- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Article Processing Charges ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Recently Accepted Articles ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Journal of Mathematics
Volume 2013 (2013), Article ID 274573, 11 pages
Restoring Poissonian Images by a Combined First-Order and Second-Order Variation Approach
1School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China
2School of Science, Huaihai Institute of Technology, Lianyungang, Jiangsu 222005, China
Received 6 December 2012; Accepted 27 January 2013
Academic Editor: Kaleem R. Kazmi
Copyright © 2013 Le Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- H. Andrew and B. Hunt, Digital Image Restoration,, Prentice Hall, Englewood Cliffs, NJ, USA, 1977.
- M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging, Institute of Physics Publishing, London, UK, 1998.
- T. F. Chan and J. Shen, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods, SIAM, Philadelphia, Pa, USA, 2005.
- M. Banham and A. Katsaggelos, “Digital image restoration,” IEEE Signal Processing Magazine, vol. 14, no. 2, pp. 24–41, 1997.
- L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physical D, vol. 60, no. 1–4, pp. 259–268, 1992.
- A. Chambolle and P. L. Lions, “Image recovery via total variation minimization and related problems,” Numerische Mathematik, vol. 76, no. 2, pp. 167–188, 1997.
- L. Rudin, P. L. Lions, and S. Osher, “Multiplicative denoising and deblurring: theory and algorithms,” in Geometric Level Set Methods in Imaging, Vision, and Graphics, S. Osher and N. Paragios, Eds., pp. 103–119, Springer, New York, NY, USA, 2003.
- G. Aubert and J. F. Aujol, “A variational approach to removing multiplicative noise,” SIAM Journal on Applied Mathematics, vol. 68, no. 4, pp. 925–946, 2008.
- Y. M. Huang, M. K. Ng, and Y. W. Wen, “A new total variation method for multiplicative noise removal,” SIAM Journal on Imaging Sciences, vol. 2, no. 1, pp. 20–40, 2009.
- B. Chen, J. L. Cai, W. S. Chen, and Y. Li, “A multiplicative noise removal approach based on partial differential equation model,” Mathematical Problems in Engineering, vol. 2012, Article ID 242043, 14 pages, 2012.
- S. Correia, M. Carbillet, P. Boccacci, M. Bertero, and L. Fini, “Restoration of interferometric images—I. The software package AIRY,” Astronomy and Astrophysics, vol. 387, no. 2, pp. 733–743, 2002.
- A. Chambolle, “An algorithm for total variation minimization and applications,” Journal of Mathematical Imaging and Vision, vol. 20, no. 1-2, pp. 89–97, 2004.
- T. Chan, S. Esedoglu, and F. Park, “Image decomposition combining staircase reduction and texture extraction,” Journal of Visual Communication and Image Representation, vol. 18, no. 6, pp. 464–486, 2007.
- S. Becker, J. Bobin, and E. J. Candès, “NESTA: a fast and accurate first-order method for sparse recovery,” SIAM Journal on Imaging Sciences, vol. 4, no. 1, pp. 1–39, 2011.
- A. Beck and M. Teboulle, “Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems,” IEEE Transactions on Image Processing, vol. 18, no. 11, pp. 2419–2434, 2009.
- Y. L. Wang, J. F. Yang, W. T. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM Journal on Imaging Sciences, vol. 1, no. 3, pp. 248–272, 2008.
- J. M. Bioucas-Dias and M. A. T. Figueiredo, “Multiplicative noise removal using variable splitting and constrained optimization,” IEEE Transactions on Image Processing, vol. 19, no. 7, pp. 1720–1730, 2010.
- S. Yun and H. Woo, “A new multiplicative denoising variational model based on th root transformation,” IEEE Transactions on Image Processing, vol. 21, no. 5, pp. 2523–2533, 2012.
- H. Woo and S. Yun, “Alternating minimization algorithm for speckle reduction with a shifting technique,” IEEE Transactions on Image Processing, vol. 21, no. 4, pp. 1701–1714, 2012.
- M. Bertero, P. Boccacci, G. Desiderà, and G. Vicidomini, “Image deblurring with Poisson data: from cells to galaxies,” Inverse Problems, vol. 25, no. 12, Article ID 123006, 26 pages, 2009.
- J. M. Bardsley and A. Luttman, “Total variation-penalized Poisson likelihood estimation for ill-posed problems,” Advances in Computational Mathematics, vol. 31, no. 1-3, pp. 35–59, 2009.
- M. A. T. Figueiredo and J. M. Bioucas-Dias, “Restoration of Poissonian images using alternating direction optimization,” IEEE Transactions on Image Processing, vol. 19, no. 12, pp. 3133–3145, 2010.
- S. Setzer, G. Steidl, and T. Teuber, “Deblurring Poissonian images by split Bregman techniques,” Journal of Visual Communication and Image Representation, vol. 21, no. 3, pp. 193–199, 2010.
- D. Q. Chen and L. Z. Cheng, “Deconvolving Poissonian images by a novel hybrid variational model,” Journal of Visual Communication and Image Representation, vol. 22, no. 7, pp. 643–652, 2011.
- I. Csiszár, “Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems,” The Annals of Statistics, vol. 19, no. 4, pp. 2032–2066, 1991.
- J. M. Bardsley and J. Goldes, “Techniques for regularization parameter and hyper-parameter selection in PET and SPECT imaging,” Inverse Problems in Science and Engineering, vol. 19, no. 2, pp. 267–280, 2011.
- J. M. Bardsley and J. Goldes, “Regularization parameter selection methods for ill-posed Poisson maximum likelihood estimation,” Inverse Problems, vol. 25, no. 9, Article ID 095005, 18 pages, 2009.
- D. Q. Chen and L. Z. Cheng, “Spatially adapted regularization parameter selection based on the local discrepancy function for Poissonian image deblurring,” Inverse Problems, vol. 28, no. 1, Article ID 015004, 24 pages, 2012.
- G. Landi and E. L. Piccolomini, “NPTool: a Matlab software for nonnegative image restoration with Newton projection methods,” Numerical Algorithms, 2012.
- A. Tikhonov and V. Arsenin, Solution of Ill-Poised Problems, Winston, Washington, DC, USA, 1977.
- G. Landi and E. L. Piccolomini, “An improved Newton projection method for nonnegative deblurring of Poisson-corrupted images with Tikhonov regularization,” Numerical Algorithms, vol. 60, no. 1, pp. 169–188, 2012.
- J. M. Bardsley and C. R. Vogel, “A nonnegatively constrained convex programming method for image reconstruction,” SIAM Journal on Scientific Computing, vol. 25, no. 4, pp. 1326–1343, 2003/04.
- J. M. Bardsley and N. Laobeul, “Tikhonov regularized Poisson likelihood estimation: theoretical justification and a computational method,” Inverse Problems in Science and Engineering, vol. 16, no. 2, pp. 199–215, 2008.
- W. H. Richardson, “Bayesan-based iterative methods of image restoration,” Journal of the Optical Society of America, vol. 62, no. 1, pp. 55–59, 1972.
- L. B. Lucy, “An iterative technique for the rectification of observed images,” The Astronomical Journal, vol. 79, pp. 745–754, 1974.
- V. Agarwal, A. V. Gribok, and M. A. Abidi, “Image restoration using norm penalty function,” Inverse Problems in Science and Engineering, vol. 15, no. 8, pp. 785–809, 2007.
- G. Landi and E. L. Piccolomini, “An efficient method for nonnegatively constrained total variation-based denoising of medical images corrupted by Poisson noise,” Computerized Medical Imaging and Graphics, vol. 36, no. 1, pp. 38–46, 2012.
- S. Lefkimmiatis, A. Bourquard, and M. Unser, “Hessian-based norm regularization for image restoration with biomedical applications,” IEEE Transactions on Image Processing, vol. 21, no. 3, pp. 983–995, 2012.
- F. Li, C. M. Shen, J. S. Fan, and C. L. Shen, “Image restoration combining a total variational filter and a fourth-order filter,” Journal of Visual Communication and Image Representation, vol. 18, no. 4, pp. 322–330, 2007.
- M. Lysker and X. C. Tai, “Iterative image restoration combining total varition minimization and a second-order functional,” International Journal of Computer Vision, vol. 66, no. 1, pp. 5–18, 2006.
- K. Papafitsoros and C. B. Schönlieb, “A combined first and second order variational approach for image reconstruction,” http://arxiv.org/abs/1202.6341.
- T. Chan, A. Marquina, and P. Mulet, “High-order total variation-based image restoration,” SIAM Journal on Scientific Computing, vol. 22, no. 2, pp. 503–516, 2000.
- M. Lysaker, A. Lundervold, and X. C. Tai, “Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time,” IEEE Transactions on Image Processing, vol. 12, no. 12, pp. 1579–1590, 2003.
- G. Steidl, “A note on the dual treatment of higher-order regularization functionals,” Computing, vol. 76, no. 1-2, pp. 135–148, 2006.
- H. Z. Chen, J. P. Song, and X. C. Tai, “A dual algorithm for minimization of the LLT model,” Advances in Computational Mathematics, vol. 31, no. 1–3, pp. 115–130, 2009.
- T. Le, R. Chartrand, and T. J. Asaki, “A variational approach to reconstructing images corrupted by Poisson noise,” Journal of Mathematical Imaging and Vision, vol. 27, no. 3, pp. 257–263, 2007.
- W. F. Zhou and Q. G. Li, “Poisson noise removal scheme based on fourth-order PDE by alternating minimization algorithm,” Abstract and Applied Analysis, vol. 2012, Article ID 965281, 14 pages, 2012.
- Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Transactions on Image Processing, vol. 13, no. 4, pp. 600–612, 2004.
- J. G. Nagy, K. Palmer, and L. Perrone, “Iterative methods for image deblurring: a Matlab object-oriented approach,” Numerical Algorithms, vol. 36, no. 1, pp. 73–93, 2004.
- P. C. Hansen, J. G. Nagy, and D. P. O'Leary, Deblurring Images: Matrices, Spectra, and Filtering, vol. 3 of Fundamentals of Algorithms, SIAM, Philadelphia, Pa, USA, 2006.