Spatiotemporal imaging with diffeomorphic optimal transportation

Abstract

Motivated by the image reconstruction in spatiotemporal imaging, we introduce a concept named diffeomorphic optimal transportation (DOT), which combines the Wasserstein distance with Benamou–Brenier formula in optimal transportation and the flow of diffeomorphisms in large deformation diffeomorphic metric mapping. Using DOT, we propose a new variational model for joint image reconstruction and motion estimation, which is suitable for spatiotemporal imaging involving mass-preserving large diffeomorphic deformations. We further get its equivalent PDE-constrained optimal control formulation. The proposed model is easy-to-implement and solved by an alternating gradient descent algorithm, which is compared against existing alternatives theoretically and numerically. The performance is validated by several numerical experiments in spatiotemporal tomography, where the projection data is time-dependent sparse and/or high-noise. Moreover, we present several extensions based on DOT. Under appropriate conditions, the proposed algorithm can be adapted as a new algorithm to solve the models using quadratic Wasserstein distance.