The importance of boundary conditions in the phase retrieval problem

In this paper, the importance of the boundary values of a two dimensional sequence in the phase retrieval problem is investigated. Specifically, it is shown that, given the boundary values, phase retrieval becomes a linear problem. Therefore, the interior points of the two-dimensional sequence may easily be deduced from the boundary values of the sequence and the magnitude of its Fourier transform or, equivalently, the autocorrelation function of the sequence. Furthermore, although the determination of the boundary values from only Fourier transform magnitude information is, in general, a non-trivial problem, it is shown that the shape of the region of support of the two-dimensional sequence determines the ease with which the boundary values may be determined.