Computing the time-continuous Optimal Mass Transport Problem without Lagrangian techniques

TL;DR

This paper introduces a novel algorithm for solving the time-continuous optimal mass transport problem using a fixed point and least squares approach, avoiding traditional Lagrangian methods, with applications in medical image analysis.

Contribution

The paper presents a new fixed point and least squares based algorithm for optimal mass transport that does not rely on Lagrangian techniques, demonstrated in 2D medical imaging.

Findings

Efficient algorithm for continuous mass transport without Lagrangian methods

Numerical results show the method's effectiveness in 2D image tracking

First order finite element implementation confirms practical viability

Abstract

This work originates from a heart's images tracking which is to generate an apparent continuous motion, observable through intensity variation from one starting image to an ending one both supposed segmented. Given two images p0 and p1, we calculate an evolution process p(t, \cdot) which transports p0 to p1 by using the optimal extended optical flow. In this paper we propose an algorithm based on a fixed point formulation and a time-space least squares formulation of the mass conservation equation for computing the optimal mass transport problem. The strategy is implemented in a 2D case and numerical results are presented with a first order Lagrange finite element, showing the efficiency of the proposed strategy.