CAREER: Integral Geometry: Theory, Implementations, and Applications

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A typical problem of integral geometry is to determine the distribution of some physical quantity inside the body (say, density) from the known averages of this quantity over a given family of curves (say, along straight rays of different directions). Several problems in imaging sciences are modeled as integral geometric problems like the one just mentioned. Examples include: the X-ray/Radon transform used in Computerized Tomography, with applications to medical imaging and homeland security; the travel-time tomography problem in seismology or geophysical prospection; and more recently, the Neutron Spin Tomography problem, where a magnetic field inside a material must be recovered from its 'non-abelian' integrals. In this project, the Principal Investigator will address core questions in integral geometry, on both theoretical and applied levels. For such problems where the complexity lies in the geometry of propagation, the representation of the unknown parameters, and sometimes the non-linear aspects of the problem, the feasibility of reconstruction from available data will be first assessed at the level of the continuous models. In cases where inversion is possible, theoretical inversions will be validated by proof-of-concept implementations which will address practical issues of noisy data, sampling/resolution, regularization and Uncertainty Quantification. This research project is integrated with opportunities for training, research experience and career development for undergraduate and graduate students, as well as for a postdoctoral researcher. This project focuses on the theoretical and applied analysis of integral geometric problems, where unknowns are modeled as functions, tensors, connections over bundles, or sections of these bundles. The measurements are linear or non-linear integral functionals of said unknowns. In addition to the usual injectivity, stability and inversion assessments, focus will be turned toward sharpening the mapping properties of integral geometric operators and obtaining range characterizations for certain non-linear operators. The proposed methods combine deep theoretical tools (microlocal analysis, harmonic analysis, Clifford analysis and partial differential equations on manifolds) with a concern to produce explicit routes toward the reconstruction of the unknowns, some of which already exist in non-explicit form. To complement this theoretical agenda, addressing practical issues such as noisy data, sampling and Uncertainty Quantification will require methods from theoretical statistics and signal processing. Numerical validations will be provided to implement the theoretical derivations and uncover the next challenges toward real-life applications.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.