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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

 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.
Citation: Nam-Yong Lee, Bradley J. Lucier. Preconditioned conjugate gradient method for boundary artifact-free image deblurring. Inverse Problems and 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. [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. [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. [4] M. Bertalmio, G. Sapiro, V. Caselles and C. Ballester, Image inpainting, SIGGRAPH, (2000), 417-424. doi: 10.1145/344779.344972. [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. [6] D. Calvetti, J. P. Kaipio and E. Someralo, Aristotelian prior boundary conditions, Inter. J. Mathematics and Computer Science, 1 (2006), 63-81. [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. [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. [9] C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Integral Equations of the First Kind, Pitman, Boston, 1984. [10] M. Hanke and P. C. Hansen, Regularization methods for large-scale problems, Surveys Math. Indust., 3 (1993), 253-315. [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. [12] A. K. Jain, Fundamentals of Digital Image Processing, Prentice-Hall, Englewood Cliffs, NJ, 1989. [13] R. Liu and J. Jia, Reducing boundary artifacts in image deconvolution, in Proceedings of ICIP, 2008, 505-508. [14] L. B. Lucy, An iterative techniques for the rectification of observed distributions, Astronomical Journal, 79 (1974), 745-754. doi: 10.1086/111605. [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. [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. [17] Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM Publications, Philadelphia, 2003. doi: 10.1137/1.9780898718003. [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. [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. [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. [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.

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##### 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. [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. [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. [4] M. Bertalmio, G. Sapiro, V. Caselles and C. Ballester, Image inpainting, SIGGRAPH, (2000), 417-424. doi: 10.1145/344779.344972. [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. [6] D. Calvetti, J. P. Kaipio and E. Someralo, Aristotelian prior boundary conditions, Inter. J. Mathematics and Computer Science, 1 (2006), 63-81. [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. [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. [9] C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Integral Equations of the First Kind, Pitman, Boston, 1984. [10] M. Hanke and P. C. Hansen, Regularization methods for large-scale problems, Surveys Math. Indust., 3 (1993), 253-315. [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. [12] A. K. Jain, Fundamentals of Digital Image Processing, Prentice-Hall, Englewood Cliffs, NJ, 1989. [13] R. Liu and J. Jia, Reducing boundary artifacts in image deconvolution, in Proceedings of ICIP, 2008, 505-508. [14] L. B. Lucy, An iterative techniques for the rectification of observed distributions, Astronomical Journal, 79 (1974), 745-754. doi: 10.1086/111605. [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. [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. [17] Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM Publications, Philadelphia, 2003. doi: 10.1137/1.9780898718003. [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. [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. [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. [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.
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