Singular value decompositions and inversion methods for the exterior Radon transform and a spherical transform

Abstract

The classical Radon transform, R, maps an integrable function in R n to its integrals over all n − 1 dimensional hyperplanes, and the exterior Radon transform is the transform R restricted to hyperplanes that do not intersect a given disc. A singular value decomposition for the exterior transform is given for spaces of square integrable functions on the exterior of the disc. This decomposition in orthogonal functions explicitly produces the null space and range of the exterior transform and gives a new method for inverting the transform modulo the null space. A modification of this method is given that will exactly invert functions of compact support. These results generalize theorems of R. M. Perry and the author. A singular value decomposition for the Radon transform that integrates over spheres in R n containing the origin is also given. This follows from the singular value decomposition for R and yields the null space and a new inversion method for this transform.