We give a detailed microlocal study of X-ray transforms over geodesic-like families of
curves with conjugate points of fold type. We show that the normal operator is the
sum of a pseudodifferential operator and a Fourier integral operator. We
compute the principal symbol of both operators and the canonical relation
associated to the Fourier integral operator. In two dimensions, for the geodesic
transform, we show that there is always a cancellation of singularities to some
order, and we give an example where that order is infinite; therefore the
normal operator is not microlocally invertible in that case. In the case of three
dimensions or higher if the canonical relation is a local canonical graph we show
microlocal invertibility of the normal operator. Several examples are also
studied.
Keywords
caustics, conjugate points, geodesic X-ray transform,
integral geometry