Inversion formula for an integral geometry problem over surfaces of revolution

Abstract

An integral geometry problem is considered on a family of $n$-dimensional surfaces of revolution whose vertices lie on a hyperplane and directions of symmetry axes are fixed and orthogonal to this plane, in $R^{n+1}$. More precisely, the reconstruction of a function $f(x,y)$, $x\in R^n$, $y\in R$, from the integrals of the form $f(x,y)dx$ 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 $n=1$ and $n=2$ are provided.