Deconvolution under Poisson noise using exact data fidelity and synthesis or analysis sparsity priors

Article, Preprint English OPEN
Dupé , François-Xavier ; Fadili , Jalal M. ; Starck , Jean-Luc (2012)
  • Publisher: Elsevier
  • Journal: STATISTICAL METHODOLOGY (issn: 1572-3127)
  • Related identifiers: doi: 10.1016/j.stamet.2011.04.008
  • Subject: Statistics - Applications | Deconvolution | [ INFO.INFO-TS ] Computer Science [cs]/Signal and Image Processing | Iterative thresholding | Sparse representations | Proximal iteration | [ INFO.INFO-TI ] Computer Science [cs]/Image Processing | Poisson noise | [ SPI.SIGNAL ] Engineering Sciences [physics]/Signal and Image processing | [ STAT.AP ] Statistics [stat]/Applications [stat.AP]

International audience; In this paper, we propose a Bayesian MAP estimator for solving the deconvolution problems when the observations are corrupted by Poisson noise. Towards this goal, a proper data fidelity term (log-likelihood) is introduced to reflect the Poisson statistics of the noise. On the other hand, as a prior, the images to restore are assumed to be positive and sparsely represented in a dictionary of waveforms such as wavelets or curvelets. Both analysis and synthesis-type sparsity priors are considered. Piecing together the data fidelity and the prior terms, the deconvolution problem boils down to the minimization of non-smooth convex functionals (for each prior). We establish the well-posedness of each optimization problem, characterize the corresponding minimizers, and solve them by means of proximal splitting algorithms originating from the realm of non-smooth convex optimization theory. Experimental results are conducted to demonstrate the potential applicability of the proposed algorithms to astronomical imaging datasets.
  • References (32)
    32 references, page 1 of 4

    [1] Andrews, H. C., Hunt, B. R., 1977. Digital Image Restoration. Prentice-Hall.

    [2] Anscombe, F. J., 1948. The Transformation of Poisson, Binomial and Negative-Binomial Data. Biometrika 35, 246-254.

    [3] Bertero, M., Boccacci, P., Desidera`, G., Vicidomini, G., 2009. Image deblurring with Poisson data: from cells to galaxies. Inverse Problems 25 (123006).

    [4] Cand`es, E. J., Donoho, D. L., 1999. Curvelets - a surprisingly effective nonadaptive representation for objects with edges. In: Cohen, A., Rabut, C., Schumaker, L. (Eds.), Curve and Surface Fitting: Saint-Malo 1999. Vanderbilt University Press, Nashville, TN.

    [5] Cand`es, E. J., Wakin, M. B., Boyd, S. P., 2008. Enhancing sparsity by reweighted L1 minimization. Journal of Fourier Analysis and Applications 14 (5), 877-905.

    [6] Ciarlet, P., 1982. Introduction `a l'Analyse Num´erique Matricielle et `a l'Optimisation. Paris.

    [7] Combettes, P. L., Pesquet, J.-., 2007. A Douglas-Rachford splittting approach to nonsmooth convex variational signal recovery. IEEE Journal of Selected Topics in Signal Processing 1 (4), 564-574.

    [8] Combettes, P. L., Pesquet, J.-C., November 2007. Proximal thresholding algorithm for minimization over orthonormal bases. SIAM Journal on Optimization 18 (4), 1351-1376.

    [9] Combettes, P. L., Pesquet, J.-C., 2008. A proximal decomposition method for solving convex variational inverse problems. Inverse Problems 24 (6).

    [10] Combettes, P. L., Wajs, V. R., 2005. Signal recovery by proximal forward-backward splitting. SIAM Multiscale Model. Simul. 4 (4), 1168-1200.

  • Metrics
    No metrics available
Share - Bookmark