Redundancy and inversion of the Compton transform

Data acquired with a Compton gamma-camera are empirical measures of the Compton transform. This integral transform consists to calculate integrals of the intensity distribution of the source on conical surfaces. In this work, we analyze the direct problem and we show that under the non-realistic hypothesis of an infinite extent detector it is possible to isolate classes of Compton projections that are related together via the Radon transform. A reduction of dimensionality may be operated in the data space, leading to a new transform that reveals to be invertible. The invertibility of the Compton transform follows as a consequence and takes the form of a particular filtered backprojection tomographical reconstruction formula. In practical situations where the acquired data allows a rather poor estimation of the Compton projections, averaging projections that belong to the same class allows to reduce the noise in the reconstructed images. In the case of a finite extent camera, numerical experiments show that the reconstructed images present artifacts due to absence of data in some projections. Selection of non-truncated projections may be a way to partially get around this issue.