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

Abstract

We propose a new variational model for joint image reconstruction and motion estimation applicable to spatiotemporal imaging, which is investigated along a general framework that we present with shape theory. This model consists of two parts, one that conducts modified image reconstruction in a static setting and the other that estimates the motion by sequentially indirect image registration. For the latter, we generalize the large deformation diffeomorphic metric mapping framework into the sequentially indirect registration setting. The proposed model is compared theoretically against alternative approaches (optical flow based model and diffeomorphic motion models), and we demonstrate that the proposed model has desirable properties in terms of the optimal solution. The theoretical derivations and efficient algorithms are also presented for a time-discretized scenario of the proposed model, which show that the optimal solution of the time-discretized version is consistent with that of the time-continuous one, and most of the computational components are the easily implemented linearized deformations. The complexity of the algorithm is analyzed as well. This work is concluded by some numerical examples in two-dimensional space + time tomography with very sparse and/or highly noisy data.