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 of this paper presents the experimental aspects of the technique.