The lnvertibility of Rotation Invariant Radon Transforms

Let R, denote the Radon transform on R” that integrates a function over hyperplanes in given smooth positive measures p depending on the hyperplane. We characterize the measures ,u for which R, 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, Li(R”). We prove the hole theorem: iffE Li(R”) and Ryf = 0 for hyperplanes not intersecting a ball centered at the origin, thenfis 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”. 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.