使用最优控制进行图像重建与运动估计联合的数学研究
A mathematical study of joint image reconstruction and motion estimation using optimal control
AI总结：
本文从最优控制角度，严格证明了前文提出的用于联合图像重建与运动估计的变分模型时间离散版本的解存在性，通过验证输运方程约束的闭包性质得出结论。
摘要：
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。为解决这类时空成像问题（本质上是一个时间依赖的动态逆问题），前文 [C. Chen, B. Gris, and O. Öktem, 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，用于联合图像重建和运动估计，该模型适用于反演具有大微分同胚形变的运动目标的时间依赖稀疏采样数据。然而，该模型解的存在性尚未给出。为了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容许的 Hilbert 空间中，未知模板图像在有界变差函数空间中进行建模。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的解的存在性。具体而言，在等价的最优控制模型中存在一个输运方程的约束。我们严格证明了该方程的闭包性质，包括解的存在性和唯一性、相关非线性解算子的稳定性以及收敛性。最后，可以得出该模型的解存在性结论。
英文摘要：
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. Öktem, 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.
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