The adjoint operator of the Radon transform on rotating V-lines and its role in image reconstruction

Integral transformations over conical sets appear as the natural mathematical models in scattered radiation imaging. Compton cameras are emblematic examples of these imaging techniques. The concept of Compton camera originates from the need to improve sensitivity in Single Photon Emission Imaging, which uses hole collimators. It advocates electronic collimation which registers radiation emitted by the radiating object and scattered by a scattering detector placed before an absorption detector. The data consists of three dimensional conical projections of the activity density of a radio-tracer. We consider here a particular two-dimensional Compton camera in which the collected data consists of the set of integrals of the density on rotating V-lines. We present approximate reconstructions obtained both from an adequate back-projection procedure, and from the action of the adjoint operator of this transform. The relation between these operators is also established.