Applied Computational Intelligence and Soft Computing

Volume 2017, Article ID 2520301, 9 pages

https://doi.org/10.1155/2017/2520301

## A Regular -Shrinkage Thresholding Operator for the Removal of Mixed Gaussian-Impulse Noise

School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai, China

Correspondence should be addressed to Han Pan; nc.ude.utjs@napnah

Received 5 April 2017; Accepted 12 June 2017; Published 12 July 2017

Academic Editor: Ridha Ejbali

Copyright © 2017 Han Pan 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

The removal of mixed Gaussian-impulse noise plays an important role in many areas, such as remote sensing. However, traditional methods may be unaware of promoting the degree of the sparsity adaptively after decomposing into low rank component and sparse component. In this paper, a new problem formulation with regular spectral -support norm and regular -support norm is proposed. A unified framework is developed to capture the intrinsic sparsity structure of all two components. To address the resulting problem, an efficient minimization scheme within the framework of accelerated proximal gradient is proposed. This scheme is achieved by alternating regular -shrinkage thresholding operator. Experimental comparison with the other state-of-the-art methods demonstrates the efficacy of the proposed method.

#### 1. Introduction

Image restoration [1–4] attempts to recover a clear image from the observations of real scenes. As a fundamental procedure, it has been applied to various application areas, such as image fusion [5] and action recognition [6]. However, typically, the noise characteristics of imaging camera is completely or partially unknown. Among these, the removal of mixed noise has not been investigated because the noise model is not easy to establish accurately.

Recently, a patch based method [7] for video restoration has attracted much attention [8–10]. This method also is extended to video in-painting for archived films. However, the mechanisms of modeling the sparsity level of the grouping patches remain unclear.

To deal with the lack of adaptivity in sparsity level [7], a robust video restoration algorithm is proposed. The main idea of the proposed method is to model the sparsity levels of the low rank component by regular spectral -support norm and sparse component by regular -support norm. Specially, a new problem formulation is presented, where the objective function is minimized under an upper bound constraint on the regularization term. However, it is not easy to solve the resulting problem. Some recent progress [11] in the theory of optimization on iterative shrinkage thresholding method is considered. And, an efficient alternating minimization scheme is proposed to solve the new objective.

##### 1.1. Related Works

Recently, the problem of denoising image corrupted by mixed Gaussian-impulse noise has been studied in many different contexts [8–10, 12, 13]. These methods fall into three categories: variational methods [9, 12], sparse representation [8, 10], and patch based method [7].

Variational methods are a new class of the solutions to promote edge-preservation, such as total variation [14]. These methods first utilized some spatial filters to detect and remove the corrupted pixels, for example, adaptive center-weighted median filter [15] (ACWMF) or rank order absolute differences [16] (ROAD) detector. In [12], Cai et al. employed Mumford-Shah regularization term to encourage sparsity in gradient domain. In [9], Rodríguez et al. presented a novel optimization method for the generalized total variation regularization method. It can be seen that the denoised performance of these methods relies on the detection for the damaged candidates. The adaptivity of sparsity level of the regularization terms has not been investigated carefully.

Sparse representation based methods have been extended to this problem. In the main idea of this scheme, it is assumed that the signal can be described by linear combination of a spare number of elements or atoms of an overcomplete dictionary. In [8], an efficient image reconstruction method by posing norm on the error, and norm on image patches in learned dictionary, was proposed. In [10], Filipovic and Jukic reformulated a new problem formulation by enforcing - sparsity constraints. The resulting problem is solved by a mixed soft-hard thresholding method. However, it should be noted that these methods are time-consuming.

Patch based method is proven to be a state-of-the-art denoising scheme. In [7], Ji et al. approximated the patch stack by reformulating the problem as a low rank matrix completion problem. Despite its efficacy, one of the limitations of patch based method in [7] is that the degree of sparsity has not been considered carefully. When the underlying sparsity level is unknown, we may obtain a bias estimate, considerably. To alleviate these issues in a unified formulation, a new problem formulation is proposed.

##### 1.2. Contributions

The main idea of this paper is to deal with the weakness of the approach in [7]. Existing methods, such as norm and trace norm, can not promote the sparsity level of all two components adaptively. The details or local fine content can not be represented and described well. To deal with these issues, a new problem formulation incorporating correlated and adaptive sparsity is proposed.

Our contributions can be summarized as follows.(1)A new problem formulation to model the sparsity level of the patches is proposed. A new norm extended from -support norm and ordered norm is presented.(2)An efficient minimization scheme with regular -shrinkage thresholding operator is proposed, which is based on the optimization framework of accelerated proximal gradient (APG) method.(3)Numerical experiments, compared to other state-of-the-art methods, demonstrate that the proposed method outperforms the related restoration methods.

##### 1.3. Organization

The remainder of this paper is presented as follows. In Section 2, some basic notations are provided. In Section 3, a detailed description about the proposed objective function is given. In Section 4, an efficient minimization scheme within the framework of APG is proposed. Then, some experiments are conducted to validate the effectiveness of the proposed method in Section 5. Finally, we conclude the paper in Section 6.

#### 2. Preliminaries

There are some notations presented for the simplicity of discussions. Frobenius norm and norm of a matrix are defined by and , respectively. For a scalar , the shrinkage operator [17] for norm minimization problem is defined as follows: where is a signum function; calculates the absolute value.

Assuming that is of rank , the singular value decomposition (SVD) of with nonnegative singular values is defined by , where denotes a diagonal matrix with the singular values. Based on the SVD computation, the nuclear norm is defined in the following way:where is the th largest singular value of . A solution with shrinkage operator to the nuclear norm is singular value shrinkage operator [18] , which can be expressed as follows:where is defined as follows:However, it should be noted that the shrinkage operator for norm is different from singular value shrinkage operator for nuclear norm. There operators play an important role in joint sparse and low rank matrix approximation.

In this paper, ordered -norm [19] is provided as follows:where is sorted in decreasing order. denotes the th largest element of the magnitude vector . is a trade-off vector in nonincreasing order. When is a constant vector, (5) reduces to norm. When and , then (5) reduces to -norm.

-support norm [20] is defined as follows:where denotes the bound of the sparsity level of , which is a positive integer. The details of -support norm are in traduced in [20, 21]. It has been extended to the case of matrix, which named by regular spectral -support norm. Although this norm provides the number of elements of the sparsity level, it lacks of efficient mechanism to promote the sparsity adaptively.

After taking advantage of both ordered norm and -support norm, regular -support norm is defined as follows:where is a positive regularization vector in nonincreasing order. And , where is the size of vector .

For the case of matrix, regular spectral -support norm is proposed, which can be expressed as follows:where . It can be noted that the singular values are arranged in nonincreasing order.

#### 3. Problem Setup

This section introduces the objective function in detail. For each reference patch , similar patches in the spatial- and temporal-domain are obtained by utilizing the patch matching algorithm. The matched patches are denoted as , where stands for the size. It can be noted that each patch is rearranged as a vector with size through concatenating all columns into a column vector. At last, a matrix is generated after considering all the patches, which can be represented as follows:In this paper, we assumed that the observed patch matrix can be decomposed into three components:where stands for low rank component, is sparse component, and is additive noise. There are some regularization methods for (10), such as with nuclear norm (also known as the trace norm) and with norm. The problem formulation in [7] can be expressed as follows:where is nuclear norm, and for norm.

However, these norms may lead to a large estimation bias [19] but can not promote the sparsity level adaptively. For example, the limitations of norm have been investigated in [22, 23]. Similarly, some alternative cases, such as the quasinorm, also have been discussed in [24–26]. Thus, a suitable solution is required to recover these components.

To alleviate these limitations, a new problem formulation is proposed to model the sparsity levels both on and . Moreover, a unified formulation to describe the correlated variables is considered. To estimate the underlying structures of and , we focus on the following minimization function:where and are two positive regularization vectors in a nondecreasing order. stands for regular spectral -support norm on . denotes regular -support norm on . And is the standard deviation of noise .

The above formulation amounts to the constraint , which is considered more natural than usual formulation because it stands for the tolerance on the error.

After choosing a suitable , (12) can be reformulated as follows:where is a suitable positive value. It can be seen that these are some challenges to solve (13). First, regular -support -norm is posed on sparse component . Second, the low rank component is penalized by regular spectral -support norm.

There are several properties of our problem formulation in (13). First, the proposed regular spectral -support norm on and regular -support norm on aim to reconstruct the local structures clearly. It should be noted that these modeling strategies can adaptively promote the sparsity level with an upper bound. Second, to the best of our knowledge, this is the first time of combining the advantages of both ordered norm and -support norm to yield a robust subspaces estimation against noise. Third, although the optimization method in [27] is very similar to the proposed method, the proposed method can deal with more complex situations. Moreover, the proposed method can adopt the more challenging situations, such as the removal of mixed Gaussian, salt-and-pepper noise, and random value impulse noise. It should be noted that this noisy situation has not been explored in [7].

#### 4. Proposed Method

##### 4.1. Proposed Framework

In this section, an optimization framework using regular -shrinkage thresholding operator is presented. First, accelerated proximal gradient method (APG) is applied to the resulting problem because of its simplicity and popularity in imaging applications [28, 29]. Second, the proposed regular -shrinkage thresholding operator is applied to the two resulting subproblems. As showed in [22, 23], nonconvex regularization functions have been shown both theoretically and experimentally to provide better results than norm. Then, some explicit proximal mappings are developed.

APG based scheme aims to solve an unconstrained minimization problem bywhere is assumed to be a nonsmooth function and for a smooth function. Here, denotes the Lipschitz constant of the gradient of .

Applying the framework of APG to problem (13), we have the following expressions:Setting , then we have the following objective function:where the proximal points and in the framework of APG are defined in Algorithm 1. It can be noted that both the variables and are separable. Thus, there are two subproblems, which can be represented as follows:It can be seen that (17) is a minimization problem with regular spectral -support norm and (18) with regular -support -norm. To deal with these problems, regular -shrinkage thresholding operator (RK) is defined as follows:where denotes an operation of direct minus in nonincreasing order. denotes the sparsity level of the input vector. also is a vector in a nonincreasing order.