Analytic and statistical aspects of geodesic X-ray transforms 

In this talk, we will study the inverse problem of reconstructing a function from the collection of its integrals along a certain family of curves, specifically, geodesics for a fixed, known Riemannian metric. This problem and several of its integral geometric cousins have received a lot of attention in recent years, as they arise in medical imaging in media with variable index of refraction, in seismology and several other fields of application. 

 

The qualitative properties of this inverse problem (injectivity, stability, invertibility....) depend on geometric features of the underlying metric, in particular the presence (or not) of caustics and/or infinite-length geodesics. In some cases, the transform is stably invertible, and dealing with noisy data has a solid regularization theory, proved statistically optimal via a Bernstein-von Mises theorem. In other cases, stability breaks down while injectivity questions still remain open. We will discuss the state of the art on each of these cases as well as the methods to obtain the results, namely, energy estimates for certain transport PDEs as well as microlocal methods. Numerical examples will be given throughout.