Phase Retrieval with Application to Optical Imaging: A contemporary overview

The problem of phase retrieval, i.e., the recovery of a function given the magnitude of its Fourier transform, arises in various fields of science and engineering, including electron microscopy, crystallography, astronomy, and optical imaging. Exploring phase retrieval in optical settings, specifically when the light originates from a laser, is natural since optical detection devices [e.g., charge-coupled device (CCD) cameras, photosensitive films, and the human eye] cannot measure the phase of a light wave. This is because, generally, optical measurement devices that rely on converting photons to electrons (current) do not allow for direct recording of the phase: the electromagnetic field oscillates at rates of ~1015 Hz, which no electronic measurement device can follow. Indeed, optical measurement/detection systems measure the photon flux, which is proportional to the magnitude squared of the field, not the phase. Consequently, measuring the phase of optical waves (electromagnetic fields oscillating at 1015 Hz and higher) involves additional complexity, typically by requiring interference with another known field, in the process of holography.

Historical Background

Algorithmic phase retrieval offers an alternative means for recovering the phase of optical images without requiring sophisticated measuring setups as in holography. These approaches typically rely on some advanced information to facilitate recovery. In 1952, Sayre envisioned, in the context of crystallography, that the phase information of a scattered wave may be recovered if the intensity pattern at and between the Bragg peaks of the diffracted wave is finely measured [1]. In crystallography, the material structure under study is usually periodic (a crystal); hence, the far-field information contains strong peaks reflecting the Fourier transform of the usually periodic information. Measuring the fine features in the Fourier transform enabled the recovery of the phase in some simple cases. In 1978, 26 years later, Fienup developed algorithms for retrieving phases of two-dimensional (2-D) images from their Fourier modulus and constraints such as nonnegativity and a known support of the image [2] (see Figure 1). Fig 1

A numerical 2-D phase-retrieval example adapted from Fienup's 1978 paper [2]: (a) test object, (b) Fourier magnitude, and (c) reconstruction results [using hybrid input-output (HIO)—see Figure 3(b) for details]. (Images used with permission from [2].)