An algorithm is described for incorporating symmetry information into reconstruction of an image from the amplitude of its Fourier transform. The symmetry is used to compensate for the loss of information due to sampling of the Fourier amplitude below the Nyquist density. The study is motivated by an image reconstruction problem in x-ray crystallography. Application of the algorithm to a simulated crystallographic problem shows that it converges to the correct solution, with no initial phase information, where algorithms currently used in crystallography fail. The results lend support to the possibility of ab initio phase retrieval in macromolecular crystallography when sufficient ta priori information is available.