329 research outputs found
Non-Convex Phase Retrieval from STFT Measurements
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
STFT Phase Retrieval: Uniqueness Guarantees and Recovery Algorithms
The problem of recovering a signal from its Fourier magnitude is of paramount importance in various fields of engineering and applied physics. Due to the absence of Fourier phase information, some form of additional information is required in order to be able to uniquely, efficiently, and robustly identify the underlying signal. Inspired by practical methods in optical imaging, we consider the problem of signal reconstruction from the short-time Fourier transform (STFT) magnitude. We first develop conditions under, which the STFT magnitude is an almost surely unique signal representation. We then consider a semidefinite relaxation-based algorithm (STliFT) and provide recovery guarantees. Numerical simulations complement our theoretical analysis and provide directions for future work
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