LMS Workshop on Geometric Inverse Problems
Speaker: Prof Gabriel Paternain
Title: Effective inversion of the attenuated X-ray transform associated with a connection.
Abstract: I will discuss inversion formulas (up to compact error) for the attenuated X-ray transform on a non-trapping surface, when the attenuation is determined by a complex matrix of 1-forms (a GL(n,C)-connection). The formulas lead naturally to a filtered-backprojection algorithm, and if time allows I'll discuss some numerical simulations. This is joint work with Francois Monard.
Speaker: Prof Bill Lionheart
Speaker: Dr Hanming Zhou
Title:The weighted X-ray transform and applications.
Abstract:There is a geophysical question of determining the inner structure of the Earth from the measurements of travel times of seismic waves at the surface. The mathematical formulation of the question consists of recovering a function or more generally a Riemannian metric from the distance or lens data, which is known as the boundary or lens rigidity problem. The linearization of the problem is concerned with the inversion of the X-ray transform of scalar functions or tensor fields, and has important applications in medical imaging techniques. In this talk, I will present recent progress on inverting X-ray transforms with a possible weight through a local-to-global approach. I will also discuss their direct applications to the rigidity problems, as well as to other non-linear inverse problems. Part of the talk is based on joint work with Gabriel Paternain, Mikko Salo and Gunther Uhlmann.
Speaker: Dr Alden Waters
Title:Recovery of scatterers from phaseless measurements.
Abstract:This talk is based on recent work "Recovery of the sound speed for the Acoustic wave equation from phaseless measurements". We introduce a new form of boundary value measurements which measure the energy density of a scattered wave. We show the recovery of scatterers for the acoustic wave and generalized Helmholtz equation is reduced to a question of stability for the Hamiltonian flow transform, in contrast to the classic Calderón problem.