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Abstract.A new circular arc Radon transform arising from the mathematical modeling of image formation in a new modality of Compton scattering tomography is introduced. We describe some of its properties and establish its analytic inverse formula. This result demonstrates the feasibility of image reconstruction from Compton scattered radiation in Compton scattering tomography. We also show that it belongs to a larger class of Radon transforms on algebraic curves, which remain invariant under a specific geometric inversion.

Radon transforms defined on smooth curves are well known and extensively studied in the literature. In this paper, we consider a Radon transform defined on a discontinuous curve formed by a pair of half-lines forming the vertical letter V. If the classical two-dimensional Radon transform has served as a work horse for tomographic transmission and/or emission imaging, we show that this V-line Radon transform is the backbone of scattered radiation imaging in two dimensions. We establish its analytic inverse formula as well as a corresponding filtered back projection reconstruction procedure. These theoretical results allow the reconstruction of two-dimensional images from Compton scattered radiation collected on a one-dimensional collimated camera. We illustrate the working principles of this imaging modality by presenting numerical simulation results.

Integral transforms which map functions on R 3 onto their integrals on circular cones having fixed axis direction and variable opening angle are introduced and studied as generalizations of the known Radon transform. Besides their intrinsic mathematical interest, they serve as backbone support to emission imaging based on Compton scattered radiation, the way the standard Radon transform does for emission imaging based on non-scattered radiation. In this work, we establish its basic properties and prove analytically its invertibility. Formulae to express it in terms of the standard Radon transform (or vice versa) are given. We also discuss some extensions as applications.

Radon transforms on piecewise smooth curves in R 2 are rather unfamiliar and have not been so far widely investigated. In this paper we consider three types of Radon transforms defined on a pair of half-lines in the shape a V-letter, with a fixed axis direction. These three Radon transforms arise from recently suggested tomographic procedures. Our main result consists in obtaining their analytic inverse formulas, which may serve as mathematical foundation for new imaging systems in engineering and physics.