Mathematical Problems in Engineering
Volume 2016 (2016), Article ID 6132356, 11 pages
http://dx.doi.org/10.1155/2016/6132356

Research Article

Solving Adaptive Image Restoration Problems via a Modified Projection Algorithm

Hao Yang,1,2 Xueping Luo,³ and Leiting Chen1,2

¹School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China
²Sichuan Provincial Key Laboratory of Digital Media, Chengdu, Sichuan 611731, China
³College of Computer Science and Technology, Southwest University for Nationalities, Chengdu, Sichuan 610041, China

Received 26 January 2016; Accepted 9 March 2016

Academic Editor: Daniel Zaldivar

Copyright © 2016 Hao Yang 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.

Abstract.

We introduce a new general TV regularizer, namely, generalized TV regularization, to study image denoising and nonblind image deblurring problems. In order to discuss the generalized TV image restoration with solution-driven adaptivity, we consider the existence and uniqueness of the solution for mixed quasi-variational inequality. Moreover, the convergence of a modified projection algorithm for solving mixed quasi-variational inequalities is also shown. The corresponding experimental results support our theoretical findings.

1. Introduction

Digital image restoration plays an important role in many applications of sciences and engineering such as medical and astronomical imaging, film restoration, and image and video coding. Recovering an image from a degraded image is usually an ill-posed inverse problem and it should be dealt with through selecting a suitable regularizer. Since the work of Rudin, Osher, and Fatemi (ROF) in [1], the regularization methods based on total variation (TV) have known a success, mostly due to their ability to preserve edges in the image. In recent years, a number of research works have been proposed in the field of TV regularization approaches, which are used for the task of image denoising and nonblind image deblurring. Aujol et al. [2] replaced norm of the data fidelity term by norm to modify the ROF functional model. The TV regularization approaches in [3, 4] can be described by means of locally dependent constraint sets; that is, the functional is adaptive to the input data. Another class of approaches are the nonlocal methods [5, 6] including nonlocal variants of TV regularization. [7–9] extended TV regularization to second- or higher-order cases. These works mentioned above considered TV regularization approaches for solving the image denoising problems. In addition, these approaches can also be utilized for the image deblurring; see, for example, [10–12]. Chambolle [10] proposed algorithm for minimizing the TV model and applied the algorithm to image zooming. A TV deblurring approach with adaptive choice of the regularization parameter was presented in [12]. In all these literatures, the image restoration problem is always regarded as optimization problem using discrete TV regularization. How to solve such optimization problem with a TV regularization, which is fundamental and crucial, is the core problem in our discussion.

It is well known that the theory of variational inequality has been developed as a class of important tools for the study of minimization problems; see, for example, [13]. Among such variational inequalities, inverse variational inequalities, mixed variational inequalities, and quasi-variational inequalities are very significant generalizations, which have been applied to a wide range of problems, such as mechanics, economics, finance, optimal control, and transportation. References [14, 15] proposed Tikhonov regularization method and a general regularization method for solving inverse variational inequality problems. Luo and Yang [16, 17] further extended the results of [14, 15] to the inverse mixed variational inequality problems. The generalized quasi-variational inequality problem was introduced in [18]. However, to our knowledge, a few works implemented quasi-variational inequality to deal with image restoration problem. Recently, Lenzen et al. [19–21] firstly considered a class of quasi-variational inequalities for studying adaptive image restoration, where adaptivity is solution-driven adaptivity. Moreover, they showed that a lot of experimental results support their theoretical findings.

Inspired and motivated by the works of [19, 20], in this paper, we introduce a general TV regularization which includes TV regularization of the classical ROF model [1] as its special case. For solving the minimization problem with generalized TV regularization, we discuss its dual problem, which is like the following formulation:

where is a convex constraint set. For generalizing the regularization approach to solution-driven adaptivity, we find a fixed point of the following mapping:

The above fixed point problem is equivalent to solving a mixed quasi-variational inequality [22]. We provide the existence and uniqueness of a fixed point for the mixed quasi-variational inequality for adaptive image restoration. Thus, our theoretical results generalize the research works of [19]. Meanwhile, we propose a modified projection algorithm for solving mixed quasi-variational inequality and prove its convergence. Finally, we give improved experimental results compared to the experiments presented in [19]. Moreover, our experimental results show that the solution-driven adaptive generalized TV model produces excellent restoration effects for different test images.

The rest of this paper is organized as follows. In Section 2, we recall some notations concerned with generalized -projection operator. In Section 3, we introduce the generalized TV regularization which covers other TV regularizers given in literature. Our model of solution-driven adaptivity described by means of mixed quasi-variational inequalities is shown in Section 4. We consider the theoretical results in Section 5, where we prove the existence and uniqueness of the solution for mixed quasi-variational inequality. In Section 6, we present a modified projection algorithm and its convergence. We give a lot of numerical experiments supporting our theoretical results and showing our better improvement in Section 7. Finally, we conclude this paper in Section 8.

2. Preliminaries

In this section, we recall the concept of the generalized -projection operator, together with its properties.

Let be a nonempty closed convex subset of Let be a function defined as follows:

where ,, is a positive number, and is a proper, convex, and lower semicontinuous function.

Definition 1. The generalized -projection operator is defined as

From Lemmas and of [25], we know that is a single valued and nonexpansive mapping and if and only if

3. Generalized TV Regularization

In this section, we introduce a new variational approach for image denoising and nonblind image deblurring that is based on total variation regularization. Our general approach covers various adaptive and anisotropic types of TV regularization approaches.

The image deblurring problem formulation is as follows. Let be a degraded noisy image, which is obtained from a noise-free image by convolution with a blurring kernel , followed by an addition of Gaussian noise; that is,

where is invertible matrix and is a Gaussian random variable with zero mean. The above problem is a typical inverse problem. In order to recover from , assuming that is a mapping from , we aim at considering the following optimization problem:

In particular, if , the minimization problem (7) reduces to image denoising problem, where denotes an identity mapping.

Now we denote by the discretization of the divergence operator div and denote by norm. Let us define the following generalized total variation regularizer:

where

for with and is -dimensional closed ball with radius centered at 0. Then the optimization problem is given as

Problem (10) includes a large variety of problems as its special cases:

(i)If , then (10) reduces to problem (2.12) of [19].(ii)If and , then (10) reduces to the classical ROF model of [1].

We derive the corresponding dual problem of (10) as follows. The optimality condition for reads

It follows from (11) that

where Using the abbreviation , from (11) and (12) we obtain the dual problem

When maximizing over , the constant term can be omitted without changing the optimum. Moreover, the maximization of equals the minimization of , and we can formulate the dual problem of (10) as

with , where each local constraint set is a -dimensional closed ball.

From a solution of the dual problem, we can retrieve the solution of the primal problem by Therefore, the key issue in our discussion of generalized TV image restoration problem is to solve the above minimization problem (15).

4. Solution-Driven Adaptivity

In [20], Lenzen et al. proposed a kind of adaptivity, where the constraint set depends on the unknown solution of the problem. Naturally, the adaptivity is determined by the noise-free image , which can be obtained by Moreover, the experimental results of [19] showed that the adaptivity was improved by solution-driven model. In the following, in order to study generalized TV image restoration with solution-driven adaptivity, we generalize problem (15) by introducing a dependency of on the dual variable: find a fixed point of the mapping

Having found a fixed point , the corresponding constraint set is ; that is, the adaptivity is solution-driven. Solving problem (16) is equivalent to considering the following mixed quasi-variational inequality (MQVI):

where and

From (5), it is easy to see that the MQVI (17) is equivalent to the following projection equation:

5. Theory for MQVI

In this section, we provide the existence and uniqueness results for the MQVI (17).

5.1. Existence of Solutions

Theorem 2. Let and , where is a linear operator. Let be defined as follows:

where each , , has the following properties:

(i)For fixed the set is a closed convex subset of (ii)There exists such that, for all , one has (iii)There exists such that, for every and every , one has In particular, is nonempty.(iv)The generalized -projection operator of onto for a fixed is continuous with respect to

Then mixed quasi-variational inequality (17) has a solution.

Proof. Firstly, we know from the definition of that

Thus, from assumption (ii), we immediately derive that

Since is a bounded closed convex ball, is compact convex.
Assumptions (i) and (iii) imply that is a nonempty closed convex valued mapping on Moreover, is a proper, convex, and lower semicontinuous function. Hence the generalized -projection operator is well defined. By and (iv), we obtain that is continuous with respect to
Now we define the mapping by

It follows from the continuity of and that is continuous. Hence by (18) and the Brouwer fixed point theorem, we have that problem (17) has a solution.

In the case that , a similar result of Theorem 2 was obtained (see Proposition of [19]). Therefore, Theorem 2 can also be considered as a generalization of Proposition of [19].

5.2. Uniqueness Result of the Proposed Approach

In this subsection, let us consider the uniqueness results for MQVI (17). In [19], Lenzen et al. discussed the uniqueness of solution of quasi-variational inequality on only a subspace of , because is not strongly monotone on the null space of (). On the other hand, our main aim is to find , which does not depend on the component of in In view of these reasons mentioned, [19] restricted the problem of quasi-variational inequality to the complement of Now we utilize similar method to study problem (17). Firstly, we give the following search model that is restricted to :

Theorem 3. Assume that the set depends only on

(i)Let be a solution to the restricted problem (23), and then any is a solution to the original problem (17).(ii)Let be a solution to the unrestricted problem (17), and then any is a solution to the restricted problem (23).

Proof. Let be a solution to the restricted problem (23); that is,

For any such that , it holds that Now let be arbitrary. We decompose into , where and Then it follows from (25) and and that

Thus, is a solution of (17).
Let be a solution of problem (17). In particular, We consider the decomposition , where and Then

Let be arbitrary. There exists such that

It follows from (28) that

where the last inequality holds since due to and solves (17). Thus is a solution of (23).

From Section 3, we can see that the final purpose of finding the existence and uniqueness of is to solve the optimal problem (10), because which does not depend on We therefore focus on , which depends only on the component of in ; that is, we only need to consider the restricted problem (23). Based on Theorem 3, the restricted problem has a solution if and only if the original problem has a solution. Here we specify In the following discussion, denotes a nonempty, closed, convex set such that

Before showing the uniqueness, let us define such that if and only if and is a solution to the following MQVI:

Theorem 4. Under all the assumption conditions of Theorem 2 and the assumptions

(i) is Lipschitz continuous with Lipschitz constant , that is,

(ii) the generalized -projection operator is Lipschitz continuous with respect to with the variation rate , that is,

(iii),

if is a solution of mixed quasi-variational inequality (17), then is unique.

Proof. Fix Let

(i)If , we obtain

(ii)If , since , , solve , the following MQVI

holds; that is, for any ,

In particular, , and it follows from (35) and (ii) that

On the other hand, implies that

Therefore, from (34) and (37), we have

that is,

and, by dividing by , because ,

Since is arbitrarily large, we find

By (i), we have Thus, it follows from (36) that

that is,

Therefore, we can see from (iii) that is a contractive mapping. Moreover, the Banach fixed point theorem implies that there exists a unique fixed point of in

We already mentioned that, in the considered applications for image restoration, we are actually interested in the variable Obviously, it follows from Theorem 4 that is unique.

6. Numerics

Throughout this section, assume that all the assumption conditions of Theorems 2 and 4 are satisfied. Next, we propose an iterative algorithm to solve the MQVI (17) and show convergence of the proposed algorithm.

6.1. Proposed Iterative Algorithm

See Algorithm 1.

Input: The maximal number of iterations ; the constraint set ; the starting point

; the parameter number and such that .

Output: .

begin

        for to do

             Step  1.   ;

             Step  2.   ;

             Step  3.  If , for all , then, stop; otherwise, and go the Step .

        end

end

Algorithm 1: Modified projection method for solving the MQVI (17).

6.2. Convergence of Algorithm 1

Theorem 5. Let all the assumptions of Theorems 2 and 4 be satisfied. Moreover, assume that is monotone; that is,

Then the sequence generated by Algorithm 1 converges to the unique solution of MQVI (17).

Proof. For the proof, see Theorem   in [26].

7. Experiment Results

7.1. Improvement of Solution-Driven Adaptive TV Regularization

In this section, we show that the adaptivity is improved by switching from a solution-driven adaptive TV regularization to a solution-driven adaptive generalized TV model. To this end, we consider the image Cameraman. We generate test data for the denoising problem by adding Gaussian noise with zero mean and standard deviation 0.1 and for the deblurring problem by applying a blurring operator and adding Gaussian noise with zero mean and standard deviation 0.01. For comparison, we make use of a mean SSIM (MSSIM) index [27] to evaluate the restored image quality in Figures 1 and 2. The definitions of the similarity measures SSIM and MSSIM are given as follows:

where , are the average values of two signals and , and are the variances of and , and , are variables to stabilize the division with weak denominator

where and are the reference and the distorted images, respectively; and are the image contents at the th local window; and is the number of local windows of the image.

(a)

(b)

(c)

Figure 1: (a) Noisy input data, MSSIM = 0.327. (b) Solution-driven adaptive TV, MSSIM = 0.812. (c) Solution-driven adaptive generalized TV, MSSIM = 0.822.

(a)

(b)

(c)

Figure 2: (a) Blurred input data, MSSIM = 0.032. (b) Solution-driven adaptive TV, MSSIM = 0.866. (c) Solution-driven adaptive generalized TV, MSSIM = 0.868.

In Figure 1, we give the close-up of the results of denoising the image Cameraman with TV regularization and generalized TV regularization approaches. The values for similarity given show that generalized TV regularization approach enhances the reconstruction compared to the TV regularization.

In Figure 2, close-up of the results of deblurring the image Cameraman with TV regularization and generalized TV regularization approaches is shown. In terms of similarity to the original data, we can see that generalized TV regularization approach improves the reconstruction compared to the TV regularization.

7.2. Comparison of Image Restoration for Different Test Images

In this subsection, we report that our method can remove Gaussian noise efficiently while preserving details very well. For noise removal, we compare our method with some recently developed models, namely, the local TV [28], the NLTV [29], the anisotropic TV [30], and the isotropic TV [30], when the Split Bregman method is considered. The quality of our restoration is measured by the peak signal-to-noise ratio (PSNR) in decibels (dB):

where indicates the image size and and denote the original image and the restored one, respectively. Generally, the higher PSNR values indicate better quality of the restored images.

The test images are shown in Figure 3.

(a)

(b)

(c)

(d)

Figure 3: Test images. (a) Cameraman. (b) Lena. (c) Pepper. (d) Building.

In Figure 4, we make use of the local TV model, the NLTV model, the anisotropic TV model, the isotropic TV model, and the solution-driven adaptive generalized TV model to display the restoration results for the test image (a) corrupted by 8% Gaussian noise. Here, our proposed model performs significantly better and can suppress the Gaussian noise successfully.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4: (a) Noisy image (PSNR = 8.748 dB). (b) Local TV, Split Bregman (PSNR = 28.18 dB). (c) Nonlocal TV, Split Bregman (PSNR = 30.35 dB). (d) Anisotropic TV, Split Bregman (PSNR = 27.687 dB). (e) Isotropic TV, Split Bregman (PSNR = 28.594 dB). (f) Solution-driven adaptive generalized TV, modified projection (PSNR = 30.56 dB).

Table 1 lists the restoration results in the PSNR of different methods for test images Cameraman, Lena, Pepper, and Building corrupted by Gaussian noise.

Table 1: Comparison of restoration results in PSNR (dB) for images corrupted by Gaussian noise.

Methods
Cameraman
Lena
Pepper
Building

Local TV
28.18
27.68
26.68
27.55

NLTV
30.35
29.87
30.86
29.23

Anisotropic TV
27.687
26.1
26.591
29.3

Isotropic TV
28.594
27.902
27.661
26.603

Proposed
30.56
30.21
30.98
29.5

From Table 1, it is apparent that our proposed method generates the best restoration results for all the test images. Actually, in Table 1, the proposed method obtains the highest PSNR values for all the test images. This demonstrated that our method is more robust for images corrupted by Gaussian noise.

7.3. Expanded Experiments

Remote sensing very often deals with inverting a forward model. To this aim, one has to produce an accurate and robust model able to predict physical, chemical, geological, or atmospheric parameters from spectra, such as surface temperature, water vapour, and ozone; see, for example, [31]. Denoising images could be part of the accurate and robust model. In particular, our image restoration method could be used to remove noises in the class of satellite images.

In Figure 5, we give the close-up of the results of denoising the satellite image adding Gaussian noise with the standard derivation Though the fourth-order PDE model removes noise more thoroughly, texture detail also had been changed. Comparing rivers and streets with those in the original image, some information was lost. Our method not only suppresses the noise very well but also keeps a lot of detail and texture information of the original image.

(a)

(b)

(c)

(d)

Figure 5: (a) Original satellite image. (b) Gaussian noise () image. (c) Fourth-order PDE model. (d) Proposed method.

From Table 2, we can see that our method has better denoising effects than pure anisotropic diffusion and fourth-order PDE.

Table 2: Comparison of restoration results in PSNR (dB) for satellite images.

Methods

Pure anisotropic diffusion [23]
30.16
23.68
15.36
6.10

Fourth-order PDE [24]
31.64
25.40
17.40
11.36

Proposed
34.32
27.68
19.54
13.28

8. Conclusion

In this paper, our main contribution is to introduce the generalized TV regularization which includes various types of TV regularization as special cases. We studied the existence and uniqueness of the solution for mixed quasi-variational inequality for solution-driven adaptive image denoising and image deblurring with generalized TV regularization. The convergence of proposed algorithm for MQVI was proposed. Moreover, our experimental results showed that we improve the image restoration quality and apply the proposed model to deal with the images in many different fields. Our further work will consider utilizing the theory of other variational inequalities for solving optimization problems with other kinds of generalized TV regularizations.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work was supported by National Natural Science Foundation of China (Grants 61379019 and 11401493), Innovation Team Funds of Southwest University for Nationalities (Grant 14CXTD03), Innovative Research Team of the Education Department of Sichuan Province (Grant 15TD0050), The Science and Technology Department of Sichuan Province (Grant 2015JY0027), and Fundamental Research Funds for the Central Universities, Southwest University for Nationalities (Grant 2014NZYQN29).

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