A Proximal Iteration for Deconvolving Poisson Noisy Images Using Sparse Representations

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 as follows. First, we handle the Poisson noise properly by using the Anscombe variance stabilizing transform leading to a nonlinear 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 nonsmooth sparsity-promoting penalty over the image representation coefficients (e.g., $\ell _{1}$ -norm). An additional term is also included in the functional to ensure positivity of the restored image. Third, a fast iterative forward-backward 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.