The optimal mass transport problem is a geometric framework for how to transport masses in an optimal way. Historically it has had large impact in economic theory and operations research, and recently it has also gained significant interest in application areas such as signal processing, image processing, and machine learning.
The optimal mass transport problem can be formulated as a linear programming problem, however when computing the distance between two images the size of this linear program becomes prohibitively large. A recently development to address this builds on using an entropic barrier term and solving the resulting optimization problem using so called Sinkhorn iterations. This allows for an approximate solution of large optimal mass transport problems. In this work we show how these results can be used and extended in order to use optimal mass transport for solving inverse problems in, e.g., computerized tomography.