A mathematical study of joint image reconstruction and motion estimation using optimal control

Spatiotemporal dynamic medical imaging is critical in clinical applications, such as tomographic imaging of the heart or lung. To address such kind of spatiotemporal imaging problems, essentially, a time-dependent dynamic inverse problem, the variational model with intensity, edge feature and topology preservations was proposed for joint image reconstruction and motion estimation in the previous paper [C. Chen, B. Gris, and O. \"Oktem, SIAM J. Imaging Sci., 12 (2019), pp. 1686--1719], which is suitable to invert the time-dependent sparse sampling data for the motion target with large diffeomorphic deformations. However, the existence of solution to the model has not been given yet. In order to preserve its topological structure and edge feature of the motion target, the unknown velocity field in the model is restricted into the admissible Hilbert space, and the unknown template image is modeled in the space of bounded variation functions. Under this framework, this paper analyzes and proves the solution existence of its time-discretized version from the point view of optimal control. Specifically, there exists a constraint of transport equation in the equivalent optimal control model. We rigorously demonstrate the closure of the equation, including the solution existence and uniqueness, the stability of the associated nonlinear solution operator, and the convergence. Finally, the solution existence of that model can be concluded.