The maximum entropy on the mean method for image deblurring: applying Fenchel-Rockafellar duality in finite and infinite dimensions
Gabriel Rioux · eScholarship@McGill (McGill) · 2020
Image deblurring is an inverse problem which has seen a surge of activity in recent years due to the advent of machine learning-based approaches. As such, traditional methods consisting of optimizing a fidelity term coupled with a regularizer over the set of all possible images have fallen in popularity. In the following, a novel approach to this problem is proposed, based upon the maximum entropy on the mean method. It consists of optimizing at the level of the set of probability distributions on the set of all images and employing an entropic regularization. The theory is first described in the context of barcode deblurring and, subsequently, for the deblurring of general images. The problem afforded by the principle of maximum entropy on the mean is intractable (it is finite-dimensional, but prohibitively large in the former case and infinite-dimensional in the latter). Nevertheless, a judicious application of the Fenchel-Rockafellar duality theorem affords a finite-dimensional dual problem which can be solved using standard optimization software, as well as a formula to recover a solution of the original problem from that of its dual counterpart. Numerical experiments are provided to demonstrate the strength of this method