X-ray Transforms and Tensor Tomography on Surfaces

Speaker Name: 
Francois Monard
Speaker Title: 
Professor
Start Time: 
Monday, May 22, 2017 - 4:00pm
End Time: 
Monday, May 22, 2017 - 5:00pm
Location: 
Engineering 2, Room 180

Abstract: In this talk, we will study what can be reconstructed of a function (or a tensor) on a surface, from the knowledge of its integrals along a given family of geodesic curves, that is, its X-ray transform. The "straight-line" version of this question was first answered by J. Radon in 1917 and its solution makes the theoretical backbone of Computerized Tomography since the 1960's. In practice, variations of refractive index do occur and bend photon paths in optics-based imaging, and this requires that this problem be studied for general curves. In a geometric setting beyond that of straight lines, the family of integration curves may have qualitative features (e.g. conjugate points, trapped curves) which deteriorate the invertibility and the stability of the problem. We will discuss positive and negative theoretical results occurring when one considers cases with the features above, covering numerical aspects along the way.

Bio: Francois Monard received his M.S. in Applied Mathematics at the Universite Paul Sabatier in 2007. He received his M.S. an Ph.D. In Applied Mathetics at Columbia University, in 2007 and 2012 respectively. Monard worked as an Acting Assistant Professor at the Universtiy of Washington and a Postdoctoral Assistant Professor at the University of Michigan, before joining UCSC as an Assistant Professor in the Department of Mathematics in July 2016. Monard's research interests span from inverse problems involving partial differential equations to the analysis and integral geometry being applied to imaging sciences.