof presented in [8].

3.3. Complexity Analysis

In the following, we will give some simple analysis of complexity for the proposed method.

For our proposed method, the cost of updating is at each iteration, where is the total number of grid points. If a root operation is equal to 12 times addition or subtraction or multiplication or division, at each iteration it will spend ( times root operations), , , and operations on updating , respectively. Therefore, the total cost of one step is , which gives rise to the complexity.

4. Numerical Experiments

In this section, we give some experimental results for the proposed model (10) in terms of our proposed algorithm. For simplicity of presentation, we call it SATVL in the following. We will compare our results with those of the previous algorithm for solving the spatially adaptive TV and high-order TV regularization model (3), which we call . It should be noted that the authors in [8] proposed the same ADMM for solving model (3). All numerical experiments are conducted in MATLAB environment on a PC with a 3.85 GHz CPU processor. To assess the denoising performance qualitatively, the peak signal to noise ratio (PSNR) and the structural similarity (SSIM) [40] are adopted. Their definitions are expressed aswhere and represent the original clear image and the restored image, respectively, and denote the local mean values of images and , and express the standard deviations, signifies the covariance value between and , and are two constants which can avoid near-zero denominator values. In general, higher PSNR and SSIM values signify better performance in image denoising.

The relative error of the solution is defined by

It can give some important information about the convergence of iterations. Therefore, we track its value during iteration process. In experiments, the stopping criterion is or the iteration reaches 500.

The test images are shown in Figure 1, which includes one synthetic and four natural images. Since the proximal terms are added for convergence analysis, we set .

(a)

(b)

(c)

(d)

(e)

The values of parameters significantly affect the convergence rates and other characteristics of the methods. In our experiments, we study the influence of parameters on the . Concretely speaking, after taking a reasonable guess for the parameters: , we consider different values of . We observe which value of could contribute to the biggest value of PSNR, and then we set it to be the value of . And in turn, we use the same strategy for the choice of other parameters. Similar strategy is adopted for tuning the parameters of our SATVL. For the parameter tuning, the algorithms are terminated after the stopping criterion is satisfied.

Experiment 1. In this example, we use the shape image as the test image. We first consider to restore the image corrupted by Gaussian white noise with . For the , we set . For our SATVL, we use . Besides, the Gaussian white noises at levels and are added for comparison, respectively. The numerical results are arranged in Table 1, while the corresponding visual results for noise level are shown in Figure 2. In Table 1, Iter. and CPU (s) denote the number of iterations and the CPU time in seconds required for the whole denoising process.

(a)

(b)

(c)

(d)

(e)

Experiment 2. In our second example, the pepper image is adopted as the test image. The original image is degraded by Gaussian white noise with . The parameters are set to be for the and for the SATVL. Then, we consider more Gaussian noises at and , respectively. Here, the values of parameters need to be reset. Numerical results are also listed in Table 1. The visual results about noise level are presented in Figure 3.

(a)

(b)

(c)

(d)

(e)

Experiment 3. In our third example, we test the cameraman image. The original image is corrupted by Gaussian white noise with . Here, parameters for the and our SATVL are set to be and , respectively. In order to show better visual results, we also test the cases of noise levels and . Results of this experiment are given in Table 1. The visual results about the noise level are presented in Figure 4.

(a)

(b)

(c)

(d)

(e)

(f)

Experiment 4. In this example, we choose the Barbara image. Same to the above three experiments, we test the noisy images with different noise levels. Related numerical results are reported in Table 1, and the visual results about the noise level are displayed in Figure 5.

(a)

(b)

(c)

(d)

(e)

(f)

Experiment 5. In the last example, the owl image is used. To show the advantage of our proposed method, we also test noisy images with three different noise levels. For each algorithms, related parameters have been tuned to obtain the best restoration performance uniformly.
In terms of PSNR and SSIM, the denoising results by these two methods are similar. It is easy to see from Figures 2–5 that the visual results of the SATVL are comparable to the . As shown in Table 1, our SATVL is faster than the . Experiments have shown this situation at different noise levels. From the plots of PSNR and relative error versus CPU time in Figures 2–5, we can also draw this conclusion. Besides, it is shown from these plots that the SATVL has converged, just as the did. Actually, it is clear that the energy and PSNR have converged to a steady state.

5. Conclusion

In this paper, we have proposed a new relaxation model based on mean curvature for adaptive image denoising. We analyzed the connections to the existing models. Furthermore, we proposed an efficient ADMM to solve the proposed relaxation model and studied its convergence and complexity. Compared with the previous method for an adaptive first- and second-order regularization-based image denoising model, experimental results show that our proposed method is fast. Obviously, the proposed regularization term can be applied to other image processing problems, such as image deblurring, and image segmentation.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Science and Technology Project of Jiangxi Provincial Department of Education (GJJ171015), the Science Foundation for Post Doctorate of China (2020M672484), the Natural Science Foundation of Jiangxi Province (20192BAB211005 and 20192BAB217003), and the NNSF of China (61865012 and 61861032).

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Copyright © 2020 Mingxi Ma et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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