Optimal Matching of 3D Movies by "Time/Space" Deformations

本课题主要研究动态变形形状3D电影对的最优微分同胚匹配问题。这包括开发和分析时空中适当的4D变形的数学类别，以及实施有效的算法工具来确定近似最优的微分同胚匹配。这类问题在医学成像中具有重要意义，例如比较心脏跳动等软器官的超声图像，研究人员将与卫理公会医院(休斯顿)的心脏病专家合作，在真实患者数据上测试他们的数字和数学方法。这一项目的结果将有助于医学研究的一个非常活跃的领域。扩展了Trouve，Youes，Miller，GLaunes关于静态3D形状的最优微分同胚匹配的显著结果，研究人员将未知的时空微分同胚视为由未知的矢量场V流产生的时空微分同胚流的终点。然后，他们的策略是在适当的Hilbert空间中最小化允许向量场流V上的适当的代价泛函。代价泛函结合了测量变形电影和目标电影之间距离的视差泛函和能量泛函，即V的动能。将研究解的存在性以及近似代价泛函，其最小化通过梯度下降方法在计算上是可行的。这些技术需要求解高维常微分方程组，其有效积分将是数值挑战之一。将深入探索一种以操作员值控制为特征的最优控制方法。这些模型和数值工具将通过与心脏病学专家合作的超声心动图电影上的选定医学案例研究来验证。患者心脏的超声电影现在是心脏病学众多临床方案的一部分。医生经常对超声心动图电影进行视觉比较，以评估患者的治疗效果，或比较不同患者的情况。在这些比较中，生物心脏周期在时间上并不相同，只能通过心理时间扭曲来匹配。不同患者的心脏在细节上形状和体积不同，处于恒定的弹性变形中。为了直观地比较它们的形状，需要对它们的形状进行几何失真，这是由心脏病专家的视觉系统隐式实现的。在这个项目中，数学家试图在非常一般的背景下模拟这些比较任务，通过计算时间扭曲和逐帧几何失真来实现两部电影的最佳匹配。然后，这些失真的大小被用使一个跳动的心脏物理变形以匹配另一个心脏所需的想象能量来量化。这个问题的解决需要复杂的数学理论，并提出了在标准计算机上实现有效的数值计算的严峻挑战。该项目的结果影响了比较医学诊断，例如在心脏病学和胎儿发育方面，随着新的工具来指定和可视化软变形器官的时间动态之间的关键差异。它们还将提供通用工具来从技术上比较变形动力学，并广泛应用于复杂可变形材料和软对象的高科技制造中的性能评估和优化。

This project focuses on the optimal diffeomorphic matching for pairs of 3D movies of dynamic deformable shapes. This includes the development and analysis of adequate mathematical classes of 4D-deformations in time-space, as well as the implementation of efficient algorithmic tools to determine an approximately optimal diffeomorphic matching. Such problems are of significant relevance in medical imaging, for instance to compare echographic movies of soft organs such as beating hearts, and the investigators will collaborate with cardiologists at The Methodist Hospital (Houston) to test their numerical and mathematical approaches on real patients data. The results of this project will thus contribute to a very active field in medical research.Extending remarkable results of Trouve, Younes, Miller, Glaunes for optimal diffeomorphic matching of static 3D- shapes, the investigators consider the unknown time-space diffeomorphism as the endpoint of the flow of time-space diffeomorphisms generated by an unknown flow of vector field V . Their strategy is then to minimize a suitable cost functional over admissible vector field flows V in a suitable Hilbert space. The cost functional combines a disparity functional, measuring the distance between a deformed movie and the target movie, and an energy functional , namely the kinetic energy of V. The existence of solutions will be studied as well as approximate cost functionals whose minimization is computationally feasible by gradient-descent methods. These techniques require the solution of high dimensional systems of ODEs whose efficient integration will be one of the numerical challenges. An optimal control approach featuring operator-valued controls will be intensively explored. The models and numerical tools will be validated by selected medical case studies on echocardiographic movies, in collaboration with cardiology specialists.Echographic movies of patients hearts are now part of numerous clinical protocols in cardiology. Visual comparison of echocardiographic movies by medical doctors is frequent, to evaluate the effect of treatment on a patient, or to compare the cases of different patients. In these comparisons, biological heart cycles are not identical in time and can only be matched by a mental time warping. The hearts of distinct patients are dissimilar in detailed shape and volume , and are in constant elastic deformation. To compare them visually requires a geometric distortion of their shapes which is implicitly realized by the vision system of expert cardiologists.In this project, mathematicians seek to emulate these comparison tasks, in a very generic context, by computing the time warping and the frame by frame geometric distortions which achieve the best matching of two movies. The size of these distortions is then quantified by the imaginary energy which would be needed to physically deform one beating heart to match the other.The solution of this problem requires sophisticated mathematical theory and presents a serious challenges to reach an efficient numerical computation on standard computers.The results of this project impact comparative medical diagnosis, for instance in cardiology and foetus development, with new tools to specify and visualize key differences between the time dynamics of soft deformable organs. They will also provide generic tools to technically compare deformation dynamics, with a wide range of applications to performance evaluation and optimization in high tech manufacturing of sophisticated deformable materials and soft objects.