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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.

This work is dedicated to the reduction of reconstruction artefacts due to motion occurring during the acquisition of computerized tomographic projections. This problem has to be solved when imaging moving organs such as the lungs or the heart. The proposed method belongs to the class of motion compensation algorithms, where the model of motion is included in the reconstruction formula. We address two fundamental questions. First what conditions on the deformation are required for the reconstruction of the object from projections acquired sequentially during the deformation, and second how do we reconstruct the object from those projections. Here we answer these questions in the particular case of 2D general time-dependent affine deformations, assuming the motion parameters are known. We treat the problem of admissibility conditions on the deformation in the parallel-beam and fan-beam cases. Then we propose exact reconstruction methods based on rebinning or sequential FBP formulae for each of these geometries and present reconstructed images obtained with the fan-beam algorithm on simulated data.