In this work, a unified representation of all the time-varying dynamics is accomplished with a Lagrangian framework for analyzing  regularized dynamical optimal mass transport (OMT) derived flows. Our Lagrangian framework is also applicable to the Fisher-Rao regularization because, as we show, it is equivalent to our regularized dynamical OMT formulation . The advantage of the Lagrangian framework is that the time-varying trajectories and particle attributes are displayed in a single visualization. This provides a natural capability to identify and distinguish flows under different conditions. Applying our Lagrangian analysis to the glymphatic system , we successfully distinguish between flow patterns under two different anesthetics . We should also note that the Fisher-Rao regularization makes direct contact with some of the very nice recent work on entropic regularization for the computation of optimal mass transport as well as the Schr\"odinger bridge theory. The latter provides a stochastic aspect to the approach described in the present work .
In this work, a unified representation of all the time-varying dynamics is accomplished with a Lagrangian framework for analyzing Fisher-Rao regularized dynamical optimal mass transport (OMT) derived flows. While formally equivalent to the Eulerian based Schr\"odinger bridge OMT regularization scheme, the Fisher-Rao approach allows a simple and interpretable methodology for studying the flows of interest in the present work . The advantage of the proposed Lagrangian technique is that the time-varying particle trajectories and attributes are displayed in a single visualization. This provides a natural capability to identify and distinguish flows under different conditions. The Lagrangian analysis applied to the glymphatic system (brain waste removal pathway associated with Alzheimer's Disease) successfully captures known flows and distinguishes between flow patterns under two different anesthetics , providing deeper insights into altered states of waste drainage .
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Fisher-Rao
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Our Lagrangian framework is also applicable to the Fisher-Rao regularization because, as we show, it is equivalent to our regularized dynamical OMT formulation
While formally equivalent to the Eulerian based Schr\"odinger bridge OMT regularization scheme, the Fisher-Rao approach allows a simple and interpretable methodology for studying the flows of interest in the present work
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Lagrangian framework
proposed Lagrangian technique
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trajectories and particle
particle trajectories and
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Applying our Lagrangian analysis
The Lagrangian analysis applied
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, we successfully distinguish
(brain waste removal pathway associated with Alzheimer's Disease) successfully captures known flows and distinguishes
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. We should also note that the Fisher-Rao regularization makes direct contact with some of the very nice recent work on entropic regularization for the computation of optimal mass transport as well as the Schr\"odinger bridge theory. The latter provides a stochastic aspect to the approach described in the present work
, providing deeper insights into altered states of waste drainage