Mathematical Considerations for the Problem of Fourier Transform Phase Retrieval from Magnitude

In this paper, we deal with the problem of retrieving a finite-extent function from the magnitude of its Fourier transform. This so-called phase retrieval problem will first be posed under its different underlying models. We will present a brief review of the main results known in this area for both discrete and continuous phase retrieval models giving special emphasis to the algebraic problem of the uniqueness of the solution. Several important issues which are yet unresolved will be pointed out and discussed. We will then consider the discrete phase retrieval problem as a special case of a more general problem which consists of recovering a real-valued sequence x from the magnitude of the output of a linear distortion: $| Hx | ( j ),\, j = 1, \cdots ,n$. A number of important results will be obtained for this general setting by means of algebraic-geometric techniques. In particular, the problems of the existence of a solution for phase retrieval, number of feasible solutions, stability of the (essential...