TY - JOUR
T1 - Non-convex phase retrieval from STFT measurements
AU - Bendory, Tamir
AU - Eldar, Yonina C.
AU - Boumal, Nicolas
N1 - Publisher Copyright:
© 2017 IEEE.
PY - 2018/1
Y1 - 2018/1
N2 - 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.
AB - 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.
KW - Least-squares
KW - Non-convex optimization
KW - Optimization on manifolds
KW - Phase retrieval
KW - Ptychography
KW - Short-time Fourier transform
KW - Spectral initialization
KW - Ultra-short laser pulse characterization
UR - http://www.scopus.com/inward/record.url?scp=85028532255&partnerID=8YFLogxK
U2 - 10.1109/TIT.2017.2745623
DO - 10.1109/TIT.2017.2745623
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AN - SCOPUS:85028532255
SN - 0018-9448
VL - 64
SP - 467
EP - 484
JO - IEEE Transactions on Information Theory
JF - IEEE Transactions on Information Theory
IS - 1
ER -