On the stability of the phase problem

Abstract

Phase retrieval of a signal given its intensity is considered as a problem of statistically estimating a set of unknown parameters, the Zernike coefficients. Specifically, the phase problem is presented in the context of classical wave optics in the Fresnel approximation. Investigating the stability in this case suggests first learning if the Zernike coefficients can be restored in principle. If this is indeed the case, it then suggests determining the accuracy of their estimation. The stability of a solution to the phase problem depends, as it does for the other inverse problems, on the spectrum of the Fisher information matrix. An explicit representation of the Fisher matrix is given, and its spectrum is calculated for in-focus and out-of-focus images of a pointlike source. Simulations show that the solutions in the latter case are generally stable, so the coefficients of the Zernike series can be determined with an acceptable accuracy. The principal components, the mutually independent combinations of aberrations that are a generalization of the coefficients of the well-known Karhunen-Loeve decomposition, are calculated. As an example of this approach, the maximum-likelihood method is used to determine the aberrations of the optical system.