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

François-Xavier Dupé (1, 2) , Jalal M. Fadili (2) , Jean-Luc Starck (1)

1 DEDIP - Département d'Electronique, des Détecteurs et d'Informatique pour la Physique (ex SEDI)
2 Equipe Image - Laboratoire GREYC - UMR6072

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.

Mots clés: Iterative thresholding; Sparse representations; Deconvolution; Poisson noise; Proximal iteration

Domaines: Traitement du signal et de l'image [eess.SP]; Traitement du signal et de l'image [eess.SP]; Traitement des images [eess.IV]; Applications [stat.AP]

François-Xavier Dupé, Jalal M. Fadili, Jean-Luc Starck. Deconvolution under Poisson noise using exact data fidelity and synthesis or analysis sparsity priors. Statistical Methodology, 2012, 9 (1-2), pp.4-18. ⟨10.1016/j.stamet.2011.04.008⟩. ⟨hal-00575429⟩