Generalized optimal transport with singular sources

TL;DR

This paper introduces a generalized optimal transport model that allows for local density modulation and singular sources by relaxing the mass-preserving constraint with a source term, enabling applications like image warping.

Contribution

It extends the optimal transport framework to include singular sources and sinks using a linear growth functional, ensuring well-posedness and providing a numerical scheme.

Findings

Model supports singular sources and sinks on points or lines.

Numerical tests show different behaviors compared to traditional models.

Ensures well-posedness and existence of geodesic paths.

Abstract

We present a generalized optimal transport model in which the mass-preserving constraint for the $L^2$-Wasserstein distance is relaxed by introducing a source term in the continuity equation. The source term is also incorporated in the path energy by means of its squared $L^2$-norm in time of a functional with linear growth in space. This extension of the original transport model enables local density modulation, which is a desirable feature in applications such as image warping and blending. A key advantage of the use of a functional with linear growth in space is that it allows for singular sources and sinks, which can be supported on points or lines. On a technical level, the $L^2$-norm in time ensures a disintegration of the source in time, which we use to obtain the well-posedness of the model and the existence of geodesic paths. Furthermore, a numerical scheme based on the proximal splitting approach (Papadakis et al., 2014) is presented. We compare our model with the corresponding model involving the $L^2(L^2)$-norm of the source, which merges the metamorphosis approach and the optimal transport approaches in imaging. Selected numerical test cases show strikingly different behaviour.