Mathematical Problems in Engineering

Volume 2018, Article ID 9620754, 11 pages

https://doi.org/10.1155/2018/9620754

## Image Denoising Based on Adaptive Fractional Order with Improved PM Model

^{1}Chongqing University of Posts and Telecommunications, Chongqing 400065, China^{2}Chongqing Key Lab of Computer Network and Communication Technology, Chongqing 400065, China^{3}Chongqing University, Chongqing 400044, China

Correspondence should be addressed to Shangbo Zhou; nc.ude.uqc@uohzbhs

Received 3 December 2017; Revised 19 March 2018; Accepted 26 March 2018; Published 20 May 2018

Academic Editor: Zhen-Lai Han

Copyright © 2018 Jimin Yu 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

In order to improve the image quality, in this paper, we propose an improved PM model. In the proposed model, we introduce two novel diffusion coefficients and a residual error term and replace the integer differential operator with the fractional differential operator in the PM model. The diffusion coefficients can be used effectively for edge detection and noise removal. The residual error term can help to prevent image distortion. Fractional order differential operator has a good characteristic that it can enhance image texture information while removing image noise. Additionally, in the two new diffusion coefficients, a novel method is proposed for automatically setting parameter* k*, and it does not need to do any experiments to get the value of . For the computing fractional order diffusion coefficient, we employ the discrete Fourier transform, and an iterative scheme is carried out in the frequency domain. In the proposed model, not only is the integer differential operator replaced with the fractional differential operator, but also the order of the fractional differentiation is determined adaptively with the local variance. Comparing with some existing models, the experimental results show that the proposed algorithm can not only better suppress noise, but also better preserve edge and texture information. Moreover, the running time is greatly reduced.

#### 1. Introduction

Image denoising is a critical task in image processing, and the denoising algorithm is widely researched in recent years. The anisotropic diffusion filter has received much attention since it was first proposed by Perona and Malik in 1990, which is called Perona–Malik model (PM model) [1]. In the classical PM model, the gradient value is used in the direction of east, west, south, and north to distinguish variations which are caused by noise or edge in a corrupted image [2]. The shortcomings of PM model are that the model is easy to lose contrasting information and texture information and produce staircase effects [3]. To solve these shortcomings, some improved algorithms have been proposed. For image denoising, Ma and Nie [4] proposed an edge fusion scheme based on anisotropic diffusion models. Zhao et al. proposed an improved PM model based on local entropy [5]. H. Zhang and Y. Zhang [6] proposed an adaptive diffusion coefficient scheme based on anisotropic diffusion. Prasath and Delhibabu [7] proposed a fuzzy diffusion coefficient which takes into account local pixel variability for better denoising and selective smoothing of edges. Sun et al. [8] have investigated the edges and details blurring issues. To avoid diffusion perpendicular to edge direction, Wang et al. [9] introduced a modified PM model using directional Laplacian. Prasath and Vorotnikov [10] suggested a weighted anisotropic diffusion to reduce blurring and staircasing effects. Xu et al. [11] suggested an adaptive thresholding in PM diffusion coefficient to better handle the diffusion as time elapsed. A new diffusion coefficient has been proposed by Tebini et al. [12, 13] for better control of the diffusion process in regions containing edge. Wang et al. [14] proposed new second- and fourth-order anisotropic equations for denoising. All of these methods mentioned above are gradient dependent where the gradient controls the diffusion process and therefore degrades texture and fine details.

At present, fractional calculus has been widely applied in many fields of science and engineering. As a result, differential equations with arbitrary orders have been large investigated for different applications in physics, fluid mechanics, physiology, engineering, potential theory, elasticity, and so on [15–18]. Yang et al. [19] proposed an adaptive image denoising model of anisotropic diffusion based on fractional derivative. Bai and Feng [20] proposed a fractional order anisotropic diffusion for image denoising. Yuan and Liu [21] proposed an anisotropic diffusion model based on a new diffusion coefficient and fractional order differential for image denoising. Bai and Feng [20] proposed a fractional order anisotropic diffusion model for image denoising, and the model can not only excellently remove noise but also save the edge information. To automatically select the optional order of fractional differentiation, Che et al. [22] bring forward a denoising model in which the complexity of the local image texture was reflected by local variance and the order of fractional differentiation was decided adaptively. The above-mentioned image processing algorithms based on fractional order partial differential equation have made improvement on keeping detailed image information, texture information, good visual effects, and image denoising.

The PM model mainly has three disadvantages as follows: staircase effects exist, the texture information is lacked, and the diffusion coefficient threshold is set by carrying out many experiments. The optimal order of traditional fractional differentiation is often preset manually. In this paper, in order to solve these problems, by combining adaptive fractional differential operator with improved PM model, a novel model is proposed. Additionally, in the proposed method, according to [1, 25], we proposed two new diffuse coefficients, and automatically setting diffusion coefficient parameter is developed. Based on above measurement, a denoising algorithm based on adaptive fractional order with improved PM model is put forward. In addition, in the improved model, the local image texture complexity and the fractional order can be reflected by the local variance.

The paper is organized as follows. In Section 2, the fractional order definition and PM model are introduced. In Section 3, analysis of image denoising is shown based on adaptive fractional order with PM model. In Section 4, the experimental results and analysis are illustrated. Finally, conclusions are drawn in Section 5.

#### 2. Introduction of Fractional Order and PM Model

##### 2.1. PM Denoising Model

In the classical Perona–Malik equation [1], proposed by Perona and Malik, the diffusion process is defined by a partial differential equation. The PM equation is defined as follows:where div is the divergence operator,* ∇* is the gradient operator, is the amplitude, is the diffusion coefficient equation, is the original noise image, and

*t*is the introduced time, used to exhibit diffusion duration. For the diffusion coefficient function , Perona et al. gave two classic options as (2) and (3), respectively.where is diffusion coefficient threshold parameter, is the gradient of the image, and in PM model, the diffusion coefficient is a nonnegative function on the magnitude of local gradient information. Its value is generally inversely proportional to the gradient magnitude of the image. In the isotropic flat area, the gradient modulus is generally small, and the value of the diffusion coefficient is close to 1. The diffusion behavior of the model is close to the thermal diffusion pattern, which achieves the purpose of smoothing flat area of the image. In the edge region of the image, the gradient modulus is very large, and the diffusion coefficient is close to zero. The diffusion rate of the model is almost zero.

##### 2.2. Introduction of Fractional Order

The commonly used fractional order differentiation is Grünwald-Letnikov (G-L) definition and Riemann-Liouville (R-L) definition [26, 27]. However, the definition in frequency domain is easier to implement and a simple definition, so we use the Fourier transform to compute the fractional derivative in this paper. According to the basic theory of signal processing, the Fourier transform of energy signal can be obtained; based on the Fourier transform, we can draw the amplitude frequency characteristic. Figure 1 shows the amplitude frequency characteristic curves of fractional order differentiation with different fractional order.