Deformation of tomograms for curvilinear tomography problems

Earlier in our works, it was proposed to apply the method of a fan-beam mapping into a set of parallel lines in the problems of fan-beam tomography. This was achieved by special deformation of the reconstracted tomogram at the stage of back projection of the measured and filtered projections, followed by the operation of reverse deformation. The deformation of the tomogram for each direction of observation will be different, but the one-to-one nature of these deformations allows you to return to the original coordinate system. In this paper, the method is generalized to a family of plane curvilinear trajectories that allow one-to-one transitions to parallel rays. For each back projection, the image is modulated by a known function following from the path differential of the given trajectory. The results of generalization of the FBP algorithm widely used in two-dimensional tomography methods are demonstrated by examples of parabolic, sinusoidal and fan-beam ray trajectories.