Study on the convergence property of the hybrid input–output algorithm used for phase retrieval

Even though the hybrid input–output algorithm (HIO) has been recognized empirically to be one of the most successful versions of the iterative Fourier transform algorithm used for phase retrieval, its behavior is not yet well understood. Therefore a theoretical investigation on the convergence property of the HIO with an infinitesimally small feedback parameter is presented, although on a rather intuitive level, and it is shown that, until a solution is found, this algorithm continues to travel among the objects seeking those that satisfy the Fourier-domain constraint and for which the object-domain error has a locally minimum value. The concept of the territory is introduced with use of the algorithm constructed by modifying the HIO, and then the results are presented of the computer simulations for 2×2 objects with L-shaped support that were carried out to test the validity of our theory and to gain insight into the case in which the value of the feedback parameter is finite.