Refractive dynamic tensor field tomography: towards a holistic approach

• Applicants: Professor Dr. Alfred Karl Louis (†); Professor Dr. Thomas Schuster
• Subject Area: Mathematics
• Term: from 2019 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 411005946

Over the past decades tomographic methods became an integral part of natural science research, engineering, biology and medicine, Earth, atmosphere and ocean monitoring. The relevance of such methods for solving complex problems is proven by the rapid development of tomography and its numerous practical applications. The project is concerned with fundamental theoretical and numerical research in the field of tensor tomography in fairly general and complex settings that are essential for many practical applications, but where satisfying models and numerical solvers are not available yet. The resarch comprehends the construction of adequate mathematical models which take time dependence, absorption and refraction into account, a rigorous analysis followed by the development of problem adapted numerical solution schemes. Tensor tomography means the inverse problem of computing a tensor field from integral data that are taken along geodesic curves of an associated Riemannian metric. If the medium shows no refraction, then the metric is the Euclidean one and the geodesics are straight lines as in classical computerized tomography. Whereas refraction is modeled by an appropriate Riemannian metric, absorption implies an attenuation term. So the mathematical model is given by the geodesic, respectively exponential geodesic, ray transform for tensor fields. If the sought for tensor field is time-dependent, which, e.g. is the case if the object is moving, we even face the problem of dynamic tensor tomography. The inverse problems can be described either by suitable integral transforms in Riemannian manifolds or as parameter identification problems for a transport equation where the integral data act as boundary values. Especially the last perception is to shed new light on this demanding inverse problem and its solution. The objectives of the project comprehend all the way from modeling, analysis to the construction of stable numerical solution methods for refractive dynamic tensor field tomography (RDTT). The comprehensive and challenging tasks in this research field demand for a close cooperation with our Russian partners from Novosibirsk who are well-known experts in this area. A successful outcome of the project will have a deep impact to the entire tomographic community.

• DFG Programme: Research Grants
• International Connection: Russia
• Partner Organisation: Russian Foundation for Basic Research
• Cooperation Partners: Professor Dr. Evgeny Derevtsov; Professorin Dr. Bernadette Hahn-Rigaud