  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.
