Analytical inversion of the Compton transform using the full set of available projections

The detection of gamma rays using a camera based on Compton scattering has become widespread because of its sensitivity and efficiency. Apart from the challenge arising from the Compton camera design, another price must be paid: the complex image reconstruction process of the object under study from the measured Compton scattered data. In this paper, a mathematical model of projections generated by photons scattered in a generic Compton camera is defined, relating the projections to the object by an integral transform, namely the Compton transform. The set of projections can be split into subsets following the slope α of the direction of the scattered photons. We show that each subset allows us to invert the Compton transform and exactly recover the image of the object. In practical applications, where the number of acquired photons is low and each reconstructed image is noisy, averaging improves dramatically the signal-to-noise ratio. Finally, the validity of the approach is proved by numerical experiments.