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