Spatiotemporal Imaging with Diffeomorphic Optimal Transportation

TL;DR

This paper introduces a novel variational model combining diffeomorphic optimal transportation with image reconstruction and motion estimation, suitable for large deformations in spatiotemporal imaging, and demonstrates its effectiveness through numerical experiments.

Contribution

It develops a new model integrating Wasserstein distance and diffeomorphic mappings for improved spatiotemporal image analysis, with an efficient algorithm and theoretical comparisons.

Findings

Effective in handling large deformations in space-time tomography

Outperforms existing methods in noisy and sparse data scenarios

Provides a new computational scheme for Wasserstein-based models

Abstract

We propose a variational model with diffeomorphic optimal transportation for joint image reconstruction and motion estimation. The proposed model is a production of assembling the Wasserstein distance with the Benamou--Brenier formula in optimal transportation and the flow of diffeomorphisms involved in large deformation diffeomorphic metric mapping, which is suitable for the scenario of spatiotemporal imaging with large diffeomorphic and mass-preserving deformations. Specifically, we first use the Benamou--Brenier formula to characterize the optimal transport cost among the flow of mass-preserving images, and restrict the velocity field into the admissible Hilbert space to guarantee the generated deformation flow being diffeomorphic. We then gain the ODE-constrained equivalent formulation for Benamou--Brenier formula. We finally obtain the proposed model with ODE constraint following the framework that presented in our previous work. We further get the equivalent PDE-constrained optimal control formulation. The proposed model is compared against several existing alternatives theoretically. The alternating minimization algorithm is presented for solving the time-discretized version of the proposed model with ODE constraint. Several important issues on the proposed model and associated algorithms are also discussed. Particularly, we present several potential models based on the proposed diffeomorphic optimal transportation. Under appropriate conditions, the proposed algorithm also provides a new scheme to solve the models using quadratic Wasserstein distance. The performance is finally evaluated by several numerical experiments in space-time tomography, where the data is measured from the concerned sequential images with sparse views and/or various noise levels.