  Recovering a signal from auto-correlations or, equivalently, retrieving the
phase linked to a given Fourier modulus, is a wide-spread problem in imaging.
This problem has been tackled in a number of experimental situations, from
optical microscopy to adaptive astronomy, making use of assumptions based on
constraints and prior information about the recovered object. In a similar
fashion, deconvolution is another common problem in imaging, in particular
within the optical community, allowing high-resolution reconstruction of
blurred images. Here we address the mixed problem of performing the
auto-correlation inversion while, at the same time, deconvolving its current
estimation. To this end, we propose an I-divergence optimization, driving our
formalism into a widely used iterative scheme, inspired by Bayesian-based
approaches. We demonstrate the method recovering the signal from blurred
auto-correlations, further analysing the cases of blurred objects and
band-limited Fourier measurements.
