An Efficient Variational Model for Restoring Noisy Images with Gamma Multiplicative Noise

Multiplicative noise removal has been a focus of research in recent years. Aiming at solving the problem that the total variation regularization method can remove noise well but sometimes produce stair-case effect, this paper proposes an efficient variational model and gives the iterative algorithm to remove Gamma multiplicative noise. This paper also shows that the iterative sequence converges to the optimal solution of the model. Through simulation experiments the proposed model has proved highly effective. That is, this model can preserve the edges of the image well and significantly reduce the stair-case effect in the smooth regions while removing Gamma multiplicative noise effectively.


INTRODUCTION
It is well known that images deteriorate during formation, transmission, recording processing. Image denoising is of momentous significance in coherent imaging systems and various image processing applications [1]. Therefore, the image noise removal problems have attracted much attention in recent years [2,3].
Image noise can be roughly divided into additive noise and multiplicative noise. Models for removing additive noise have been studied extensively. Since additive noises are dominant for high frequencies, their effects can be removed using some kinds of low-pass filters [4]. Other kinds of methods, such as variational methods, have also been proposed. In [5], for example, Rudin, Osher and Fatemi proposed a total variation regularization model (ROF model), which has been widely accepted as a reliable tool for removing additive noise. Over the last few years, various research efforts have been devoted to studying, solving and extending the ROF model [6][7][8][9]. Multiplicative noises are commonly found in many real world image processing applications, such as in synthetic aperture radar, medical ultrasound images, and particle tomography [10,11]. Therefore, how to effectively remove multiplicative noises becomes a significant study in recent years [12][13][14][15].
In general, the recorded image g, defined in Du dxdy dxdy dxdy u u This model is called RLO model, it was first used to remove Gaussian noise, but the experimental results show that the texture detail of the restored image has been destroyed. When the multiplicative noise is out of the Gauss distribution, the above RLO model is no longer available.
In 2008, Aubert and Aujol [13] used the maximum a posteriori (MAP) regularization approach and derived a functional whose minimizer corresponds to the denoised image to be recovered. Their model is called AA model which can be described as follows: The numerical experiments showed that the texture details in the images restored by HL model were kept and the 'stair-case effect' was suppressed at some level.
In order to obtain an efficient denoising method for Gamma noise, it is a natural choice to build a novel model by introducing an appropriate coordinating term and regularization term. In this paper we propose a novel denoising model which not only ensures that our model removes Gamma noise very well, but also it can effectively protect the edge details and texture features of the image. Our experimental results show that the quality of images restored by the proposed method is better than that by several popular models.
The rest of this paper is organized as follows. In next section, we introduce our model briefly. In Section 3, the minimizing algorithm and the convergence of the proposed method are given.
In Section 4, we show experimental results to demonstrate the performance of our proposed model. Finally, concluding remarks are given in section 5.

THE PROPOSED MODEL
What is usually referred to as multiplicative noise removal is of course nothing but the estimation of the reflectance of the underlying scene in imaging systems. This is an inverse problem calling for regularization, which usually consists in assuming that the underlying reflectance image is piecewise smooth. This assumption has been formalized, in a Bayesian estimation framework and variational approaches. Both the variational and the Bayesian MAP formulations to image denoising (under multiplicative, Gaussian, or other noise models) lead to optimization problems with two terms: a data fidelity term and a regularizer.
The first term of model (2.2) is called the loyalty term which ensures recovering image u to retain the main features from the virtual image log g. The second term is the coordination term which measures the influence between the fitting term and the regularization term and plays a role in undertaking and tradeoffs. The final term is the regularization term which ensures the smooth of the denoising image w, and prevents stair-case effects.
Before removing image noise by using our model, we first do a convolution operation for the original degraded image so as to avoid mistaking some noise gradient for the edge gradient.

THE ITERATIVE ALGORITHM
Inspired by the idea of the literature [14,19], we use an adaptive alternating iterative algorithm to solve the model (2.2). It is divided into two optimization problems. Starting from the initial data w (0) , we solve the following variational problems Cycling alternately, we can get the following iterative sequence Step 1. To solve the variational problem (3.1). we need to solve its discretization: . , the function f is convex strictly with respect to z, so we can find has a unique solution of the system (3.4) by using Newton iterative method This is the solution of the problem (3.1).
Step 2. To find the solution of the variational problem (3.2). Letting  In this paper, Dw i j is the gradient at the location   , we can obtain the numerical solution of (3.7) as follows and iterative formula , ε is an arbitrary small positive constant, dt is a constant, we take dt = 0.12 in this paper.
Step 3. Repeat the above process until a stopping criterion is satisfied. The cyclic condition is as follows Proof . This proof please refers to Appendix.

EXPERIMENT RESULTS AND ANALYSIS
In this section, numerical results are presented to demonstrate the performance of our proposed model. The results are compared with several popular models such as AA model, HNW model and HL model. We used three images with the size 256 × 256, Lena, Barbara and Butterfly, in our experiments; see Fig. 1-7. Image Lena is a good test image because it has a nice mixture of detail, flat regions, shading area, and texture. Both Barbara and Butterfly also consist of complex components in different scales, with different patterns and under inhomogeneous illuminations. The three images are suited for our experiments.
In the tests, each pixel of an original image is degraded by a Gamma noise with mean one, and the noise level is controlled by the value of L in the experiments. The experiments are performed in MATLAB under the same software and hardware conditions. The original image Lena and Barbara are shown in Fig. 1. The Lena image in Fig. 1 (a) is distorted by a Gamma noise with L = 20 and L = 10, respectively. The noisy images are shown in Fig. 2 (a) and Fig. 3 (a), respectively. The Barbara image in Fig. 1 (b) is distorted by a Gamma noise with L = 5 and L =10 respectively.
The noisy images are shown in Fig. 4 (a) and Fig. 5 (a), respectively. As L is smaller, the pictures are more noisy. In Fig. 2, we show how the four models behave with a small noisy image. Notice that in this case all the models perform well. Besides, the proposed model is superior to the other three models for preserving the textures and fine details of the images. In Figs. 3-5, we see that the proposed model gets a very good visual effect and works well in the case of heavy noise.
In addition to visual comparison, we use the peak signal to noise ratio (PSNR) and relative error rate (ReErr) of the images to assess the quality of the restored images [14]. Greater PSNR or the smaller ReErr is, the denoising effect is better. From Table 1, we can see that the PSNR obtained by our model is the biggest in those by all four models for the same noisy image. Similarly the ReErr obtained by the proposed model is smallest.
Both denoising visual effect and objective index have showed that the proposed model behaves very well. The model can remove Gamma noise effectively, but also it can protect the edge details and texture features of the image, prevent from producing stair-case phenomenon at some level.

CONCLUSION
In this paper, we propose an efficient total variation model for Gamma multiplicative noise removal. Strictly convex objective function not only ensures that the new model has a unique solution, but also improves the quality of the restored image. Our model can reduce stair-case effect significantly while removing Gamma multiplicative noise quite well. Our experimental results also show that the method is superior to some of existing total variation methods. T z -T z T z -T z z -z  (4.2)