Generalized Sinkhorn Iterations for Regularizing Inverse Problems Using Optimal Mass Transport

The optimal mass transport problem gives a geometric framework for optimal allocation and has recently attracted significant interest in application areas such as signal processing, image processing, and computer vision. Even though it can be formulated as a linear programming problem, it is in many cases intractable for large problems due to the vast number of variables. A recent development addressing this builds on an approximation with an entropic barrier term and solves the resulting optimization problem using Sinkhorn iterations. In this work we extend this methodology to a class of inverse problems. In particular we show that Sinkhorn-type iterations can be used to compute the proximal operator of the transport problem for large problems. A splitting framework is then used to solve inverse problems where the optimal mass transport cost is used for incorporating a priori information. We illustrate this method on problems in computerized tomography. In particular we consider a limited-angle computerized tomography problem, where a priori information is used to compensate for missing measurements.