Phase Retrieval: An Overview of Recent Developments

In many physical measurement systems, one can only measure the power spectral density, i.e., the magnitudesquare 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], blind deconvolution [8] and more. Reconstructing a signal from its Fourier magnitude alone is generally a very difficult task. It is well known that Fourier phase is quite often more important than Fourier magnitude in reconstructing a signal from its Fourier transform [9]. To demonstrate this fact, a synthetic example, courtesy of [10], is provided in Figure 1. The figure shows the result of the following numerical simulation: Two images are Fourier transformed, their Fourier phases are swapped and then they are inverse Fourier transformed. The result clearly demonstrates the importance of Fourier phase. Therefore, simply ignoring the phase and performing an inverse Fourier transform does not lead to satisfactory recovery. Instead, algorithmic phase retrieval can be used, offering a means for recovering the phase from the given magnitude measurements and possibly additional prior knowledge, providing an alternative to sophisticated measurement setups as in holography which attempt to directly measure the phase by requiring interference with another known field. To set up the phase retrieval problem mathematically, we focus on the discretized one-dimensional (1D) setting. Let x = (x[0], x[1], . . . , x[N − 1]) be a signal of length N such that it has non-zero values only within the interval [0, N−1]. Denote by y = (y[0], y[1], . . . , y[N−1])T its N point discrete Fourier transform (DFT) and let z = (z[0], z[1], . . . , z[N − 1]) be the Fourier magnitude-square measurements z[m] = ∣∣y[m]∣∣2. Phase retrieval can be mathematically stated as: