Radon transforms on generalized Cormack’s curves and a new Compton scatter tomography modality

Abstract

In his seminal work of 1981, Cormack established that Radon transforms defined on two remarkable families of curves in the plane are invertible and admit explicit inversion formulas via circular harmonic decomposition. A sufficient condition for finding larger classes of curves enjoying the same property is given in this paper. We show that these generalized Cormack’s curves are given by the solutions of a nonlinear first-order differential equation, which is invariant under geometric inversion. A derivation of the analytic inverse formula of the corresponding Radon transforms, as well as some of their main properties, are worked out. Interestingly, among these generalized Cormack’s curves are circles orthogonal to a circle of fixed radius centered at the origin of coordinates. It is suggested that a novel Compton scatter tomography modality may be modeled by a Radon transform defined on these circles.