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

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

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.
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.

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