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The Cauchy problem for a nonlinear elliptic equation: Nash-game approach and application to image inpainting
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 |
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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