7. Integral Geometry and Tomography

The problems of integral geometry are to determine a function given (weighted) integrals of this function over a “rich” family of manifolds. These problems are of importance in medical applications (tomography), and they are quite useful for dealing with inverse problems in hyperbolic differential equations (integrals of unknown coefficients over ellipsoids or lines can be obtained from the first terms of the asymptotic expansion of rapidly oscillating solutions and an information about first-arrival times of a wave). There has been significant progress in the classical Radon problem when manifolds are hyperplanes and the weight function is the unity; there are interesting results in the plane case when a family of curves is regular (resembling locally the family of straight lines) or in case of the family of straight lines with an arbitrary regular attenuation. Still there are many interesting open questions about the problem with local data and simultaneous recovery of density of a source and of attenuation. We give a brief review of this area, referring for more information to the books of Natterer [Nat] and Sharafutdinov [Sh].