The Compton scattering camera (sometimes called the electronically collimated camera) has been shown by others to have the potential to better the photon counting statistics and the energy resolution of the Anger camera for imaging in SPECT. By using coincident detection of Compton scattering events on two detecting planes, a photon can be localized to having been sourced on the surface of a cone. New algorithms are needed to achieve fully three-dimensional reconstruction of the source distribution from such a camera. If a complete set of cone-surface projections are collected over an infinitely extending plane, it is shown that the reconstruction problem is not only analytically solvable, but also overspecified in the absence of measurement uncertainties. Two approaches to direct reconstruction are proposed, both based on the photons which travel perpendicularly between the detector planes. Results of computer simulations are presented which demonstrate the ability of the algorithms to achieve useful reconstructions in the absence of measurement uncertainties (other than those caused by quantization). The modifications likely to be required in the presence of realistic measurement uncertainties are discussed.