Partitioning algorithms for 1-D and 2-D discrete phase-retrieval problems with disconnected support

The discrete phase-retrieval problem with disconnected support is to reconstruct a discrete-time signal whose support is the union of disjoint intervals from the magnitude of its discrete Fourier transform. We use the Good-Thomas (1958) FFT mapping to transform the one-dimensional (1-D) version of this problem into a two-dimensional (2-D) discrete phase-retrieval problem. We then solve the latter problem by partitioning it into a set of 1-D phase-retrieval problems. The discrete and modulated Radon transforms are used to formulate two coupled 1-D problems, the solution to which specifies solutions to the other 1-D problems. This effectively partitions the original disconnected support 1-D problem into smaller 1-D problems, which may be solved in parallel. Small amounts of noise in the data can be rejected by using more than two coupled 1-D problems, The 2-D phase-retrieval problem with disconnected support is also considered. Numerical examples are provided, including comparison with the hybrid I/O algorithm