Single scattering tomography with curved detectors

Classical computed tomography assumes that x-rays propagate without scattering. The data is modeled as the two-dimensional Radon transform of the attenuation coefficient of the scanned object. To reconstruct the image, the Radon transform has to be inverted. Recently, several new imaging methods that use singly scattered rays have been suggested. The data in these cases is modeled by integrals of a function along V-lines (two rays joined at a common vertex). In this paper we study the V-line transform arising in Single Scattering Optical Tomography in the case of a single concave detector. In this setup the direction of one of the rays constituting a V-line depends on the common vertex. We derive an explicit inversion formula which allows the recovery of the absorption coefficient. Our results are illustrated with numerical simulations.