Deconvolved Image Restoration from Auto-Correlations

The recovery of a real signal from its auto-correlation is a wide-spread problem in computational imaging, and it is equivalent to retrieve the phase linked to a given Fourier modulus. Image-deconvolution, on the other hand, is a funda- mental aspect to take into account when we aim at increasing the resolution of blurred signals. These problems are addressed separately in a large number of experimental situations, ranging from adaptive astronomy to optical microscopy. Here, instead, we tackle both at the same time, performing auto-correlation inversion while deconvolving the current object estimation. To this end, we propose a method based on ${I}$ -divergence optimization, turning our formalism into an iterative scheme inspired by Bayesian-based approaches. We demonstrate the method by recovering sharp signals from blurred auto-correlations, regardless of whether the blurring acts in auto-correlation, object, or Fourier domain.