The geodesic ray transform on Riemannian surfaces with conjugate points

Abstract

We study the geodesic X-ray transform $X$ on compact Riemannian surfaces with conjugate points. Regardless of the type of the conjugate points, we show that we cannot recover the singularities and therefore, this transform is always unstable (ill-posed). We describe the microlocal kernel of $X$ and relate it to the conjugate locus. We present numerical examples illustrating the cancellation of singularities. We also show that the attenuated X-ray transform is well posed if the attenuation is positive and there are no more than two conjugate points along each geodesic; but still ill-posed, if there are three or more conjugate points. Those results follow from our analysis of the weighted X-ray transform.

new section about the attenuated X-ray transform added

References in corpus (2)

• The inverse problem for the local geodesic ray transform
• Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map

Cited by in corpus (2)

• The local magnetic ray transform of tensor fields
• Semi-classical limit of an inverse problem for the Schrödinger equation