Explicit Methods for Linear and Non-Linear Tomography

University Of California-Santa Cruz, Santa Cruz CA

Investigators

Abstract

Several inverse problems with applications to medical imaging, geophysical imaging and non-destructive material testing are modeled as integral geometric problems, which consist of reconstructing internal features of materials from their cumulated integrals along a given family of trajectories. Examples include the X-ray/Radon transform used in Computerized Tomography, the travel-time tomography problem in seismology or geophysical prospection, and the Neutron Spin Tomography problem, where a magnetic field inside a material must be recovered from its "non-abelian" integrals. In this project, the investigator and his collaborators focus on integral geometric problems where the complexity lies in the type of object to be reconstructed, in the fact that the geometry of propagation of information is curved, and in the nonlinear character of some of these problems. For each of the problems considered, the task is to put in mathematical terms the answer to the following questions: (i) Is the unknown reconstructible from the given measurements and, if yes, how stable is the inversion? (ii) How to reconstruct the unknown in practice? (iii) How to deal with imperfections in the model and in the measurements (due to instrumental noise for example) and how to quantify the uncertainty induced on the proposed reconstructions? This project focuses on some linear and nonlinear integral geometric problems where unknowns are modeled as functions, tensors, connections over bundles or sections of these bundles, and the measurements are integral functionals of the unknowns. For various "solvable" settings, explicit reconstruction algorithms will be derived and implemented whenever possible, assessing injectivity, stability, implementation, and uncertainty quantification aspects of the inverse problems at play. The proposed methods combine deep theoretical tools (microlocal analysis, harmonic analysis, Clifford analysis and partial differential equations on manifolds) with a concern for implementability, to produce explicit answers, some of which already exist in non-explicit form. Numerical validations will be provided to confirm the implementability of the derivation 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.