Energy-spectral Compton scatter imaging .1. Theory and mathematics

A technique for tomographic imaging of electron density using the energy spectrum of Compton scattered gamma-rays is described. The energy-angle relationship for Compton scattering is utilized to determine the direction of scattered photons, A single-scattering ''forward mapping'' model is constructed to relate electron density to the detector response, The nonlinearity of the model, caused by accounting for pre- and post-scattering radiation attenuation, is dealt with by solving the ''inverse mapping'' iteratively, In order to assure convergence of the image reconstruction problem to a nonnegative and bounded solution, a variety of ''regularization'' algorithms are examined. The capabilities and limitations of these algorithms are demonstrated by reconstructing a number of images from Monte Carlo simulated measurements. Part II [1] of this paper presents the experimental aspects of the technique.