Two-Dimensional Phase Restoration

The constraints laid on the phase of a Fourier transform by its intensity are reviewed in the contexts of well known phase problems. The considerable differences between phase problems involving one-dimensional and multi-dimensional images, and finite-sized (as arise in astronomy, for instance) and periodic (as occur in crystallography) images, are explained. The crucial importance, for uniqueness questions, of the concept of the image-form (and also its most compact manifestation) is emphasised, as is the almost always unique connection between the image-form of a positive multi-dimensional image and the intensity of its Fourier transform. The current status of phase recovery algorithms, as regards Fourier transforms of finite-sized images, is assessed. The necessity for composite algorithms, incorporating simple but powerful constructions, is pleaded and reinforced by computational examples illustrating our previously reported defogging routine and a new procedure called fringe magnification.