Abstract

The problem of reconstructing an object function from the modulus of its Fourier transform arises in many disciplines. Several algorithms have been proposed to solve this problem. One of the most successful of those algorithms is the iterative transform algorithm, but is has a disadvantage in that it stagnates before it reaches a solution in some cases. Here, we propose a multigrid adaptation of this algorithm to speed up convergence. We also propose a second algorithm that attempts to solve the problem entirely in the space domain by using the support constraint and the nonlinear correlation equations. This algorithm is based on an iterative global linearization of the correlation equations. For both algorithms, several cycles of the basic iterative procedure are performed until stagnation is detected or until the number of cycles reaches a predetermined upper limit. At this point, the error and the current object-function estimate are transferred to a coarser grid, and the same basic iterative algorithm is then used to compute a grid correction to the last estimate that was computed on the fine grid. We show that a multigrid generalization of this two-grid procedure greatly improves the convergence of both algorithms.

© 1990 Optical Society of America