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May  2008, 2(2): 187-204. doi: 10.3934/ipi.2008.2.187

Two-phase approach for deblurring images corrupted by impulse plus gaussian noise

Jian-Feng Cai 1, , Raymond H. Chan 2, and  Mila Nikolova 3,

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

Received  January 2008 Revised  March 2008 Published  April 2008

The restoration of blurred images corrupted with impulse noise is a difficult problem which has been considered in a series of recent papers. These papers tackle the problem by using variational methods involving an L1-shaped data-fidelity term. Because of this term, the relevant methods exhibit systematic errors at the corrupted pixel locations and require a cumbersome optimization stage. In this work we propose and justify a much simpler alternative approach which overcomes the above-mentioned systematic errors and leads to much better results. Following a theoretical derivation based on a simple model, we decouple the problem into two phases. First, we identify the outlier candidates---the pixels that are likely to be corrupted by the impulse noise, and we remove them from our data set. In a second phase, the image is deblurred and denoised simultaneously using essentially the outlier-free data. The resultant optimization stage is much simpler in comparison with the current full variational methods and the outlier contamination is more accurately corrected. The experiments show that we obtain a 2 to 6 dB improvement in PSNR. We emphasize that our method can be adapted to deblur images corrupted with mixed impulse plus Gaussian noise, and hence it can address a much wider class of practical problems.
Keywords: two-phase approach, Mumford-Shah functional., impulse noise, Image deblurring.
Mathematics Subject Classification: Primary: 94A08; Secondary: 49N45, 68U1.
Citation: Jian-Feng Cai, Raymond H. Chan, Mila Nikolova. Two-phase approach for deblurring images corrupted by impulse plus gaussian noise. Inverse Problems & Imaging, 2008, 2 (2) : 187-204. doi: 10.3934/ipi.2008.2.187
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.  Google Scholar

[2]

J. Astola and P. Kuosmanen, "Fundamentals of Nonlinear Digital Filtering," Boca Rator, CRC, 1997. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. Blake and A. Zisserman, "Visual Reconstruction," The MIT Press, Cambridge, 1987.  Google Scholar

[9]

A. Bovik, "Handbook of Image and Video Processing," Academic Press, 2000. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

NASA, Help for DESPIKE, The VICAR Image Processing System, http://www-mipl.jpl.nasa.gov/vicar/vicar260/ html/vichelp/despike.html, 1999. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

A. Tarantola, "Inverse Problem Theory. Methods for Data Fitting and Model Parameter Estimation," Elsevier Science Publishers, 1987.  Google Scholar

[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.  Google Scholar

[31]

C. Vogel, "Computational Methods for Inverse Problems," SIAM (Frontiers in Applied Mathematics Series, Number 23), Philadelphia, PA, 2002.  Google Scholar

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.  Google Scholar

[2]

J. Astola and P. Kuosmanen, "Fundamentals of Nonlinear Digital Filtering," Boca Rator, CRC, 1997. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. Blake and A. Zisserman, "Visual Reconstruction," The MIT Press, Cambridge, 1987.  Google Scholar

[9]

A. Bovik, "Handbook of Image and Video Processing," Academic Press, 2000. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

NASA, Help for DESPIKE, The VICAR Image Processing System, http://www-mipl.jpl.nasa.gov/vicar/vicar260/ html/vichelp/despike.html, 1999. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

A. Tarantola, "Inverse Problem Theory. Methods for Data Fitting and Model Parameter Estimation," Elsevier Science Publishers, 1987.  Google Scholar

[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.  Google Scholar

[31]

C. Vogel, "Computational Methods for Inverse Problems," SIAM (Frontiers in Applied Mathematics Series, Number 23), Philadelphia, PA, 2002.  Google Scholar

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