Direct phase retrieval

A direct, noniterative approach to retrieving a multidimensional complex image (i.e., its phase can vary from pixel to pixel) from the magnitude of its Fourier transform is developed. The uniqueness of the reconstruction is shown to be a direct consequence of the existence of zero surfaces or sheets in the multidimensional z transforms of the image. The analytic properties of these zero sheets enable one to distinguish the zero sheet belonging to the image from that belonging to the complex conjugate of its reflection in the coordinate origin. It is thereby possible to recover the unique, most compact "image form" of the original image. Two-dimensional examples are presented.