  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 transform. Our key
innovations are: First, we handle the Poisson noise properly by using the
Anscombe variance stabilizing transform leading to a 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. l1-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. 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, e.g. astronomy or microscopy.
