A weighted denoising method based on Bregman iterative regularization and gradient projection algorithms

A weighted Bregman-Gradient Projection denoising method, based on the Bregman iterative regularization (BIR) method and Chambolle’s Gradient Projection method (or dual denoising method) is established. Some applications to image denoising on a 1-dimensional curve, 2-dimensional gray image and 3-dimensional color image are presented. Compared with the main results of the literatures, the present numerical results of the proposed method are improved.


Introduction
In this paper, we consider the image denoising problems. The objective is to find the unknown true image u ∈ R n from an observed image g ∈ R n formed as the follows: where n ∈ R n refers to the additive white gaussian noise. To remove the additive white gaussian noise well, Rudin, Osher and Fatemi (ROF) first proposed the total-variation (TV) regularization denoising model in []. This denoising model is actually the optimization of the ROF functional: where, for the continuous case, i.e. u ∈ L  ( ), is an open subset of R n . Here, ·  denotes the L  norm. We also denote ∇u  by TV (u). The TV regularization model had been popular from then on for it can preserve the edges and the details as denoising [, ]. There are various excellent algorithms to solve the ROF denoising model [-]. In this paper, we consider two state-of-the-art denoising methods, i.e. Chambolle Firstly, we choose a proper weight parameter β and modify the ROF functional to a modified form with a weighted taking-back-noise term: The weight parameter β ∈ (, ), maintains a balance between the Bregman iterative regularization method and the dual denoising method. The value of β varies according to the noise level and it is approximately inversely proportional to the noise level. Specially, when β = , we solve the ROF model by the gradient projection method for there is no information that is taken back to the model. As for β = , the model becomes the Bregman iterative regularization model. Secondly, we iteratively solve the modified ROF model until the end condition is met. When  < β < , we solve the modified ROF model by Chambolle's dual algorithm. The results of the numerical experiments demonstrate that the new method cannot only restore more straight edges than the dual denosing method but also restore more bent edges than the Bregman iterative regularization method.
The rest of this paper is organized as follows. In Section , we briefly review the dual denoising method and Bregman iteration denoising method. In Section , we propose our weighted gradient projection denoising method. Then, in Section , we apply our new method to -D curve, -D gray image and -D color image denoising examples, respectively, and present the numerical results. Finally, we give a conclusion.

Dual denoising method
Noticing that (.) can be rewritten as The Euler equation for (.) is which is equivalent to and equivalent to u ∈ ∂J * (gu)/λ .
The above equation can be rewritten as By Proposition ., we get ω = π K (g/λ). The solution of equation (.) can be simplified as For computing u, we just need to compute the nonlinear projection π λK (g), i.e. to solve the following problem: here, M, N represent the total number of pixels in each row and in each column. Given λ = λ  > ,  < τ <   , p  = , for any n ≥ , Chambolle's gradient projection method for the denoising problem (.) is described as below.
for n = , . . . , L out Here I out , I in denote the iterative numbers of the external iterations and the internal iterations time for u. N  is the total number of pixels. σ is the noise standard deviation. For convenience, we set the inner loop times L in and the outside loop time I out .

Bregman iterative regularization denoising method
Osher et al. proposed a Bregman iterative regularization denoising method and proved the convergence []. A simple and equivalent iterative procedure to the BRI denoising model was given in [], and the convergence of this simplified method was also analyzed. Here we consider the simplified BRI denoising model in []. The simplified BIR denoising model is as follows: where BV( ) denotes the space of functions with bounded variation on and | · | BV denotes the BV seminorm, formally given by which is also referred to as the total variation (TV) of u, and update where b k is the information taken back (we set b  = ), g is the degenerative image contaminated by additive white gaussian noise. The Bregman iteration technique has the advantage of converging quickly when applied to certain types of objective functions and the advantage of keeping a fixed value of λ as denoisings [].

A weighted denoising method
While the gradient projection and the BIR denoising methods are extremely efficient, they can either keep the straight edges or keep the bent curves well. From the denoised results

of [] and []
, we see that the bent parts of the curve do not get restored perfectly by the BIR method, while the straight edges are not be kept well by Chambolle's dual denoising method. So we plan to combine these two methods to improve the restored efficiency of the noisy images. We found that the denoising effects were not very good if we just put these two methods together. This is because too much noise was taken back if the noise level is heavy. So we propose a weighted coefficient strategy to eliminate this phenomenon.
Firstly, we use the simplified Bregman iterative regularization model: for the sake of consistency, setting the parameter μ in the above functional equal to  λ we consider the modified ROF model if we apply Chambolle's dual algorithm to each iteration of (.) by Chambolle's dual algorithm, we have By a simple derivation, we have Bregman-gradient projection method initialize: While u ku k-  > tol (tol is the tolerance) for k = , . . . , K * : Using Chambolle's dual method to compute u k+ in (.), we obtain u k+ = (g + b k )π λK (g + b k ) end and using the new update (.): For simplicity, we preset the outside recycling (i.e. the Bregman iteration) numbers and the internal recycling (i.e. the dual iteration) numbers. Usually we just need  or  steps outside recycling. It is easy to see that we just need to replace g in (.) by g +b k . This mixed denoising method is mainly based on the Bregman iterative regularization denoising and Chambolle's gradient projection denoising method, which ensures that each sub-problem has a closed-form solution. However, if we just put these two methods together, the denoising effects were not very good.
Secondly, we add a weight factor β before the taking-back-noise term of the updating iteration step, i.e.
Here, the weighted coefficient β ∈ [, ], β ≥ , is used to balance the amount of the noises taken back to the latest denoised result. The strategy is that the bigger the noise level, the smaller the β is. This is because too much noise was taken back if the noise level is heavy. Next, we will give the mixed denoising method of BIR denoising and the dual denoising method.
Weighted Bregman-gradient projection method      Table .

Numerical experiments and discussions
In this section, we will examine the effectiveness of the weighted Bregman method on TV denoising. The new method was implemented in FORTRAN and MATLAB, and compiled on a Win platform. Firstly, we test our method by denoising three kinds of images: the -D curves, the -D gray images and the -D color images.
Next, we will compare the peak signal-to-noise ratios (PSNRs) of the dual denoising method and our new denoising method. Under the same number of iterations, all the results show that the PSNRs of the new method are higher than those of the dual denoising method. Here, the PSNR is defined as follows: given a noise-free m-by-n image I and its noisy approximation K , where the mean squared error (MSE) is defined as where MAX I denotes the maximum possible pixel value of the image. When the pixels are represented using  bits per sample, MAX I is .     Figure , the second row, the denoised results for the dual method are shown, the PSNR of the left one is . dB and the right one is . dB. In the third row, the restoration results by the proposed method are displayed, with the PSNRs . dB (left) and . dB (right) severally. The denoising parameter of λ from left to right is . and ., the taking-back-noises iteration number equals one and the dual iteration number k is ten. It is clear that, no matter the noise level is light or heavy, the restored results by our method are much better than those by the dual method, especially in the contours and details, such as the hair and the hat.
In Table , we present the PSNRs of the dual algorithm and our new method. Here σ is the noise standard deviation.

Conclusions
In this paper, we proposed a weighted Bregman-gradient projection denoising method. Several kinds of images are denoised by the new method. Numerical results indicate that the new method is more accurate than the dual denoising method and Bregman iteration regularized method.