On the microlocal analysis of the geodesic X-ray transform with conjugate points

TL;DR

This paper analyzes the microlocal properties of the geodesic X-ray transform on manifolds with conjugate points, showing a decomposition of the normal operator and mild ill-posedness in higher dimensions.

Contribution

It provides a microlocal decomposition of the normal operator for the geodesic X-ray transform with conjugate points, extending understanding of its invertibility.

Findings

Normal operator decomposes into pseudodifferential and Fourier integral parts.

Inversion is only mildly ill-posed in dimensions three and higher.

Assumes no self-intersecting geodesics and nonsingular conjugate pairs.

Abstract

We study the microlocal properties of the geodesic X-ray transform $\mathcal{X}$ on a manifold with boundary allowing the presence of conjugate points. Assuming that there are no self-intersecting geodesics and all conjugate pairs are nonsingular we show that the normal operator $\mathcal{N} = \mathcal{X}^t \circ \mathcal{X}$ can be decomposed as the sum of a pseudodifferential operator of order $-1$ and a sum of Fourier integral operators. We also apply this decomposition to prove inversion of $\mathcal{X}$ is only mildly ill-posed in dimension three or higher.