The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform

This paper addresses two fundamental issues involved in the reconstruction of a multidimensional sequence from either the phase or magnitude of its Fourier transform. The first issue relates to the uniqueness of a multidimensional sequence in terms of its phase or magnitude. Although phase or magnitude information alone is not sufficient, in general, to uniquely specify a sequence, a large class of sequences are shown to be recoverable from their phase or magnitude. The second issue which is addressed in this paper concerns the actual reconstruction of a multidimensional sequence from its phase or magnitude. For those sequences which are uniquely specified by their phase, several practical algorithms are described which may be used to reconstruct a sequence from its phase. Several examples of phase-only reconstruction are also presented. Unfortunately, however, even for those sequences which are uniquely defined by their magnitude, it appears that a practical algorithm is yet to be developed for reconstructing a sequence from only its magnitude. Nevertheless, an iterative procedure which has been proposed is briefly discussed and evaluated.