The geodesic X-ray transform with fold caustics

TL;DR

This paper analyzes the microlocal properties of the geodesic X-ray transform with fold caustics, revealing conditions for invertibility and illustrating the behavior of the normal operator in different dimensions.

Contribution

It provides a detailed microlocal analysis of the normal operator for the X-ray transform with fold caustics, including symbol computations and invertibility conditions.

Findings

Normal operator is a sum of pseudodifferential and Fourier integral operators.

In 2D, singularities can cancel, preventing microlocal invertibility.

In higher dimensions, invertibility depends on the canonical relation being a local graph.

Abstract

We give a detailed microlocal study of X-ray transforms over geodesics-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.