Analytical image reconstruction of cone-beam projections from limited-angle Compton camera data

Since the integration of gamma-rays on a cone is measured with Compton cameras, some sort of image reconstruction method is necessary. Parra (2000) developed an analytical reconstruction algorithm based on a spherical harmonics expansion of projection data that covers the entire scattering-angle range. The measurable scattering angle range is limited due to the electrical noise of the detector and to the finite detector configuration. In this article, a reconstruction algorithm from the projection of a limited scattering angle range is presented. The algorithm was based on an inversion of the summed backprojection. The variance of the reconstructed image of a uniform source was calculated and found proportional to the third power of the number of expansion terms. Thus, window functions were introduced to suppress noise at the cost of spatial resolution. The dependency of the spatial resolution on the number of expansion terms was analyzed. From this, the dependency of the variance on the spatial resolution was found to be variance∝(spatial resolution)^-3.