Reconstruction of continuous object distributions from Fourier magnitude

The phase retrieval from Fourier magnitude problem has traditionally been considered from the viewpoint of the analytic properties of Fourier transforms of finite support object distributions, [1]. It is well known that intrinsic phase ambiguities arise if the product representation of the Fourier transform has one or more non-self conjugate factors. In one dimension there can be an infinity of such factors but in two or more the exact number is difficult to ascertain. Recently there has been a trend towards considering object distributions comprising a set of discrete points,[2]. For several applications, e.g. astronomy, this is not unreasonable. This model has the great advantage that one can adopt a discrete Fourier transform representation for the Fourier data or, equivalently, a z-transform representation leading to a finite degree polynomial model. Such a model can be generally assumed to lead to a unique relationship between Fourier magnitude and phase [3] and has given confidence to the use of iterative methods which had been observed to succeed in the recovering Fourier phase from magnitude or vice-versa,[4].