The invertibility of rotation invariant Radon transforms

Abstract

Let R μ denote the Radon transform on R n that integrates a function over hyperplanes in given smooth positive measures μ depending on the hyperplane. We characterize the measures μ for which R μ is rotation invariant. We prove rotation invariant transforms are all one-to-one and hence invertible on the domain of square integrable functions of compact support, L 0 2( R n ). We prove the hole theorem: if f ϵ L 0 2 R n and R μ f = 0 for hyperplanes not intersecting a ball centered at the origin, then f is zero outside of that ball. Using the theory of Fourier integral operators, we extend these results to the domain of distributions of compact support on R n . Our results prove invertibility for a mathematical model of positron emission tomography and imply a hole theorem for the constantly attenuated Radon transform as well as invertibility for other Radon transforms.