Compressed sensing performance bounds under poisson noise

IEEE Transactions on Signal Processing

Volumn 58, Issue 8, 2010, Pages 3990-4002

  Raginsky, Maxim ^a   Willett, Rebecca M ^a   Harmany, Zachary T ^a   Marcia, Roummel F ^b  

^a Duke University   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Complexity regularization; compressive sampling; nonparametric estimation; photon limited imaging; sparsity

Indexed keywords

BOUNDED NOISE; COMPRESSED SENSING; COMPRESSIVE SAMPLING; ERROR BOUND; LOG LIKELIHOOD; LOW-INTENSITY; MATRIX; NON-ADDITIVE; NON-NEGATIVITY; NON-PARAMETRIC ESTIMATIONS; OBJECTIVE FUNCTIONS; OVERFITTING; PENALTY TERM; PERFORMANCE BOUNDS; PHOTON FLUX; PHOTON-LIMITED IMAGING; POISSON DATA; POISSON NOISE; RECONSTRUCTION ERROR; SIGNAL INTENSITIES; UPPER BOUND;

SIGNAL RECONSTRUCTION;

PHOTONS;

EID: 77954591196     PISSN: 1053587X     EISSN: None     Source Type: Journal    
DOI: 10.1109/TSP.2010.2049997     Document Type: Article

References (39)

1. E. J. Candès and T. Tao, "Near-optimal signal recovery from random projections: Universal encoding strategies?," IEEE Trans. Inf. Theory, vol.52, pp. 5406-5425, Dec. 2006.
2. D. Donoho, "Compressed sensing," IEEE Trans. Inf. Theory, vol.52, pp. 1289-1306, Apr. 2006.
3. D. Lingenfelter, J. Fessler, and Z. He, "Sparsity regularization for image reconstruction with Poisson data," in Proc. SPIE Computat. Imag. VII, 2009, vol.7246.
4. Z. Harmany, R. Marcia, and R. Willett, "Sparse Poisson intensity reconstruction algorithms," in Proc. IEEE Workshop on Statist. Signal Process., 2009 [Online]. Available: http://arxiv.org/abs/0905.0483
5. M. Figueiredo and J. Bioucas-Dias, "Deconvolution of Poissonian images using variable splitting and augmented Lagrangian optimization," in Proc. IEEE Workshop on Statist. Signal Process., 2009 [Online]. Available: http://arxiv.org/abs/0904.4868
6. J.-L. Starck and J. Bobin, "Astronomical data analysis and sparsity: From wavelets to compressed sensing," Proc. IEEE: Special Issue on Appl. Sparse Represent. Compress. Sensing, 2010, to be published.
7. E. Candès, J. Romberg, and T. Tao, "Stable signal recovery from incomplete and inaccurate measurements," Commun. Pure Appl. Math., vol.59, no.8, pp. 1207-1223, 2006.
8. J. Haupt and R. D. Nowak, "Signal reconstruction from noisy random projections," IEEE Trans. Inf. Theory, vol.52, pp. 4036-4048, Sep. 2006.
9. D. L. Snyder, A.M. Hammond, and R. L. White, "Image recovery from data acquired with a charge-coupled-device camera," J. Opt. Soc. Amer. A, vol.10, pp. 1014-1023, 1993.
10. R.Willett and R. Nowak, "Multiscale Poisson intensity and density estimation," IEEE Trans. Inf. Theory, vol.53, pp. 3171-3187, Sep. 2007.
11. 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, pp. 332-350, 2003.
12. E. J. Candès and T. Tao, "Decoding by linear programming," IEEE Trans. Inf. Theory, vol.15, pp. 4203-4215, Dec. 2005.
13. J. A. Tropp, "Just relax: Convex programming methods for identifying sparse signals in noise," IEEE Trans. Inf. Theory, vol.52, pp. 1030-1051, Mar. 2006.
14. R. Tibshirani, "Regression shrinkage and selection via the lasso," J. Roy. Statist. Soc. Ser. B, vol.58, no.1, pp. 267-288, 1996.
15. W. Dai and O. Milenkovic, "Weighted superimposed codes and constrained integer compressive sensing," IEEE Trans. Inf. Theory, vol.55, pp. 2215-2229, May 2009.
16. M. A. Khajehnejad, A. G. Dimakis, W. Xu, and B. Hassibi, "Sparse recovery of positive signals with minimal expansion," 2009 [Online]. Available: http://arxiv.org/abs/0902.4045, submitted for publication
17. F. Bunea, A. Tsybakov, and M. Wegkamp, "Sparse density estimation with l1 penalties," in COLT, 2007.
18. I. Rish and G. Grabarnik, "Sparse signal recovery with exponentialfamily noise," in Proc. Allerton Conf. Commun., Control, Comput., 2009.
19. S. Mendelson, A. Pajor, and N. Tomczak-Jaegermann, "Reconstruction and sub-Gaussian operators in asymptotic geometric analysis," Geom. Funct. Anal., vol.17, pp. 1248-1282, 2007.
20. S. Mendelson, A. Pajor, and N. Tomczak-Jaegermann, "Uniform uncertainty principle for Bernoulli and sub-Gaussian ensembles," Construct. Approx., vol.28, pp. 277-289, 2008.
21. M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, "Single pixel imaging via compressive sampling," IEEE Signal Process. Mag., vol.25, no.2, pp. 83-91, 2008.
22. M. Gehm, R. John, D. Brady, R. Willett, and T. Schultz, "Single-shot compressive spectral imaging with a dual-disperser architecture," Opt. Expr., vol.15, no.21, pp. 14013-14027, 2007. (Pubitemid 47608612)
23. R. DeVerse, R. Coifman, A. Coppi, W. Fateley, F. Geshwind, R. Hammaker, S. Valenti, F.Warner, and G. Davis, "Application of spatial light modulators for new modalities in spectrometry and imaging," in Proc. SPIE, 2003, vol.4959.
24. M. A. Neifeld and J. Ke, "Optical architectures for compressive imaging," Appl. Opt., vol.46, pp. 5293-5303, 2007.
25. S. Jafarpour, R. Willett, M. Raginsky, and R. Calderbank, "Performance bounds for expander-based compressed sensing with Poisson noise," in Proc. Asilomar, 2009.
26. K. Krishnamurthy, M. Raginsky, and R. Willett, "Multiscale photonlimited hyperspectral image reconstruction," SIAM J. Imag. Sci., 2010, submitted for publication.
27. A. R. Barron, C. Huang, J. Q. Li, and X. Luo, "MDL, penalized likelihood, and statistical risk," in Proc. IEEE Inf. Theory Workshop, Porto, Portugal, May 2008, pp. 247-257.
28. I. Csiszár, "Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems," Ann. Statist., vol.19, no.4, pp. 2032-2066, 1991.
29. F.-X. Dupé, J. M. Fadili, and J.-L. Starck, "A proximal iteration for deconvolving Poisson noisy images using sparse representations," IEEE Trans. Image Process., vol.18, pp. 310-321, 2009.
30. S. Setzer, G. Steidl, and T. Teuber, "Restoration of images with rotated shapes," Numer. Alg., vol.48, no.1-3, pp. 49-66, Jul. 2008.
31. P. L. Combettes and J.-C. Pesquet, "A proximal decomposition method for solving convex variational inverse problems," Inverse Prob., vol.24, no.6, pp. 065014-065014, 2008.
32. C. Chaux, J.-C. Pesquet, and N. Pustelnik, "Nested iterative algorithms for convex constrained image recovery problems," SIAM J. Imag. Sci., vol.2, no.2, pp. 730-762, 2009.
33. A. W. van der Waat and J. A. Wellner, Weak Convergence and Empirical Processes. New York: Springer, 1996.
34. Q. Li and A. Barron, "Mixture density estimation," in Advances in Neural Information Processing. Cambridge: MIT Press, 2000, vol.12.
35. E. Kolaczyk and R. Nowak, "Multiscale likelihood analysis and complexity penalized estimation," Ann. Statist., vol.32, pp. 500-527, 2004.
36. Y. Censor and S. A. Zenios, Parallel Optimization: Theory, Algorithms, and Applications. Oxford: Oxford Univ. Press, 1997.
37. T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd ed. New York: Wiley, 2006.
38. A. Bhattacharyya, "On a measure of divergence between two statistical populations defined by probability distributions," Bull. Calcutta Math. Soc., vol.35, pp. 99-109, 1943.
39. H. Chernoff, "A measure of asymptotic efficiency of test of a hypothesis based on the sum of observations," Ann. Math. Statist., vol.23, pp. 493-507, 1952.