Generalized Sinkhorn iterations for regularizing inverse problems using optimal mass transport

TL;DR

This paper extends Sinkhorn iterations to efficiently compute the proximal operator of optimal mass transport for large-scale inverse problems, enabling improved regularization in applications like limited-angle tomography.

Contribution

It introduces a novel splitting framework that uses Sinkhorn-type iterations to incorporate optimal mass transport in inverse problems, especially for large-scale applications.

Findings

Effective in large problems with high computational efficiency

Improves image reconstruction in limited-angle tomography

Demonstrates the method's ability to incorporate prior information

Abstract

The optimal mass transport problem gives a geometric framework for optimal allocation, and has recently gained 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 to address 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 the 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.