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.

翻译：推广Benamou和Brenier的流体动力学最优质量传输（OMT）方法，正则化最优质量传输（rOMT）将传输问题表述为从初始质量分布到另一分布的过程，其最优性由动能定义，但受平流-扩散约束方程限制。rOMT与Benamou-Brenier公式均要求初始与最终总质量相等，即质量在整个传输过程中保持守恒。然而，在许多实际应用（如动态图像追踪）中，这一约束几乎无法满足。本文提出了一种非平衡版本的rOMT，以消除该约束，并详述了数值求解过程及其在大脑动态图像追踪中的应用。