A weighted denoising method based on Bregman iterative regularization and gradient projection algorithms
- Beilei Tong^{1, 2}Email authorView ORCID ID profile
https://doi.org/10.1186/s13660-017-1551-4
© The Author(s) 2017
Received: 16 September 2017
Accepted: 12 October 2017
Published: 9 November 2017
Abstract
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.
Keywords
MSC
1 Introduction
There are various excellent algorithms to solve the ROF denoising model [4–9]. In this paper, we consider two state-of-the-art denoising methods, i.e. Chambolle’s gradient projection denoising algorithm [4] and Osher et al.’s Bregman iterative regularization method [5]. Chambolle solved the ROF model in the dual field. The Bregman iterative regularization method in [5] gave a significant improvement over standard ROF models by taking back useful information to the denoising results. Yin et al. proved a more simple equivalent formation to the Bregman iterative regularization model in [6]. It is known from the numerical examples of [5] that the Bregman iterative regularization method can keep the horizontal and the vertical edges well and the bent edges badly. On the contrary, we see that Chambolle’s dual denoising method in [4] can keep the curve well and the horizontal and the vertical edges badly. Accordingly, in this paper, we give a comprehensive denoising method based on the dual denoising algorithm [4] and the Bregman iterative regularization method [5, 6]. In this paper, implicitly assumed, dual denoising just refers to Chambolle’s dual denoising algorithm or the gradient projection method.
The weight parameter \(\beta\in(0,1)\), 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 \(\beta= 0\), we solve the ROF model by the gradient projection method for there is no information that is taken back to the model. As for \(\beta = 1\), the model becomes the Bregman iterative regularization model. Secondly, we iteratively solve the modified ROF model until the end condition is met. When \(0 < \beta< 1\), 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 2, we briefly review the dual denoising method and Bregman iteration denoising method. In Section 3, we propose our weighted gradient projection denoising method. Then, in Section 4, we apply our new method to 1-D curve, 2-D gray image and 3-D color image denoising examples, respectively, and present the numerical results. Finally, we give a conclusion.
2 Preliminaries
2.1 Dual denoising method
Proposition 2.1
Here \(I_{\mathrm{out}}\), \(I_{\mathrm{in}}\) denote the iterative numbers of the external iterations and the internal iterations time for u. \(N ^{2}\) is the total number of pixels. σ is the noise standard deviation. For convenience, we set the inner loop times \(L_{\mathrm{in}}\) and the outside loop time \(I_{\mathrm{out}}\).
Lemma 2.1
([4])
Let \(0 < \tau< \frac{1}{8}\) and \(p^{0} = 0\), then, for any \(\lambda_{0} > 0\), \(\lambda \operatorname{div}p^{n}\) converges to \(\pi_{\lambda_{ n} K}(g)\) as \(n \to+ \infty\).
2.2 Bregman iterative regularization denoising method
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 [8].
3 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 [5] and [4], 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.
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 1 or 2 steps outside recycling. It is easy to see that we just need to replace g in (2.7) 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.
Here, the weighted coefficient \(\beta\in[0,1]\), \(\beta\ge0\), 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.
Values of the parameter β
Dimension | 1-D | 2-D | 3-D | |
---|---|---|---|---|
σ = 12 | σ = 25 | |||
β | 0.4 | 0.1 | 0.1 | 0.05 |
4 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 Win7 platform.
Firstly, we test our method by denoising three kinds of images: the 1-D curves, the 2-D gray images and the 3-D color images.
Example 1
Example 2
Example 3
PSNR results of Chambolle’s dual method and our new method
Dimension | σ | Dual (dB) | Ours (dB) |
---|---|---|---|
1-D | σ = 9.4544 | 36.3555 | 40.1584 |
σ = 25 | 31.1127 | 32.8930 | |
2-D | σ = 12 | 33.9067 | 34.3631 |
σ = 25 | 31.1647 | 31.2651 | |
3-D | σ = 12 | 32.1209 | 32.7622 |
σ = 25 | 28.7921 | 29.6473 |
5 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.
Declarations
Acknowledgements
The author was supported by the Grant No. 17ZB0447 of the Scientific Research Fund of Sichuan Provincial Education Department. Also, this work was partly support by the State Key Laboratory of Science and Engineering Computing of the Chinese Academy of Sciences (LSEC of CAS). I would like to thank Dr. Chong Chen (LSEC of CAS) for his friendly inviting me to visit LSEC of CAS, and I would like to thank Prof. Gang Li (QDU) and Prof. Kelong Zhen (SWUST) for their help too.
Authors’ contributions
The author of the manuscript has read and agreed to its content and is accountable for all aspects of the accuracy and integrity of the manuscript.
Competing interests
The author declares that she has no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Rudin, LI, Osher, S, Fatemi, E: Nonlinear total variation based noise removal algorithms. Phys. D, Nonlinear Phenom. 60(1-4), 259-268 (1992) View ArticleMATHMathSciNetGoogle Scholar
- Aubert, G, Kornprobst, P: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations. Springer, Berlin (2002) MATHGoogle Scholar
- Chan, T, Shen, J: Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods. SIAM, Philadelphia (2005) View ArticleMATHGoogle Scholar
- Chambolle, A: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20(1), 89-97 (2004) MATHMathSciNetGoogle Scholar
- Osher, S, Burger, M, Goldfarb, D, Xu, J, Yin, W: An iterated regularization method for total variation-based image restoration. In: IEEE International Conference on Imaging Systems and Techniques, pp. 170-175 (2011) Google Scholar
- Yin, W, Osher, S, Goldfarb, D, Darbon, J: Bregman iterative algorithms for l1-minimization with applications to compressed sensing. SIAM J. Imaging Sci. 1(1), 143-168 (2008) View ArticleMATHMathSciNetGoogle Scholar
- Chang, Q, Chern, IL: Acceleration methods for total variation-based image denoising. SIAM J. Sci. Comput. 25, 982-994 (2003) View ArticleMATHMathSciNetGoogle Scholar
- Goldstein, T, Osher, S: The split Bregman method for L1 regularized problems. SIAM J. Imaging Sci. 2(2), 323-343 (2009) View ArticleMATHMathSciNetGoogle Scholar
- Jia, R, Zhao, H: A fast algorithm for the total variation model of image denoising. Adv. Comput. Math. 33(2), 231-241 (2010) View ArticleMATHMathSciNetGoogle Scholar
- Rockafellar, RT: Convex Analysis, 2nd edn. Princeton University Press, Princeton (1972) MATHGoogle Scholar
- Boyd, S, Vandenberghe, L: Convex Optimization. Cambridge University Press, Cambridge (2004) View ArticleMATHGoogle Scholar
- Ekeland, I, Temam, R: Convex Analysis and Variational Problems. SIAM, Philadelphia (1999) View ArticleMATHGoogle Scholar