Unbiased Filtered Back-Projection in $4\pi$ Compton Imaging With 3D Position Sensitive Detectors

In Compton imaging, iterative methods provide good performance but are usually computationally intensive. Thus, direct inverse algorithms such as filtered back-projection (FBP) are preferable when computation time is limited. The conventional FBP method assumes that the point spread function of a back-projection image is isotropic and invariant to incident direction, as required by frequency spectrum analysis. However, most of the time this assumption is not true because the detector geometry is rarely uniform. Therefore the conventional FBP reconstructed image is biased. To solve the geometry non-uniformity problem, this paper proposes an unbiased FBP algorithm that groups Compton events having the same Compton rings using a system matrix decomposition strategy. The proposed method produces more isotropic resolution, and preserves the capability to use frequency spectrum analysis. The algorithm has been applied to data from a 3D position-sensitive detector array with 4 crystals and a digital readout system. The resulting images had more isotropic resolution than standard FBP.