Mitigating the Effect of Noise in Iterative Projection Phase Retrieval

Since the renowned papers [1, 2] by Fienup, many methods of iterative phase retrieval have been devised that address the problem where a two dimensional image is the object of interest and only the magnitude of its Fourier transform is known. Thus the phase of its Fourier transform needs to be estimated to recover the image. Building upon Gerchberg and Saxton’s Error Reduction method [3], Fienup’s Hybrid Input-Output (HIO) method serves as the basis for comparison of most other methods. It projects the estimate of the image from the image domain to the Fourier domain repeatedly and imposes various constraints upon each projection in an attempt to find the domains’ intersection. While the only viable method, the projective approach has a major caveat: it is susceptible to stagnation. Explicitly, the error metric contains local minima which cause the minimization process to halt before reaching the global minimum. Much research after the development of the HIO method has focused on modifying the constraints imposed with the goal of finding the global minimum. Not much attention, however, has been placed on filtering noise in the modulus data. Directly addressing the issue of noise in the Fourier modulus data introduces a change to the problem statement which has a severe effect, as will be shown later. Where the ideal phase retrieval problem statement is to find the intersection of the Fourier and image domains via successive projections, noise in the modulus data may cause the two domains not to intersect. The problem statement must be reformulated such that the minimum distance between the domains is sought, not the intersection. The examples shown here reveal that not having an intersection can cause many phase retrieval methods to diverge. This discussion expands on the work in [4] and includes applications to phase retrieval methods beyond the HIO. The formal problem statement here is to devise a projective phase retrieval algorithm capable of filtering noise from the Fourier modulus data and converging to the true image. This expands on some previous attention to noise in the modulus data [5, 6] in that not only is the algorithm required not to diverge, the noise must be filtered. Some papers have already approached this problem from the viewpoint of mitigating divergence at high noise levels; however, they did not have the end goal of filtering the noise. Comparisons will be presented to these methods. A useful compilation of some of the best phase retrieval methods was published by Marchesini [7]. He performed a side-by-side comparison of some phase retrieval methods in a simple two degree-of-freedom example which will be used extensively in the work here. The example problem consists of finding the intersection between two domains in two dimensional space. This example allows for a visualization of the projections and constraints at each iteration.