A reweighted l^2 method for image restoration with Poisson and mixed Poisson-Gaussian noise

Previous Article
Oracle-type posterior contraction rates in Bayesian inverse problems
IPI Home
This Issue
Next Article
The Cauchy problem for a nonlinear elliptic equation: Nash-game approach and application to image inpainting

August 2015, 9(3): 875-894. doi: 10.3934/ipi.2015.9.875

A reweighted $l^2$ method for image restoration with Poisson and mixed Poisson-Gaussian noise

Jia Li ^1, , Zuowei Shen ^2, , Rujie Yin ^3, and Xiaoqun Zhang ^4,

1.	Department of mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, 119076, Singapore
2.	Dept. of Math., National Univ. of Singapore, 119076
3.	Zhiyuan College, Shanghai Jiao Tong University, 800, Dongchuan Road, Shanghai, 200240, China
4.	Department of Mathematics, MOE-LSC and Institute of Natural Sciences, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240

Received June 2013 Revised March 2014 Published July 2015

Abstract
Related Papers

We study weighted $l^2$ fidelity in variational models for Poisson noise related image restoration problems. Gaussian approximation to Poisson noise statistic is adopted to deduce weighted $l^2$ fidelity. Different from the traditional weighted $l^2$ approximation, we propose a reweighted $l^2$ fidelity with sparse regularization by wavelet frame. Based on the split Bregman algorithm introduced in [21], the proposed numerical scheme is composed of three easy subproblems that involve quadratic minimization, soft shrinkage and matrix vector multiplications. Unlike usual least square approximation of Poisson noise, we dynamically update the underlying noise variance from previous estimate. The solution of the proposed algorithm is shown to be the same as the one obtained by minimizing Kullback-Leibler divergence fidelity with the same regularization. This reweighted $l^2$ formulation can be easily extended to mixed Poisson-Gaussian noise case. Finally, the efficiency and quality of the proposed algorithm compared to other Poisson noise removal methods are demonstrated through denoising and deblurring examples. Moreover, mixed Poisson-Gaussian noise tests are performed on both simulated and real digital images for further illustration of the performance of the proposed method.

Keywords: inverse problem, Poisson-Gaussian mixed noise, splitting algorithm, framelets., regularization, Poisson noise.

Mathematics Subject Classification: Primary: 65K10, 65F22; Secondary: 94A08, 65T6.

Citation: Jia Li, Zuowei Shen, Rujie Yin, Xiaoqun Zhang. A reweighted $l^2$ method for image restoration with Poisson and mixed Poisson-Gaussian noise. Inverse Problems and Imaging, 2015, 9 (3) : 875-894. doi: 10.3934/ipi.2015.9.875

References:

[1]	F. J. Anscombe, The transform of poisson, binomial and negative-binomial data, Biometrika, 35 (1948), 246-254. doi: 10.1093/biomet/35.3-4.246.
[2]	M. Bertero, P. Boccacci, G. Desidera 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.
[3]	M. Bertero, P. Boccacci, G. Talenti, R. Zanella and L. Zanni, A discrepancy principle for Poisson data, Inverse Problems, 26 (2010), 105004, 20pp. doi: 10.1088/0266-5611/26/10/105004.
[4]	P. J. Bickel and E. Levina, Regularized estimation of large covariance matrices, Ann. Statistics, 36 (2008), 199-227. doi: 10.1214/009053607000000758.
[5]	C. Brune, A. Sawatzky and M. Burger, Bregman-EM-TV methods with application to optical nanoscopy, in Scale Space and Variational Methods in Computer Vision, Lecture Notes in Computer Science, 5567, Springer, Berlin-Heidelberg, 2009, 235-246. doi: 10.1007/978-3-642-02256-2_20.
[6]	C. Brune, A. Sawatzky and M. Burger, Primal and dual Bregman methods with application to optical nanoscopy, Int. J. Comput. Vis., 92 (2011), 211-229. doi: 10.1007/s11263-010-0339-5.
[7]	C. Brune, M. Burger, A. Sawatzky, T. Kösters and Frank Wübberling, Forward-Backward EM-TV methods for inverse problems with Poisson noise, preprint, 2009.
[8]	J.-F. Cai, S. Osher and Z. Shen, Linearized Bregman iterations for compressed sensing, Mathematics of Computation, 78 (2009), 1515-1536. doi: 10.1090/S0025-5718-08-02189-3.
[9]	J.-F. Cai, S. Osher and Z. Shen, Split bregman methods and frame based image restoration,, Multiscale Modeling & Simulation, 8 (): 337. doi: 10.1137/090753504.
[10]	J.-F. Cai, B. Dong, S. Osher and Z. Shen, Image restoration: Total variation; wavelet frames; and beyond, J. Amer. Math. Soc., 25 (2012), 1033-1089. doi: 10.1090/S0894-0347-2012-00740-1.
[11]	I. Csiszár, Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems, Ann. Statist., 19 (1991), 2032-2066. doi: 10.1214/aos/1176348385.
[12]	I. Daubechies, B. Han, A. Ron and Z. Shen, Framelets: MRA-based constructions of wavelet frames, Applied and Computational Harmonic Analysis, 14 (2003), 1-46. doi: 10.1016/S1063-5203(02)00511-0.
[13]	B. Dong and Z. Shen, MRA-based wavelet frames and applications, in Mathematics in Image Processing, IAS/Park City Math. Ser., 19, Amer. Math. Soc., Providence, RI, 2013, 9-158.
[14]	B. Dong and Y. Zhang, An efficient algorithm for $l^0$ minimization in wavelet frame based image restoration, Journal of Scientific Computing, 54 (2013), 350-368. doi: 10.1007/s10915-012-9597-4.
[15]	D. Donoho, Compressed sensing, IEEE Trans. Inform. Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582.
[16]	E. Esser, Applications of Lagrangian-Based Alternating Direction Methods and Connections to Split Bregman, UCLA CAM report 09-31, 2009.
[17]	E. Esser, X. Zhang and T.-F. Chan, A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science, SIAM J. Imag. Sci., 3 (2010), 1015-1046. doi: 10.1137/09076934X.
[18]	M. Figueiredo and J. 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.
[19]	J. Friedman, T. Hastie and R. Tibshirani, Sparse inverse covariance estimation with the graphical lasso, Biostatistics, 9 (2008), 432-441. doi: 10.1093/biostatistics/kxm045.
[20]	T. Goldstein, B. O'Donoghue and S. Setzer, Fast alternating direction optimization methods, SIAM J. Imaging Sci., 7 (2015), 1588-1623. doi: 10.1137/120896219.
[21]	T. Goldstein and S. Osher, The split bregman method for $l^1$ regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343. doi: 10.1137/080725891.
[22]	Z. Gong, Z. Shen and K.-C. Toh, Image restoration with mixed or unknown noises, Multiscale Model. Simul., 12 (2014), 458-487. doi: 10.1137/130904533.
[23]	K. Lange and R. Carson, EM reconstruction algorithms for emission and transmission tomography, J. Comput. Assist. Tomogr., 8 (1984), 306-316.
[24]	F. Luisier, T. Blu and M. Unser, Image denoising in mixed Poisson-Gaussian niose, IEEE Trans. Image Process., 20 (2011), 696-708. doi: 10.1109/TIP.2010.2073477.
[25]	Y. Nesterov, A method of solving a convex programming problem with convergence rate $o(1 / k^2)$, (Russian) Dokl. Akad. Nauk SSSR, 269 (1983), 543-547.
[26]	R. Rockafellar, Augmented lagrangians and applications of the proximal point algorithm in convex programming, Mathematics of Operations Research, 1 (1976), 97-116. doi: 10.1287/moor.1.2.97.
[27]	A. Ron and Z. Shen, Affine systems in $L_2(R^d)$: The analysis of the analysis operator, Journal of Functional Analysis, 148 (1997), 408-447. doi: 10.1006/jfan.1996.3079.
[28]	L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physics D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.
[29]	S. Setzer, Split bregman algorithm, douglas-rachford splitting and frame shrinkage, in Scale Space and Variational Methods in Computer Vision, Lecture Notes in Computer Science, 5567, Springer, Berlin-Heidelberg, 2009, 464-476. doi: 10.1007/978-3-642-02256-2_39.
[30]	S. Setzer, G. Steidl and T. Teuber, Deblurring Poissonian images by split Bregman techniques, Journal of Visual Communication and Image Representation, 21 (2010), 193-199. doi: 10.1016/j.jvcir.2009.10.006.
[31]	L. A. Shepp and Y. Vardi, Maximum Likelihood Reconstruction in Positron Emission Tomography, IEEE Transactions on Medical Imaging, 1 (1982), 113-122.
[32]	D. L. Snyder, A. M. Hammoud and R. L. White, Image recovery from data acquired with a charge-coupled-device camera, J. Opt. Soc. Am. A, 10 (1993), 1014-1023. doi: 10.1364/JOSAA.10.001014.
[33]	A. Staglianò, P. Boccacci and M. Bertero, Analysis of an approximate model for Poisson data reconstruction and a related discrepancy principle, Inverse Problems, 27 (2011), 125003, 20pp. doi: 10.1088/0266-5611/27/12/125003.
[34]	C. Tomasi and R. Manduchi, Bilateral filtering for gray and color images, in Computer Vision, 1998. Sixth International Conference on, IEEE, 1998, 839-846. doi: 10.1109/ICCV.1998.710815.
[35]	H. Yu and G. Wang, Compressed sensing based interior tomography, Physics in Medicine and Biology, 54 (2009), p2791.
[36]	X. Zhang, M. Burger and S. Osher, A unified primal-dual algorithm framework based on bregman iteration, Journal of Scientific Computing, 46 (2011), 20-46. doi: 10.1007/s10915-010-9408-8.

show all references

References:

[1]	F. J. Anscombe, The transform of poisson, binomial and negative-binomial data, Biometrika, 35 (1948), 246-254. doi: 10.1093/biomet/35.3-4.246.
[2]	M. Bertero, P. Boccacci, G. Desidera 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.
[3]	M. Bertero, P. Boccacci, G. Talenti, R. Zanella and L. Zanni, A discrepancy principle for Poisson data, Inverse Problems, 26 (2010), 105004, 20pp. doi: 10.1088/0266-5611/26/10/105004.
[4]	P. J. Bickel and E. Levina, Regularized estimation of large covariance matrices, Ann. Statistics, 36 (2008), 199-227. doi: 10.1214/009053607000000758.
[5]	C. Brune, A. Sawatzky and M. Burger, Bregman-EM-TV methods with application to optical nanoscopy, in Scale Space and Variational Methods in Computer Vision, Lecture Notes in Computer Science, 5567, Springer, Berlin-Heidelberg, 2009, 235-246. doi: 10.1007/978-3-642-02256-2_20.
[6]	C. Brune, A. Sawatzky and M. Burger, Primal and dual Bregman methods with application to optical nanoscopy, Int. J. Comput. Vis., 92 (2011), 211-229. doi: 10.1007/s11263-010-0339-5.
[7]	C. Brune, M. Burger, A. Sawatzky, T. Kösters and Frank Wübberling, Forward-Backward EM-TV methods for inverse problems with Poisson noise, preprint, 2009.
[8]	J.-F. Cai, S. Osher and Z. Shen, Linearized Bregman iterations for compressed sensing, Mathematics of Computation, 78 (2009), 1515-1536. doi: 10.1090/S0025-5718-08-02189-3.
[9]	J.-F. Cai, S. Osher and Z. Shen, Split bregman methods and frame based image restoration,, Multiscale Modeling & Simulation, 8 (): 337. doi: 10.1137/090753504.
[10]	J.-F. Cai, B. Dong, S. Osher and Z. Shen, Image restoration: Total variation; wavelet frames; and beyond, J. Amer. Math. Soc., 25 (2012), 1033-1089. doi: 10.1090/S0894-0347-2012-00740-1.
[11]	I. Csiszár, Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems, Ann. Statist., 19 (1991), 2032-2066. doi: 10.1214/aos/1176348385.
[12]	I. Daubechies, B. Han, A. Ron and Z. Shen, Framelets: MRA-based constructions of wavelet frames, Applied and Computational Harmonic Analysis, 14 (2003), 1-46. doi: 10.1016/S1063-5203(02)00511-0.
[13]	B. Dong and Z. Shen, MRA-based wavelet frames and applications, in Mathematics in Image Processing, IAS/Park City Math. Ser., 19, Amer. Math. Soc., Providence, RI, 2013, 9-158.
[14]	B. Dong and Y. Zhang, An efficient algorithm for $l^0$ minimization in wavelet frame based image restoration, Journal of Scientific Computing, 54 (2013), 350-368. doi: 10.1007/s10915-012-9597-4.
[15]	D. Donoho, Compressed sensing, IEEE Trans. Inform. Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582.
[16]	E. Esser, Applications of Lagrangian-Based Alternating Direction Methods and Connections to Split Bregman, UCLA CAM report 09-31, 2009.
[17]	E. Esser, X. Zhang and T.-F. Chan, A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science, SIAM J. Imag. Sci., 3 (2010), 1015-1046. doi: 10.1137/09076934X.
[18]	M. Figueiredo and J. 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.
[19]	J. Friedman, T. Hastie and R. Tibshirani, Sparse inverse covariance estimation with the graphical lasso, Biostatistics, 9 (2008), 432-441. doi: 10.1093/biostatistics/kxm045.
[20]	T. Goldstein, B. O'Donoghue and S. Setzer, Fast alternating direction optimization methods, SIAM J. Imaging Sci., 7 (2015), 1588-1623. doi: 10.1137/120896219.
[21]	T. Goldstein and S. Osher, The split bregman method for $l^1$ regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343. doi: 10.1137/080725891.
[22]	Z. Gong, Z. Shen and K.-C. Toh, Image restoration with mixed or unknown noises, Multiscale Model. Simul., 12 (2014), 458-487. doi: 10.1137/130904533.
[23]	K. Lange and R. Carson, EM reconstruction algorithms for emission and transmission tomography, J. Comput. Assist. Tomogr., 8 (1984), 306-316.
[24]	F. Luisier, T. Blu and M. Unser, Image denoising in mixed Poisson-Gaussian niose, IEEE Trans. Image Process., 20 (2011), 696-708. doi: 10.1109/TIP.2010.2073477.
[25]	Y. Nesterov, A method of solving a convex programming problem with convergence rate $o(1 / k^2)$, (Russian) Dokl. Akad. Nauk SSSR, 269 (1983), 543-547.
[26]	R. Rockafellar, Augmented lagrangians and applications of the proximal point algorithm in convex programming, Mathematics of Operations Research, 1 (1976), 97-116. doi: 10.1287/moor.1.2.97.
[27]	A. Ron and Z. Shen, Affine systems in $L_2(R^d)$: The analysis of the analysis operator, Journal of Functional Analysis, 148 (1997), 408-447. doi: 10.1006/jfan.1996.3079.
[28]	L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physics D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.
[29]	S. Setzer, Split bregman algorithm, douglas-rachford splitting and frame shrinkage, in Scale Space and Variational Methods in Computer Vision, Lecture Notes in Computer Science, 5567, Springer, Berlin-Heidelberg, 2009, 464-476. doi: 10.1007/978-3-642-02256-2_39.
[30]	S. Setzer, G. Steidl and T. Teuber, Deblurring Poissonian images by split Bregman techniques, Journal of Visual Communication and Image Representation, 21 (2010), 193-199. doi: 10.1016/j.jvcir.2009.10.006.
[31]	L. A. Shepp and Y. Vardi, Maximum Likelihood Reconstruction in Positron Emission Tomography, IEEE Transactions on Medical Imaging, 1 (1982), 113-122.
[32]	D. L. Snyder, A. M. Hammoud and R. L. White, Image recovery from data acquired with a charge-coupled-device camera, J. Opt. Soc. Am. A, 10 (1993), 1014-1023. doi: 10.1364/JOSAA.10.001014.
[33]	A. Staglianò, P. Boccacci and M. Bertero, Analysis of an approximate model for Poisson data reconstruction and a related discrepancy principle, Inverse Problems, 27 (2011), 125003, 20pp. doi: 10.1088/0266-5611/27/12/125003.
[34]	C. Tomasi and R. Manduchi, Bilateral filtering for gray and color images, in Computer Vision, 1998. Sixth International Conference on, IEEE, 1998, 839-846. doi: 10.1109/ICCV.1998.710815.
[35]	H. Yu and G. Wang, Compressed sensing based interior tomography, Physics in Medicine and Biology, 54 (2009), p2791.
[36]	X. Zhang, M. Burger and S. Osher, A unified primal-dual algorithm framework based on bregman iteration, Journal of Scientific Computing, 46 (2011), 20-46. doi: 10.1007/s10915-010-9408-8.

[1]	Jie Zhang, Yuping Duan, Yue Lu, Michael K. Ng, Huibin Chang. Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal. Inverse Problems and Imaging, 2021, 15 (2) : 339-366. doi: 10.3934/ipi.2020071
[2]	Linghai Kong, Suhua Wei. A variational method for Abel inversion tomography with mixed Poisson-Laplace-Gaussian noise. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022007
[3]	Mingchao Zhao, You-Wei Wen, Michael Ng, Hongwei Li. A nonlocal low rank model for poisson noise removal. Inverse Problems and Imaging, 2021, 15 (3) : 519-537. doi: 10.3934/ipi.2021003
[4]	Jianbin Yang, Cong Wang. A wavelet frame approach for removal of mixed Gaussian and impulse noise on surfaces. Inverse Problems and Imaging, 2017, 11 (5) : 783-798. doi: 10.3934/ipi.2017037
[5]	Bruno Sixou, Tom Hohweiller, Nicolas Ducros. Morozov principle for Kullback-Leibler residual term and Poisson noise. Inverse Problems and Imaging, 2018, 12 (3) : 607-634. doi: 10.3934/ipi.2018026
[6]	Thorsten Hohage, Mihaela Pricop. Nonlinear Tikhonov regularization in Hilbert scales for inverse boundary value problems with random noise. Inverse Problems and Imaging, 2008, 2 (2) : 271-290. doi: 10.3934/ipi.2008.2.271
[7]	Xueqin Li, Chao Tang, Tianmin Huang. Poisson $S^2$-almost automorphy for stochastic processes and its applications to SPDEs driven by Lévy noise. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3309-3345. doi: 10.3934/dcdsb.2018282
[8]	Leonid Shaikhet. Stability of delay differential equations with fading stochastic perturbations of the type of white noise and poisson's jumps. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3651-3657. doi: 10.3934/dcdsb.2020077
[9]	Xiao Ai, Guoxi Ni, Tieyong Zeng. Nonconvex regularization for blurred images with Cauchy noise. Inverse Problems and Imaging, 2022, 16 (3) : 625-646. doi: 10.3934/ipi.2021065
[10]	Johnathan M. Bardsley. A theoretical framework for the regularization of Poisson likelihood estimation problems. Inverse Problems and Imaging, 2010, 4 (1) : 11-17. doi: 10.3934/ipi.2010.4.11
[11]	Yan Wang, Lei Wang, Yanxiang Zhao, Aimin Song, Yanping Ma. A stochastic model for microbial fermentation process under Gaussian white noise environment. Numerical Algebra, Control and Optimization, 2015, 5 (4) : 381-392. doi: 10.3934/naco.2015.5.381
[12]	Boris P. Belinskiy, Peter Caithamer. Energy estimate for the wave equation driven by a fractional Gaussian noise. Conference Publications, 2007, 2007 (Special) : 92-101. doi: 10.3934/proc.2007.2007.92
[13]	Boris P. Belinskiy, Peter Caithamer. Energy of an elastic mechanical system driven by Gaussian noise white in time. Conference Publications, 2001, 2001 (Special) : 39-49. doi: 10.3934/proc.2001.2001.39
[14]	C. Davini, F. Jourdan. Approximations of degree zero in the Poisson problem. Communications on Pure and Applied Analysis, 2005, 4 (2) : 267-281. doi: 10.3934/cpaa.2005.4.267
[15]	Hanwool Na, Myeongmin Kang, Miyoun Jung, Myungjoo Kang. Nonconvex TGV regularization model for multiplicative noise removal with spatially varying parameters. Inverse Problems and Imaging, 2019, 13 (1) : 117-147. doi: 10.3934/ipi.2019007
[16]	Luca Rondi. On the regularization of the inverse conductivity problem with discontinuous conductivities. Inverse Problems and Imaging, 2008, 2 (3) : 397-409. doi: 10.3934/ipi.2008.2.397
[17]	Yuying Shi, Zijin Liu, Xiaoying Wang, Jinping Zhang. Edge detection with mixed noise based on maximum a posteriori approach. Inverse Problems and Imaging, 2021, 15 (5) : 1223-1245. doi: 10.3934/ipi.2021035
[18]	Marius Ghergu, Gurpreet Singh. On a class of mixed Choquard-Schrödinger-Poisson systems. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 297-309. doi: 10.3934/dcdss.2019021
[19]	Wolfgang Arendt, Daniel Daners. Varying domains: Stability of the Dirichlet and the Poisson problem. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 21-39. doi: 10.3934/dcds.2008.21.21
[20]	Daniel Gerth, Andreas Hofinger, Ronny Ramlau. On the lifting of deterministic convergence rates for inverse problems with stochastic noise. Inverse Problems and Imaging, 2017, 11 (4) : 663-687. doi: 10.3934/ipi.2017031

2020 Impact Factor: 1.639

Tools

Metrics

Other articles
by authors

on AIMS
Jia Li Zuowei Shen Rujie Yin Xiaoqun Zhang
on Google Scholar
Jia Li Zuowei Shen Rujie Yin Xiaoqun Zhang

[Back to Top]