Toward fast and provably accurate near-field ptychographic phase retrieval

Ptychography is an imaging technique that involves a sample being illuminated by a coherent, localized probe of illumination. When the probe interacts with the sample, the light is diffracted and a diffraction pattern is detected. Then the sample (or probe) is shifted laterally in space to illuminate a new area of the sample whilst ensuring sufficient overlap. Similarly, in Fourier ptychography a sample is illuminated at different angles of incidence (effectively shifting the sample's Fourier transform) after which a lens acts as a low-pass filter, thereby effectively providing localized Fourier information about the sample around frequencies dictated by each angle of illumination. Mathematically, one therefore obtains a similar set of overlapping measurements of the sample in both Fourier ptychography and ptychography, except in the different domains (Fourier for the former, and physical for the latter). In either case, one is then able to reconstruct an image of the sample from the measurements using similar methods. Near-Field (Fourier) Ptychography (NFP) (see, e.g., Stockmar et al. (Sci Rep 3(1):1–6, 2013), Stockmar et al. (Phys Rev Appl 3(1): 014005, 2015) and Zhang et al. (Optics Exp 27(5): 7498–7512, 2019)) occurs when the sample is placed at a short defocus distance having a large Fresnel number. In this paper, we prove that certain NFP measurements are robustly invertible (up to an unavoidable global phase ambiguity) for specific Point Spread Functions (PSFs) and physical masks which lead to well-conditioned lifted linear systems. We then apply a block phase retrieval algorithm using weighted angular synchronization and prove that the proposed approach accurately recovers the measured sample for these specific PSF and mask pairs. Finally, we also propose using a Wirtinger Flow for NFP problems and numerically evaluate that alternate approach both against our main proposed approach, as well as with NFP measurements for which our main approach does not apply.