Statistics Seminar, "Optimal mass transport: a metric framework for image reconstruction with a priori information", Johan Karlsson, KTH
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.
The first part of the talk we review the area and consider examples where the optimal mass transport is used as a distance measure for image classification and robust system identification.In the second part of the talk we consider large optimization problems where the optimal mass transport cost is used for incorporating a priori information. The optimal mass transport problem can be formulated as a linear programming problem, but 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.