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An efficient computational method for total variation-penalized Poisson likelihood estimation
Two-phase 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, F-94230 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), 999-1036.
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. Springer-Verlag, New York, 2002. |
[4] |
L. Bar, A. Brook, N. Sochen and N. Kiryati, Deblurring of color images corrupted by salt-and-pepper noise, IEEE Transactions on Image Processing, 16 (2007), 1101-1111.
doi: 10.1109/TIP.2007.891805. |
[5] |
L. Bar, N. Sochen and N. Kiryati, Image deblurring in the presence of salt-and-pepper noise, in "Proceeding of 5th International Conference on Scale Space and PDE methods in Computer Vision'', LNCS, 3439 (2005), 107-118.
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), 279-298.
doi: 10.1007/s11263-006-6468-1. |
[7] |
A. Ben Hamza and H. Krim, Image denoising: a nonlinear robust statistical approach, IEEE Transactions on Signal Processing, 49 (2001), 3045-3054.
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, Salt-and-pepper noise removal by median-type noise detector and edge-preserving regularization, IEEE Transactions on Image Processing, 14 (2005), 1479-1485.
doi: 10.1109/TIP.2005.852196. |
[11] |
R. H. Chan, C. Hu and M. Nikolova, An iterative procedure for removing random-valued impulse noise, IEEE Signal Processing Letters, 11 (2004), 921-924.
doi: 10.1109/LSP.2004.838190. |
[12] |
P. Charbonnier, L. Blanc-Féraud, G. Aubert and M. Barlaud, Deterministic edge-preserving regularization in computed imaging IEEE Transactions on Image Processing, 6 (1997), 298-311.
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), 2024-2036.
doi: 10.1109/29.45551. |
[14] |
S. Esedoglu and J. Shen, Digital inpainting based on the Mumford-Shah-Euler image model European Journal of Applied Mathematics, 13 (2002), 353-370.
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), 1747-1754.
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), 367-383.
doi: 10.1109/34.120331. |
[17] |
D. Geman and C. Yang, Nonlinear image recovery with half-quadratic regularization, IEEE Transactions on Image Processing, 4 (1995), 932-946.
doi: 10.1109/83.392335. |
[18] |
J. G. Gonzalez and G. R. Arce, Optimality of the myriad filter in practical impulsive-noise environments, IEEE Transactions on Signal Processing, 49 (2001), 438-441.
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), 192-206.
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), 499-502.
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), 984-993.
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), 577-685.
doi: 10.1002/cpa.3160420503. |
[23] |
NASA, Help for DESPIKE, The VICAR Image Processing System, http://www-mipl.jpl.nasa.gov/vicar/vicar260/ html/vichelp/despike.html, 1999. |
[24] |
M. Nikolova, Minimizers of cost-functions involving nonsmooth data-fidelity terms. Application to the processing of outliers, SIAM Journal on Numerical Analysis, 40 (2002), 965-994 (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), 99-120.
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 least-squares, SIAM Journal on Multiscale Modeling and Simulation, 4 (2005), 960-991.
doi: 10.1137/040619582. |
[27] |
M. Nikolova and R. H. Chan, The equivalence of half-quadratic minimization and the gradient linearization iteration, IEEE Transactions on Image Processing, 16 (2007), 1623-1627.
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), 259-268.
doi: 10.1016/0167-2789(92)90242-F. |
[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 Ill-Posed 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 York-Toronto, 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), 999-1036.
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. Springer-Verlag, New York, 2002. |
[4] |
L. Bar, A. Brook, N. Sochen and N. Kiryati, Deblurring of color images corrupted by salt-and-pepper noise, IEEE Transactions on Image Processing, 16 (2007), 1101-1111.
doi: 10.1109/TIP.2007.891805. |
[5] |
L. Bar, N. Sochen and N. Kiryati, Image deblurring in the presence of salt-and-pepper noise, in "Proceeding of 5th International Conference on Scale Space and PDE methods in Computer Vision'', LNCS, 3439 (2005), 107-118.
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), 279-298.
doi: 10.1007/s11263-006-6468-1. |
[7] |
A. Ben Hamza and H. Krim, Image denoising: a nonlinear robust statistical approach, IEEE Transactions on Signal Processing, 49 (2001), 3045-3054.
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, Salt-and-pepper noise removal by median-type noise detector and edge-preserving regularization, IEEE Transactions on Image Processing, 14 (2005), 1479-1485.
doi: 10.1109/TIP.2005.852196. |
[11] |
R. H. Chan, C. Hu and M. Nikolova, An iterative procedure for removing random-valued impulse noise, IEEE Signal Processing Letters, 11 (2004), 921-924.
doi: 10.1109/LSP.2004.838190. |
[12] |
P. Charbonnier, L. Blanc-Féraud, G. Aubert and M. Barlaud, Deterministic edge-preserving regularization in computed imaging IEEE Transactions on Image Processing, 6 (1997), 298-311.
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), 2024-2036.
doi: 10.1109/29.45551. |
[14] |
S. Esedoglu and J. Shen, Digital inpainting based on the Mumford-Shah-Euler image model European Journal of Applied Mathematics, 13 (2002), 353-370.
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), 1747-1754.
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), 367-383.
doi: 10.1109/34.120331. |
[17] |
D. Geman and C. Yang, Nonlinear image recovery with half-quadratic regularization, IEEE Transactions on Image Processing, 4 (1995), 932-946.
doi: 10.1109/83.392335. |
[18] |
J. G. Gonzalez and G. R. Arce, Optimality of the myriad filter in practical impulsive-noise environments, IEEE Transactions on Signal Processing, 49 (2001), 438-441.
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), 192-206.
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), 499-502.
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), 984-993.
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), 577-685.
doi: 10.1002/cpa.3160420503. |
[23] |
NASA, Help for DESPIKE, The VICAR Image Processing System, http://www-mipl.jpl.nasa.gov/vicar/vicar260/ html/vichelp/despike.html, 1999. |
[24] |
M. Nikolova, Minimizers of cost-functions involving nonsmooth data-fidelity terms. Application to the processing of outliers, SIAM Journal on Numerical Analysis, 40 (2002), 965-994 (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), 99-120.
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 least-squares, SIAM Journal on Multiscale Modeling and Simulation, 4 (2005), 960-991.
doi: 10.1137/040619582. |
[27] |
M. Nikolova and R. H. Chan, The equivalence of half-quadratic minimization and the gradient linearization iteration, IEEE Transactions on Image Processing, 16 (2007), 1623-1627.
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), 259-268.
doi: 10.1016/0167-2789(92)90242-F. |
[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 Ill-Posed 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 York-Toronto, 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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