Blind Deconvolution Problems: A Literature Review and Practitioner’s Guide

Blind deconvolution is the process of deconvolving a known acquired signal with an unknown point spread function (PSF) or channel encoding, by jointly estimating the system’s input as well as the system’s PSF. Models for this problem often present as the convolution or product of the underlying signal and the system’s PSF. The acquired signal is bi-linear in these two unknown signals (i.e. linear in one if the other is constant), which results in an inverse problem that is non-convex. In addition, the number of observations to uniquely reconstruct the signal and the system’s PSF is often unattainable as the number of degrees of freedom grows as the product of the size of the signal with the size of the PSF, thus the problem is often underdetermined. While the task seems daunting, there has been a plethora of work over the past twenty years with the aim of performing this task with exact recovery, similar to that of sparse vector recovery in compressed sensing and of low rank matrix recovery in matrix completion. There are a variety of works in the area of exact recovery to pose convex and non-convex problem alike as well as theory regarding under what conditions this is possible. We shall see that advances in these fields compliment the blind deconvolution problem as it can be posed as a linear inverse problem with rank constraint and relaxed to a convex program. In this literature review the point of view as a practitioner in the areas of signal processing and computational imaging (or computational sensing) is taken. This review is laid out as follows: section 2 goes over into several applications and importance, section 3 reviews the technique of lifting as well as several convex and non-convex problem formulations, section 4 explains some of the theoretical guarantees that accompany convex and non-convex problem formulations, and finally section 5 discusses a broader picture regarding computation and memory for large scale problems.