Spatiotemporal Imaging with Diffeomorphic Optimal Transportation

by   Chong Chen, et al.

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.


page 19

page 21

page 25

page 27


A New Variational Model for Joint Image Reconstruction and Motion Estimation in Spatiotemporal Imaging

We propose a new variational model for joint image reconstruction and mo...

Diffeomorphic Image Registration with An Optimal Control Relaxation and Its Implementation

Image registration has played an important role in image processing prob...

Constrained H^1-regularization schemes for diffeomorphic image registration

We propose regularization schemes for deformable registration and effici...

Ocean Mover's Distance: Using Optimal Transport for Analyzing Oceanographic Data

Modern ocean datasets are large, multi-dimensional, and inherently spati...

Computational performance studies for space-time phase-field fracture optimal control problems

The purpose of this work are computational demonstations for a newly dev...

Regularized Wasserstein Means Based on Variational Transportation

We raise the problem of regularizing Wasserstein means and propose sever...

Sparse dynamic tomography. A shearlet-based approach for iodine perfusion in plant stems

In this paper we propose a motion-aware variational approach to reconstr...