Abstract

Transforms mapping a function to its integrals over a collection of smooth surfaces are ubiquitous in tomography. In the case that the surfaces are flat with codimension 1, this is known as the Radon transform and the inversion is classical with many well-known applications. When the hypersurfaces are allowed to be curved we obtain a generalised Radon transform; such transforms appear in geophysics and radar. Generalised Radon transforms correspond to a double fibration of a larger space and this concept leads to analytic results including the well-known Bolker condition for microlocal invertibility. In this talk, I will review the double fibration framework, discuss the Bolker condition in a number of specific cases where it holds and fails, and discuss invertibility of certain transforms in the spirit of Palamodov.