Mathematical Problems in Engineering

Volume 2016, Article ID 3151303, 16 pages

http://dx.doi.org/10.1155/2016/3151303

## Variable Splitting Based Method for Image Restoration with Impulse Plus Gaussian Noise

College of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, China

Received 9 June 2016; Revised 7 October 2016; Accepted 23 October 2016

Academic Editor: Bogdan Dumitrescu

Copyright © 2016 Tingting Wu. 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

Image denoising is a fundamental problem in realm of image processing. A large amount of literature is dedicated to restoring an image corrupted by a certain type of noise. However, little literature is concentrated on the scenario of mixed noise removal. In this paper, based on the model of two-phase method for image denoising proposed by Cai et al. (2008) and the idea of variable splitting, we are capable of decomposing the image denoising problem into subproblems with closed form. Numerical results illustrate the validity and robustness of the proposed algorithms, especially for restoring the images contaminated by impulse plus Gaussian noise.

#### 1. Introduction

When recording a real-world scene by camera, we desire to acquire an ideal image which is a faithful representation of the scene. However, most observed images are more or less involved in blurry and noise. Hence, deblurring and denoising to the observed image are fundamental aspects in image processing. Let be an ideal image scaled in . Hereafter, we represent an image as () by stacking its columns as a vector. is matrix representation of spatially invariant point spread function (PSF) which characterizes the blurry effects on imaging system and is additive noise with some statistical distributions (e.g., Gaussian noise follows normal distribution, impulse noise follows binomial distribution or uniform distribution, and uniform white noise follows uniform distribution). Accordingly, the observed image involved in the blurring matrix and the additive noise can be boiled down to

The main task of image restoration is to recover the ideal image from the observed image . There are two difficulties in finishing this task. The first one is that the singular values of decay asymptotically to zero which results in a large condition number, and as a consequence (1) is an ill-posed inverse problem in mathematical sense. The second difficulty is that the inevitably additive noise in the observed image makes (1) difficult to handle. These two difficulties make the linear least squares solvers generally fail in deriving ideal image and regularization strategy is therefore introduced to improve the numerical performances, for example, Tikhonov regularization [1], Mumford-Shah regularization [2], and total variation (TV) regularization [3]. The main superiority of TV regularization is that it can regularize images without blurring their edges. The common models utilizing TV regularization for image restoration are the constrained model and it is equivalent to (or rather equivalent under a proper , see [4]) the unconstrained modelwhere , are parameters depending on noise variance; is first-order derivative operator and is the TV-norm (details in Section 2); denotes the -norm in Euclidean space (if , herein should be comprehended as ). Commonly, is hinged on the statistical distribution of the additive noise ; for example, for impulse noise removal, for Gaussian noise removal, and for uniform white noise removal. It is therefore pivotal to figure out statistical information of before opting for mathematical model.

Recently, Huang et al. [5] proposed a modified TV regularization model for Gaussian noise removalwhere , are positive parameters. Intuitively, a new fitting term is added to the unconstrained model (3). Essentially, by smoothing of the nonsmooth function via the Moreau approximation, the objective function of the following problem can be replaced bywhich is identical with (4). They employed the alternating minimization (AM) algorithm to handle model (4) in variables and , respectively. The numerical results therein indicated that the model was attractive. Lately, they extended model (4) to the mixed noise (impulse plus Gaussian noise) removal in [6]. The extended model readswhere contains the noise-free pixels (pixels are not contaminated by impulse noise); is the characteristic function of set . Specifically, for any , , can be defined as which is convex and nonsmooth. Then, by using the incomplete data set in , TV minimization method is constructed for image restoration and suitable values can be filled in the detected noisy pixel locations.

As interposition, we expand in this paragraph about mixed noise removal. Gaussian noise [3] generally results from the analog-to-digital conversion of measured voltages and it corrupts an image by exerting subtle perturbations on the gray scale of all pixels. Impulse noise [7] typically arises from the malfunctioning arrays in camera sensors or faulty memory locations in hardware and it affects an image by seriously devastating the gray scales of some pixels (but keeping the other pixels noise-free). Salt-and-pepper noise [8] and random-valued noise [9] are two categories of impulse noise. Let be the dynamic interval of image . For given noise level (also the so-called noise intensity) , salt-and-pepper noise and random-valued noise contaminate image , respectively, by the following:(i)Salt-and-pepper noise (see [10]) (ii)Random-valued noise (see [11]) where represents the procedure of contaminating image with impulse noise and satisfies uniform distribution in interval .Figure 1 shows distinctions of image contaminated by salt-and-pepper noise and random-valued noise. Visually, the salt-and-pepper noise corrupted image possesses black-white speckles while the random-valued noise corrupted image has speckles with various gray scales.