  Image deblurring is a notoriously challenging ill-posed inverse problem. In
recent years, a wide variety of approaches have been proposed based upon
regularization at the level of the image or on techniques from machine
learning. We propose an alternative approach, shifting the paradigm towards
regularization at the level of the probability distribution on the space of
images. Our method is based upon the idea of maximum entropy on the mean
wherein we work at the level of the probability density function of the image
whose expectation is our estimate of the ground truth. Using techniques from
convex analysis and probability theory, we show that the method is
computationally feasible and amenable to very large blurs. Moreover, when
images are imbedded with symbology (a known pattern), we show how our method
can be applied to approximate the unknown blur kernel with remarkable effects.
While our method is stable with respect to small amounts of noise, it does not
actively denoise. However, for moderate to large amounts of noise, it performs
well by preconditioned denoising with a state of the art method.
