Polynomial system of equations and its applications to the study of the effect of noise on multidimensional Fourier transform phase retrieval from magnitude

In this paper we deal with the problem of retrieving a finite-extent signal from the magnitude of its Fourier transform. We will present a brief review of the algebraic problem of the uniqueness of the solution for both discrete and continuous phase retrieval models. 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 signal x from the magnitude of the output of a linear distortion: |Hx|(j), j = 1, ..., n . An important result concerning the conditioning of this problem 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, conditioning of the problem and stability of the (essentially) unique solution will be addressed.

[1]  A. Walther.  The Question of Phase Retrieval in Optics , 1963 .

[2]  H. Ferwerda,et al.  The Phase Reconstruction Problem for Wave Amplitudes and Coherence Functions , 1978 .

[3]  R. H. T. Bates,et al.  Composite two-dimensional phase-restoration procedure , 1983 .

[4]  L. G. Sodin,et al.  On the ambiguity of the image reconstruction problem , 1979 .

[5]  J. Fienup.  Fine Resolution Imaging of Space Objects. , 1981 .

[6]  EDWARD M. HOFSTETTER,et al.  Construction of time-limited functions with specified autocorrelation functions , 1964, IEEE Trans. Inf. Theory.

[7]  L. Hörmander.  Linear Partial Differential Operators , 1963 .

[8]  Fernando Cukierman,et al.  Stability of unique Fourier-transform phase reconstruction , 1983 .

[9]  Wayne Lawton.  Uniqueness results for the phase-retrieval problem for radial functions , 1981 .

[10]  L. Ronkin,et al.  Introduction to the Theory of Entire Functions of Several Variables , 1974 .

[11]  Thomas S. Huang,et al.  A note on iterative fourier transform phase reconstruction from magnitude , 1984 .

[12]  A. Oppenheim,et al.  Signal reconstruction from phase or magnitude , 1980 .

[13]  Thomas S. Huang,et al.  Unique reconstruction of a band-limited multidimensional signal from its phase or magnitude , 1983 .

[14]  M. Hayes,et al.  Reducible polynomials in more than one variable , 1982, Proceedings of the IEEE.

[15]  M. Hayes.  The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform , 1982 .