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We propose a variational model with diffeomorphic optimal transportation forjoint image reconstruction and motion estimation. The proposed model is aproduction of assembling the Wasserstein distance with the Benamou--Brenierformula in optimal transportation and the flow of diffeomorphisms involved inlarge deformation diffeomorphic metric mapping, which is suitable for thescenario of spatiotemporal imaging with large diffeomorphic and mass-preservingdeformations. Specifically, we first use the Benamou--Brenier formula tocharacterize the optimal transport cost among the flow of mass-preservingimages, and restrict the velocity field into the admissible Hilbert space toguarantee the generated deformation flow being diffeomorphic. We then gain theODE-constrained equivalent formulation for Benamou--Brenier formula. We finallyobtain the proposed model with ODE constraint following the framework thatpresented in our previous work. We further get the equivalent PDE-constrainedoptimal control formulation. The proposed model is compared against severalexisting alternatives theoretically. The alternating minimization algorithm ispresented for solving the time-discretized version of the proposed model withODE constraint. Several important issues on the proposed model and associatedalgorithms are also discussed. Particularly, we present several potentialmodels based on the proposed diffeomorphic optimal transportation. Underappropriate conditions, the proposed algorithm also provides a new scheme tosolve the models using quadratic Wasserstein distance. The performance isfinally evaluated by several numerical experiments in space-time tomography,where the data is measured from the concerned sequential images with sparseviews and/or various noise levels.

Spatiotemporal Imaging with Diffeomorphic Optimal Transportation