Algorithms for the Phase Retrieval Problem: an automatic Way to Overcome Stagnation

This work is concerned with the Phase Retrieval (PR) problem [3]. In its general formulation, the PR problem consists of retrieving the Fourier phase for a signal f whose Fourier transform F is known only in magnitude. In the applications, some additional information on f is available, e.g. the magnitude |f|. We shall be interested in the 2-D case where the additional information is the nonnegativity of f and the autocorrelation support. Iterative algorithms are commonly used to solve this problem [4]. Whereas these algorithms are the most successful in the applications, they do not enjoy ensured convergence, and often do not converge to a solution. Quite often they wind up either oscillating between two non-solutions, or converging very slowly towards a non- solution. In [2] a geometric characterization of the solution set is presented as well as experimental results suggesting an automatic way to avoid stagnation in the context of the retrieval algorithms mentioned above. For concreteness, let us consider the Error Reduction (ER) method. We observe that oscillations for the ER always occur between one fixed toroid [2] and the nonnegative orthant. The point x′ of stagnation on the toroid is external to the (convex) orthant. It is reasonable, as a means to avoid stagnation, to move from x′ to the point x” on the toroid which is “antipodal” to x′, thereby (roughly speaking) overrelaxing the operation of (convex) projection onto the orthant. More precisely, let c be the center of the toroid in question. It has been observed before [2] that c is in the positive orthant. We set x” = 2c − x′. We then use x” as an initial value for the ER algorithm. To summarize, we propose the following method : suppose that the algorithm stagnated in an image g k , that has a projection g k ′ onto the toroids set; then we get a new initial estimate for the algorithm by setting g o =−g k ′+2c with c representing the center of the toroid. This new starting point automatically satisfies the Fourier magnitude constraints, but not necessarily the positivity constraint, unless it happens to be a solution. In principle, the new initial point may lead to a renewed stagnation. However, in practice the phase retrieval improves considerably from one stagnation to another, leading to a good approximate solution. We have performed numerous experiments, using the ER algorithm with this stagnation breaker, and in all the cases it was not necessary to repeat the stagnation breaker more than twice per example. One advantage of the suggested method is that it is automatic, i.e. there is no need to identify on-line the stagnation type. A typical experiment using our new method is shown below. Two overrelaxations were necessary in this case, in order to obtain an acceptable approximation to the solution. The starting image that begins the process was taken random. Figure 1 is the image to be retrieved, Figure 2, the result after 100 iterations of ER.