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February  2016, 10(1): 195-225. doi: 10.3934/ipi.2016.10.195

Preconditioned conjugate gradient method for boundary artifact-free image deblurring

Nam-Yong Lee 1, and  Bradley J. Lucier 2,

1. 

Department of Applied Mathematics, Inje University, Gimhae, Gyeongnam 621-749, South Korea
2. 

Department of Mathematics, Purdue University, West Lafayette, IN47906, United States

Received  May 2013 Revised  September 2015 Published  February 2016

Several methods have been proposed to reduce boundary artifacts in image deblurring. Some of those methods impose certain assumptions on image pixels outside the field-of-view; the most important of these assume reflective or anti-reflective boundary conditions. Boundary condition methods, including reflective and anti-reflective ones, however, often fail to reduce boundary artifacts, and, in some cases, generate their own artifacts, especially when the image to be deblurred does not accurately satisfy the imposed condition. To overcome these difficulties, we suggest using free boundary conditions, which do not impose any restrictions on image pixels outside the field-of-view, and preconditioned conjugate gradient methods, where preconditioners are designed to compensate for the non-uniformity in contributions from image pixels to the observation. Our simulation studies show that the proposed method outperforms reflective and anti-reflective boundary condition methods in removing boundary artifacts. The simulation studies also show that the proposed method can be applicable to arbitrarily shaped images and has the benefit of recovering damaged parts in blurred images.
Keywords: Boundary artifacts, image deblurring, convolution, preconditioner..
Mathematics Subject Classification: Primary: 65F08, 65K10, 68U1.
Citation: Nam-Yong Lee, Bradley J. Lucier. Preconditioned conjugate gradient method for boundary artifact-free image deblurring. Inverse Problems & Imaging, 2016, 10 (1) : 195-225. doi: 10.3934/ipi.2016.10.195
References:
[1]

F. Aghdasi and R. K. Ward, Reduction of boundary artifacts in image restoration, IEEE Trans. Image Processing, 5 (1996), 611-618. doi: 10.1109/83.491337.  Google Scholar

[2]

M. S. C. Almeida and M. A. T. Figueiredo, Frame-based image deblurring with unknown boundary conditions using the alternating direction method of multipliers, in Proceedings of ICIP, (2013), 582-585. doi: 10.1109/ICIP.2013.6738120.  Google Scholar

[3]

A. Arićo, M. Donatelli and S. Serra-Capizzano, Spectral analysis of the anti-reflective algebra, Linear Algebra Appl., 428 (2008), 657-675. doi: 10.1016/j.laa.2007.08.020.  Google Scholar

[4]

M. Bertalmio, G. Sapiro, V. Caselles and C. Ballester, Image inpainting, SIGGRAPH, (2000), 417-424. doi: 10.1145/344779.344972.  Google Scholar

[5]

M. Bertero and P. Boccacci, A simple method for the reduction of boundary effects in the Richardson-Lucy approach to image deconvolution, Astron. Astrophys., 437 (2005), 369-374. doi: 10.1051/0004-6361:20052717.  Google Scholar

[6]

D. Calvetti, J. P. Kaipio and E. Someralo, Aristotelian prior boundary conditions, Inter. J. Mathematics and Computer Science, 1 (2006), 63-81.  Google Scholar

[7]

M. Donatelli, C. Estatico, A. Martinelli and S. Serra-Capizzano, Improved image deblurring with anti-reflective boundary conditions and re-blurring, Inverse Problems, 22 (2006), 2035-2053. doi: 10.1088/0266-5611/22/6/008.  Google Scholar

[8]

Y. W. Fan and J. G. Nagy, Synthetic boundary conditions for image deblurring, Linear Algebra Appl., 434 (2011), 2244-2268. doi: 10.1016/j.laa.2009.12.021.  Google Scholar

[9]

C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Integral Equations of the First Kind, Pitman, Boston, 1984.  Google Scholar

[10]

M. Hanke and P. C. Hansen, Regularization methods for large-scale problems, Surveys Math. Indust., 3 (1993), 253-315.  Google Scholar

[11]

M. R. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, Journal of Research of the National Bureau of Standards, 49 (1952), 409-436. doi: 10.6028/jres.049.044.  Google Scholar

[12]

A. K. Jain, Fundamentals of Digital Image Processing, Prentice-Hall, Englewood Cliffs, NJ, 1989. Google Scholar

[13]

R. Liu and J. Jia, Reducing boundary artifacts in image deconvolution, in Proceedings of ICIP, 2008, 505-508. Google Scholar

[14]

L. B. Lucy, An iterative techniques for the rectification of observed distributions, Astronomical Journal, 79 (1974), 745-754. doi: 10.1086/111605.  Google Scholar

[15]

M. K. Ng, R. H. Chan and W.-C. Tang, A fast algorithm for deblurring models with Neumann boundary conditions, SIAM J. Sci. Comput., 21 (1999), 851-866. doi: 10.1137/S1064827598341384.  Google Scholar

[16]

W. H. Richardson, Bayesian-based iterative method of image restoration, J. Opt. Soc. Am., 62 (1972), 55-59. doi: 10.1364/JOSA.62.000055.  Google Scholar

[17]

Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM Publications, Philadelphia, 2003. doi: 10.1137/1.9780898718003.  Google Scholar

[18]

S. Serra-Capizzano, A note on anti-reflective boundary conditions and fast deblurring models, SIAM J. Sci. Comput., 25 (2003), 1307-1325. doi: 10.1137/S1064827502410244.  Google Scholar

[19]

A. M. Tekalp and M. I. Sezan, Quantitative analysis of artifacts in linear space-invariant image restoration, Multidimensional Syst. Signal Processing, 1 (1990), 143-177. doi: 10.1007/BF01816547.  Google Scholar

[20]

R. Vio, J. Bardsley, M. Donatelli and W. Wamsteker, Dealing with edge effects least-squares image deconvolution problems, Astron. Astrophys, 442 (2005), 397-403. doi: 10.1051/0004-6361:20053414.  Google Scholar

[21]

J. W. Woods, J. Biemond and A. M. Kekalp, Boundary value problem in image restoration, Proc. Sixth Int. Conf. Acoust. Speech Signal Processing, 10 (1985), 692-695. doi: 10.1109/ICASSP.1985.1168354.  Google Scholar

show all references

References:
[1]

F. Aghdasi and R. K. Ward, Reduction of boundary artifacts in image restoration, IEEE Trans. Image Processing, 5 (1996), 611-618. doi: 10.1109/83.491337.  Google Scholar

[2]

M. S. C. Almeida and M. A. T. Figueiredo, Frame-based image deblurring with unknown boundary conditions using the alternating direction method of multipliers, in Proceedings of ICIP, (2013), 582-585. doi: 10.1109/ICIP.2013.6738120.  Google Scholar

[3]

A. Arićo, M. Donatelli and S. Serra-Capizzano, Spectral analysis of the anti-reflective algebra, Linear Algebra Appl., 428 (2008), 657-675. doi: 10.1016/j.laa.2007.08.020.  Google Scholar

[4]

M. Bertalmio, G. Sapiro, V. Caselles and C. Ballester, Image inpainting, SIGGRAPH, (2000), 417-424. doi: 10.1145/344779.344972.  Google Scholar

[5]

M. Bertero and P. Boccacci, A simple method for the reduction of boundary effects in the Richardson-Lucy approach to image deconvolution, Astron. Astrophys., 437 (2005), 369-374. doi: 10.1051/0004-6361:20052717.  Google Scholar

[6]

D. Calvetti, J. P. Kaipio and E. Someralo, Aristotelian prior boundary conditions, Inter. J. Mathematics and Computer Science, 1 (2006), 63-81.  Google Scholar

[7]

M. Donatelli, C. Estatico, A. Martinelli and S. Serra-Capizzano, Improved image deblurring with anti-reflective boundary conditions and re-blurring, Inverse Problems, 22 (2006), 2035-2053. doi: 10.1088/0266-5611/22/6/008.  Google Scholar

[8]

Y. W. Fan and J. G. Nagy, Synthetic boundary conditions for image deblurring, Linear Algebra Appl., 434 (2011), 2244-2268. doi: 10.1016/j.laa.2009.12.021.  Google Scholar

[9]

C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Integral Equations of the First Kind, Pitman, Boston, 1984.  Google Scholar

[10]

M. Hanke and P. C. Hansen, Regularization methods for large-scale problems, Surveys Math. Indust., 3 (1993), 253-315.  Google Scholar

[11]

M. R. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, Journal of Research of the National Bureau of Standards, 49 (1952), 409-436. doi: 10.6028/jres.049.044.  Google Scholar

[12]

A. K. Jain, Fundamentals of Digital Image Processing, Prentice-Hall, Englewood Cliffs, NJ, 1989. Google Scholar

[13]

R. Liu and J. Jia, Reducing boundary artifacts in image deconvolution, in Proceedings of ICIP, 2008, 505-508. Google Scholar

[14]

L. B. Lucy, An iterative techniques for the rectification of observed distributions, Astronomical Journal, 79 (1974), 745-754. doi: 10.1086/111605.  Google Scholar

[15]

M. K. Ng, R. H. Chan and W.-C. Tang, A fast algorithm for deblurring models with Neumann boundary conditions, SIAM J. Sci. Comput., 21 (1999), 851-866. doi: 10.1137/S1064827598341384.  Google Scholar

[16]

W. H. Richardson, Bayesian-based iterative method of image restoration, J. Opt. Soc. Am., 62 (1972), 55-59. doi: 10.1364/JOSA.62.000055.  Google Scholar

[17]

Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM Publications, Philadelphia, 2003. doi: 10.1137/1.9780898718003.  Google Scholar

[18]

S. Serra-Capizzano, A note on anti-reflective boundary conditions and fast deblurring models, SIAM J. Sci. Comput., 25 (2003), 1307-1325. doi: 10.1137/S1064827502410244.  Google Scholar

[19]

A. M. Tekalp and M. I. Sezan, Quantitative analysis of artifacts in linear space-invariant image restoration, Multidimensional Syst. Signal Processing, 1 (1990), 143-177. doi: 10.1007/BF01816547.  Google Scholar

[20]

R. Vio, J. Bardsley, M. Donatelli and W. Wamsteker, Dealing with edge effects least-squares image deconvolution problems, Astron. Astrophys, 442 (2005), 397-403. doi: 10.1051/0004-6361:20053414.  Google Scholar

[21]

J. W. Woods, J. Biemond and A. M. Kekalp, Boundary value problem in image restoration, Proc. Sixth Int. Conf. Acoust. Speech Signal Processing, 10 (1985), 692-695. doi: 10.1109/ICASSP.1985.1168354.  Google Scholar

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