Unbalanced Regularized Optimal Mass Transport with Applications to Fluid Flows in the Brain

Generalizing the fluid dynamical optimal mass transport (OMT) approach of Benamou and Brenier, regularized optimal mass transport (rOMT) formulates a transport problem from an initial mass configuration to another with the optimality defined by the kinetic energy, but subject to an advection-diffusion constraint equation. Both rOMT and the Benamou and Brenier's formulation require the total initial and final mass to be equal; mass is preserved during the entire transport process. However, for many applications, e.g., in dynamic image tracking, this constraint is rarely if ever satisfied. Here we introduce an unbalanced version of rOMT to remove this constraint together with a detailed numerical solution procedure and applications to dynamic image tracking in the brain.