Divide-and-conquer 2-D phase retrieval using subband decomposition and filter banks

The two-dimensional (2-D) discrete phase-retrieval problem is to reconstruct a discrete-time signal whose support is known and compact from the magnitude of its discrete Fourier transform. We show how a subband decomposition of this problem can be performed using filter banks. The result is a series of smaller 2-D phase-retrieval problems whose solutions are the phases of the original problem in different frequency bands. The 2-D phase-retrieval problem is first mapped to a one-dimensional (1-D) phase-retrieval problem, and the subband decomposition is applied to this problem to decompose it into smaller 1-D problems, each of which can be viewed as a smaller 2-D problem. This is more flexible than performing the decomposition directly in 2-D. The filter must have finite support, zero phase, and be approximately halfband. While these results can also perform subband decomposition of any 1-D phase-retrieval problem, the general lack of uniqueness for 1-D phase retrieval creates problems in reassembling the solutions to the smaller 1-D problems