Inversion formula for an integral geometry problem over surfaces of revolution

Abstract

AbstractAn integral geometry problem is considered on a family of ‐dimensional surfaces of revolution whose vertices lie on a hyperplane and directions of symmetry axes are fixed and orthogonal to this plane, in . More precisely, the reconstruction of a function , , , from the integrals of the form extended over a chosen side of all surfaces of revolution of a given family is investigated. Unlike the usual Radon transform, the integrals considered here are not taken with respect to the surface area element. A Fourier slice identity and a backprojection‐type inversion formula are obtained with a method based on the Fourier and Hankel transforms. The reconstruction procedure and some analytical and numerical implementations of the obtained inversion formulas in the cases of and are provided.