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An Efficient Total Variation Minimization Method for Image Restoration
Volume 31, Issue 3 (2020), pp. 539–560
Pub. online: 15 April 2020      Type: Research Article      Open accessOpen Access
Received
1 April 2019
Accepted
1 February 2020
Published
15 April 2020

Abstract

In this paper, we present an effective algorithm for solving the Poisson–Gaussian total variation model. The existence and uniqueness of solution for the mixed Poisson–Gaussian model are proved. Due to the strict convexity of the model, the split-Bregman method is employed to solve the minimization problem. Experimental results show the effectiveness of the proposed method for mixed Poisson–Gaussion noise removal. Comparison with other existing and well-known methods is provided as well.

References

 
Aubert, G., Aujol, J. (2008). A variational approach to remove multiplicative noise. SIAM Journal on Applied Mathematics, 68(4), 925–946.
 
Barcelos, C.A.Z., Chen, Y. (2000). Heat flows and related minimization problem in image restoration. Computers & Mathematics with Applications, 39(5–6), 81–97.
 
Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J. (2010). Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends in Machine Learning, 3, 1–122.
 
Calatroni, L., De Los Reyes, J.C., Schönlieb, C.B. (2017). Infimal convolution of data discrepancies for mixed noise removal. SIAM Journal on Imaging Sciences, 10(3), 1196–1233.
 
Chambolle, A. (2004). An algorithm for total variation minimization and applications. Journal of Mathematical Imaging and Vision, 20, 89–97.
 
Chambolle, A., Caselles, V., Novaga, M., Cremers, D., Pock, T. (2010). An introduction to total variation for image analysis. In: Theoretical Foundations and Numerical Methods for Sparse Recovery, Radon Series on Computational and Applied Mathematics, Vol. 9, pp. 263–340.
 
Chan, T.F., Shen, J. (2005). Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods. Society for Industrial and Applied Mathematics. 400 pp.
 
Chouzenoux, E., Jezierska, A., Pesquet, J.C., Talbot, H. (2015). A convex approach for image restoration with exact Poisson–Gaussian likelihood. SIAM Journal on Imaging Sciences, 8(4), 2662–2682.
 
De Los Reyes, J.C., Schönlieb, C.B. (2013). Image denoising: learning the noise model via nonsmooth PDE-constrained optimization. Inverse Problems & Imaging, 7(4), 1183–1214.
 
Dong, Y., Zeng, T. (2013). A convex variational model for restoring blurred images with multiplicative noise. SIAM Journal on Imaging Sciences, 6(3), 1598–1625.
 
Foi, A., Trimeche, M., Katkovnik, V., Egiazarian, K. (2008). Practical Poissonian–Gaussian noise modeling and fitting for single-image raw-data. IEEE Transactions on Image Processing, 17(10), 1737–1754.
 
Goldstein, T., Osher, S. (2009). The split Bregman method for L1-regularized problems. SIAM Journal on Imaging Sciences, 2(2), 323–343.
 
Jezierska, A., Chaux, C., Pesquet, J.C., Talbot, H. (2011). An EM approach for Poisson–Gaussian noise modeling In: 19th European Signal Processing Conference (EUSIPCO), Barcelona, Spain, pp. 2244–2248.
 
Kornprobst, P., Deriche, R., Aubert, G. (1999). Image sequence analysis via partial differential equations. Journal of Mathematical Imaging and Vision, 11, 5–26.
 
Lanza, A., Morigi, S., Sgallari, F., Wen, Y.W. (2014). Image restoration with Poisson–Gaussian mixed noise. Computer Methods in Biomechanics and Biomedical Engineering: Imaging and Visualization, 2(1), 12–24.
 
Le Montagner, Y., Angelini, E.D., Olivo-Marin, J.C. (2014). An unbiased risk estimator for image denoising in the presence of mixed Poisson–Gaussian noise. IEEE Transactions on Image Processing, 23(3), 1255–1268.
 
Le, T., Chartrand, R., Asaki, T.J. (2007). A variational approach to reconstructing images corrupted by Poisson noise. Journal of Mathematical Imaging and Vision, 27, 257–263.
 
Li, J., Shen, Z., Yin, R., Zhang, X. (2015). A reweighted ${l^{2}}$ method for image restoration with Poisson and mixed Poisson–Gaussian noise. Inverse Problems & Imaging, 9(3), 875–894.
 
Li, J., Luisier, F., Blu, T. (2018). PURE-LET image deconvolution. IEEE Transactions on Image Processing, 27(1), 92–105.
 
Liu, L., Chen, L. Chen, C.L.P., Tang, Y.Y., Man Pun, C. (2017). Weighted joint sparse representation for removing mixed noise in image. IEEE Transactions on Cybernetics, 47(3), 600–611.
 
Luisier, F., Blu, T., Unser, M. (2011). Image denoising in mixed Poisson–Gaussian noise. IEEE Transactions on Image Processing, 20(3), 696–708.
 
Makitalo, M., Foi, A. (2013). Optimal inversion of the generalized Anscombe transformation for Poisson–Gaussian noise. IEEE Transactions on Image Processing, 22(1), 91–103.
 
Marnissi, Y., Zheng, Y., Pesquei, J.C. (2016). Fast variational Bayesian signal recovery in the presence of Poisson–Gaussian noise. In: IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 3964–3968.
 
Micchelli, C.A., Shen, L., Xu, Y. (2011). Proximity algorithms for image models: denoising. Inverse Problems, 27(4), 045009.
 
Pham, C.T., Kopylov, A. (2015). Multi-quadratic dynamic programming procedure of edge-preserving denoising for medical images. ISPRS-International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, XL–5/W6, 101–106.
 
Pham, C.T., Gamard, G., Kopylov, A., Tran, T.T.T. (2018). An algorithm for image restoration with mixed noise using total variation regularization. Turkish Journal of Electrical Engineering and Computer Sciences, 26(6), 2831–2845.
 
Pham, C.T., Tran, T.T.T., Phan, T.D.K., Dinh, V.S., Pham, M.T., Nguyen M.H. (2019). An adaptive algorithm for restoring image corrupted by mixed noise. Journal Cybernetics and Physics, 8(2), 73–82.
 
Rudin, L., Osher, S., Fatemi, E. (1992). Nonlinear total variation-based noise removal algorithms. Physica D, 60, 259–268.
 
Sciacchitano, F., Dong, Y., Zeng, T. (2015). Variational approach for restoring blurred images with Cauchy noise. SIAM Journal on Imaging Sciences, 8(3), 1894–1922.
 
Tran, T.T.T., Pham, C.T., Kopylov, A.V., Nguyen, V.N. (2019). An adaptive variational model for medical images restoration. ISPRS – International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, XLII–2/W12, 219–224.
 
Wang, Z., Bovik, A.C. (2006). Modern Image Quality Assessment, Synthesis Lectures on Image, Video, and Multimedia Processing. Morgan and Claypool Publishers. 156 pp.
 
Wang, Y., Yang, J., Yin, W., Zhang, Y. (2008). A new alternating minimization algorithm for total variation image reconstruction. SIAM Journal on Imaging Sciences, 1(3), 248–272.
 
Yan, M. (2013). Restoration of images corrupted by impulse noise and mixed Gaussian impulse noise using blind inpainting. SIAM Journal on Imaging Sciences, 6(3), 1227–1245.
 
Zhang, B., Fadili, M.J., Starck, J.L., Olivo-Marin, J.C. (2007). Multiscale variance-stabilizing transform for mixed-Poisson–Gaussian processes and its applications in bioimaging. In: IEEE International Conference on Image Processing (ICIP), Vol. 6, 233–236.

Biographies

Pham Cong Thang
pcthang@dut.udn.vn

C.T. Pham received his PhD degree in engineering sciences from Tula State University, Tula, Russia, in 2016. He did a postdoctoral fellowship at Centre of Deep Learning and Bayesian Methods, National Research University Higher School of Economics, Moscow, Russia. He is currently a lecturer at the Faculty of Information Technology, The University of Danang – University of Science and Technology, Danang, Vietnam. His research interests include image processing, machine learning.

Tran Thi Thu Thao
thaotran@due.udn.vn

T.T.T. Tran received a MS degree in applied mathematics and computer science from Tula State University, Tula, Russia, in 2018. She is currently a lecturer at the Faculty of Statistics and Informatics, The University of Danang – University of Economics. Her research interests include fractal, percolation, image processing, machine learning.

Gamard Guilhem
guilhem.gamard@normale.fr

G. Gamard received his PhD degree in computer science from University of Montpellier, in 2017. He is currently a member of the ENS of Lyon. His main research interests are discrete mathematics, dynamical systems, sampled signals, data and image processing.


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Copyright
© 2020 Vilnius University
Open access article under the CC BY license.

Keywords
total variation image restoration mixed Poisson–Gaussian noise convex optimization split-Bregman method

Funding
This research is funded by Funds for Science and Technology Development of the University of Danang under project number B2019-DN02-62.

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