A convex nonlocal total variation regularization algorithm for multiplicative noise removal
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Abstract
This study proposes a nonlocal total variation restoration method to address multiplicative noise removal problems. The strictly convex, objective, nonlocal, total variation effectively utilizes prior information about the multiplicative noise and uses the maximum a posteriori estimator (MAP). An efficient iterative multivariable minimization algorithm is then designed to optimize our proposed model. Finally, we provide a rigorous convergence analysis of the alternating multivariable minimization iteration. The experimental results demonstrate that our proposed model outperforms other currently related models both in terms of evaluation indices and image visual quality.
Keywords
Multiplicative noise Nonlocal total variation Alternating minimization problem Maximum a posteriori estimationAbbreviations
- AA
Aubert and Aujol
- HNW
Huang, Ng, and Wen
- MAP
Maximum a posteriori estimator
- NLM
Nonlocal means filter
- NLTV
Nonlocal total variation
- OCT
Optical coherence tomography
- PSNR
Peak signal-to-noise ratio
- SAR
Synthetic aperture radar
- SO
Shi and Osher
- SSIM
Structural similarity index
- TV
Total variation
1 Introduction
Image deblurring is an important task with numerous applications in both mathematics and image processing. Image deblurring is an inverse problem that determines the unknown original image u from the noisy image f. Total variation (TV) regularization methods are efficient for smoothing a noisy image while effectively preserving the image textures and edges [1, 2]. In recent years, a large number of TV methods have been extensively studied for additive noise removal [3, 4], most of which are convex variation models. The convex models can be optimized using simple and reliable numerical methods, such as the gradient descent [5], primal-dual formulation [6], alternating direction method of multipliers [7], and Bregmanized operator splitting [8].
Multiplicative noise often exists in many coherent imaging systems, such as ultrasonic imaging, optical coherence tomography (OCT), synthetic aperture radar (SAR), and so on [9, 10, 11]. Speckle is the most essential characteristic of noisy images that are corrupted by multiplicative noise. For example, a radar sends coherent waves, and then the reflected scattered waves are captured by the radar sensor. The scattered waves are correlative and interfere with one another, resulting in the obtained image, which is degraded by speckle noise. Owing to the coherent characteristics of multiplicative noise, despeckle is more difficult than additive noise removal. If the statistical properties of multiplicative noise are known, multiplicative noise can be removed effectively. According to forming mechanism of multiplicative noise, many statistical distribution patterns of noise are found, such as Rayleigh noise model [12], Poisson noise model [13], Gaussian noise model [14], and Gamma noise model [15].
Over the last decade, some famous local TV approaches have been successfully used to remove multiplicative noise because of the edge-preserving property of the local TV regularizer. Rudin, Lions, and Osher (RLO) [14] proposed the first local total variational method for multiplicative Gaussian noise removal. Aubert and Aujol (AA) [15] composed a novel local TV model based on multiplicative noise and use the maximum a posteriori (MAP) to remove multiplicative Gamma noise. Shi and Osher (SO) [16] discussed the statistical characteristics of multiplicative noise and proposed a general local TV model for different multiplicative noise reduction, but the fidelity term in the model is not strictly convex. In order to overcome this particular drawback, Huang, Ng, and Wen (HNW) [17] utilized a log transformation and constructed a strictly convex local TV model that can be easily solved by the global optimal solution. Furthermore, reference [18] integrated a quadratic penalty function into local TV model and proposed a new convex variational model for low multiplicative noise removal. Reference [19] designed a convex model that is quite suitable for high multiplicative noise removal by combining a data fitting term, a quadratic penalty term, and a TV regularizer.
Unfortunately, owing to the local total variation regularization framework, smeared textures and numerous staircase effects frequently occur in the denoised image [20, 21]. Exploiting nonlocal correlation information of the image can improve performance of total variation and achieve better image denoising results [22, 23]. One of the well-known nonlocal-based methods is the nonlocal means filter (NLM), which restores the image by using the nonlocal similarity patches. Nonlocal convex functions that were recently utilized as the regularization terms have been successfully used for multiplicative noise reduction [24, 25]. Reference [26] applied the nonlocal total variation (NLTV) norm to the AA model and proposed a new NLTV-based method for multiplicative noise reduction. Unfortunately, this model was nonconvex. Therefore, it is usually difficult to obtain a global solution. Dong et al. proposed a convex nonlocal TV model for multiplicative noise and introduced minimization iterative algorithms corresponding to the model [27]. Since the NLTV makes full use of self-similarity and redundancy within images, it has good image despeckling and denoising performance. However, the NLTV for multiplicative noise reduction is still an open area of research.
In this study, we concentrate on the Gamma-distributed noise and propose a new NLTV-based model for multiplicative noise removal to overcome the drawbacks in current NLTV-based models. First, we utilize prior information regarding multiplicative noise and use the MAP estimation to formulate a novel, strictly convex NLTV model. To efficiently optimize our proposed model, we use split Bregman iteration method to design an alternating multivariable minimization iteration to optimize the convex model. We also provide a rigorous convergence analysis of the alternating iteration method. The experimental results demonstrate that the proposed NLTV model has better performance than some other NLTV-based models for multiplicative noise deblurring.
The following sections are organized as follows. The related NLTV methods are reviewed in Section 2. In Section 3, we propose a new NLTV-based model for multiplicative noise deblurring and design an alternating algorithm for optimizing our proposed model. In Section 4, we applied the proposed model to image deblurring to present its good performance. Finally, conclusions are provided in Section 5.
2 Overview of NLTV algorithms for multiplicative noise reduction
Image denoising obtains the denoised image u^{∗} by minimizing the above bounded energy function (2), which is composed of total variation term and fidelity term. Reducing the total variation term smooths the noisy image and minimizing the fidelity term makes denoised image similar to the original image. λ is the regularization parameter that adjusts the balance between the two terms above. To date, NLTV methods for additive noise reduction have been extensively studied. However, multiplicative noise reduction by NLTV methods is still an open area of research. In this study, we provide the definitions of NLTV and review NLTV models for multiplicative noise reduction.
2.1 Nonlocal total variation
2.2 NLTV method for multiplicative noise reduction
The AA model is efficient for multiplicative noise removal. However, it has some problems since the local total variation regularization framework is exploited, such as smeared textures and the occurrence of staircase effects.
We note that the above TV function is strictly convex. It is easy to obtain a global optimal solution and find the unique minimizer z for the minimization problem. This TV model is referred to as the exponential nonlocal-SO model [27].
3 The proposed method—multiplicative denoising nonlocal total variation model
In our study, we obtain a strictly convex NLTV model for multiplicative noise removal and employ Bregman iteration to optimize it.
3.1 The proposed model
The initial data satisfy u(0) = f, and H(u) has a minimum at u = u(0). We can obtain c = a + b.
3.2 Bregman iteration for the proposed model
All of these equations are combined and summarized in the algorithm that follows:
3.3 Bregman iteration for NLTV minimization
Initialization: u^{0} = log f, p^{0} = z^{0}, b^{0} = d^{0} = 0, k = 0 and λ, μ, γ, tol
End
3.3.1 Convergence analysis
We first analyze the convexity of the objective function to simplify our proof for the convergence of the minimization iteration schemes of our proposed model. We then prove that the sequence generated by the alternative iteration scheme converges to the minimum point of (21).
For the transformation z = log u, it is obvious that the second derivative of the fidelity term in (21) is af exp(−z) + bf^{2} exp(−2z), which is always greater than zero. Therefore, this term is strictly convex in z.Next, we prove that the first term |∇_{NL}z| is also convex.
This implies that \( \left(\tilde{z},\tilde{p}\right) \) is the minimizers of E_{2μ}. This signifies that \( \tilde{z}=R\left(\tilde{p}\right)=R\left(S\left(\tilde{z}\right)\right) \) and \( \tilde{p}=S\left(\tilde{z}\right)=S\left(R\left(\tilde{p}\right)\right) \). Therefore, \( \tilde{z} \) and \( \tilde{p} \) are the fixed points.
In Eq. (45), we conclude z^{k} converges to z^{∗}, which is the unique minimizer of E_{1}(z).
4 Experiment results and discussions
4.1 Experimental setting
In this subsection, we present some experimental results to demonstrate the effectiveness of our proposed model. We experiment on classical grayscale images and coherent imaging images contaminated by artificial multiplicative Gamma noise. Our proposed model is compared with several recent NLTV-based models, namely the nonlocal-AA and nonlocal-SO model. All simulations are performed in MATLAB9.0 on an Intel I7 PC with 4 GB of memory.
4.2 Results on classical grayscale images with artificial noise
Comparisons of the results using different models based on different images
Image | σ | Nonlocal-AA | Nonlocal-SO | Our method | |||
---|---|---|---|---|---|---|---|
PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | ||
Lena | 0.02 | 28.9757 | 0.7974 | 29.6365 | 0.8355 | 29.6522 | 0.8377 |
0.05 | 26.3467 | 0.7530 | 27.3785 | 0.7750 | 27.4042 | 0.7768 | |
0.1 | 24.6380 | 0.6816 | 25.8112 | 0.7279 | 25.8365 | 0.7310 | |
Woman | 0.02 | 27.9584 | 0.7741 | 28.5441 | 0.8044 | 28.5547 | 0.8132 |
0.05 | 25.5996 | 0.7014 | 26.6301 | 0.7477 | 26.6229 | 0.7436 | |
0.1 | 24.8337 | 0.6884 | 25.0348 | 0.7002 | 25.1303 | 0.7059 | |
Cameraman | 0.02 | 28.4807 | 0.8027 | 28.4787 | 0.7969 | 28.7461 | 0.8172 |
0.05 | 26.1229 | 0.7584 | 26.0631 | 0.7380 | 26.3612 | 0.7693 | |
0.1 | 24.0989 | 0.6549 | 24.3225 | 0.6911 | 24.6099 | 0.7308 | |
Baboon | 0.02 | 25.1895 | 0.6904 | 26.2700 | 0.7380 | 26.3582 | 0.7442 |
0.05 | 23.9535 | 0.5718 | 24.3048 | 0.6257 | 24.2040 | 0.6154 | |
0.1 | 21.9093 | 0.4388 | 23.0813 | 0.5216 | 23.1622 | 0.5324 |
4.3 Results on images acquired by coherent imaging system
Since multiplicative Gamma noise often occurs in the coherent imaging systems, we compare the performance of our proposed model with other models on more complicated images acquired by coherent imaging technique where it is not easy to discern the foreground from the background. In this section, we use ultrasonic image, OCT image, and SAR image to verify the effectiveness of our proposed method.
5 Conclusion
This study utilizes prior information and proposes a strictly convex NLTV-based multiplicative noise removal model based on the maximum prior estimate framework. Based on the split Bregman iteration algorithm, we design an efficient alternating minimization iteration to optimize our proposed NLTV model. We also prove that the alternative minimization iteration converges to a fixed point, which is the unique solution of the original minimization problem. Finally, results compared with related NLTV-based multiplicative noise removal models indicate that our proposed NLTV method effectively removes multiplicative noise and outperforms other related NLTV models.
The proposed method is suitable for multiplicative noise removal and successfully implements the coherent imaging system. However, a large number of predefined constants and parameters involved in the alternative iteration algorithm, values of these constants, as well as their parameters are important factors influencing the denoising result of the proposed method. In our experiment, these values are manually set. In later works, adaptively adjusting these parameters to obtain better denoising results will be a future research direction.
It is worth mentioning that the proposed method cannot be directly applied to other types of noise removal problems, such as mixed noise. For example, in the electronic microscopy imaging system, the captured images are usually contaminated by Gaussian and Poisson noises, which are combined as a superposition. Future research is required to make use of NLTV for different types noise removing, especially for mixed noise. On the other hand, the proposed method can be successfully implemented on video sequences, which is not present in this paper due to space limitations. However, we have simply focused on utilizing the correlation information in a single image for noise removal and have not considered similar content and correlation information in different images. There is a great correlation and a large number of redundant information existing between the adjacent frames in the video sequences. Accordingly, utilizing similar and redundant information in the video sequences to improve our proposed method is another direction for future research.
Notes
Acknowledgements
The authors thank the editor and anonymous reviewers for their helpful comments and valuable suggestions.
Funding
This research was supported by the Open Fund Project of the Artificial Intelligence Key Laboratory of Sichuan Province (Grant no. 2016RYY02), and the Scientific Research Project of Sichuan University of Science and Engineering (Grant no. 2018RCL17 and no. 2015RC16).
Availability of data and materials
Request for authors.
Authors’ contributions
All authors take part in the discussion of the work described in this paper. The author MC conceived the idea, optimized the model, and did the experiments of the paper. HZ, QH, and CH were involved in the extensive discussions and evaluations, and all authors read and approved the final manuscript.
Authors’ information
Mingju Chen (1982-) received the M.S. degree in College of Communication Engineering from Chongqing University of Posts and Telecommunications in 2007. He is currently pursuing the Ph.D. degree in Southwest University of Science and Technology. His research interests include machine vision inspection systems and image processing.
Hua Zhang (1969-) received his PhD degree in College of Communication Engineering from Chongqing University in 2006. He is currently a professor in School of Information Engineering of Southwest University of Science and Technology. His research interests include nuclear detection technology, robot technology, and machine vision inspection systems.
Qiang Han(1987-) received the B.S degree from Ocean university of China in 2010, and M.S. degree from Sichuan University of Science and Engineering in 2013. Now, he is currently pursuing his PhD degree in Southwest University of Science and Technology. His current research interests include consensus and coordination in multi-agent systems, networked control system theory, and its application.
Chencheng Huang(1984-) received BS degree in applied mathematics from Shijiazhuang Tiedao University in 2007, Master degree in applied mathematics from Chongqing University in 2011, and a PhD degree from Chongqing University in 2015. He is currently a lecturer with School of Automation and Information Engineering of Sichuan University of Science and Engineering. His research interests are image processing.
Competing interests
The authors declare that they have no competing interests.
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