Complexity-guided Fourier phase retrieval from noisy data

Reconstruction of a stable and good quality solution from noisy single-shot Fourier intensity data is a challenging problem for phase retrieval algorithms. We examine behavior of the solution provided by the hybrid input–output (HIO) algorithm for noisy data, from the perspective of the complexity guidance methodology that was introduced by us in an earlier paper [ J. Opt. Soc. Am. A 36 , 202 ( 2019 ) JOAOD6 0740-3232 10.1364/JOSAA.36.000202 ]. We find that for noisy data, the complexity of the solution outside the support keeps increasing as the HIO iterations progress. Based on this observation, a strategy for controlling the solution complexity within and outside the support during the HIO iterations is proposed and tested. In particular, we actively track and control the growth of complexity of the solution outside the support region with iterations. This in turn provides us with guidance regarding the level to which the complexity of the solution within the support region needs to be adjusted, such that the total solution complexity is equal to that estimated from raw Fourier intensity data. In our studies, Poisson noise with mean photon counts per pixel in the Fourier intensity data ranges over four orders of magnitude. We observe that the performance of the proposed strategy is noise robust in the sense that with increasing noise, the quality of the phase solution degrades gradually. For higher noise levels, the solution loses textural details while retaining the main object features. Our numerical experiments show that the proposed strategy can uniformly handle pure phase objects, mixed amplitude-phase objects, and the case of dc blocked Fourier intensity data. The results may find a number of applications where single-shot Fourier phase retrieval is critical to the success of corresponding applications.