1. Introduction

As mentioned in this book's preface, microlocal analysis (MA) is very useful in inverse problems in determining singularities of the medium parameters. In this chapter we survey several such applications, including an elaboration of some applications of MA to tomography that were already mentioned in Section 5 of Faridani's chapter (pages 11-14). We recall below the general setting.

While in two-dimensional tomography it is often possible to irradiate an unknown object from all directions, in three dimensions it is usually not practical to obtain this many data. Moreover, since the manifold of lines in ℝ3 is four dimensional, while the object under investigation is a function of three variables, it should suffice to restrict the measurements to a three-dimensional submanifold of lines. Of course, in practice one has only finitely many measurements, but the considerations of the continuous case can be used to guide the design of algorithms and sampling geometries.

Reconstruction from line integral data is never local. That is, to reconstruct a function f at a point x ∈ ℝn requires more than the data of the line integrals of f over all lines passing through a neighborhood of x. However, if the full line integral transform is composed with its adjoint, the resulting operator is an elliptic pseudodifferential operator, which preserves singular supports.