Mathematical Problems in Engineering

Volume 2018, Article ID 7975248, 11 pages

https://doi.org/10.1155/2018/7975248

## Removal of Salt and Pepper Noise in Corrupted Image Based on Multilevel Weighted Graphs and IGOWA Operator

^{1}School of Computer Science and Technology, Anhui University, Hefei 230601, China^{2}School of Business, Anhui University, Hefei 230601, China^{3}Edward P. Fitts Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC 27695, China

Correspondence should be addressed to Qin Xu; moc.nuyila@3102niqux

Received 29 December 2017; Accepted 8 April 2018; Published 15 May 2018

Academic Editor: Peide Liu

Copyright © 2018 Qin Xu 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

This paper proposes a novel iterative two-stage method to suppress salt and pepper noise. In the first phase, a multilevel weighted graphs model for image representation is built to characterize the gray or color difference between the pixels and their neighbouring pixels at different scales. Then the noise detection is cast into finding the node with minimum node strength in the graphs. In the second phase, we develop a method to determine the order-inducing variables and weighted vectors of the induced generalized order weighted average (IGOWA) operator to restore the detected noise candidate. In the proposed method, the two stages are not separate, but rather alternate. Simulated experiments on gray and color images demonstrate that the proposed method can remove the noise effectively and keep the image details well in comparison to other state-of-the-art methods.

#### 1. Introduction

Images are easily corrupted by impulse noise during the signal acquisition, transmission, or storage [1]. There are two types of impulse noise, salt and pepper noise and random valued noise. Salt and pepper noise randomly alters a certain amount of pixels into two extremes, either 0 or 255, for an 8-bit image. The noise significantly damages the image information which leads to difficulties in succeeding image processing tasks such as edge detection or image segmentation and image recognition tasks. Because the noise pixel differs from most of its local neighbours, it has large gradient value the same as the edge pixel [2]. How to effectively restore the noisy image is still a challenging problem.

Numerous techniques have been proposed to suppress the salt and pepper noise. The most simple and well-known methods are the standard median filter [3] and its variants [4–6] which are nonlinear filters whose responses are based on reordering the intensity values in the neighbourhood of the corrupted pixel. The nonlinear operation exhibits good denoising power [7], but when the noise level increases, the edges and other details of the image cannot be restored. This is because the median filter simply replaces every pixel’s value. Switching or decision based filters are proposed and have become a hotspot of removal of salt and pepper noise; their strategy in common is using a two-stage technique which first detects the possible noise pixels in an image and then replaces the noise pixels [8–19]. For example, Chan et al. used the adaptive median filter to detect the noise pixels and an objective function with an data-fidelity term and edge-preserved regularization term to denoise, while preserving the edges [9]. The NAFSMF method uses the image histogram to find the salt and pepper noise pixels and computes the number of noise-free pixels to avoid selecting a noise pixel as the median pixel for restoration [10]. The MDBUTMF method regards the maximum or minimum gray level as the noise pixel and replaces the noise pixel with the median of remaining elements in the selected window or with the mean of the selected window if the selected window contains all 0’s or 255’s or both [11]. The AWMF uses a size alterable window to judge whether the given pixel is noise pixel or not and computes the weighted mean of the current window to restore the noise pixel [12].

Most recently, Veerakumar et al. used the Rank Ordered Logarithmic Difference which is the logarithmic function on the absolute difference to detect the noise and restore the noise pixels by the adaptive anisotropic diffusion filter [13]. Ahmed proposed a two-stage iterative filter which detects the noisy pixels with an adaptive fuzzy detector and uses weighted mean filter on the uncorrupted pixels [14]. Vijaykumar et al. developed a fast switching filter which identifies the extreme minimum value and extreme maximum value as the noise pixels and replaced the noise pixels by either median value or mean value [15]. Li et al. developed an image block-based noise density estimation method to guide noise detection and noise restoration and used the global image information for noise rectification [16]. Wang et al. explored the similar patches with repeat patterns in images and proposed an iterative nonlocal means filter (INLM) for salt and pepper noise removal [17]. The ANCLPVMF judges the noise pixels based on the linear prediction error and uses adaptive window based vector median filtering operation to restore the noise [18]. In [19], two approaches have been provided for detecting the noise pixels based on means and variance, and a novel type-1 fuzzy approach is presented for denoising.

It can be concluded that two aspects of problems in salt and pepper noise removal are how to retain the uncorrupted pixels and how to use the uncorrupted pixels to estimate the corrupted pixel precisely [20]. In order to solve the two aspects of problems effectively, in this paper, we investigate the graph theory for image noise detection and information aggregation operators for image restoration. Studies indicate that the image processing and analysis performance are dependent on the choice of data representation on which they are applied [21]. Recently, graph-based representation for image has been applied to image processing and recognition successfully [22–26]. Inspired by these, on the one hand, we build a multilevel weighted graphs model for image representation to characterize the relationship between the pixels and their neighbouring pixels at different scales. Because the gray or color difference between the noise pixel and its neighbours is large, the node strength of the weighted graphs is utilized to detect the noise candidates. On the other hand, the ordered weighted averaging (OWA) operators provide a parameterized family of mean type aggregation operators [27–32]. They learn the associated weighting vector from observational data to obtain an optimal aggregation result. The generalized OWA (GOWA) [28] operator uses generalized means in the OWA operator. The induced generalized OWA (IGOWA) [29] operator is an extension of the GOWA operator, with the difference that the reordering step of the IGOWA operator is not defined by the values of the arguments, but rather by order-inducing variables, where the ordered position of the arguments depends upon the values of the order-inducing variables. It is a more general formulation of the reordering process that is able to consider more complex situations. Thus, we develop a restoration method based on the IGOWA operator to aggregate the uncorrupted pixels. It has high flexibility and could restore the noise pixel accurately even when the 0’s and 255’s are not balanced in the local neighbourhood of the noise pixel. The proposed noise detection and restoration are not separate, but rather alternate, and iterative, which can deal with the single-point (single-pixel) and noise patches. This is the first attempt of employing the graph theory for solving the noise detection and the information aggregation operator for removing salt and pepper noise. Hopefully, this is an initial work of graph theory and information aggregation operator for image restoration.

The rest of the paper is structured as follows. The proposed method and the flow diagram of the proposed algorithm are described in Section 2. Section 3 provides the simulation results with different images in comparison with the six state-of-the-art methods. Finally conclusions are drawn is Section 4.

#### 2. The Proposed Method

##### 2.1. Multilevel Weighted Graphs Modeling for Noise Detection

To detect the noise in an image we model the image as multilevel weighted graphs , which are a series of graphs . At each level , each corresponds to a pixel in the image, an edge connects vertices and , represents the weight on that edge connecting and , and denotes the level version. The edges are created by connecting the nodes, whose corresponding pixels are in the neighbour pixel sets.

In this paper, we define three different neighbour pixel sets given a pixel, which is illustrated in Figure 1, for a given pixel , the pixels labeled “1” are the 8-neighbour pixels, labeled “2” are the 12-neighbour pixels, and labeled “3” are the 16-neighbour pixels. We denote the pixel set as , , and , respectively. For a pixel, we connect its corresponding node and the nodes corresponding to its 8-neighbour pixels to create the edges in , connect its corresponding node and the nodes corresponding to its 12-neighbour pixels to create the edges in , and connect its corresponding node and the nodes corresponding to its 16-neighbour pixels to create the edges in . The weight connecting two nodes should reflect the difference of the corresponding pixels’ gray or color in , , and . The edge weight connecting node and node in the multilevel weighted graphs is defined aswhere denotes the gray or RGB value of pixel , and represent the maximum and minimum gray or RGB value of the image, in , in , and in . The weight in all the weighted graphs satisfies .