Optimal Transport and Field Alignment methods are related, e.g., Haker et al. (IJCV 2004) use it for warping, and it was used earlier in EMD in Computer Vision. See also Rachev and Ruschendorf, 1998 (https://www.springer.com/la/book/9780387983523) for several others. Field Alignment methods can be related to Optimal Transport in the sense of Benamou and Bernier's work. While our original form is related to diffeomorphic alignment, subsequent ones (e.g., scale cascaded alignment) differ. When Optimal Transport is defined as being a divergence-free deformation between the source and destination mass distributions, then correspondence with nondivergent solutions of Field Alignment emerge.  However, Field Alignment is a step within the Bayesian coupling of Amplitude Adjustments and Deformation (``transport").  This is more general in the sense that the difficulties identifying appropriate densities and maintaining their mass conservation in the presence of noisy mesurements, regularizing the correspondence map between the domains when one domain is very sparsely observed are overcome.  

Despite the recent interest in Optimal Transport in Data Assimilation, we posit that extant methods (e.g. Ravela et al. 2007) have previously considered flows (transport) of fields in a variational setting for data assimilation. We have further extended this to: FABle (Field Alignment Blending), AROMA (Adaptive Reduced Order Modeling By Alignment), Field Coalescence, Coherent Random Fields, Principal Appearance and Geometry Modes. While the two variants of Field Alignment can be related to Transport, it is important to note that the variational and ensemble formalism with Bayesian inference provide substantial generalization.

We posit that  our approach to the Bayesian Coupling of Field Geometry and Amplitudes for Inference, Samplng, Reduced Modeling, Uncertainty Quantification of Random Fields and Processes offers a solution to the conundrum of whether amplitude adjustment or geometric adjustment is to be preffered and how. This is not how optimal transport was even framed though there is interest.