- Research
- Open Access
Hybrid variational model based on alternating direction method for image restoration
- Jianguang Zhu^{1},
- Kai Li^{1} and
- Binbin Hao^{2}Email author
https://doi.org/10.1186/s13662-018-1908-0
© The Author(s) 2019
- Received: 23 March 2018
- Accepted: 2 December 2018
- Published: 28 January 2019
Abstract
The total variation model is widely used in image deblurring and denoising process with the features of protecting the image edge. However, this model usually causes some staircase effects. To overcome the shortcoming, combining the second-order total variation regularization and the total variation regularization, we propose a hybrid total variation model. The new improved model not only eliminates the staircase effect, but also well protects the edges of the image. The alternating direction method of multipliers (ADMM) is employed to solve the proposed model. Numerical results show that our proposed model can get more details and higher image visual quality than some current state-of-the-art methods.
Keywords
- Total variation
- Image restoration
- Staircase effect
- Alternating direction method of multipliers
1 Introduction
Although the total variation regularization can preserve sharp edges very well, it also causes some staircase effects [31, 32]. To overcome this kind of staircase effect, some high-order total variational models [33–39] and fractional-order total variation models [40–44] are introduced. It has been proved that the high-order TV norm can remove the staircase effect and preserve the edges well in the process of image restoration.
The rest of this paper is organized as follows. In Sect. 2, we propose our alternating iterative algorithm to solve model (1.5). In Sect. 3, we give some numerical results to demonstrate the effectiveness of the proposed algorithm. Finally, concluding remarks are given in Sect. 4.
2 The alternating iterative algorithm
2.1 The deblurring step
2.2 The denoising step
Subproblem (2.2) is a classical TV regularization process for image denoising, which can be solved by the Chambolle projection algorithm. However, it is well known that the Chambolle projection algorithm has large amount of calculations in the process of experiment and causes numerical instability. To overcome the disadvantage of numerical instability and large amount of calculations of the Chambolle projection algorithm, in this paper, we adopt the alternating direction multiplier method to solve subproblem (2.2).
The variables u, f, v are coupled together, so we separate this problem into two subproblems and adopt the alternating iteration minimization method. The two subproblems are given as follows.
3 Numerical experiments
Experimental results for different images and different blur kernels, \(\mathrm{BSNR}=35\)
Image | Blur kernels | Fast-TV [26] | FNDTV [27] | Our | |||
---|---|---|---|---|---|---|---|
PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | ||
Cameraman | Gaussian(5, 5) | 27.0656 | 0.4399 | 27.2341 | 0.4417 | 27.8678 | 0.4562 |
Gaussian(7, 7) | 26.1232 | 0.3992 | 26.8021 | 0.4155 | 27.0689 | 0.4383 | |
Gaussian(9, 9) | 24.9719 | 0.3807 | 25.6502 | 0.4024 | 26.4150 | 0.4057 | |
Couple | Gaussian(5, 5) | 31.3219 | 0.7337 | 31.6776 | 0.7595 | 32.8470 | 0.7889 |
Gaussian(7, 7) | 29.9460 | 0.6767 | 30.7103 | 0.6989 | 31.3003 | 0.7321 | |
Gaussian(9, 9) | 29.2731 | 0.6694 | 29.8778 | 0.6739 | 30.6027 | 0.6963 | |
Lenna | average(7) | 31.3460 | 0.6673 | 31.8335 | 0.6916 | 32.6287 | 0.7256 |
average(9) | 30.5242 | 0.6415 | 31.0531 | 0.6541 | 31.6273 | 0.6737 | |
average(11) | 29.5574 | 0.6134 | 30.4395 | 0.6376 | 30.9392 | 0.6481 | |
Goldhill | average(7) | 28.3139 | 0.6077 | 29.2712 | 0.6188 | 30.0464 | 0.6330 |
average(9) | 28.1314 | 0.5816 | 28.3268 | 0.5990 | 28.5740 | 0.6023 | |
average(11) | 26.8336 | 0.5244 | 27.4211 | 0.5576 | 27.8857 | 0.5744 | |
Man | motion(20, 20) | 29.8667 | 0.6258 | 30.1235 | 0.6622 | 30.8716 | 0.6864 |
motion(10, 100) | 30.5363 | 0.6839 | 31.2202 | 0.7130 | 32.5163 | 0.7314 | |
Baboon | motion(20, 20) | 27.4672 | 0.7968 | 27.8722 | 0.8334 | 28.5560 | 0.8508 |
motion(10, 100) | 28.8783 | 0.8213 | 28.9383 | 0.8621 | 29.3343 | 0.8778 |
Experimental results for different images and different blur kernels, \(\mathrm{BSNR}=40\)
Image | Blur kernels | Fast-TV [26] | FNDTV [27] | Proposed | |||
---|---|---|---|---|---|---|---|
PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | ||
cameraman | Gaussian(5, 5) | 28.6897 | 0.4841 | 29.2310 | 0.5061 | 29.6541 | 0.5208 |
Gaussian(7, 7) | 27.4559 | 0.4427 | 27.0594 | 0.4369 | 27.6590 | 0.4568 | |
Gaussian(9, 9) | 25.6219 | 0.4068 | 26.2835 | 0.4128 | 27.3563 | 0.4435 | |
couple | Gaussian(5, 5) | 32.0301 | 0.7718 | 32.8019 | 0.7942 | 33.5144 | 0.8202 |
Gaussian(7, 7) | 31.4577 | 0.7401 | 32.0764 | 0.7610 | 32.6735 | 0.7797 | |
Gaussian(9, 9) | 30.1657 | 0.6786 | 30.9071 | 0.7011 | 31.5687 | 0.7426 | |
lenna | average(7) | 32.0735 | 0.7049 | 32.5344 | 0.7279 | 33.3416 | 0.7526 |
average(9) | 31.0432 | 0.6500 | 31.8694 | 0.6853 | 32.7793 | 0.7312 | |
average(11) | 30.7127 | 0.6404 | 31.0552 | 0.6586 | 31.3851 | 0.6724 | |
goldhill | average(7) | 30.3156 | 0.6356 | 31.1560 | 0.6576 | 32.0284 | 0.6920 |
average(9) | 29.4280 | 0.6183 | 30.2251 | 0.6301 | 31.4493 | 0.6656 | |
average(11) | 28.3722 | 0.6082 | 29.5978 | 0.6202 | 30.6778 | 0.6464 | |
man | motion(20, 10) | 30.7569 | 0.6720 | 31.3522 | 0.7031 | 32.6930 | 0.7559 |
motion(11, 100) | 31.6579 | 0.7234 | 32.1598 | 0.7398 | 33.3935 | 0.8044 | |
baboon | motion(20, 10) | 30.3003 | 0.8837 | 30.6833 | 0.8993 | 31.5710 | 0.9128 |
motion(11, 100) | 31.4377 | 0.9082 | 31.9844 | 0.9245 | 32.5230 | 0.9323 |
The numerical results of three different methods in terms of PSNR and SSIM are shown in the following tables. From Tables 1 and 2 it is not difficult to see that the PSNR and SSIM of the restored image by our proposed method are higher than those obtained by FastTV and FNDTV.
4 Conclusion
In this paper, we propose a hybrid total variation model. In addition, we employ the alternating direction method of multipliers to solve it. Experimental results demonstrate that the proposed model can obtain better results than those restored by some existing restoration methods. It also shows that the new model can obtain a better visual resolution than the other two methods.
Declarations
Acknowledgements
The authors would like to thank the referees for their valuable comments and suggestions.
Funding
This work was supported by National Key Research and Development Program of China (No. 2017YFC1405600), by the Training Program of the Major Research Plan of National Science Foundation of China (No. 91746104), by National Science Foundation of China (Nos. 61101208, 11326186), Qindao Postdoctoral Science Foudation (No. 2016114), Project of Shandong Province Higher Educational Science and Technology Program (No. J17KA166), Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources, Shandong Province of China and SDUST Research Fund (No. 2014TDJH102).
Authors’ contributions
All authors worked together to produce the results and read and approved the final manuscript.
Competing interests
The authors declare that there is no conflict of interest regarding the publication of this paper.
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
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