# A proximal iteration for deconvolving Poisson noisy images using sparse representations

1 Equipe Image - Laboratoire GREYC - UMR6072
GREYC - Groupe de Recherche en Informatique, Image et Instrumentation de Caen
Abstract : We propose an image deconvolution algorithm when the data is contaminated by Poisson noise. The image to restore is assumed to be sparsely represented in a dictionary of waveforms such as the wavelet or curvelet transforms. Our key contributions are: First, we handle the Poisson noise properly by using the Anscombe variance stabilizing transform leading to a {\it non-linear} degradation equation with additive Gaussian noise. Second, the deconvolution problem is formulated as the minimization of a convex functional with a data-fidelity term reflecting the noise properties, and a non-smooth sparsity-promoting penalties over the image representation coefficients (e.g. $\ell_1$-norm). Third, a fast iterative backward-forward splitting algorithm is proposed to solve the minimization problem. We derive existence and uniqueness conditions of the solution, and establish convergence of the iterative algorithm. Finally, a GCV-based model selection procedure is proposed to objectively select the regularization parameter. Experimental results are carried out to show the striking benefits gained from taking into account the Poisson statistics of the noise. These results also suggest that using sparse-domain regularization may be tractable in many deconvolution applications with Poisson noise such as astronomy and microscopy.
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Cited literature [56 references]

https://hal.archives-ouvertes.fr/hal-00264972
Contributor : François-Xavier Dupé Connect in order to contact the contributor
Submitted on : Wednesday, August 27, 2008 - 3:42:07 PM
Last modification on : Monday, December 13, 2021 - 9:14:55 AM
Long-term archiving on: : Tuesday, September 21, 2010 - 5:41:12 PM

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François-Xavier Dupé, Jalal M. Fadili, Jean-Luc Starck. A proximal iteration for deconvolving Poisson noisy images using sparse representations. IEEE Transactions on Image Processing, Institute of Electrical and Electronics Engineers, 2009, 18 (2), pp.310-321. ⟨10.1109/TIP.2008.2008223⟩. ⟨hal-00264972v2⟩

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