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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
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show all references

References:
[1]

Communications on Pure and Applied Mathematics, 43 (1990), 999-1036. doi: 10.1002/cpa.3160430805.  Google Scholar

[2]

Boca Rator, CRC, 1997. Google Scholar

[3]

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]

IEEE Transactions on Image Processing, 16 (2007), 1101-1111. doi: 10.1109/TIP.2007.891805.  Google Scholar

[5]

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]

International Journal of Computer Vision, 70 (2006), 279-298. doi: 10.1007/s11263-006-6468-1.  Google Scholar

[7]

IEEE Transactions on Signal Processing, 49 (2001), 3045-3054. doi: 10.1109/78.969512.  Google Scholar

[8]

The MIT Press, Cambridge, 1987.  Google Scholar

[9]

Academic Press, 2000. Google Scholar

[10]

IEEE Transactions on Image Processing, 14 (2005), 1479-1485. doi: 10.1109/TIP.2005.852196.  Google Scholar

[11]

IEEE Signal Processing Letters, 11 (2004), 921-924. doi: 10.1109/LSP.2004.838190.  Google Scholar

[12]

IEEE Transactions on Image Processing, 6 (1997), 298-311. doi: 10.1109/83.551699.  Google Scholar

[13]

IEEE Transactions on Acoustics, Speech, and Signal Processing, 37 (1989), 2024-2036. doi: 10.1109/29.45551.  Google Scholar

[14]

European Journal of Applied Mathematics, 13 (2002), 353-370. doi: 10.1017/S0956792502004904.  Google Scholar

[15]

IEEE Transactions on Image Processing, 14 (2005), 1747-1754. doi: 10.1109/TIP.2005.857261.  Google Scholar

[16]

IEEE Transactions on Pattern Analysis and Machine Intelligence, 14 (1992), 367-383. doi: 10.1109/34.120331.  Google Scholar

[17]

IEEE Transactions on Image Processing, 4 (1995), 932-946. doi: 10.1109/83.392335.  Google Scholar

[18]

IEEE Transactions on Signal Processing, 49 (2001), 438-441. doi: 10.1109/78.902126.  Google Scholar

[19]

IEEE Transactions on Image Processing, 3 (1994), 192-206. doi: 10.1109/83.277900.  Google Scholar

[20]

IEEE Transactions on Image Processing, 4 (1995), 499-502. doi: 10.1109/83.370679.  Google Scholar

[21]

IEEE Transactions on Circuits and Systems, 38 (1991), 984-993. doi: 10.1109/31.83870.  Google Scholar

[22]

Communications on Pure and Applied Mathematics, 42 (1989), 577-685. doi: 10.1002/cpa.3160420503.  Google Scholar

[23]

http://www-mipl.jpl.nasa.gov/vicar/vicar260/ html/vichelp/despike.html, 1999. Google Scholar

[24]

SIAM Journal on Numerical Analysis, 40 (2002), 965-994 (electronic). doi: 10.1137/S0036142901389165.  Google Scholar

[25]

Journal of Mathematical Imaging and Vision, 20 (2004), 99-120. doi: 10.1023/B:JMIV.0000011920.58935.9c.  Google Scholar

[26]

SIAM Journal on Multiscale Modeling and Simulation, 4 (2005), 960-991. doi: 10.1137/040619582.  Google Scholar

[27]

IEEE Transactions on Image Processing, 16 (2007), 1623-1627. doi: 10.1109/TIP.2007.896622.  Google Scholar

[28]

Physica D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[29]

Elsevier Science Publishers, 1987.  Google Scholar

[30]

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]

SIAM (Frontiers in Applied Mathematics Series, Number 23), Philadelphia, PA, 2002.  Google Scholar

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