A proximal iteration for deconvolving Poisson noisy images using sparse representations

IEEE Transactions on Image Processing

Volumn 18, Issue 2, 2009, Pages 310-321

  Dupe, François Xavier ^a   Fadili, Jalal M ^a   Starck, Jean Luc ^b  

^a Centre National de la Recherche Scientifique   (France)

^b UNIVERSITÉ PARIS DIDEROT   (France)

Author keywords

Deconvolution; Forward backward splitting; Iterative thresholding; Poisson noise; Proximal iteration; Sparse representations

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; CONVOLUTION; DECONVOLUTION; FOURIER TRANSFORMS; ITERATIVE METHODS; MATHEMATICAL TRANSFORMATIONS; NONLINEAR EQUATIONS; POISSON DISTRIBUTION; RESTORATION; WAVELET TRANSFORMS;

FORWARD-BACKWARD SPLITTING; ITERATIVE THRESHOLDING; POISSON NOISE; PROXIMAL ITERATION; SPARSE REPRESENTATIONS;

POISSON EQUATION;

ALGORITHM; ARTICLE; ARTIFACT; COMPUTER ASSISTED DIAGNOSIS; IMAGE ENHANCEMENT; METHODOLOGY; POISSON DISTRIBUTION; REPRODUCIBILITY; SENSITIVITY AND SPECIFICITY; STATISTICAL ANALYSIS; STATISTICAL MODEL;

ALGORITHMS; ARTIFACTS; DATA INTERPRETATION, STATISTICAL; IMAGE ENHANCEMENT; IMAGE INTERPRETATION, COMPUTER-ASSISTED; MODELS, STATISTICAL; POISSON DISTRIBUTION; REPRODUCIBILITY OF RESULTS; SENSITIVITY AND SPECIFICITY;

EID: 59649106568     PISSN: 10577149     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIP.2008.2008223     Document Type: Article

References (59)

1. P. Sarder and A. Nehorai, "Deconvolution method for 3-D fluorescence microscopy images," IEEE Signal Process Mag., vol. 23, pp. 32-45, 2006.
2. J. Pawley, Handbook of Confocal Microscopy. New York: Plenum, 2005.
3. J.-L. Starck and F. Murtagh, Astronomical Image and Data Analysis. New York: Springer, 2006.
4. H. C. Andrews and B. R. Hunt, Digital Image Restoration. Englewood Cliffs, NJ: Prentice-Hall, 1977.
5. P. Jansson, Image Recovery: Theory and Application. New York: Academic, 1987.
6. P. Jansson, Deconvolution of Images and Spectra. New York: Academic, 1997.
7. N. Dey, L. Blanc-Féraud, C. Zimmer, Z. Kam, J.-C. Olivo-Marin, and J. Zerubia, "A deconvolution method for confocal microscopy with total variation regularization," presented at the IEEE ISBI, 2004.
8. J.-L. Starck and F. Murtagh, "Image restoration with noise suppression using the wavelet transform," Astron. Astrophys., vol. 288, pp. 343-348, 1994.
9. J.-L. Starck, A. Bijaoui, and F. Murtagh, "Multiresolution support applied to image filtering and deconvolution," CVGIP: Graph. Models Image Process., vol. 57, pp. 420-431, 1995.
10. G. Jammal and A. Bijaoui, "Dequant: A flexible multiresolution restoration framework," Signal Process., vol. 84, pp. 1049-1069, 2004.
11. J. B. de Monvel et al., "Image restoration for confocal microscopy: Improving the limits of deconvolution, with application of the visualization of the mammalian hearing organ," Biophys. J., vol. 80, pp. 2455-2470, 2001.
12. C. Vonesch and M. Unser, "A fast iterative thresholding algorithm for wavelet-regularized deconvolution," presented at the IEEE ISBI, 2007.
13. I. Daubechies, M. Defrise, and C. D. Mol, "An iterative thresholding algorithm for linear inverse problems with a sparsity constraints," Comm. Pure Appl. Math., vol. 112, pp. 1413-1541, 2004.
14. R. Nowak and E. Kolaczyk, "A statistical multiscale framework for poisson inverse problems," IEEE Trans. Inf. Theory, vol. 46, pp. 2794-2802, 2000.
15. L. Cavalier and J.-Y. Koo, "Poisson intensity estimation for tomographic data using a wavelet shrinkage approach," IEEE Trans. Inf. Theory, vol. 48, pp. 2794-2802, 2002.
16. D. L. Donoho, "Nonlinear solution of inverse problems by wavelet-vaguelette decomposition," Appl. Comput. Harmon. Anal., vol. 2, pp. 101-126, 1995.
17. A. Antoniadis and Bigot, "Poisson inverse problems," Ann. Statist. vol. 34, pp. 1811-1825, 2006.
18. A. Cohen, M. Hoffman, and M. Reiss, "Adaptive wavelet-galerkin methods for inverse problems," SIAM J. Numer. Anal., vol. 42, pp. 1479-1501, 2004.
19. M. Figueiredo and R. Nowak, "An em algorithm for wavelet-based image restoration," IEEE Trans. Image Process., vol. 12, pp. 906-916, 2003.
20. P. L. Combettes and V. R. Wajs, "Signal recovery by proximal forward-backward splitting," SIAM Multiscale Model. Simul., vol. 4, no. 4, pp. 1168-1200, 2005.
21. G. Teschke, "Multi-frame representations in linear inverse problems with mixed multi-constraints," Appl. Comput. Harmon. Anal., vol. 22, no. 1, pp. 43-60, 2007.
22. C. Chaux, P. L. Combettes, J.-C. Pesquet, and V. R. Wajs, "A variational formulation for frame-based inverse problems," Inv. Prob., vol. 23, pp. 1495-1518, 2007.
23. M. J. Fadili and J.-L. Starck, "Sparse representation-based image deconvolution by iterative thresholding," in ADA IV. New York: Elsevier, 2006.
24. 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.
25. C. Chaux, L. Blanc-Féraud, and J. Zerubia, "Wavelet-based restoration methods: Application to 3D confocal microscopy images," presented at the SPIE Wavelets XII, 2007.
26. C. Lemaréchal and J.-B Hiriart-Urruty, Convex Analysis and Minimization Algorithms I, 2nd ed. New York: Springer, 1996.
27. P. G. Ciarlet, Introduction a l'analyse Numérique Matricielle et à l'optimisation. Paris, France: Masson-Paris, 1985.
28. P. L. Combettes, "Solving monotone inclusions via compositions of nonexpansive averaged operators," Optimization, vol. 53, pp. 475-504, 2004.
29. F. J. Anscombe, "The transformation of Poisson, binomial and negative-binomial data," Biometrika, vol. 35, pp. 246-254, 1948.
30. S. G. Mallat, A Wavelet Tour of Signal Processing, 2nd ed. NewYork: Academic, 1998.
31. E. J. Candès and D. L. Donoho, "Curvelets - a surprisingly effective nonadaptive representation for objects with edges," in Curve and Surface Fitting, A. Cohen, C. Rabut, and L. Schumaker, Eds. Nashville, TN: Vanderbilt Univ. Press, 1999.
32. E. Candès, L. Demanet, D. Donoho, and L. Ying, "Fast discrete curvelet transforms," SIAM Multiscale Model. Simul., vol. 5, pp. 861-899, 2005.
33. L. Demanet and L. Ying, "Wave atoms and sparsity of oscillatory patterns," Appl. Comput. Harmon Anal., vol. 23, no. 3, pp. 368-387, 2007.
34. M. Jansen, "Multiscale poisson data smoothing," J. Roy. Statist. Soc. B., vol. 68, no. 1, pp. 27-48, 2006.
35. J.-J. Moreau, "Fonctions convexes duales et points proximaux dans un espace hilbertien," CRAS Sér. A Math., vol. 255, pp. 2897-2899, 1962.
36. J. Eckstein and D. P. Bertsekas, "On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators," Math. Programm., vol. 55, pp. 293-318, 1992.
37. P.-L. Lions and B. Mercier, "Splitting algorithms for the sum of two nonlinear operators." SIAM J. Numer. Anal., vol. 16, pp. 964-979, 1979.
38. M. Fadili, J.-L. Starck, and F. Murtagh, "Inpainting and zooming using sparse representations," The Comput. J., 2006, to be published.
39. P. L. Combettes and J.-C. Pesquet, "Proximal thresholding algorithm for minintization over orthonormal bases," SIAM J. Optim., vol. 18, no. 4, pp. 1351-1376, Nov. 2007.
40. P. Tseng, "A modified forward-backward splitting method for maximal monotone mappings," SIAM J. Control Optim., vol. 38, pp. 431-446, 2000.
41. G. H. Golub, M. Heath, and G. Wahba, "Generalized cross-validation as a method for choosing a good ridge parameter," Technometrics, vol. 21, no. 2, pp. 215-223, 1979.
42. H. Akaike, B. N. Petrox, and F. Caski, "Information theory and an extension of the maximum likelihood principle," in Proc. 2nd Int. Symp. Information Theory, 1973, pp. 267-281.
43. G. Schwarz, "Estimation of the dimension of a model," Ann. Statist. vol. 6, pp. 461-464, 1978.
44. C. Stein, "Estimation of the mean of a multivariate normal distribution," Ann. Statist., vol. 9, pp. 1135-1151, 1981.
45. B. Efron, "How biased is the apparent error rate of a prediction rule," J. Amer. Statist. Assoc., vol. 81, pp. 461-470, 1981.
46. M. Jansen, M. Malfait, and A. Bultheel, "Generalized cross validation for wavelet thresholding," Signal Process., vol. 56, no. 1, pp. 33-44, 1997.
47. B. Efron, "The estimation of prediction error: Covariance penalties and cross-validation," J. Amer. Statist. Assoc., vol. 99, pp. 619-642, 2004.
48. J. Ye, "On measuring and correcting the effects of data mining and model selection," J. Amer. Statist. Assoc., vol. 93, pp. 120-131, 1998.
49. X. Shen and J. Ye, "Adaptive model selection," J. Amer. Statist. Assoc., vol. 97, pp. 210-221, 2002.
50. H. Zou, T. Hastie, and R. Tibsturani, "On the degrees of freedom of the lasso," Ann. Staist., vol. 35, pp. 2173-2192, 2007.
51. 1 Mmintization 2007 [Online]. Available: http:// hal.archives-ouvertes.fr/hal-00164738/en/
52. A. Barron and T. Cover, "Minimum complexity density estimation," IEEE Trans. Inf. Theory, vol. 37, pp. 1034-1054, 1991.
53. Rebecca Willett Homepage [Online]. Available: http://www.ee.duke.edu/ willett/
54. ImageJ website [Online]. Available: http://rsb.info.nih.gov/ij/
55. R. Willett and R. Nowak, "Fast multiresolution photon-limited image reconstruction," presented at the IEEE ISBI 2004.
56. B. Zhang, J. Zerubia, and J.-C. Olivo-Marin, "Gaussian approximations of fluorescence microscope PSF models," Appl. Opt., vol. 46, no. 10, pp. 1819-1829, 2007.
57. J. Buckheit and D. Donoho, Wavelab and reproducible research," in Wavelets and Statistics, A. Antoniadis, Ed. New York: Springer, 1995.
58. F. Murtagh, J.-L. Starck, and A. Bijaoui, "Image restoration with noise suppression using a multiresolution support," Astron., Astrophys., Supplement Ser., vol. 112, pp. 179-189, 1995.
59. P. L. 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, Apr. 2007.