A HYBRID REGULARIZERS MODEL FOR MULTIPLICATIVE NOISE REMOVAL

In this paper, we propose a variational method for restoring images corrupted by multiplicative noise. Computationally, we employ the alternating minimization method to solve our minimization problem. We also study the existence and uniqueness of the proposed problem. Finally, experimental results are provided to demonstrate the superiority of our proposed hybrid model and algorithm for image denoising in comparison with state-of-the-art methods.


Introduction
Image denoising is an important topic in digital image processing. In this field, the main task is to reconstruct a good approximation of original u from observed image f and to preserve local image features for accurate and effective subsequent analysis [Pham and Kopylov, 2015].
Images are corrupted by noise due to several causes including quality of transceivers, influence of light sources or environment condition . There are many types of noise such as Gaussian noise, Poisson noise, impulse noise, mixed noise, gamma noise etc. In this paper, we focus on the multiplicative Gamma noise removal problem.
where f is a corrupted image, and multiplicative Gamma noise η follows the Gamma distribution with its probability density function given by [Aubert and Aujol, 2008;Rudin, Lions, and Osher, 2003]: L is the positive parameter and Γ (.) is the Gamma function, the mean value of the noise η is 1 and the variance is 1 L .
Many approaches have been considered for the multiplicative noise removal [Liu and Fan, 2016;Ullah et al., 2017;Zhao, Wang, and Ng, 2014;Dong et al., 2017]. Among of them, Total variation (TV) based approaches have achieved great success [Li, Wang, and Zhao, 2016;Li, Lou , and Zeng, 2016;Zhou et al., 2015;Yao et al., 2019;Bai, 2019;Aubert and Aujol, 2008;Dong and Zeng, 2013]. In [Aubert and Aujol, 2008], the authors proposed a multiplicative noise removal model as follows (M1 model): where u is the original image, f is the corrupted image, x ∈ Ω, S(Ω) = {u ∈ BV (Ω), u > 0} is the image space, BV (Ω) is the space of functions of bounded variation, λ is a positive parameter, the operator |∇u| is defined later in (5). However,the model (1) is non-convex, and it is difficult to find its global minimal solution. To avoid the drawback, authors in [Dong and Zeng, 2013] proposed a convex variational model by adding a quadratic penalty term as follows (M2 model) : where u is the original image, f is the corrupted image, x ∈ Ω, S(Ω) = {u ∈ BV (Ω), u > 0} is the image space, λ and α are positive parameters. The mentioned models allow us to get the good image denoising results with significantly sharp edges. However, the TV based models tend to create piecewiseconstant in restored image. It leads to undesirable problem usually called the staircase effect. To overcome the staircase effect, higher-order regularization have been considered [Liu et al., 2013;Lefkimmiatis, Bourquard, and Unser, 2012;Chen et al., 2009;Chen and Wunderli, 2002;Lysaker and Tai, 2006;Liu, Yao, and Ke 2007;Li et al., 2007;Papafitsoros and Schonlieb, 2014]. Therefore, authors in [Jiang et al., 2014] proposed an adaptive model of (1) by combinating the TV norm with a secondorder regularizer as follows (M3 model): where x ∈ Ω, S(Ω) = {u ∈ BV (Ω) ∩ BV 2 (Ω), u > 0} is the image space, λ and γ are positive parameters, the operator |∇ 2 u| is defined later in (6).
Inspired of the above studies, we propose a hybrid total variational minimization model to solve the multiplicative noise removal problem. We modify the model (2) by adding a high-order functional into the objective function and investigate an adaptive model as follows: is the image space, λ and γ are positive parameters. In this paper, our main contributions can be summarized as follows. We propose the hybrid model combining the advantages of the TV regularization and the high-order TV model. It allows to avoid the staircase effect with edge-preserving image denoising. We study the issues of existence and uniqueness of a minimizer for the proposed model. Moreover, we employ the well-known alternating minimization method to solve the minimization problem in (4). Several numerical experiments are given to show the performance of our model. In particular, a comparison with related approaches in terms of the peak signal-to-noise ratio and structural similarity index is provided as well.
The rest of the paper is organized as follows. In Section (2), we study existence and uniqueness of solution for the proposed model and present the optimization framework. Next, in Section (3), we show some numerical results of our proposed method and we compare them with the results obtained with other existing and well-known methods. Finally, some conclusions are drawn in Section (4).

The Proposed Model and Method
We can rewrite the optimization problem (4) as follows: Definitions and notations of the spaces BV and BV 2 space can be found in [Chen et al., 2009;Li et al., 2007;Chen and Wunderli, 2002;Lysaker and Tai, 2006;Liu, Yao, and Ke 2007;Papafitsoros and Schonlieb, 2014;Aubert and Kornprobst, 2006]. The discrete gradient ∇u and the second-order derivatives ∇ 2 u of an image u for the pixel location (i, j) in u (i = 1..M ; j = 1..N ) are defined as follows: Motivated by [Aubert and Aujol, 2008;Dong and Zeng, 2013], we have the following theorem to show the existence and uniqueness of the optimization solution to the problem (4). First, we show that E(·) is a convex functional. Second, we show that E(·) has a lower bound. These two facts together imply the existence and uniqueness of solution for the minimization problem (4).
There are many methods which can be employed to obtain the solution of the optimization problem (4), for instance, the primal-dual algorithm, the split-Bregman algorithm, alternating minimization method [Chambolle, 2004;Chan et al., 2011;Goldstein and Osher, 2008;Pham et al., 2019]. In this article, we solve the optimization problem (4) via the alternating direction algorithm which is a variant of the classical augmented Lagrangian multiplier method [Wu and Tai, 2010].
Following the popular alternating minimization method [Chan et al., 2011;Wang et al., 2008;Tai Hahn, and Chung, 2011], we introduce three new variables (d, g, z) and rewrite (4) in the constrained discrete optimization problem as follows: The augmented Lagrangian functional for the constrained optimization problem (7) is defined as: where η 1 , η 2 , η 3 -positive parameters; ρ 1 , ρ 2 , ρ 3 -with Lagrangian multipliers. The minimization method to solve the optimization problem (8) can be expressed as follows: The u subproblem is given by: Thus, we get: We can rewrite the equation as follows: It is obvious that system (10) is linear and symmetric positive definite, therefore z (k+1) can be efficiently solved by fast Fourier transform (FFT) [Wang et al., 2008;Pham, Tran, and Gamard, 2020], under the periodic boundary conditions: where F and F −1 are the forward and inverse Fourier transform operators, and The d and g subproblems are given by: Similarly to [Goldstein and Osher, 2008], generalized shrinkage formula can be employed for solving the d and g subproblems as follows: The z subproblem is given by: Therefore, we get: Applying the Newton's Method, we obtain: Compute d (k+1) according to (12) 5: Compute g (k+1) according to (13) 6: Compute z (k+1) according to (14) 7: by (9) 8: The complete method is summarized in Algorithm (1).
We need a stopping criterion for the iteration: we end the loop if the maximum number of allowed outer iterations N has been carried out (to guarantee an upper bound on running time) or the following condition is satisfied for some prescribed tolerance ς: where ς is a small positive parameter. For our experiments, we set tolerance in (15): ς = 0.00001 and N = 200.

Experimental Results
In this section, we present some numerical results to illustrate the competitive performance of the proposed model for multiplicative noise removal. We compared our recovered results with those of the M1 model (1), the M2 model (2) and the M3 model (3). The compared models are implemented by the state-of-the-art alternating minimization algorithm. Empirically, all images are processed with the equivalent parameters η 1 = 0.01, η 2 = 0.01, η 3 = 1 in our numerical implementation. All experiments were carried out in Windows 10 and Matlab running on a desktop equipped with an Intel Core−i5, 2.4 GHz and 8 GB of RAM.
To assess quality of the restoration results, we use PSNR (Peak Signal-to-Noise Ratio), SSIM (Structural Similarity Index Measure) [Wang and Bovik, 2006] and visual quality. The test images of size 256×256 are shown in Figure (1). In our example, our images are corrupted by multiplicative gamma noise with L = 25 and L = 10. In Figures (2) and (4), we show the results of compared methods for noise levels L = 25, while in Figures (3) and (5), we show the results of compared methods for noise levels L = 10. In Figures (2)a, (3)a, (4)a and (5)a, we represent the noisy images. In the others, Figures (2) (b)-(5)(e), we show respectively the reconstructions given by compared methods.
For a better visual comparison, we have presented the zoomed details of the restored images in Figures (6), (7), (8) and (9). In these Figures, we include zoomed details of the original and noisy images in the first and second column, respectively. From the details in Figures  (6), (7), (8) and (9), we can see that the our model can get better visual improvement than the others.
For quantitative performance comparison, we compare the denoised results in terms of SSIM and PSNR reported in Tables (1) and (2) for noise level L = 25, in Tables (3) and (4) for noise level L = 10 (the best results are highlighted in bold). We can clearly see that our proposed method gets better results than other relative methods in the vast majority of cases. It again demonstrates effectiveness and efficiency of the proposed approach for suppressing multiplicative noise in terms of restoration accuracy and visual quality.

Conclusions
In this paper, we have researched the hybrid regularizers model, combining the first and second-order TV for denoising image corrupted by the multiplicative Gamma   noise. Computationally, an improved highly efficient alternating minimization algorithm is employed for solving the proposed optimization problem. Finally, compared with the existing state-of-the-art TV based models, the experimental results demonstrate that the our proposed approach outperforms other related approaches for removing multiplicative noise both in quantitative and   qualitative terms. The proposed method can be appiled for multiplicative noise removal in some practical applications, e.g. Optical coherence tomography, Laser Doppler Vibration applications, etc. Optical coherence tomography (OCT) is an imaging technique that depends fundamentally on the coherence of the light used in the imaging process, and multiplicative noise is a significant issue in OCT [Liu, Zaki, and Renaud, 2018;Goodman, 2020]. In applications of Laser Doppler Vibrations, speckles noise generated by the relative in-plane motion between the Laser Doppler Vibrometry (LDV) and the target damages the quality of the LDV-captured signal severely [Tabatabai et al., 2013;Lv et al., 2019;Zhu and Baets, 2019].