Reconstruction of cone-beam projections from Compton scattered data

The problem of reconstructing a 3D source distribution from Compton scattered data can be separated into two tasks. First, the angular distribution of line projections at different observation points within the detector volume are reconstructed. Then, reconstruction techniques are applied to the resulting cone-beam projections to synthesize the 3D source distribution. This paper describes an analytic solution for the first, yet unsolved, task. Building on the convolution theorem in spherical coordinates, a backprojection and inverse filtering technique in terms of spherical harmonics is formulated. The rotation invariance of the point response of the backprojection in spherical coordinates is proved; and the corresponding inverse filter function is derived. The resulting filtered backprojection algorithm then consists of a summation over all detected events of fixed and known event response functions. Measurement errors, which for Compton scatter detectors are typically different for each detected event, can easily be accounted for in the proposed algorithm. The computational cost of the algorithm is O(NT²), where N is the number of detected events and π/T is the desired angular resolution