Non-Convex Phase Retrieval From STFT Measurements

Abstract

Abstract-The problem of recovering a one-dimensional signal from its Fourier transform magnitude, called Fourier phase retrieval, is ill-posed in most cases. We consider the closely-related problem of recovering a signal from its phaseless short-time Fourier transform (STFT) measurements. This problem arises naturally in several applications, such as ultra-short laser pulse characterization and ptychography. The redundancy offered by the STFT enables unique recovery under mild conditions. We show that in some cases the unique solution can be obtained by the principal eigenvector of a matrix, constructed as the solution of a simple least-squares problem. When these conditions are not met, we suggest using the principal eigenvector of this matrix to initialize non-convex local optimization algorithms and propose two such methods. The first is based on minimizing the empirical risk loss function, while the second maximizes a quadratic function on the manifold of phases. We prove that under appropriate conditions, the proposed initialization is close to the underlying signal. We then analyze the geometry of the empirical risk loss function and show numerically that both gradient algorithms converge to the underlying signal even with small redundancy in the measurements. In addition, the algorithms are robust to noise.

Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview

“…• relax the stringent assumptions on the statistical models underlying the data; for example, a few recent works studied nonconvex phase retrieval under more physically-meaningful measurement models [201,202];…”

Gauss–Newton Optimization for Phase Recovery From the Bispectrum

“…More recently, the problem has been of interest in multiple applications including longrange horizontal and slant path imaging [9], [10], [11] and phase recovery in underwater imaging [12], [13]. The general problem of phase retrieval from Fourier measurements also continues to be an active area of research in the signal processing community [14], [15], [16], [17], [18], [19], [20], [21], [22]. Many phase retrieval applications considered in the literature are based on the relationship between an object's phase and higher order statistical moments collected from the data (such as the bispectrum) and require solving constrained nonlinear least-squares problems; see, e.g., [16], [21], [22].…”

“…The general problem of phase retrieval from Fourier measurements also continues to be an active area of research in the signal processing community [14], [15], [16], [17], [18], [19], [20], [21], [22]. Many phase retrieval applications considered in the literature are based on the relationship between an object's phase and higher order statistical moments collected from the data (such as the bispectrum) and require solving constrained nonlinear least-squares problems; see, e.g., [16], [21], [22]. Thus while we consider only the problem of phase recovery from the bispectrum, the content of this paper is more broadly applicable to the phase retrieval problem in general.…”

Deep Ptych: Subsampled Fourier Ptychography Using Generative Priors

“…Generally, phase retrieval algorithms fall into two categories. The first approach is to introduce redundancy into the measurement system, where more measurements are taken than the dimension of true signal usually in the form of oversampled Fourier transform [6], short-time Fourier transform [7], structured illuminations [8], etc. The second approach is to exploit the structural assumptions on the true signal as a prior, such as sparsity [9] or non-negativity [10].…”