June  2021, 15(3): 519-537. doi: 10.3934/ipi.2021003

A nonlocal low rank model for poisson noise removal

1. 

Key Laboratory of Computing and Stochastic Mathematics (LCSM), (Ministry of Education of China), School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, China

2. 

The Department of Mathematics, The University of Hong Kong, Hong Kong, China

3. 

Beijing Advanced Innovation Center for Imaging Theory and Technology, Capital Normal University, Beijing, China

* Corresponding author: You-Wei Wen(wenyouwei@gmail.com)

Received  June 2020 Revised  September 2020 Published  June 2021 Early access  December 2020

Fund Project: This work is supported by NSFC Grant No. 11871210, 11971215, the Construct Program of the Key Discipline in Hunan Province, the SRF of Hunan Provincial Education Department (No.17A128), the Hunan Province Graduate Research and Innovation Project (No. CX20190336), the HKRGC GRF 12306616, 12200317, 12300218 and 12300519, and HKU 104005583

Patch-based methods, which take the advantage of the redundancy and similarity among image patches, have attracted much attention in recent years. However, these methods are mainly limited to Gaussian noise removal. In this paper, the Poisson noise removal problem is considered. Unlike Gaussian noise which has an identical and independent distribution, Poisson noise is signal dependent, which makes the problem more challenging. By incorporating the prior that a group of similar patches should possess a low-rank structure, and applying the maximum a posterior (MAP) estimation, the Poisson noise removal problem is formulated as an optimization one. Then, an alternating minimization algorithm is developed to find the minimizer of the objective function efficiently. Convergence of the minimizing sequence will be established, and the efficiency and effectiveness of the proposed algorithm will be demonstrated by numerical experiments.

Citation: Mingchao Zhao, You-Wei Wen, Michael Ng, Hongwei Li. A nonlocal low rank model for poisson noise removal. Inverse Problems & Imaging, 2021, 15 (3) : 519-537. doi: 10.3934/ipi.2021003
References:
[1]

M. AharonM. Elad and A. Bruckstein, KSVD: An algorithm for designing overcomplete dictionaries for sparse representation, IEEE Transactions on Signal Processing, 54 (2006), 4311-4322.   Google Scholar

[2]

F. J. Anscombe, The transformation of poisson, binomial and negative-binomial data, Biometrika, 35 (1948), 246-254.  doi: 10.1093/biomet/35.3-4.246.  Google Scholar

[3]

R. Abergel, C. Louchet, L. Moisan and T. Zeng, Total variation restoration of images corrupted by poisson noise with iterated conditional expectations, in Scale Space and Variational Methods in Computer Vision (eds. J.F. Aujol, M. Nikolova, N. Papadakis), Academic Press, 9087 (2015), 178–190. doi: 10.1007/978-3-319-18461-6_15.  Google Scholar

[4]

S. BabacanR. Molina and A. Katsaggelos, Parameter estimation in TV image restoration using variational distribution approximation, IEEE Transactions on Signal Processing, 17 (2008), 326-339.  doi: 10.1109/TIP.2007.916051.  Google Scholar

[5]

S. BabacanR. Molina and A. Katsaggelos, Variational bayesian blind deconvolution using a total variation prior, IEEE Transactions on Signal Processing, 18 (2009), 12-26.  doi: 10.1109/TIP.2008.2007354.  Google Scholar

[6]

M. Bertero, P. Boccacci, G. Desiderà and G. Vicidomini, Image deblurring with Poisson data: From cells to galaxies, Inverse Problems, 25 (2009), 123006, 26pp. doi: 10.1088/0266-5611/25/12/123006.  Google Scholar

[7]

D. Bertsekas, A. Nedic and E. Ozdaglar, Convex Analysis and Optimization, Athena Scientific, Belmont, 2003.  Google Scholar

[8]

A. BuadesB. CollJ. Morel and J. M. Morel, A review of image denoising algorithms, with a new one, Multiscale Modeling and Simulation, 4 (2006), 490-530.  doi: 10.1137/040616024.  Google Scholar

[9]

A. BuadesB. Coll and J. M. Morel, Image denoising methods. A new nonlocal principle, SIAM Review, 52 (2010), 113-147.  doi: 10.1137/090773908.  Google Scholar

[10]

J. CaiE. Candes and Z. Shen, A singular value thresholding algorithm for matrix completion, SIAM Journal on Optimization, 20 (2010), 1956-1982.  doi: 10.1137/080738970.  Google Scholar

[11]

R. Chan and K. Chen, Multilevel algorithm for a Poisson noise removal model with total-variation regularization, International Journal of Computer Mathematics, 84 (2007), 1183-1198.  doi: 10.1080/00207160701450390.  Google Scholar

[12]

H. Chang and S. Marchesini, Denoising Poisson phaseless measurements via orthogonal dictionary learning, Optics Express, 26 (2018), 19773-19794.  doi: 10.1364/OE.26.019773.  Google Scholar

[13]

P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Modeling and Simulation, 4 (2005), 1168-1200.  doi: 10.1137/050626090.  Google Scholar

[14]

K. DabovA. FoiV. Katkovnik and K. Egiazarian, Image denoising by sparse 3-d transform-domain collaborative filtering, IEEE Transactions on Image Processing, 16 (2007), 2080-2095.  doi: 10.1109/TIP.2007.901238.  Google Scholar

[15]

C. Deledalle, F. Tupin and L. Denis, Poisson NL means: Unsupervised non local means for Poisson noise, in Proceedings of the IEEE International Conference on Image Processin, (2010), 801–804. doi: 10.1109/ICIP.2010.5653394.  Google Scholar

[16]

C. Eckart and G. Young, The approximation of one matrix by another of lower rank, Psychometrika, 1 (1936), 211-218.  doi: 10.1007/BF02288367.  Google Scholar

[17]

M. A. T. Figueiredo and J. M. Bioucas-Dias, Restoration of Poissonian images using alternating direction optimization, IEEE Transactions on Image Processing, 19 (2010), 3133-3145.  doi: 10.1109/TIP.2010.2053941.  Google Scholar

[18]

N. Galatsanos and A. Katsaggelos, Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation, IEEE Transactions on Signal Processing, 1 (1992), 322-336.  doi: 10.1109/83.148606.  Google Scholar

[19]

G. H. GolubM. Heath and G. Wahba, Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics, 21 (1979), 215-223.  doi: 10.1080/00401706.1979.10489751.  Google Scholar

[20]

R. Giryes and M. Elad, Sparsity-based Poisson denoising with dictionary learning, IEEE Transactions on Image Processing, 23 (2014), 5057-5069.  doi: 10.1109/TIP.2014.2362057.  Google Scholar

[21]

S. Gu, L. Zhang, W. Zuo and X. Feng, Weighted nuclear norm minimization with application to image denoising, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (2014), 2862–2869. doi: 10.1109/CVPR.2014.366.  Google Scholar

[22]

P. C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve, SIAM Review, 34 (1992), 561-580.  doi: 10.1137/1034115.  Google Scholar

[23]

P. C. Hansen, Rank-deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, SIAM Monographs on Mathematical Modeling and Computation. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. doi: 10.1137/1.9780898719697.  Google Scholar

[24]

H. Ji, C. Liu, Z. Shen and Y. Xu, Robust video denoising using low rank matrix completion, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (2010), 1791–1798. doi: 10.1109/CVPR.2010.5539849.  Google Scholar

[25]

Q. JinO. Miyashita and F. Tama, Poisson image denoising by piecewise principal component analysis and its application in single-particle X-ray diffraction imaging, IET Image Processing, 12 (2018), 2264-2274.  doi: 10.1049/iet-ipr.2018.5145.  Google Scholar

[26]

A. KucukelbirF. Sigworth and H. Tagare, A Bayesian adaptive basis algorithm for single particle reconstruction, Journal of Structural Biology, 179 (2012), 56-67.   Google Scholar

[27]

T. LeR. Chartrand and T. J. Asaki, A variational approach to reconstructing images corrupted by Poisson noise, Journal of Mathematical Imaging and Vision, 27 (2007), 257-263.  doi: 10.1007/s10851-007-0652-y.  Google Scholar

[28]

L. Lucy, An Iterative Technique for The Rectification of Observed Distributions, The Astronomical Journal, 79 (1974), 745-754.   Google Scholar

[29]

L. MaL. Xu and T. Zeng, Low rank prior and total variation regularization for image deblurring, Journal of Scientific Computing, 70 (2017), 1336-1357.  doi: 10.1007/s10915-016-0282-x.  Google Scholar

[30]

S. MaD. Goldfarb and L. Chen, Fixed point and Bregman iterative methods for matrix rank minimization, Mathematical Programming, 128 (2011), 321-353.  doi: 10.1007/s10107-009-0306-5.  Google Scholar

[31]

M. M$\ddot{a}$kitalo and A. Foi, Optimal inversion of the Anscombe transformation in low-count Poisson image denoising, IEEE Transactions on Image Processing, 20 (2011), 99-109.  doi: 10.1109/TIP.2010.2056693.  Google Scholar

[32]

V. Morozov, Methods for Solving Incorrectly Posed Problems, , Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5280-1.  Google Scholar

[33]

F. Murtagh and J. L. Starck, Astronomical Image and Data Analysis, Springer-Verlag, New York, 2006. Google Scholar

[34]

J. P. OliveiraJ. M. Bioucas-Dias and M. A. T. Figueiredo, Adaptive total variation image deblurring: A majorization–minimization approach, Signal Processing, 89 (2009), 1683-1693.   Google Scholar

[35]

Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bulletin of the American Mathematical Society, 73 (1967), 591-597.  doi: 10.1090/S0002-9904-1967-11761-0.  Google Scholar

[36]

V. Papyan and M. Elad, Multi-scale patch-based image restoration, IEEE Transactions on Image Processing, 25 (2016), 249-261.  doi: 10.1109/TIP.2015.2499698.  Google Scholar

[37]

G. PrashanthKumar and R. R. Sahay, Low rank poisson denoising (LRPD): A low rank approach using split bregman algorithm for poisson noise removal from images, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops, (2019). Google Scholar

[38]

R. PuetterT. Gosnell and A. Yahil, Digital image reconstruction: Deblurring and denoising, Annual Review of Astronomy and Astrophysics, 43 (2005), 139-194.   Google Scholar

[39]

A. RajwadeA. Rangarajan and A. Banerjee, Image denoising using the higher order singular value decomposition, IEEE Transactions on Pattern Analysis and Machine Intelligence, 35 (2013), 849-862.   Google Scholar

[40]

W. H. Richarson, Bayesian-based iterative method of image restoration, Journal of the Optical Society of America, 62 (1972), 55-59.   Google Scholar

[41]

L. RudinS. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[42]

S. H. W. Scheres, A bayesian view on cryo-EM structure determination, Journal of Molecular Biology, 415 (2012), 406-418.   Google Scholar

[43]

S. SetzerG. Steidl and T. Teuber, Deblurring poissonian images by split bregman techniques, Journal of Visual Communication and Image Representation, 21 (2010), 193-199.   Google Scholar

[44]

K. E. Timmermann and R. D. Nowak, Multiscale modeling and estimation of Poisson processes with application to photon-limited imaging, IEEE Transactions on Information Theory, 45 (1999), 846-862.  doi: 10.1109/18.761328.  Google Scholar

[45]

Y. Xiao and T. Zeng, Poisson noise removal via learned dictionary, in 2010 IEEE International Conference on Image Processing, (2010), 1177-1180. Google Scholar

[46]

Y. WenR. Chan and T. Zeng, Primal-dual algorithms for total variation based image restoration under poisson noise, Science China Mathematics, 59 (2016), 141-160.  doi: 10.1007/s11425-015-5079-0.  Google Scholar

[47]

M. N. Wernick and J. N. Aarsvold, Emission Tomography: The Fundamentals of PET and SPECT, Academic Press, 2004. Google Scholar

[48]

R. Willett and R. Nowak, Platelets: A multiscale approach for recovering edges and surfaces in photon-limited medical imaging, IEEE Transactions on Medical Imaging, 22 (2003), 332-350.   Google Scholar

[49]

Y. XieS. GuY. LiuW. ZuoW. Zhang and L. Zhang, Weighted schatten $p$ -norm minimization for image denoising and background subtraction, IEEE Transactions on Image Processing, 25 (2016), 4842-4857.  doi: 10.1109/TIP.2016.2599290.  Google Scholar

[50]

J. Xu, L. Zhang, D. Zhang and X. Feng, Multi-channel weighted nuclear norm minimization for real color image denoising, in Proceedings of the IEEE International Conference on Computer Vision, (2017), 1105–1113. Google Scholar

[51]

R. Zanella, P. Boccacci, L. Zanni and M. Bertero, Efficient gradient projection methods for edge-preserving removal of Poisson noise, Inverse Problems, 25 (2009), 045010, 24 pp. doi: 10.1088/0266-5611/25/4/045010.  Google Scholar

[52]

Y. Zheng, G. Liu, S. Sugimoto, S. Yan and M. Okutomi, Practical low-rank matrix approximation under robust l1-norm, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (2012), 1410–1417. Google Scholar

[53]

W. ZhouA. BovikH. Sheikh and E. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Transactions on image processing, 13 (2004), 600-612.   Google Scholar

show all references

References:
[1]

M. AharonM. Elad and A. Bruckstein, KSVD: An algorithm for designing overcomplete dictionaries for sparse representation, IEEE Transactions on Signal Processing, 54 (2006), 4311-4322.   Google Scholar

[2]

F. J. Anscombe, The transformation of poisson, binomial and negative-binomial data, Biometrika, 35 (1948), 246-254.  doi: 10.1093/biomet/35.3-4.246.  Google Scholar

[3]

R. Abergel, C. Louchet, L. Moisan and T. Zeng, Total variation restoration of images corrupted by poisson noise with iterated conditional expectations, in Scale Space and Variational Methods in Computer Vision (eds. J.F. Aujol, M. Nikolova, N. Papadakis), Academic Press, 9087 (2015), 178–190. doi: 10.1007/978-3-319-18461-6_15.  Google Scholar

[4]

S. BabacanR. Molina and A. Katsaggelos, Parameter estimation in TV image restoration using variational distribution approximation, IEEE Transactions on Signal Processing, 17 (2008), 326-339.  doi: 10.1109/TIP.2007.916051.  Google Scholar

[5]

S. BabacanR. Molina and A. Katsaggelos, Variational bayesian blind deconvolution using a total variation prior, IEEE Transactions on Signal Processing, 18 (2009), 12-26.  doi: 10.1109/TIP.2008.2007354.  Google Scholar

[6]

M. Bertero, P. Boccacci, G. Desiderà and G. Vicidomini, Image deblurring with Poisson data: From cells to galaxies, Inverse Problems, 25 (2009), 123006, 26pp. doi: 10.1088/0266-5611/25/12/123006.  Google Scholar

[7]

D. Bertsekas, A. Nedic and E. Ozdaglar, Convex Analysis and Optimization, Athena Scientific, Belmont, 2003.  Google Scholar

[8]

A. BuadesB. CollJ. Morel and J. M. Morel, A review of image denoising algorithms, with a new one, Multiscale Modeling and Simulation, 4 (2006), 490-530.  doi: 10.1137/040616024.  Google Scholar

[9]

A. BuadesB. Coll and J. M. Morel, Image denoising methods. A new nonlocal principle, SIAM Review, 52 (2010), 113-147.  doi: 10.1137/090773908.  Google Scholar

[10]

J. CaiE. Candes and Z. Shen, A singular value thresholding algorithm for matrix completion, SIAM Journal on Optimization, 20 (2010), 1956-1982.  doi: 10.1137/080738970.  Google Scholar

[11]

R. Chan and K. Chen, Multilevel algorithm for a Poisson noise removal model with total-variation regularization, International Journal of Computer Mathematics, 84 (2007), 1183-1198.  doi: 10.1080/00207160701450390.  Google Scholar

[12]

H. Chang and S. Marchesini, Denoising Poisson phaseless measurements via orthogonal dictionary learning, Optics Express, 26 (2018), 19773-19794.  doi: 10.1364/OE.26.019773.  Google Scholar

[13]

P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Modeling and Simulation, 4 (2005), 1168-1200.  doi: 10.1137/050626090.  Google Scholar

[14]

K. DabovA. FoiV. Katkovnik and K. Egiazarian, Image denoising by sparse 3-d transform-domain collaborative filtering, IEEE Transactions on Image Processing, 16 (2007), 2080-2095.  doi: 10.1109/TIP.2007.901238.  Google Scholar

[15]

C. Deledalle, F. Tupin and L. Denis, Poisson NL means: Unsupervised non local means for Poisson noise, in Proceedings of the IEEE International Conference on Image Processin, (2010), 801–804. doi: 10.1109/ICIP.2010.5653394.  Google Scholar

[16]

C. Eckart and G. Young, The approximation of one matrix by another of lower rank, Psychometrika, 1 (1936), 211-218.  doi: 10.1007/BF02288367.  Google Scholar

[17]

M. A. T. Figueiredo and J. M. Bioucas-Dias, Restoration of Poissonian images using alternating direction optimization, IEEE Transactions on Image Processing, 19 (2010), 3133-3145.  doi: 10.1109/TIP.2010.2053941.  Google Scholar

[18]

N. Galatsanos and A. Katsaggelos, Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation, IEEE Transactions on Signal Processing, 1 (1992), 322-336.  doi: 10.1109/83.148606.  Google Scholar

[19]

G. H. GolubM. Heath and G. Wahba, Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics, 21 (1979), 215-223.  doi: 10.1080/00401706.1979.10489751.  Google Scholar

[20]

R. Giryes and M. Elad, Sparsity-based Poisson denoising with dictionary learning, IEEE Transactions on Image Processing, 23 (2014), 5057-5069.  doi: 10.1109/TIP.2014.2362057.  Google Scholar

[21]

S. Gu, L. Zhang, W. Zuo and X. Feng, Weighted nuclear norm minimization with application to image denoising, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (2014), 2862–2869. doi: 10.1109/CVPR.2014.366.  Google Scholar

[22]

P. C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve, SIAM Review, 34 (1992), 561-580.  doi: 10.1137/1034115.  Google Scholar

[23]

P. C. Hansen, Rank-deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, SIAM Monographs on Mathematical Modeling and Computation. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. doi: 10.1137/1.9780898719697.  Google Scholar

[24]

H. Ji, C. Liu, Z. Shen and Y. Xu, Robust video denoising using low rank matrix completion, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (2010), 1791–1798. doi: 10.1109/CVPR.2010.5539849.  Google Scholar

[25]

Q. JinO. Miyashita and F. Tama, Poisson image denoising by piecewise principal component analysis and its application in single-particle X-ray diffraction imaging, IET Image Processing, 12 (2018), 2264-2274.  doi: 10.1049/iet-ipr.2018.5145.  Google Scholar

[26]

A. KucukelbirF. Sigworth and H. Tagare, A Bayesian adaptive basis algorithm for single particle reconstruction, Journal of Structural Biology, 179 (2012), 56-67.   Google Scholar

[27]

T. LeR. Chartrand and T. J. Asaki, A variational approach to reconstructing images corrupted by Poisson noise, Journal of Mathematical Imaging and Vision, 27 (2007), 257-263.  doi: 10.1007/s10851-007-0652-y.  Google Scholar

[28]

L. Lucy, An Iterative Technique for The Rectification of Observed Distributions, The Astronomical Journal, 79 (1974), 745-754.   Google Scholar

[29]

L. MaL. Xu and T. Zeng, Low rank prior and total variation regularization for image deblurring, Journal of Scientific Computing, 70 (2017), 1336-1357.  doi: 10.1007/s10915-016-0282-x.  Google Scholar

[30]

S. MaD. Goldfarb and L. Chen, Fixed point and Bregman iterative methods for matrix rank minimization, Mathematical Programming, 128 (2011), 321-353.  doi: 10.1007/s10107-009-0306-5.  Google Scholar

[31]

M. M$\ddot{a}$kitalo and A. Foi, Optimal inversion of the Anscombe transformation in low-count Poisson image denoising, IEEE Transactions on Image Processing, 20 (2011), 99-109.  doi: 10.1109/TIP.2010.2056693.  Google Scholar

[32]

V. Morozov, Methods for Solving Incorrectly Posed Problems, , Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5280-1.  Google Scholar

[33]

F. Murtagh and J. L. Starck, Astronomical Image and Data Analysis, Springer-Verlag, New York, 2006. Google Scholar

[34]

J. P. OliveiraJ. M. Bioucas-Dias and M. A. T. Figueiredo, Adaptive total variation image deblurring: A majorization–minimization approach, Signal Processing, 89 (2009), 1683-1693.   Google Scholar

[35]

Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bulletin of the American Mathematical Society, 73 (1967), 591-597.  doi: 10.1090/S0002-9904-1967-11761-0.  Google Scholar

[36]

V. Papyan and M. Elad, Multi-scale patch-based image restoration, IEEE Transactions on Image Processing, 25 (2016), 249-261.  doi: 10.1109/TIP.2015.2499698.  Google Scholar

[37]

G. PrashanthKumar and R. R. Sahay, Low rank poisson denoising (LRPD): A low rank approach using split bregman algorithm for poisson noise removal from images, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops, (2019). Google Scholar

[38]

R. PuetterT. Gosnell and A. Yahil, Digital image reconstruction: Deblurring and denoising, Annual Review of Astronomy and Astrophysics, 43 (2005), 139-194.   Google Scholar

[39]

A. RajwadeA. Rangarajan and A. Banerjee, Image denoising using the higher order singular value decomposition, IEEE Transactions on Pattern Analysis and Machine Intelligence, 35 (2013), 849-862.   Google Scholar

[40]

W. H. Richarson, Bayesian-based iterative method of image restoration, Journal of the Optical Society of America, 62 (1972), 55-59.   Google Scholar

[41]

L. RudinS. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[42]

S. H. W. Scheres, A bayesian view on cryo-EM structure determination, Journal of Molecular Biology, 415 (2012), 406-418.   Google Scholar

[43]

S. SetzerG. Steidl and T. Teuber, Deblurring poissonian images by split bregman techniques, Journal of Visual Communication and Image Representation, 21 (2010), 193-199.   Google Scholar

[44]

K. E. Timmermann and R. D. Nowak, Multiscale modeling and estimation of Poisson processes with application to photon-limited imaging, IEEE Transactions on Information Theory, 45 (1999), 846-862.  doi: 10.1109/18.761328.  Google Scholar

[45]

Y. Xiao and T. Zeng, Poisson noise removal via learned dictionary, in 2010 IEEE International Conference on Image Processing, (2010), 1177-1180. Google Scholar

[46]

Y. WenR. Chan and T. Zeng, Primal-dual algorithms for total variation based image restoration under poisson noise, Science China Mathematics, 59 (2016), 141-160.  doi: 10.1007/s11425-015-5079-0.  Google Scholar

[47]

M. N. Wernick and J. N. Aarsvold, Emission Tomography: The Fundamentals of PET and SPECT, Academic Press, 2004. Google Scholar

[48]

R. Willett and R. Nowak, Platelets: A multiscale approach for recovering edges and surfaces in photon-limited medical imaging, IEEE Transactions on Medical Imaging, 22 (2003), 332-350.   Google Scholar

[49]

Y. XieS. GuY. LiuW. ZuoW. Zhang and L. Zhang, Weighted schatten $p$ -norm minimization for image denoising and background subtraction, IEEE Transactions on Image Processing, 25 (2016), 4842-4857.  doi: 10.1109/TIP.2016.2599290.  Google Scholar

[50]

J. Xu, L. Zhang, D. Zhang and X. Feng, Multi-channel weighted nuclear norm minimization for real color image denoising, in Proceedings of the IEEE International Conference on Computer Vision, (2017), 1105–1113. Google Scholar

[51]

R. Zanella, P. Boccacci, L. Zanni and M. Bertero, Efficient gradient projection methods for edge-preserving removal of Poisson noise, Inverse Problems, 25 (2009), 045010, 24 pp. doi: 10.1088/0266-5611/25/4/045010.  Google Scholar

[52]

Y. Zheng, G. Liu, S. Sugimoto, S. Yan and M. Okutomi, Practical low-rank matrix approximation under robust l1-norm, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (2012), 1410–1417. Google Scholar

[53]

W. ZhouA. BovikH. Sheikh and E. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Transactions on image processing, 13 (2004), 600-612.   Google Scholar

Figure 1.  The test images, from left to right, top to bottom: Monarch, Cameraman, House, Peppers, Lena, Man
Figure 2.  The distribution of the residual patches $ E_{\ell} $, fitting Gaussian distribution for Monarch image
Figure 3.  $ b = 1 $, $ P = 15 $. From left to right, top to bottom, original image and noise image, restored images by methods: PIDSplit, PD, NLM, KSVD, WNNM, LRPD, LRS, LRH
Figure 4.  $ b = 1 $, $ P = 15 $. From left to right, top to bottom, original image and noise image, restored images by methods: PIDSplit, PD, NLM, KSVD, WNNM, LRPD, LRS, LRH
Figure 5.  $ b = 1 $, $ P = 15 $. From left to right, top to bottom, original image and noise image, restored images by methods: PIDSplit, PD, NLM, KSVD, MC-WNNM, LRPD, LRS, LRH
Figure 6.  $ b = 1 $, $ P = 15 $. From left to right, top to bottom, original image and noise image, restored images by methods: PIDSplit, PD, NLM, KSVD, MC-WNNM, LRPD, LRS, LRH
Table 1.  SNR and SSIM for restoration image under Poisson noise. The best results are in bold
Image b P PIDSplit PD NLM KSVD WNNM LRPD LRS LRH
SNR
Monarch 1 15 17.86 17.92 18.26 18.83 18.71 17.21 $ \bf{{19.56}} $ 19.48
30 20.13 20.23 20.38 20.65 21.09 17.57 $ \bf{{21.35}} $ 21.27
45 21.27 21.48 21.40 21.86 22.26 18.07 22.45 $ \bf{{22.47}} $
60 22.21 22.45 22.17 22.68 23.11 18.71 23.28 $ \bf{{23.38}} $
5 15 17.27 17.33 17.11 17.34 17.75 17.89 $ \bf{{18.53}} $ 18.25
30 19.62 19.81 19.81 20.03 20.39 18.71 $ \bf{{20.98}} $ 20.85
45 20.83 21.14 21.20 21.41 21.89 18.77 22.20 $ \bf{{22.27}} $
60 21.84 22.19 21.82 22.27 22.70 19.22 22.98 $ \bf{{23.06}} $
Cameraman 1 15 19.26 19.29 19.72 19.73 19.68 18.46 20.11 $ \bf{{20.59}} $
30 21.00 21.00 21.54 21.75 21.76 18.61 21.64 $ \bf{{22.03}} $
45 21.90 22.00 22.43 22.77 22.71 19.19 22.68 $ \bf{{23.12}} $
60 22.61 22.73 22.97 23.40 23.41 19.87 23.33 $ \bf{{23.82}} $
5 15 18.28 18.33 18.32 18.87 18.69 18.50 $ \bf{{19.35}} $ 19.08
30 20.32 20.38 20.95 21.22 21.28 19.70 21.27 $ \bf{{21.50}} $
45 21.31 21.41 21.98 22.19 22.35 19.86 22.32 $ \bf{{22.52}} $
60 22.15 22.27 22.62 22.94 23.11 20.32 23.03 $ \bf{{23.34}} $
SSIM
Monarch 1 15 0.9867 0.9869 0.9881 0.9888 0.9895 0.9863 $ \bf{{0.9906}} $ 0.9903
30 0.9788 0.9789 0.9813 0.9823 0.9830 0.9697 $ \bf{{0.9834}} $ 0.9833
45 0.9724 0.9727 0.9745 0.9763 0.9772 0.9551 0.9773 $ \bf{{0.9774}} $
60 0.9673 0.9674 0.9698 0.9721 0.9737 0.9448 0.9732 $ \bf{{0.9738}} $
5 15 0.9854 0.9854 0.9848 0.9856 0.9871 0.9815 $ \bf{{0.9883}} $ 0.9872
30 0.9768 0.9771 0.9784 0.9795 0.9809 0.9701 $ \bf{{0.9823}} $ 0.9817
45 0.9699 0.9706 0.9727 0.9741 0.9763 0.9545 $ \bf{{0.9766}} $ $ \bf{{0.9766}} $
60 0.9644 0.9653 0.9668 0.9696 0.9709 0.9437 0.9713 $ \bf{{0.9715}} $
Cameraman 1 15 0.9882 0.9883 0.9893 0.9893 0.9894 0.9863 0.9906 $ \bf{{0.9915}} $
30 0.9776 0.9778 0.9811 0.9816 0.9820 0.9678 0.9801 $ \bf{{0.9823}} $
45 0.9685 0.9688 0.9727 0.9734 0.9730 0.9530 0.9716 $ \bf{{0.9745}} $
60 0.9601 0.9598 0.9634 0.9641 0.9635 0.9412 0.9634 $ \bf{{0.9662}} $
5 15 0.9852 0.9852 0.9844 0.9863 0.9870 0.9818 $ \bf{{0.9887}} $ 0.9878
30 0.9742 0.9746 0.9781 0.9790 0.9802 0.9692 0.9792 $ \bf{{0.9803}} $
45 0.9645 0.9652 0.9700 0.9700 0.9713 0.9531 0.9705 $ \bf{{0.9714}} $
60 0.9560 0.9568 0.9606 0.9611 0.9620 0.9416 0.9618 $ \bf{{0.9630}} $
Image b P PIDSplit PD NLM KSVD WNNM LRPD LRS LRH
SNR
Monarch 1 15 17.86 17.92 18.26 18.83 18.71 17.21 $ \bf{{19.56}} $ 19.48
30 20.13 20.23 20.38 20.65 21.09 17.57 $ \bf{{21.35}} $ 21.27
45 21.27 21.48 21.40 21.86 22.26 18.07 22.45 $ \bf{{22.47}} $
60 22.21 22.45 22.17 22.68 23.11 18.71 23.28 $ \bf{{23.38}} $
5 15 17.27 17.33 17.11 17.34 17.75 17.89 $ \bf{{18.53}} $ 18.25
30 19.62 19.81 19.81 20.03 20.39 18.71 $ \bf{{20.98}} $ 20.85
45 20.83 21.14 21.20 21.41 21.89 18.77 22.20 $ \bf{{22.27}} $
60 21.84 22.19 21.82 22.27 22.70 19.22 22.98 $ \bf{{23.06}} $
Cameraman 1 15 19.26 19.29 19.72 19.73 19.68 18.46 20.11 $ \bf{{20.59}} $
30 21.00 21.00 21.54 21.75 21.76 18.61 21.64 $ \bf{{22.03}} $
45 21.90 22.00 22.43 22.77 22.71 19.19 22.68 $ \bf{{23.12}} $
60 22.61 22.73 22.97 23.40 23.41 19.87 23.33 $ \bf{{23.82}} $
5 15 18.28 18.33 18.32 18.87 18.69 18.50 $ \bf{{19.35}} $ 19.08
30 20.32 20.38 20.95 21.22 21.28 19.70 21.27 $ \bf{{21.50}} $
45 21.31 21.41 21.98 22.19 22.35 19.86 22.32 $ \bf{{22.52}} $
60 22.15 22.27 22.62 22.94 23.11 20.32 23.03 $ \bf{{23.34}} $
SSIM
Monarch 1 15 0.9867 0.9869 0.9881 0.9888 0.9895 0.9863 $ \bf{{0.9906}} $ 0.9903
30 0.9788 0.9789 0.9813 0.9823 0.9830 0.9697 $ \bf{{0.9834}} $ 0.9833
45 0.9724 0.9727 0.9745 0.9763 0.9772 0.9551 0.9773 $ \bf{{0.9774}} $
60 0.9673 0.9674 0.9698 0.9721 0.9737 0.9448 0.9732 $ \bf{{0.9738}} $
5 15 0.9854 0.9854 0.9848 0.9856 0.9871 0.9815 $ \bf{{0.9883}} $ 0.9872
30 0.9768 0.9771 0.9784 0.9795 0.9809 0.9701 $ \bf{{0.9823}} $ 0.9817
45 0.9699 0.9706 0.9727 0.9741 0.9763 0.9545 $ \bf{{0.9766}} $ $ \bf{{0.9766}} $
60 0.9644 0.9653 0.9668 0.9696 0.9709 0.9437 0.9713 $ \bf{{0.9715}} $
Cameraman 1 15 0.9882 0.9883 0.9893 0.9893 0.9894 0.9863 0.9906 $ \bf{{0.9915}} $
30 0.9776 0.9778 0.9811 0.9816 0.9820 0.9678 0.9801 $ \bf{{0.9823}} $
45 0.9685 0.9688 0.9727 0.9734 0.9730 0.9530 0.9716 $ \bf{{0.9745}} $
60 0.9601 0.9598 0.9634 0.9641 0.9635 0.9412 0.9634 $ \bf{{0.9662}} $
5 15 0.9852 0.9852 0.9844 0.9863 0.9870 0.9818 $ \bf{{0.9887}} $ 0.9878
30 0.9742 0.9746 0.9781 0.9790 0.9802 0.9692 0.9792 $ \bf{{0.9803}} $
45 0.9645 0.9652 0.9700 0.9700 0.9713 0.9531 0.9705 $ \bf{{0.9714}} $
60 0.9560 0.9568 0.9606 0.9611 0.9620 0.9416 0.9618 $ \bf{{0.9630}} $
Table 2.  SNR and SSIM for restoration image under Poisson noise. The best results are in bold
Image PIDSplit PD NLM KSVD WNNM LRPD LRS LRH
SNR
House 23.91 23.98 24.50 25.14 25.92 22.00 $ \bf{{26.01}} $ 25.98
Peppers 21.71 21.83 21.57 21.91 22.16 20.53 22.38 $ \bf{{22.49}} $
Lena 23.29 23.32 23.68 23.63 23.92 22.23 $ \bf{{24.37}} $ 24.01
Man 21.08 20.98 20.75 20.55 20.81 18.84 $ \bf{{21.47}} $ 21.29
SSIM
House 0.9813 0.9812 0.9841 0.9864 0.9870 0.9804 0.9871 $ \bf{{0.9872}} $
Peppers 0.9783 0.9783 0.9794 0.9805 0.9812 0.9712 0.9808 $ \bf{{0.9814}} $
Lena 0.9810 0.9808 0.9837 0.9837 0.9842 0.9774 $ \bf{{0.9845}} $ 0.9840
Man 0.9701 0.9706 0.9714 0.9696 0.9702 0.9562 $ \bf{{0.9733}} $ 0.9723
Image PIDSplit PD NLM KSVD WNNM LRPD LRS LRH
SNR
House 23.91 23.98 24.50 25.14 25.92 22.00 $ \bf{{26.01}} $ 25.98
Peppers 21.71 21.83 21.57 21.91 22.16 20.53 22.38 $ \bf{{22.49}} $
Lena 23.29 23.32 23.68 23.63 23.92 22.23 $ \bf{{24.37}} $ 24.01
Man 21.08 20.98 20.75 20.55 20.81 18.84 $ \bf{{21.47}} $ 21.29
SSIM
House 0.9813 0.9812 0.9841 0.9864 0.9870 0.9804 0.9871 $ \bf{{0.9872}} $
Peppers 0.9783 0.9783 0.9794 0.9805 0.9812 0.9712 0.9808 $ \bf{{0.9814}} $
Lena 0.9810 0.9808 0.9837 0.9837 0.9842 0.9774 $ \bf{{0.9845}} $ 0.9840
Man 0.9701 0.9706 0.9714 0.9696 0.9702 0.9562 $ \bf{{0.9733}} $ 0.9723
Table 3.  SNR and SSIM for color image. The best results are in bold
Image b P PIDSplit PD NLM KSVD MC-WNNM LRPD LRS LRH
SNR
McM1 1 15 17.39 17.24 17.41 17.30 17.54 17.22 $ \bf{{18.16}} $ 17.71
30 19.29 19.09 19.44 19.37 19.58 17.21 $ \bf{{19.92}} $ 19.35
45 20.36 20.36 20.50 20.49 20.75 17.47 $ \bf{{20.93}} $ 20.53
60 21.24 21.21 21.21 21.23 21.54 17.83 $ \bf{{21.66}} $ 21.38
5 15 16.94 15.91 16.44 16.34 16.78 19.82 $ \bf{{17.66}} $ 17.34
30 18.87 18.36 18.77 18.63 18.88 18.64 $ \bf{{19.61}} $ 19.41
45 19.96 19.85 20.02 19.97 20.15 18.35 $ \bf{{20.69}} $ 20.54
60 20.85 20.77 20.84 20.82 21.07 18.44 $ \bf{{21.46}} $ 21.43
McM2 1 15 22.35 21.15 22.05 21.81 23.76 21.02 $ \bf{{24.47}} $ 24.24
30 24.29 23.65 24.39 24.48 26.26 20.93 $ \bf{{26.64}} $ 26.52
45 25.28 25.30 25.43 25.82 27.56 21.78 27.70 $ \bf{{27.71}} $
60 26.13 26.23 26.09 26.62 28.45 22.79 28.44 $ \bf{{28.69}} $
5 15 21.55 19.50 20.94 20.60 22.80 20.13 $ \bf{{23.97}} $ 23.45
30 23.93 22.83 23.87 23.88 25.63 21.65 $ \bf{{26.34}} $ 26.08
45 25.05 24.85 25.16 25.38 27.16 22.08 27.48 $ \bf{{27.50}} $
60 25.99 25.97 25.85 26.37 28.13 22.97 28.24 $ \bf{{28.37}} $
SSIM
McM1 1 15 0.9828 0.9836 0.9821 0.9821 0.9828 0.9835 $ \bf{{0.9850}} $ 0.9832
30 0.9684 0.9687 0.9698 0.9690 0.9699 0.9543 $ \bf{{0.9705}} $ 0.9663
45 0.9557 0.9584 0.9586 0.9571 0.9581 0.9224 $ \bf{{0.9574}} $ 0.9531
60 0.9469 $ \bf{{0.9501}} $ 0.9491 0.9468 0.9487 0.8939 0.9465 0.9420
5 15 0.9812 0.9780 0.9779 0.9776 0.9799 0.9900 $ \bf{{0.9829}} $ 0.9810
30 0.9650 0.9625 0.9643 0.9623 0.9646 0.9606 $ \bf{{0.9686}} $ 0.9665
45 0.9510 0.9520 0.9527 0.9512 0.9522 0.9279 $ \bf{{0.9553}} $ 0.9528
60 0.9417 0.9435 0.9437 0.9411 0.9426 0.8972 $ \bf{{0.9446}} $ 0.9429
McM2 1 15 0.9901 0.9875 0.9897 0.9894 0.9932 0.9877 $ \bf{{0.9936}} $ 0.9933
30 0.9823 0.9805 0.9830 0.9832 0.9890 0.9681 $ \bf{{0.9891}} $ 0.9890
45 0.9752 0.9773 0.9771 0.9782 0.9850 0.9576 0.9846 $ \bf{{0.9852}} $
60 0.9703 0.9739 0.9717 0.9734 0.9818 0.9529 0.9801 $ \bf{{0.9820}} $
5 15 0.9880 0.9815 0.9867 0.9862 0.9914 0.9903 $ \bf{{0.9929}} $ 0.9919
30 0.9809 0.9761 0.9807 0.9807 0.9873 0.9696 $ \bf{{0.9884}} $ 0.9879
45 0.9743 0.9745 0.9755 0.9763 0.9841 0.9569 0.9839 $ \bf{{0.9847}} $
60 0.9700 0.9723 0.9702 0.9722 0.9807 0.9521 0.9795 $ \bf{{0.9812}} $
Image b P PIDSplit PD NLM KSVD MC-WNNM LRPD LRS LRH
SNR
McM1 1 15 17.39 17.24 17.41 17.30 17.54 17.22 $ \bf{{18.16}} $ 17.71
30 19.29 19.09 19.44 19.37 19.58 17.21 $ \bf{{19.92}} $ 19.35
45 20.36 20.36 20.50 20.49 20.75 17.47 $ \bf{{20.93}} $ 20.53
60 21.24 21.21 21.21 21.23 21.54 17.83 $ \bf{{21.66}} $ 21.38
5 15 16.94 15.91 16.44 16.34 16.78 19.82 $ \bf{{17.66}} $ 17.34
30 18.87 18.36 18.77 18.63 18.88 18.64 $ \bf{{19.61}} $ 19.41
45 19.96 19.85 20.02 19.97 20.15 18.35 $ \bf{{20.69}} $ 20.54
60 20.85 20.77 20.84 20.82 21.07 18.44 $ \bf{{21.46}} $ 21.43
McM2 1 15 22.35 21.15 22.05 21.81 23.76 21.02 $ \bf{{24.47}} $ 24.24
30 24.29 23.65 24.39 24.48 26.26 20.93 $ \bf{{26.64}} $ 26.52
45 25.28 25.30 25.43 25.82 27.56 21.78 27.70 $ \bf{{27.71}} $
60 26.13 26.23 26.09 26.62 28.45 22.79 28.44 $ \bf{{28.69}} $
5 15 21.55 19.50 20.94 20.60 22.80 20.13 $ \bf{{23.97}} $ 23.45
30 23.93 22.83 23.87 23.88 25.63 21.65 $ \bf{{26.34}} $ 26.08
45 25.05 24.85 25.16 25.38 27.16 22.08 27.48 $ \bf{{27.50}} $
60 25.99 25.97 25.85 26.37 28.13 22.97 28.24 $ \bf{{28.37}} $
SSIM
McM1 1 15 0.9828 0.9836 0.9821 0.9821 0.9828 0.9835 $ \bf{{0.9850}} $ 0.9832
30 0.9684 0.9687 0.9698 0.9690 0.9699 0.9543 $ \bf{{0.9705}} $ 0.9663
45 0.9557 0.9584 0.9586 0.9571 0.9581 0.9224 $ \bf{{0.9574}} $ 0.9531
60 0.9469 $ \bf{{0.9501}} $ 0.9491 0.9468 0.9487 0.8939 0.9465 0.9420
5 15 0.9812 0.9780 0.9779 0.9776 0.9799 0.9900 $ \bf{{0.9829}} $ 0.9810
30 0.9650 0.9625 0.9643 0.9623 0.9646 0.9606 $ \bf{{0.9686}} $ 0.9665
45 0.9510 0.9520 0.9527 0.9512 0.9522 0.9279 $ \bf{{0.9553}} $ 0.9528
60 0.9417 0.9435 0.9437 0.9411 0.9426 0.8972 $ \bf{{0.9446}} $ 0.9429
McM2 1 15 0.9901 0.9875 0.9897 0.9894 0.9932 0.9877 $ \bf{{0.9936}} $ 0.9933
30 0.9823 0.9805 0.9830 0.9832 0.9890 0.9681 $ \bf{{0.9891}} $ 0.9890
45 0.9752 0.9773 0.9771 0.9782 0.9850 0.9576 0.9846 $ \bf{{0.9852}} $
60 0.9703 0.9739 0.9717 0.9734 0.9818 0.9529 0.9801 $ \bf{{0.9820}} $
5 15 0.9880 0.9815 0.9867 0.9862 0.9914 0.9903 $ \bf{{0.9929}} $ 0.9919
30 0.9809 0.9761 0.9807 0.9807 0.9873 0.9696 $ \bf{{0.9884}} $ 0.9879
45 0.9743 0.9745 0.9755 0.9763 0.9841 0.9569 0.9839 $ \bf{{0.9847}} $
60 0.9700 0.9723 0.9702 0.9722 0.9807 0.9521 0.9795 $ \bf{{0.9812}} $
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