Phase retrieval: an overview of recent developments (2016)

Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264Classic Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
Recent Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266Sparse Phase Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273Phase Retrieval Using Masks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
Uniqueness and Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275SDP Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
Wirtinger Flow Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
Combinatorial Methods (for the Noiseless Setting) . . . . . . . . . . . . . . . . 278Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
STFT Phase Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
Nonvanishing Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
Sparse Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292Introduction
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].