n w ory of 40, Ch 23 September 2013

Accepted 3 October 2013

Available online 12 November 2013 ces a emov diffu behavior of diffusion and wave propagation and thus it can preserve edges in a highly oscillatory region. On the other hand, the usual diffusion is used to reduce the noise

Es) ba tions through a linear heat equation with an anisotropic diffuresults in enhancing the edges, it is an ill-posed process in the sense that it is very sensitive to perturbations in the nd to cause the ect”. In order to r PDEs (typically 0], but they often een studied and ter vision. Cuesta r linear integrodifferential equation for image denoising as follows: where Δ denotes Laplacian operator, ΩR2 represents the α

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Signal Pro

Signal Processing 103 (2014) 6–150165-1684/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sigpro.2013.10.028image domain, and ∂t stands for the Riemann–Liouville (R–L) fractional time derivative of order α, 1oαo2. Since the model (1) interpolates a diffusion equation (for α¼ 1) and a wave equation (for α¼ 2), the solution of (1) will satisfy n Corresponding author. Tel.: þ86 2134051209.

E-mail address: zhangweii@foxmail.com (W. Zhang).sion. While the backward diffusion of the PM equation ∂αt uðt; xÞ ¼Δuðt; xÞ; ðt; xÞA ½0; T Ω; ð1Þadvantages in both theory and computation. They allow to directly handle and process visually important geometric features. In addition, they can also effectively simulate several visually meaningful dynamic processes such as linear and nonlinear diffusion, and the information transport mechanism.

In order to preserve the image structures when removing the noise, Perona and Malik [2] proposed a nonlinear equation which replaced isotropic diffusion expressed order PDEs can achieve a good tradeof removal and edge preservation, they te denoised image to exhibit “staircase eff overcome this drawback, the high-orde fourth-order PDEs) were adopted in [8–1 lead to the speckle effect.

Recently, fractional-order PDEs have b applied to the image processing and compu et al. [11,12] proposed a fractional-ordementation, etc.) have been largely studied in the literature (see [1] and references therein) due to their remarkable related to the diffusion models and the relations have been studied in the literature [7]. Although these secondf between noiseKeywords:

Diffusion-wave equation

Fractional derivatives

Fractal operator

Image denoising 1. Introduction

Partial differential equations (PD image processing (denoising, restoraimage structure. We prove that the proposed model is well-posed, and the stable and convergent numerical scheme is also given in this paper. The experimental results indicate superiority of the proposed model over the baseline diffusion models. & 2013 Elsevier B.V. All rights reserved. sed methods for , inpainting, seginitial noisy data. Since the work of Perona and Malik, a large number of nonlinear PDEs based anisotropic diffusion models have been proposed [3–5]. In addition, the well-known total variation (TV) model [6] is closelywhereas the non-local term which exhibits an anti-diffusion effect is used to enhance theA fractional diffusion-wave equatio regularization for image denoising

Wei Zhang a,n, Jiaojie Li a,b, Yupu Yang a a Department of Automation, Shanghai Jiao Tong University, and Key Laborat

Ministry of Education of China, Shanghai 200240, China b School of Electrical Engineering, Shanghai Dianji University, Shanghai 2002 a r t i c l e i n f o

Article history:

Received 31 May 2013

Received in revised form a b s t r a c t

This paper introdu ization for noise r between the heat journal homepage: wwwith non-local

System Control and Information Processing, ina novel fractional diffusion-wave equation with non-local regularal. Using the fractional time derivative, the model interpolates sion equation and the wave equation, which leads to a mixed evier.com/locate/sigpro cessing

W. Zhang et al. / Signal Processing 103 (2014) 6–15 7intermediate properties, i.e., the maximal diffusion is reached for α¼ 1 and there is no diffusion at all for α¼ 2. Cao et al. [13] proposed a similar model which replaces the fractionalorder derivative by introducing a weight parameter between the first and second order time derivatives. On the other hand,

Bai and Feng [14] proposed an anisotropic model with fractional space derivatives, i.e., ∂tuðt; xÞ ¼ ðDαx1 Þ nðcðJDαxuðt; xÞJ2ÞDαx1 Þ ðDαx2 Þ nðcðJDαxuðt; xÞJ2ÞDαx2 Þ; tA ½0; T ; x¼ ðx1; x2ÞAΩ; αA ½1;2; ð2Þ where Dαx1 and D α x2 denote the fractional spatial derivative of the order α, and ðDα Þn represents the adjoint operator of the linear operator Dα . It is observed that themodel (2) leads to an interpolation between the PM model (for α¼ 1) and the fourth-order anisotropic diffusion equation [9] (for α¼ 2).

Thus, it contains the advantages of both methods. In addition,

Zhang et al. [15,16] generalized the TV model for image denoising using the Grünwald–Letnikov fractional-order derivative. And Ren et al. [17] proposed the fractional-order TV regularization for image super-resolution. Recently, inspired by the models (1) and (2), Janev et al. [18] proposed a fully fractional anisotropic diffusion (FFAD) equation for noise removal, which contains spatial as well as time fractional derivatives, i.e., ∂tuðt; xÞ in (2) is replaced by cDβt uðt; xÞ which is the left Caputo time fractional derivative of order βA ½1;2Þ.

Thus, it can interpolate between the parabolic and the hyperbolic PDE and, at the same time, between the second and fourth order PDE. Although this model can manage to preserve edges and highly oscillatory regions, the anisotropic diffusion is based on the PMmodel which is actually ill-posed.

Another drawback of the classical PDEs-based methods is that the derivative is a local operator. Recently, the nonlocal technique have been used very successfully for many image processing tasks [19–22]. These methods exploit the image self-similarities or redundancies to reconstruct the image. However, the process of searching for the similar patches is very time-consuming. In this paper, we attempt to extend the Cuesta's model (1) by introducing a nonlocal regular term. Note that unlike the non-local methods mentioned above, here the non-local operator is defined by Fourier transform, which dose not require the search process. This non-local operator was first proposed in the field of physics, such as overdriven detonations in gases [23], anomalous diffusion in semiconductor growth [24], the morphodynamics of dunes and drumlins [25,26], etc.