Unsupervised blind-deconvolution with optimal scaling applied to astronomical data

Many blind deconvolution algorithms have been proposed for image deblurring when the instrumental point spread function (PSF) is unknown. Blind deconvolution can be stated as an inverse problem whose unknowns are the object of interest and the PSF. In that case, the direct model is bilinear and has an intrinsic scaling degeneracy: scaling one of the components can be compensated by inversely scaling the other one. Provided that homogeneous regularization functions are chosen for the object and for the PSF, the scaling degeneracy of the direct model can be exploited to reduce the number of hyper-parameters in the problem. Using this property, we propose an instance of a blind deconvolution algorithm that amounts to alternately estimating and scaling the two components of the bilinear model. Our algorithm is insensitive to the scaling of the initial estimate of the component (PSF or object) used to start the iterations. We show that this yields much faster convergence and, in practice, reduces the odds of being trapped in a bad local minimum. These features make our algorithm suitable for being embedded into a simple procedure to automatically tune the remaining hyper-parameter(s) and obtain a fully unsupervised method. We also propose a homogeneous version of an edge-preserving regularization to be used by our algorithm. Using Stein's Unbiased Risk Estimator (SURE) as a criterion to automatically tune the hyper-parameter(s), we assess the advantages of our algorithm for empirical astronomical images compared to other methods.