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

本计画主要研究动态变形形状的3D影片对的最佳同构匹配。这包括开发和分析时间-空间中的4D变形的适当数学类，以及实施有效的算法工具来确定近似最佳的几何形态匹配。这些问题在医学成像中具有重要意义，例如比较软器官（如跳动的心脏）的回声电影，研究人员将与卫理公会医院（休斯顿）的心脏病专家合作，在真实的患者数据上测试他们的数值和数学方法。该项目的结果将有助于一个非常活跃的领域在medical research. Extension显着的结果Trouve，Younes，米勒，Glaunes的静态3D形状的最佳同构匹配，研究人员认为未知的时间-空间同构作为端点的时间-空间同构流产生的一个未知的流向量场V。然后，他们的策略是在适当的希尔伯特空间中最小化容许向量场流V上的适当的成本泛函。成本泛函结合了视差泛函，测量变形的电影和目标电影之间的距离，和能量泛函，即V的动能。将研究解决方案的存在性以及近似成本泛函，其最小化通过梯度下降法在计算上是可行的。这些技术需要解决高维系统的常微分方程，其有效的集成将是数值挑战之一。一个最佳的控制方法，经营者价值的控制将深入探讨。这些模型和数值工具将通过与心脏病专家合作的超声心动图电影的选定医疗案例研究进行验证。患者心脏的超声心动图电影现在是心脏病临床协议的一部分。医生经常对超声心动图电影进行视觉比较，以评估对患者的治疗效果，或比较不同患者的病例。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形状和体积上不同，并且处于恒定的弹性变形中。为了在视觉上比较它们，需要对它们的形状进行几何变形，这是由心脏病专家的视觉系统隐含地实现的。在这个项目中，数学家们试图在一个非常通用的背景下，通过计算时间扭曲和逐帧的几何变形来模拟这些比较任务，从而实现两部电影的最佳匹配。这些失真的大小然后通过虚能量来量化，该虚能量将需要使一个跳动的心脏物理变形以匹配另一个心脏。这个问题的解决需要复杂的数学理论，并且在标准计算机上实现有效的数值计算是一个严峻的挑战。该项目的结果影响了比较医学诊断，例如心脏病学和胎儿发育，使用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，在技术上比较变形动力学，广泛应用于复杂可变形材料和软物体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.