Sparse Phase Retrieval from Short-Time Fourier Measurements

We consider the classical 1D phase retrieval problem. In order to overcome the difficulties associated with phase retrieval from measurements of the Fourier magnitude, we treat recovery from the magnitude of the short-time Fourier transform (STFT). We first show that the redundancy offered by the STFT enables unique recovery for arbitrary nonvanishing inputs, under mild conditions. An efficient algorithm for recovery of a sparse input from the STFT magnitude is then suggested, based on an adaptation of the recently proposed GESPAR algorithm. We demonstrate through simulations that using the STFT leads to improved performance over recovery from the oversampled Fourier magnitude with the same number of measurements.

[1]  A. Walther.  The Question of Phase Retrieval in Optics , 1963 .

[2]  EDWARD M. HOFSTETTER,et al.  Construction of time-limited functions with specified autocorrelation functions , 1964, IEEE Trans. Inf. Theory.

[3]  R. Gerchberg.  A practical algorithm for the determination of phase from image and diffraction plane pictures , 1972 .

[4]  J R Fienup,et al.  Phase retrieval algorithms: a comparison. , 1982, Applied optics.

[5]  Jae S. Lim,et al.  Signal reconstruction from the short-time Fourier transform magnitude , 1982, ICASSP.

[6]  Jae S. Lim,et al.  Signal estimation from modified short-time Fourier transform , 1983, ICASSP.

[7]  Jae Lim,et al.  Signal estimation from modified short-time Fourier transform , 1984 .

[8]  Robert W. Harrison,et al.  Phase problem in crystallography , 1993 .

[9]  R. Trebino.  Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses , 2000 .

[10]  øöö Blockinø.  Phase retrieval, error reduction algorithm, and Fienup variants: A view from convex optimization , 2002 .

[11]  J. Fienup,et al.  Phase retrieval with transverse translation diversity: a nonlinear optimization approach. , 2008, Optics express.

[12]  D. Kane.  Principal components generalized projections: a review [Invited] , 2008 .

[13]  Lei Tian,et al.  Compressive Phase Retrieval , 2011 .

[14]  T. Blumensath,et al.  Theory and Applications , 2011 .

[15]  Yonina C. Eldar,et al.  Sparsity Based Sub-wavelength Imaging with Partially Incoherent Light via Quadratic Compressed Sensing References and Links , 2022 .

[16]  Julius O. Smith,et al.  Estimating a Signal from a Magnitude Spectrogram via Convex Optimization , 2012, 1209.2076.

[17]  S. Sastry,et al.  Compressive Phase Retrieval From Squared Output Measurements Via Semidefinite Programming , 2011, 1111.6323.

[18]  Chandra Sekhar Seelamantula,et al.  An iterative algorithm for phase retrieval with sparsity constraints: application to frequency domain optical coherence tomography , 2012, 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[19]  Babak Hassibi,et al.  Recovery of sparse 1-D signals from the magnitudes of their Fourier transform , 2012, 2012 IEEE International Symposium on Information Theory Proceedings.

[20]  Yonina C. Eldar,et al.  Sparsity Constrained Nonlinear Optimization: Optimality Conditions and Algorithms , 2012, SIAM J. Optim..

[21]  Martin Vetterli,et al.  Phase Retrieval for Sparse Signals: Uniqueness Conditions , 2013, ArXiv.

[22]  Yonina C. Eldar,et al.  On conditions for uniqueness in sparse phase retrieval , 2013, 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[23]  Yonina C. Eldar,et al.  Phase Retrieval with Application to Optical Imaging , 2014, ArXiv.

[24]  Yonina C. Eldar,et al.  GESPAR: Efficient Phase Retrieval of Sparse Signals , 2013, IEEE Transactions on Signal Processing.

[25]  Alexandre d'Aspremont,et al.  Phase recovery, MaxCut and complex semidefinite programming , 2012, Math. Program..

[26]  Yonina C. Eldar,et al.  Phase Retrieval via Matrix Completion , 2011, SIAM Rev..