The Euclidean X-ray transform, which encodes the integrals of a function over straight lines, is a classical topic (going back to J. Radon in 1917)
and forms the basis of imaging methods such as X-ray computed tomography and PET . The geodesic X-ray transform encodes the integrals of a function over more general families of curves, such as the geodesics of a sound speed or Riemannian metric. It arises in seismic imaging and transmission ultrasound imaging as the linearization of the travel time tomography problem, in Electrical Impedance Tomography (the Calderon problem), and in inverse spectral problems.
There has been considerable recent progress in understanding the mathematical properties of the geodesic X-ray transform, based on several
different methods. In this talk we will discuss results related to energy methods for the underlying transport PDE , partly based on joint works with G. Paternain (Cambridge) and G. Uhlmann (Washington).