  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.
