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.
