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Journal of Mathematics
Volume 2013 (2013), Article ID 274573, 11 pages
http://dx.doi.org/10.1155/2013/274573
Research Article

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.

Linked References

  1. H. Andrew and B. Hunt, Digital Image Restoration,, Prentice Hall, Englewood Cliffs, NJ, USA, 1977.
  2. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging, Institute of Physics Publishing, London, UK, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. T. F. Chan and J. Shen, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods, SIAM, Philadelphia, Pa, USA, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. M. Banham and A. Katsaggelos, “Digital image restoration,” IEEE Signal Processing Magazine, vol. 14, no. 2, pp. 24–41, 1997.
  5. 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. View at Publisher · View at Google Scholar
  6. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  8. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  10. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  11. 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. View at Publisher · View at Google Scholar
  12. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  13. 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. View at Publisher · View at Google Scholar
  14. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  16. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  18. S. Yun and H. Woo, “A new multiplicative denoising variational model based on mth root transformation,” IEEE Transactions on Image Processing, vol. 21, no. 5, pp. 2523–2533, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  19. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  20. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  23. 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. View at Publisher · View at Google Scholar
  24. 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. View at Publisher · View at Google Scholar
  25. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. G. Landi and E. L. Piccolomini, “NPTool: a Matlab software for nonnegative image restoration with Newton projection methods,” Numerical Algorithms, 2012. View at Publisher · View at Google Scholar
  30. A. Tikhonov and V. Arsenin, Solution of Ill-Poised Problems, Winston, Washington, DC, USA, 1977.
  31. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  33. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  34. 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. View at Publisher · View at Google Scholar
  35. L. B. Lucy, “An iterative technique for the rectification of observed images,” The Astronomical Journal, vol. 79, pp. 745–754, 1974. View at Publisher · View at Google Scholar
  36. V. Agarwal, A. V. Gribok, and M. A. Abidi, “Image restoration using L1 norm penalty function,” Inverse Problems in Science and Engineering, vol. 15, no. 8, pp. 785–809, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  37. 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. View at Publisher · View at Google Scholar
  38. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  39. 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. View at Publisher · View at Google Scholar
  40. 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. View at Publisher · View at Google Scholar
  41. K. Papafitsoros and C. B. Schönlieb, “A combined first and second order variational approach for image reconstruction,” http://arxiv.org/abs/1202.6341.
  42. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  43. 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.
  44. G. Steidl, “A note on the dual treatment of higher-order regularization functionals,” Computing, vol. 76, no. 1-2, pp. 135–148, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  45. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  46. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  47. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  48. 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. View at Publisher · View at Google Scholar
  49. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  50. 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. View at Publisher · View at Google Scholar · View at MathSciNet