Inverse Problems with Poisson noise: Primal and Primal-Dual Splitting

François-Xavier Dupé (1, 2) , Jalal M. Fadili (3) , Jean-Luc Starck (1, 2)

1 DEDIP - Département d'Electronique, des Détecteurs et d'Informatique pour la Physique (ex SEDI)
2 AIM (UMR7158 / UMR_E_9005 / UM_112) - Astrophysique Interprétation Modélisation
3 Equipe Image - Laboratoire GREYC - UMR6072

In this paper, we propose two algorithms for solving linear inverse problems when the observations are corrupted by Poisson noise. 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. Piecing together the data fidelity and the prior terms, the solution to the inverse problem is cast as the minimization of a non-smooth convex functional. We establish the well-posedness of the optimization problem, characterize the corresponding minimizers, and solve it by means of primal and primal-dual proximal splitting algorithms originating from the field of non-smooth convex optimization theory. Experimental results on deconvolution and comparison to prior methods are also reported.

Mots clés: Inverse Problems; Poisson noise; Duality; Proximity operator; Sparsity

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. Inverse Problems with Poisson noise: Primal and Primal-Dual Splitting. 18th IEEE International Conference on Image Processing (ICIP 2011), Sep 2011, Bruxelles, Belgium. ⟨10.1109/ICIP.2011.6115841⟩. ⟨hal-00575602⟩