Gradient-based Wiener filter for image denoising

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Abstract

In this paper, we develop a new adaptive image denoising algorithm in the presence of Gaussian noise. Because the proposed method operates in the gradient domain and is close to Wiener filter, it is named as gradient-based Wiener filter (GWF). Inspired by the Perona–Malik anisotropic diffusion (PMAD), the proposed algorithm is implemented by iterations. The parameters for the GWF are studied in full detail. At the same time, the tuning method of the gradient thresholding based on noise variance for PMAD is presented. Experimental results indicate the proposed algorithm achieves higher peak signal-to-noise ratio (PSNR) and better visual effect compared to related algorithms. On the other hand, the simulation results also show the tremendous power of the given parameter tuning method for PMAD.

Graphical abstract

Highlights

► The shrinkage operates in the gradient domain. ► The scheme is formed in statistical sense. ► A new parameter tuning method for the Perona–Malik anisotropic diffusion is presented.

Introduction

Denoising has been an active area of research in image processing. In the last several decades, a lot of new methods have emerged for removing Gaussian random noise. The main aim of these methods is to reduce the noise level, while preserving the image features as much as possible.

The spatial Wiener filter [1] is one of the classical linear filtering, and is known to be the optimal estimator for the true underlying image. However, as a linear and shift invariant filter scheme, Wiener filter is often assumed to be unsuitable for images containing edges and details. In order to deal with edges and details in images, the PMAD image denoising method was proposed by Perona and Malik in 1990 [2]. This approach is a modification of the linear diffusion, and it is expressed asIt(i,j)t=div[ct(i,j)·It(i,j)]where It(i, j) is the image at time t; div is the divergence operator; ∇It(i, j) is the gradient of the image, and ct(i, j) is the diffusion coefficient. Compared with the linear diffusion, the advantage of anisotropic diffusion Eq. (1) is to adaptively set the diffusion coefficient ct(i, j) at every iteration so that the diffusion is discouraged across the boundaries while encouraged within homogenous. Two common used diffusion functions arect(i,j)=exp-|It(i,j)|/K2andct(i,j)=11+|It(i,j)|/K2which are suggested by Perona and Malik. In Eqs. (2), (3) K is a constant to be tuned for a particular application. Eq. (2) privileges high-contrast values over low-contrast ones, Eq. (3) privileges wide regions over smaller regions. Anisotropic diffusion has received ever-increasing attention since the PMAD model emerged. Many revised anisotropic diffusion models with different purposes have been proposed for image denoising. For example, the regularization method based on Gaussian kernel was used to overcome the theoretical and practical limitations of the PMAD model in [3]. The Gaussian filter acts as a pre-processing to reduce the influence of noise during the diffusion process. The regularization was also done with other kernels, see e.g. a wavelet regularization [4], a curvelet regularization [5] and a shearlet regularization [6] which lead to better performances for discontinuity-preserving denoising. In [7], Weickert developed an anisotropic regularization, in which a diffusion tensor steers the diffusion process according to the directional information contained in the image structure. In [8], a ramp preserving Perona–Malik (RPPM) model based on difference curvature was proposed. After the non-local means (NLM) approach was well established [9], the nonlocal nonlinear diffusion models for noise reductions were also proposed, analyzed and implemented [10]. They showed these models were a close relative of the celebrated PMAD model, and can be viewed as a new regularization paradigm for PMAD in a way. The above-mentioned anisotropic diffusion models were generally designed for specific applications. They only obtain good results for their domain-problems. It is not considered to achieve the high noise-reduction gain, structure and contrast preservation simultaneously. The proposed technique (GWF) aims to deal with this problem. As with anisotropic diffusion, the GWF will be carried out in the gradient domain.

Besides anisotropic diffusion, wavelet-based algorithms are also researched extensively [11], [12], [13], [14], [15], [16], [17]. The denoising process is known as wavelet shrinkage or thresholding. The basic wavelet shrinkage includes the hard and soft thresholding shrinkages, and which are asymptotically optimal in a minimax mean-square error (MSE) sense over a variety of smoothness spaces. However, for any given signal, the MSE-optimal processing is achieved by the Wiener filter [11], which produces substantially improved performance [12], [13], [14], [15]. Although image denoising methods by using the wavelet transform have show to be very effective in signal and image denoising, textures and details can be preserved very well compared to anisotropic diffusion, artifacts often arise in the denoised results. Gaussian scale mixture (GSM) denoising algorithm employing over-complete multiscale transforms achieves improved results by modeling clusters of coefficients as the product of a Gaussian random vector and a positive scaling variable [18]. Inspired by the success of NLM methods [9], Dabov et al. [19] presented a collaborative image denoising scheme by patch matching and sparse 3D transform. They searched for similar blocks in the image by using block matching and grouped those blocks into a 3D data array. A sparse 3D transform was then applied to the data array and noise was removed by applying a specially developed Wiener filtering in the transformed domain. The so-called BM3D and GSM algorithms achieve the state-of-the-art denoising performance yet their implementations are a little complex.

We notice that all the algorithms in [1], [11], [12], [13], [14], [15], [18], [19] involve the use of Wiener filter. Wiener filter has been receiving more attention because of its simplicity and effectiveness. At the same time, we also notice that the sequential anisotropic Wiener filtering (SAWF) [20], [21] was designed to dispose 3D magnetic resonance imaging (MRI) and ultrasound image data. The authors selected one of the six different spatial orientation neighborhood subsystems of centered pixel. When centered pixel is judged as boundary pixel, Wiener filter is carried out in the chosen neighborhood subsystem by discarding other neighborhood pixels; otherwise, Wiener filter is carried out in the whole neighborhood of centered pixel. The good performances are displayed, but the SAWF did not compete with wavelet-based Wiener-like methods. The details can be found in [20]. In this paper, we also focus on Wiener filter, but aiming to denoise 2D images. The traditional Wiener filter is modified. The local pixel information is made full use of. Although the good algorithms for Wiener filter usually operate in wavelet domain, this paper verifies the Wiener filter in the gradient domain can obtain better denoising performance compared to conventional Wiener filters and classical anisotropic diffusions, and the several state-of-the-art wavelet-based algorithms. The proposed method is carried out by iterations. At the same time a new gradient thresholding tuning method is introduced in PMAD.

The paper is organized as follows. Section 2 presents the proposed method and details it. Section 3 presents the results of experiments. Comparing the simulation results obtained by the proposed method with those of the others shows a noticeable improvement in the quality of denoised images evaluated objectively and quantitatively. At the same time, the capability of tuning method of the parameters for PMAD is verified. Section 4 concludes the paper.

Section snippets

The proposed method

In this paper, we assume an original image is degraded by additive noise and the noise is signal independent. The typical image degradation model at (i, j) in a two-dimensional coordinate can be written asI0(i,j)=I(i,j)+n(i,j)where I, I0, and n represent the original image, the observed image, and the additive Gaussian random noise with zero mean and variance σ2, respectively.

In the proposed GWF, the novel is in: (1) inspired by nonlinear anisotropic diffusion , the shrinkage operates in the

Experimental results

In this section, to verify the proposed GWF and tuning method of parameters for PMAD, some comparisons are presented. We measured objectively the experimental results by PSNR in decibels (dB), which is defined asPSNR=10·log102552MSEwhere MSE=(1/rs)·i=1rj=1s(I(i,j)-W(i,j))2,I is the original image, W is the estimator of I, rs is the number of pixels. The denoised image is closer to the original one when PSNR is higher.

Conclusion

This paper proposed a local adaptive denoising algorithm based on the Wiener-like shrinkage function in the gradient domain. Unlike previous gradient domain method, the proposed method is formed in statistical sense. We call it gradient-based Wiener filter (GWF). As a result of considering the anisotropic characteristic of image, the GWF improves edge preservation capability compared to Wiener filter. Because of employing statistical characteristic of removed noise at each iteration step, the

Acknowledgments

The author thanks Luisier F. and Tian J. for their respective open source codes. This work is partially supported by the National Nature Science Foundation of China (Grant No. 61102018 and 60872138) and Scientific Research Program Funded by Shaanxi Provincial Education Department (Program No. 11JK1030) and Natural Science Foundation of Xianyang Normal University (No. 11XSYK304).

Xiaobo Zhang received M.S. degree in computational mathematics from Xidian University, Xi’an, China, in 2007. He is currently pursuing the Ph.D. degree in applied mathematics from Xidian University, is also a lecturer with Xianyang Normal University, Xiangyang, China. His research interests include wavelets and partial differential equations for image processing.

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    Xiaobo Zhang received M.S. degree in computational mathematics from Xidian University, Xi’an, China, in 2007. He is currently pursuing the Ph.D. degree in applied mathematics from Xidian University, is also a lecturer with Xianyang Normal University, Xiangyang, China. His research interests include wavelets and partial differential equations for image processing.

    Xiangchu Feng received the B.S. degree in computational mathematics from the Xian JiaoTong University, Xi’an, China, in 1984, and the M.S. and PH.D degrees in applied mathematics from Xidian University, Xi’an, China, in 1989 and 1999, respectively. He is currently a Professor of applied mathematics at Xidian University. His research interests include numerical analysis, wavelets, and partial differential equations for image processing.

    Weiwei Wang received the B.S., M.S. and PH.D degrees in applied mathematics from Xidian University, Xi’an, China, in 1993, 1998 and 2001, respectively. She is currently a Professor of applied mathematics at Xidian University. Her research interests include numerical analysis, wavelets, and partial differential equations for image processing.

    Shunli Zhang received the B.S. degree in computational mathematics from Xidian University, Xi’an, china, in 1997, and the M.S. and PH.D degrees from Northwestern Polytechnical University, Xi’an, China, in 2004 and 2010, respectively. He is currently an Associate Professor at Xianyang Normal University, Xiangyang, China. His research interests include industry CT, graphics, image processing.

    Qunfeng Dong received the M.S. degree from Xidian University, Xi’an, China, in 2006 and the Ph.D. degree in Northwestern Polytechnical University, Xi’an, China, in 2011. He is currently a lecturer with Xianyang Normal University, Xiangyang, China. Her research interest is numerical analysis.

    Reviews processed and recommended for publication to Editor-in-Chief by Deputy Editor Dr. Ferat Sahin.

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