Phase retrieval by iterated projections

Several strategies in phase retrieval are unified by an iterative "difference map" constructed from a pair of elementary projections and three real parameters. For the standard application in optics, where the two projections implement Fourier modulus and object support constraints, respectively, the difference map reproduces the "hybrid" form of Fienup's input-output map when a particular choice is made for two of the parameters. The geometric construction of the difference map illuminates the distinction between its fixed points and the recovered object, as well as the mechanism whereby the form of stagnation encountered by alternating projection schemes is avoided. When support constraints are replaced by object histogram or atomicity constraints, the difference map lends itself to crystallographic phase retrieval. Numerical experiments with synthetic data suggest that structures with hundreds of atoms can be solved.