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An efficient computational method for total variationpenalized Poisson likelihood estimation
Twophase approach for deblurring images corrupted by impulse plus gaussian noise
1.  Temasek Laboratories and Department Mathematics, National University of Singapore, 2 Science Drive 2, 117543, Singapore 
2.  Department of Mathematics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong, China 
3.  CMLA, ENS Cachan, CNRS, PRES UniverSud, 61 Av. President Wilson, F94230 Cachan 
References:
[1] 
L. Ambrosio and V. M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via $\Gamma$convergence, Communications on Pure and Applied Mathematics, 43 (1990), 9991036. doi: 10.1002/cpa.3160430805. 
[2] 
J. Astola and P. Kuosmanen, "Fundamentals of Nonlinear Digital Filtering," Boca Rator, CRC, 1997. 
[3] 
G. Aubert and P. Kornprobst, "Mathematical Problems in Images Processing," Partial differential equations and the calculus of variations. With a foreword by Olivier Faugeras, Applied Mathematical Sciences, 147. SpringerVerlag, New York, 2002. 
[4] 
L. Bar, A. Brook, N. Sochen and N. Kiryati, Deblurring of color images corrupted by saltandpepper noise, IEEE Transactions on Image Processing, 16 (2007), 11011111. doi: 10.1109/TIP.2007.891805. 
[5] 
L. Bar, N. Sochen and N. Kiryati, Image deblurring in the presence of saltandpepper noise, in "Proceeding of 5th International Conference on Scale Space and PDE methods in Computer Vision'', LNCS, 3439 (2005), 107118. doi: 10.1007/11408031_10. 
[6] 
L. Bar, N. Sochen and N. Kiryati, Image deblurring in the presence of impulsive noise, International Journal of Computer Vision, 70 (2006), 279298. doi: 10.1007/s1126300664681. 
[7] 
A. Ben Hamza and H. Krim, Image denoising: a nonlinear robust statistical approach, IEEE Transactions on Signal Processing, 49 (2001), 30453054. doi: 10.1109/78.969512. 
[8] 
A. Blake and A. Zisserman, "Visual Reconstruction," The MIT Press, Cambridge, 1987. 
[9] 
A. Bovik, "Handbook of Image and Video Processing," Academic Press, 2000. 
[10] 
R. H. Chan, C. W. Ho and M. Nikolova, Saltandpepper noise removal by mediantype noise detector and edgepreserving regularization, IEEE Transactions on Image Processing, 14 (2005), 14791485. doi: 10.1109/TIP.2005.852196. 
[11] 
R. H. Chan, C. Hu and M. Nikolova, An iterative procedure for removing randomvalued impulse noise, IEEE Signal Processing Letters, 11 (2004), 921924. doi: 10.1109/LSP.2004.838190. 
[12] 
P. Charbonnier, L. BlancFéraud, G. Aubert and M. Barlaud, Deterministic edgepreserving regularization in computed imaging IEEE Transactions on Image Processing, 6 (1997), 298311. doi: 10.1109/83.551699. 
[13] 
G. Demoment, Image reconstruction and restoration : overview of common estimation structure and problems, IEEE Transactions on Acoustics, Speech, and Signal Processing, 37 (1989), 20242036. doi: 10.1109/29.45551. 
[14] 
S. Esedoglu and J. Shen, Digital inpainting based on the MumfordShahEuler image model European Journal of Applied Mathematics, 13 (2002), 353370. doi: 10.1017/S0956792502004904. 
[15] 
R. Garnett, T. Huegerich, C. Chui and W. He, A universal noise removal algorithm with an impulse detector, IEEE Transactions on Image Processing, 14 (2005), 17471754. doi: 10.1109/TIP.2005.857261. 
[16] 
D. Geman and G. Reynolds, Constrained restoration and recovery of discontinuities, IEEE Transactions on Pattern Analysis and Machine Intelligence, 14 (1992), 367383. doi: 10.1109/34.120331. 
[17] 
D. Geman and C. Yang, Nonlinear image recovery with halfquadratic regularization, IEEE Transactions on Image Processing, 4 (1995), 932946. doi: 10.1109/83.392335. 
[18] 
J. G. Gonzalez and G. R. Arce, Optimality of the myriad filter in practical impulsivenoise environments, IEEE Transactions on Signal Processing, 49 (2001), 438441. doi: 10.1109/78.902126. 
[19] 
R. C. Hardie and K. E. Barner, Rank conditioned rank selection filters for signal restoration, IEEE Transactions on Image Processing, 3 (1994), 192206. doi: 10.1109/83.277900. 
[20] 
H. Hwang and R. A. Haddad, Adaptive median filters: new algorithms and results, IEEE Transactions on Image Processing, 4 (1995), 499502. doi: 10.1109/83.370679. 
[21] 
S.J. Ko and Y. H. Lee, Center weighted median filters and their applications to image enhancement, IEEE Transactions on Circuits and Systems, 38 (1991), 984993. doi: 10.1109/31.83870. 
[22] 
D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Communications on Pure and Applied Mathematics, 42 (1989), 577685. doi: 10.1002/cpa.3160420503. 
[23] 
NASA, Help for DESPIKE, The VICAR Image Processing System, http://wwwmipl.jpl.nasa.gov/vicar/vicar260/ html/vichelp/despike.html, 1999. 
[24] 
M. Nikolova, Minimizers of costfunctions involving nonsmooth datafidelity terms. Application to the processing of outliers, SIAM Journal on Numerical Analysis, 40 (2002), 965994 (electronic). doi: 10.1137/S0036142901389165. 
[25] 
M. Nikolova, A variational approach to remove outliers and impulse noise, Special issue on mathematics and image analysis, Journal of Mathematical Imaging and Vision, 20 (2004), 99120. doi: 10.1023/B:JMIV.0000011920.58935.9c. 
[26] 
M. Nikolova, Analysis of the recovery of edges in images and signals by minimizing nonconvex regularized leastsquares, SIAM Journal on Multiscale Modeling and Simulation, 4 (2005), 960991. doi: 10.1137/040619582. 
[27] 
M. Nikolova and R. H. Chan, The equivalence of halfquadratic minimization and the gradient linearization iteration, IEEE Transactions on Image Processing, 16 (2007), 16231627. doi: 10.1109/TIP.2007.896622. 
[28] 
L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259268. doi: 10.1016/01672789(92)90242F. 
[29] 
A. Tarantola, "Inverse Problem Theory. Methods for Data Fitting and Model Parameter Estimation," Elsevier Science Publishers, 1987. 
[30] 
A. Tikhonov and V. Arsenin, "Solutions of IllPosed Problems," Translated from the Russian. Preface by translation editor Fritz John. Scripta Series in Mathematics. V. H. Winston & Sons, Washington, D.C.: John Wiley & Sons, New YorkToronto, Ont.London, 1977. 
[31] 
C. Vogel, "Computational Methods for Inverse Problems," SIAM (Frontiers in Applied Mathematics Series, Number 23), Philadelphia, PA, 2002. 
show all references
References:
[1] 
L. Ambrosio and V. M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via $\Gamma$convergence, Communications on Pure and Applied Mathematics, 43 (1990), 9991036. doi: 10.1002/cpa.3160430805. 
[2] 
J. Astola and P. Kuosmanen, "Fundamentals of Nonlinear Digital Filtering," Boca Rator, CRC, 1997. 
[3] 
G. Aubert and P. Kornprobst, "Mathematical Problems in Images Processing," Partial differential equations and the calculus of variations. With a foreword by Olivier Faugeras, Applied Mathematical Sciences, 147. SpringerVerlag, New York, 2002. 
[4] 
L. Bar, A. Brook, N. Sochen and N. Kiryati, Deblurring of color images corrupted by saltandpepper noise, IEEE Transactions on Image Processing, 16 (2007), 11011111. doi: 10.1109/TIP.2007.891805. 
[5] 
L. Bar, N. Sochen and N. Kiryati, Image deblurring in the presence of saltandpepper noise, in "Proceeding of 5th International Conference on Scale Space and PDE methods in Computer Vision'', LNCS, 3439 (2005), 107118. doi: 10.1007/11408031_10. 
[6] 
L. Bar, N. Sochen and N. Kiryati, Image deblurring in the presence of impulsive noise, International Journal of Computer Vision, 70 (2006), 279298. doi: 10.1007/s1126300664681. 
[7] 
A. Ben Hamza and H. Krim, Image denoising: a nonlinear robust statistical approach, IEEE Transactions on Signal Processing, 49 (2001), 30453054. doi: 10.1109/78.969512. 
[8] 
A. Blake and A. Zisserman, "Visual Reconstruction," The MIT Press, Cambridge, 1987. 
[9] 
A. Bovik, "Handbook of Image and Video Processing," Academic Press, 2000. 
[10] 
R. H. Chan, C. W. Ho and M. Nikolova, Saltandpepper noise removal by mediantype noise detector and edgepreserving regularization, IEEE Transactions on Image Processing, 14 (2005), 14791485. doi: 10.1109/TIP.2005.852196. 
[11] 
R. H. Chan, C. Hu and M. Nikolova, An iterative procedure for removing randomvalued impulse noise, IEEE Signal Processing Letters, 11 (2004), 921924. doi: 10.1109/LSP.2004.838190. 
[12] 
P. Charbonnier, L. BlancFéraud, G. Aubert and M. Barlaud, Deterministic edgepreserving regularization in computed imaging IEEE Transactions on Image Processing, 6 (1997), 298311. doi: 10.1109/83.551699. 
[13] 
G. Demoment, Image reconstruction and restoration : overview of common estimation structure and problems, IEEE Transactions on Acoustics, Speech, and Signal Processing, 37 (1989), 20242036. doi: 10.1109/29.45551. 
[14] 
S. Esedoglu and J. Shen, Digital inpainting based on the MumfordShahEuler image model European Journal of Applied Mathematics, 13 (2002), 353370. doi: 10.1017/S0956792502004904. 
[15] 
R. Garnett, T. Huegerich, C. Chui and W. He, A universal noise removal algorithm with an impulse detector, IEEE Transactions on Image Processing, 14 (2005), 17471754. doi: 10.1109/TIP.2005.857261. 
[16] 
D. Geman and G. Reynolds, Constrained restoration and recovery of discontinuities, IEEE Transactions on Pattern Analysis and Machine Intelligence, 14 (1992), 367383. doi: 10.1109/34.120331. 
[17] 
D. Geman and C. Yang, Nonlinear image recovery with halfquadratic regularization, IEEE Transactions on Image Processing, 4 (1995), 932946. doi: 10.1109/83.392335. 
[18] 
J. G. Gonzalez and G. R. Arce, Optimality of the myriad filter in practical impulsivenoise environments, IEEE Transactions on Signal Processing, 49 (2001), 438441. doi: 10.1109/78.902126. 
[19] 
R. C. Hardie and K. E. Barner, Rank conditioned rank selection filters for signal restoration, IEEE Transactions on Image Processing, 3 (1994), 192206. doi: 10.1109/83.277900. 
[20] 
H. Hwang and R. A. Haddad, Adaptive median filters: new algorithms and results, IEEE Transactions on Image Processing, 4 (1995), 499502. doi: 10.1109/83.370679. 
[21] 
S.J. Ko and Y. H. Lee, Center weighted median filters and their applications to image enhancement, IEEE Transactions on Circuits and Systems, 38 (1991), 984993. doi: 10.1109/31.83870. 
[22] 
D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Communications on Pure and Applied Mathematics, 42 (1989), 577685. doi: 10.1002/cpa.3160420503. 
[23] 
NASA, Help for DESPIKE, The VICAR Image Processing System, http://wwwmipl.jpl.nasa.gov/vicar/vicar260/ html/vichelp/despike.html, 1999. 
[24] 
M. Nikolova, Minimizers of costfunctions involving nonsmooth datafidelity terms. Application to the processing of outliers, SIAM Journal on Numerical Analysis, 40 (2002), 965994 (electronic). doi: 10.1137/S0036142901389165. 
[25] 
M. Nikolova, A variational approach to remove outliers and impulse noise, Special issue on mathematics and image analysis, Journal of Mathematical Imaging and Vision, 20 (2004), 99120. doi: 10.1023/B:JMIV.0000011920.58935.9c. 
[26] 
M. Nikolova, Analysis of the recovery of edges in images and signals by minimizing nonconvex regularized leastsquares, SIAM Journal on Multiscale Modeling and Simulation, 4 (2005), 960991. doi: 10.1137/040619582. 
[27] 
M. Nikolova and R. H. Chan, The equivalence of halfquadratic minimization and the gradient linearization iteration, IEEE Transactions on Image Processing, 16 (2007), 16231627. doi: 10.1109/TIP.2007.896622. 
[28] 
L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259268. doi: 10.1016/01672789(92)90242F. 
[29] 
A. Tarantola, "Inverse Problem Theory. Methods for Data Fitting and Model Parameter Estimation," Elsevier Science Publishers, 1987. 
[30] 
A. Tikhonov and V. Arsenin, "Solutions of IllPosed Problems," Translated from the Russian. Preface by translation editor Fritz John. Scripta Series in Mathematics. V. H. Winston & Sons, Washington, D.C.: John Wiley & Sons, New YorkToronto, Ont.London, 1977. 
[31] 
C. Vogel, "Computational Methods for Inverse Problems," SIAM (Frontiers in Applied Mathematics Series, Number 23), Philadelphia, PA, 2002. 
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