Phase Retrieval: An Overview of Recent Developments

Introduction In many physical measurement systems, one can only measure the power spectral density, that is, the magnitude square of the Fourier transform of the underlying signal. For example, in an optical setting, detection devices like CCD cameras and photosensitive films cannot measure the phase of a light wave and instead measure the photon flux. In addition, at a large enough distance from the imaging plane the field is given by the Fourier transform of the image (up to a known phase factor). Thus, in the far field, optical devices essentially measure the Fourier transform magnitude. Since the phase encodes a lot of the structural content of the image, important information is lost. The problem of reconstructing a signal from its Fourier magnitude is known as phase retrieval [1,2]. This reconstruction problem is one with a rich history and arises in many areas of engineering and applied physics, including optics [3], x-ray crystallography [4], astronomical imaging [5], speech processing [6], computational biology [7], and blind deconvolution [8].