Phase Retrieval Using Two Fourier Intensities

The two-dimensional discrete phase-retrieval problem is concerned with the reconstruction of a signal, or image, x(m, n), from the magnitude (intensity) of its Fourier transform, |X(ω1,ω2)|- Phase retrieval is an important problem that arises in a variety of different applications including x-ray crystallography, astronomy, electron microscopy, optics, and signal processing [1-5]. There are three issues that need to be considered in the solution of the phase retrieval problem: the uniqueness of the solution, the development of phase retrieval algorithms that reconstruct a signal from its Fourier transform intensity and any á priori information that might be available, and the sensitivity of the reconstruction to computational noise and measurements errors. It is now well known that if a two-dimensional signal with finite support has a z-transform that is an irreducible polynomial then the signal is uniquely defined to within a trivial ambiguity by the intensity of its Fourier transform [6]. This result becomes important with the fact that it has been shown that “almost all” discrete two-dimensional signals with finite support have z-transforms that are irreducible [7]. In spite of this uniqueness of the solution, however, the reconstruction of a signal from its Fourier intensity remains a difficult problem.