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 scattering-angle range is limited due to the electrical noise of the detector and due to the finite detector configuration. In this article, a reconstruction algorithm from the projection of a limited scattering-angle range is presented. The algorithm is based on an inversion of the summed back-projection. The variance of the reconstructed image of a uniform source was analyzed and is proportional to the third power of the number of expansion terms. Thus, window functions are introduced to suppress noise at the cost of spatial resolution. The dependency of the spatial resolution on the number of expansion terms was analyzed. The dependency of the variance on the spatial resolution was found to be variance ∝ (spatial resolution)^-13/.