Talk by Bill Lionheart, University of Manchester

Non linear problems in tomography

Abstract

We are used to considering x-ray, gamma ray  and neutron attenuation tomography as a linear problems, and for monochromatic attenuation tomography with  point sources and point detectors we need only take the logarithm of the data and we have a linear inverse problem. As soon as we have sources and detectors of finite extent the exponential of a linear operator applied to the image is itself averaged and a logarithm no longer linearizes the problem. In rich tomography problems, for example polarimetric neutron tomography of magnetic fields, we have a matrix ODE along rays that cannot be solved with an exponential, even a matrix exponential.

These non-linear problems are relatively mild compared to many other inverse boundary value problems for PDES. For example the Jacobians are still sparse matrices. In this talk we will explore how much of a problem the non-linearity is for reconstruction and practical strategies for solution.