Deconvolution under Poisson noise using exact data fidelity and synthesis or analysis sparsity priors
Dupé , François-Xavier
Fadili , Jalal M.
Starck , Jean-Luc
- Publisher: Elsevier
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.