MATHEMATICAL CONSIDERATIONS FOR THE PROBLEM OF FOURIER-TRANSFORM PHASE RETRIEVAL FROM MAGNITUDE

The author deals 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. He presents 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. He then considers 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: vertical Hx vertical (j), j equals 1,. . . , n. A number of important results are 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 (essentially) unique solution, differences between the one-dimensional and the multi-dimensional cases, and conditioning of the problem are addressed.