Closed-form phase retrieval of images from the zeros of 1-D signals obtained using the Good-Thomas FFT

The 2-D discrete phase retrieval problem is to reconstruct an image defined at integer coordinates and having known finite spatial extent from the magnitude of its discrete Fourier transform. Most methods for solving this problem are iterative but not POCS, and they tend to stagnate. We have shown that a subband decomposition can be used to divide a large phase retrieval problem into smaller problems defined over subbands. However, these numerous smaller phase retrieval problems must be solved efficiently. This paper uses the Good-Thomas FFT to map the 2-D problem into a 1-D problem of reconstructing the signal formed by concatenating rows of the image with zero bands. The z-transform zeros of this 1-D signal are close enough to the unit circle to be found from local minima of the (known) DFT magnitude. The 1-D signal can then be reconstructed in closed form