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.

翻译：时空动态医学成像在心脏或肺部断层成像等临床应用中至关重要。针对这类本质上是时变动态反问题的时空成像问题，前期论文[C. Chen, B. Gris, and O. \"Oktem, SIAM J. Imaging Sci., 12 (2019), pp. 1686--1719]提出了具有强度、边缘特征和拓扑保持的变分模型，用于联合图像重建与运动估计，该模型适用于对具有大微分同胚形变的运动目标进行时变稀疏采样数据的反演。然而，该模型解的存在性尚未给出。为保持运动目标的拓扑结构和边缘特征，模型中的未知速度场被限制在可容许的希尔伯特空间中，而未知模板图像则建模为有界变差函数空间中的元素。在此框架下，本文从最优控制的角度分析并证明了其时间离散版本解的存在性。具体而言，等价最优控制模型中存在输运方程约束。我们严格论证了该方程的闭包性质，包括解的存在唯一性、相关非线性解算子的稳定性以及收敛性。最终可得出该模型解的存在性结论。