Restoration of poissonian images using alternating direction optimization

IEEE Transactions on Image Processing

Volumn 19, Issue 12, 2010, Pages 3133-3145

  Figueiredo, Mário A T ^a   Bioucas Dias, José M ^a  

^a INSTITUTO SUPERIOR TÉCNICO   (Portugal)

Author keywords

Alternating direction methods; augmented Lagrangian; convex optimization; image deconvolution; image restoration; Poisson images

Indexed keywords

ALTERNATING DIRECTION METHODS; AUGMENTED LAGRANGIANS; IMAGE DECONVOLUTION; IMAGE RESTORATION; POISSON IMAGES;

DECONVOLUTION; IMAGE RECONSTRUCTION; LAGRANGE MULTIPLIERS; MAXIMUM LIKELIHOOD; MEASUREMENT THEORY; MEDICAL IMAGING; OPTIMIZATION; RESTORATION;

CONVEX OPTIMIZATION;

ALGORITHM; ARTICLE; AUTOMATED PATTERN RECOGNITION; IMAGE ENHANCEMENT; METHODOLOGY;

ALGORITHMS; IMAGE ENHANCEMENT; PATTERN RECOGNITION, AUTOMATED;

EID: 78649287919     PISSN: 10577149     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIP.2010.2053941     Document Type: Article

References (61)

1. F. Abramovich and B. Silverman, "Wavelet decomposition approaches to statistical inverse problems", Biometrika, vol. 85, pp. 115-129, 1998.
2. A. Antoniadis and J. Bigot, "Poisson inverse problems", Ann. Statist., vol. 34, pp. 1811-1825, 2006.
3. G. Aubert and J. Aujol, "A variational approach to remove multiplicative noise", SIAM J. Appl. Math., vol. 68, no. 4, pp. 925-946, 2008.
4. A. Beck and M. Teboulle, "A fast iterative shrinkage-thresholding algorithm for linear inverse problems", SIAM J. Imag. Sci., vol. 2, pp. 183-202, 2009.
5. J. Bioucas-Dias and M. Figueiredo, "A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration", IEEE Trans. Image Process., vol. 16, no. 12, pp. 2992-3004, Dec. 2007.
6. J. Bioucas-Dias and M. Figueiredo, "Multiplicative noise removal using variable splitting and constrained optimization", IEEE Trans. Image Process., vol. 19, no. 7, pp. 1720-1730, Jul. 2010.
7. E. Candès, L. Demanet, D. Donoho, and L. Ying, "Fast discrete curvelet transforms", SIAM J. Multiscale Model. Sim., vol. 5, pp. 861-899, 2005.
8. L. Cavalier and J.-Y. Koo, "Poisson intensity estimation for tomographic data using a wavelet shrinkage approach", IEEE Trans. Inf. Theory, vol. 48, no. 10, pp. 2794-2802, Oct. 2002.
9. A. Chambolle, "An algorithm for total variation minimization and applications", J. Math. Imag. Vis., vol. 20, pp. 89-97, 2004.
10. A. Chambolle and P.-L. Lions, "Image recovery via total variation minimization and related problems", Numer. Math., vol. 76, pp. 167-188, 1997.
11. T. Chan, S. Esedoglu, F. Park, and A. Yip, "Recent developments in total variation image restoration", in Mathematical Models of Computer Vision. New York: Springer-Verlag, 2005.
12. C. Chaux, P. Combettes, J.-C. Pesquet, and V. Wajs, "A variational formulation for frame-based inverse problems", Inv. Probl., vol. 23, pp. 1495-1518, 2007.
13. T. Chan, G. Golub, and P. Mulet, "A nonlinear primal-dual method for total variation-based image restoration", SIAM J. Sci. Comput., vol. 20, pp. 1964-1977, 1999.
14. P. Combettes and J.-C. Pesquet, "Proximal thresholding algorithm for minimization over orthonormal bases", SIAM J. Optim., vol. 18, pp. 1351-1376, 2007.
15. P. Combettes and J.-C. Pesquet, "A Douglas-Rachford splitting approach to nonsmooth convex variational signal recovery", IEEE J. Sel. Topics Signal Process., vol. 1, no. 4, pp. 564-574, Dec. 2007.
16. P. Combettes and V. Wajs, "Signal recovery by proximal forward-backward splitting", SIAM J. Multiscale Model. Sim., vol. 4, pp. 1168-1200, 2005.
17. J. Dahl, P. Hansen, S. Jensen, and T. Jensen, "Algorithms and software for total variation image reconstruction via first-order methods", Numer. Algorithms, vol. 53, pp. 67-92, 2010.
18. N. Dey, L. Blanc-Feraud, C. Zimmer, P. Roux, Z. Kam, J. Olivo-Marin, and J. Zerubia, "Richardson-Lucy algorithm with TV regularization for 3D confocal microscope deconvolution", Microsc. Res. Tech., vol. 69, pp. 260-266, 2006.
19. D. Donoho, "Nonlinear solution of linear inverse problems by waveletvaguelette decomposition", Appl. Comput. Harmon. Anal., vol. 2, pp. 101-126, 1995.
20. F.-X. Dupé, J. Fadili, and J.-L. Starck, "A proximal iteration for deconvolving Poisson images using sparse representations", IEEE Trans. Image Process., vol. 18, no. 2, pp. 310-321, Feb. 2009.
21. J. Eckstein and D. Bertsekas, "On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators", Math. Program., vol. 5, pp. 293-318, 1992.
22. M. Elad, P. Milanfar, and R. Rubinstein, "Analysis versus synthesis in signal priors", Inv. Probl., vol. 23, pp. 947-968, 2007.
23. E. Esser, "Applications of Lagrangian-based alternating direction methods and connections to split Bregman computational and applied math", Univ. California, Los Angeles, 2009, Tech. Rep. 09-31.
24. J. Fessler and A. Hero, III, "Penalized maximum-likelihood image reconstruction using space-alternating generalized EM algorithms", IEEE Trans. Image Process., vol. 4, no. 10, pp. 1417-29, Oct. 1995.
25. M. Figueiredo and J. Bioucas-Dias, "Deconvolution of Poissonian images using variable splitting and augmented Lagrangian optimization", in Proc. IEEE Workshop Statist. Signal Process., Cardiff, U. K., 2009, pp. 733-736.
26. M. Figueiredo and R. Nowak, "An EM algorithm for wavelet-based image restoration", IEEE Trans. Image Process., vol. 12, no. 8, pp. 906-916, Aug. 2003.
27. A. Foi, S. Alenius, M. Trimeche, V. Katkovnik, and K. Egiazarian, "A spatially adaptive Poissonian image deblurring", in Proc. IEEE Int. Conf. Image Process., Genova, Italy, 2005, vol. 1, pp. 925-928.
28. D. Gabay and B. Mercier, "A dual algorithm for the solution of nonlinear variational problems via finite-element approximations", Comput. Math. Appl., vol. 2, pp. 17-40, 1976.
29. R. Glowinski and A. Marroco, "Sur l'approximation, par elements finis d'ordre un, et la resolution, par penalisation-dualité d'une classe de problemes de dirichlet non lineares", Rev. Française d'Automatique, Inf. Recherche Opérationelle, vol. 9, pp. 41-76, 1975.
30. T. Goldstein and S. Osher, "The split Bregman method for L1 regularized problems, computational and applied math", Univ. California, Los Angeles, 2008, Tech. Rep. 08-29.
31. M. Hestenes, "Multiplier and gradient methods", J. Optim. Theory Appl., vol. 4, pp. 303-320, 1969.
32. N. Kingsbury, "Complex wavelets for shift invariant analysis and filtering of signals", J. Appl. Comput. Harmon. Anal., vol. 10, pp. 234-253, 2001.
33. M. Lang, H. Guo, J. Odegard, C. Burrus, and R. Wells, "Noise reduction using an undecimated discrete wavelet transform", IEEE Signal Process. Lett., vol. 3, no. 1, pp. 10-12, Jan. 1996.
34. T. Le, R. Chartrand, and T. Asaki, "A variational approach to reconstructing images corrupted by Poisson noise", J. Math. Imag. Vis., vol. 27, pp. 257-263, 2007.
35. M. Lustig, D. Donoho, and J. Pauly, "Sparse MRI: The application of compressed sensing for rapid MR imaging", Magnetic Resonance Med., vol. 58, pp. 1182-1195, 2007.
36. S. Mallat, A Wavelet Tour of Signal Processing. New York: Academic, 2009.
37. J.-J. Moreau, "Propriétés des applications prox", Comptes Rendus de l'Académie des Sciences (Série A, Mathématiques), vol. 256, pp. 1069-1071, 1963.
38. J.-J. Moreau, "Proximité et dualité dans un espace hilbertien", Bull. Soc. Math. France, vol. 93, pp. 273-299, 1965.
39. J. Nocedal and S. Wright, Numerical Optimization. New York: Springer-Verlag, 2006.
40. R. Nowak, "Optimal signal estimation using cross-validation", IEEE Signal Process. Lett., vol. 4, no. 1, pp. 23-25, Jan. 1997.
41. R. Nowak and R. Baraniuk, "Wavelet-domain filtering for photon imaging systems", IEEE Trans. Image Process., vol. 8, no. 5, pp. 666-678, May 1999.
42. R. Nowak and E. Kolaczyk, "A Bayesian multiscale framework for Poisson inverse problems", IEEE Trans. Inf. Theory, vol. 46, pp. 1811-1825, 2000.
43. R. Nowak and M. Thul, "Wavelet-vaguelette restoration in photon-limited imaging", in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., 1998, vol. 5, pp. 2869-2872.
44. S. Osher, L. Rudin, and E. Fatemi, "Nonlinear total variation based noise removal algorithms", Phys. D, vol. 60, pp. 259-268, 1992.
45. M. Powell, "A method for nonlinear constraints in minimization problems", in Optimization, R. Fletcher, Ed. New York: Academic, 1969, pp. 283-298.
46. P. Sarder and A. Nehorai, "Deconvolution method for 3-D fluorescence microscopy images", IEEE Signal Process. Mag., vol. 23, no. 3, pp. 32-45, May 2006.
47. I. Selesnick, "Hilbert transform pairs of wavelet bases", IEEE Signal Process. Lett., vol. 8, no. 6, pp. 170-173, Jun. 2001.
48. I. Selesnick and M. Figueiredo, "Signal restoration with overcomplete wavelet transforms: comparison of analysis and synthesis priors", Proc. SPIE, vol. 7446, pp. 0D-15D, 2009.
49. S. Setzer, "Split Bregman algorithm, Douglas-Rachford splitting, and frame shrinkage", in Proc. 2nd Int. Conf. Scale Space Variat. Meth. Comput. Vis., 2009, vol. 5567, pp. 464-476.
50. S. Setzer, G. Steidl, and T. Teuber, "Deblurring Poissonian images by split Bregman techniques", J. Vis. Commun. Image Represent., 2010.
51. J.-L. Starck and F. Murtagh, Astronomical Image and Data Analysis. New York: Springer-Verlag, 2006.
52. J.-L. Starck, A. Bijaoui, and F. Murtagh, "Multiresolution support applied to image filtering and deconvolution", Comput. Vis. Graph. Image Process., vol. 57, pp. 420-431, 1995.
53. J.-L. Starck, M. Nguyen, and F. Murtagh, "Wavelets and curvelets for image deconvolution: A combined approach", Signal Process., vol. 83, pp. 2279-2283, 2003.
54. G. Steidl and T. Teuber, "Removing multiplicative noise by Douglas-Rachford splitting methods", J. Math. Imag. Vis., vol. 36, pp. 168-184, 2010.
55. K. Timmermann and R. Nowak, "Multiscale modeling and estimation of Poisson processes with application to photon-limited imaging", IEEE Trans. Inf. Theory, vol. 45, no. 3, pp. 846-862, Apr. 1999.
56. Y. Wang, J. Yang, W. Yin, and Y. Zhang, "A new alternating minimization algorithm for total variation image reconstruction", SIAM J. Imag. Sci., vol. 1, pp. 248-272, 2008.
57. R. Willett and R. Nowak, "Platelets: A multiscale approach for recovering edges and surfaces in photon-limited medical imaging", IEEE Trans. Med. Imag., vol. 22, no. 3, pp. 332-350, Mar. 2003.
58. R. Willett and R. Nowak, "Fast multiresolution photon-limited image reconstruction", in IEEE Int. Symp. Biomed. Imag., 2004, vol. 2, pp. 1192-1195.
59. S. Wright, R. Nowak, and M. Figueiredo, "Sparse reconstruction by separable approximation", IEEE Trans. Signal Process., vol. 57, no. 7, pp. 2479-2493, Jul. 2009.
60. 1-minimization with applications to compressed sensing", SIAMJ. Imag. Sci., vol. 1, pp. 143-168, 2008.
61. C. Zalinescu, Convex Analysis in General Vector Spaces. Singapore: World Scientific, 2002.