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.

[1]  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.

[2]  Babak Hassibi,et al.  Reconstruction of integers from pairwise distances , 2012, 2013 IEEE International Conference on Acoustics, Speech and Signal Processing.

[3]  Sundeep Rangan,et al.  Compressive phase retrieval via generalized approximate message passing , 2012, Allerton Conference.

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

[5]  Dustin G. Mixon,et al.  Phase retrieval from power spectra of masked signals , 2013, ArXiv.

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

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

[8]  T. Kailath,et al.  Indefinite-quadratic estimation and control: a unified approach to H 2 and H ∞ theories , 1999 .

[9]  Mark Stefik,et al.  Inferring DNA Structures from Segmentation Data , 1978, Artif. Intell..

[10]  Rick P. Millane,et al.  Phase retrieval in crystallography and optics , 1990 .

[11]  Yonina C. Eldar,et al.  Simultaneously Structured Models With Application to Sparse and Low-Rank Matrices , 2012, IEEE Transactions on Information Theory.

[12]  O. Bunk,et al.  Coherent diffractive imaging using phase front modifications. , 2008, Physical review letters.

[13]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[14]  Yonina C. Eldar,et al.  Phase retrieval with masks using convex optimization , 2015, 2015 IEEE International Symposium on Information Theory (ISIT).

[15]  Biing-Hwang Juang,et al.  Fundamentals of speech recognition , 1993, Prentice Hall signal processing series.

[16]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[17]  M.G. Bellanger,et al.  Digital processing of speech signals , 1980, Proceedings of the IEEE.

[18]  Xiaodong Li,et al.  Sparse Signal Recovery from Quadratic Measurements via Convex Programming , 2012, SIAM J. Math. Anal..

[19]  Yonina C. Eldar,et al.  Recovering signals from the Short-Time Fourier Transform magnitude , 2015, 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

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

[21]  Emmanuel J. Candès,et al.  Decoding by linear programming , 2005, IEEE Transactions on Information Theory.

[22]  Xiaodong Li,et al.  Phase Retrieval from Coded Diffraction Patterns , 2013, 1310.3240.

[23]  Justin K. Romberg,et al.  Blind Deconvolution Using Convex Programming , 2012, IEEE Transactions on Information Theory.

[24]  R. Horstmeyer,et al.  Wide-field, high-resolution Fourier ptychographic microscopy , 2013, Nature Photonics.

[25]  Yonina C. Eldar,et al.  Phase Retrieval: Stability and Recovery Guarantees , 2012, ArXiv.

[26]  Roman Vershynin,et al.  Introduction to the non-asymptotic analysis of random matrices , 2010, Compressed Sensing.

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

[28]  Yonina C. Eldar,et al.  Sparse Phase Retrieval from Short-Time Fourier Measurements , 2014, IEEE Signal Processing Letters.

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

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

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

[32]  Buyurmaiz Baykal,et al.  Blind channel estimation via combining autocorrelation and blind phase estimation , 2004, IEEE Transactions on Circuits and Systems I: Regular Papers.

[33]  Arvind Gupta,et al.  On the turnpike problem , 2000 .

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

[35]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[36]  Weiyu Xu,et al.  Weighted ℓ1 minimization for sparse recovery with prior information , 2009, 2009 IEEE International Symposium on Information Theory.

[37]  M. Hayes,et al.  Reducible polynomials in more than one variable , 1982, Proceedings of the IEEE.

[38]  M. Hayes.  The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform , 1982 .

[39]  Stephen P. Boyd,et al.  Log-det heuristic for matrix rank minimization with applications to Hankel and Euclidean distance matrices , 2003, Proceedings of the 2003 American Control Conference, 2003..

[40]  Ali H. Sayed,et al.  A survey of spectral factorization methods , 2001, Numer. Linear Algebra Appl..

[41]  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).

[42]  Yonina C. Eldar,et al.  Phase Retrieval: An Overview of Recent Developments , 2015, ArXiv.

[43]  David P. Williamson,et al.  Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming , 1995, JACM.

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

[45]  Babak Hassibi,et al.  On robust phase retrieval for sparse signals , 2012, 2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[46]  Emmanuel J. Candès,et al.  PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming , 2011, ArXiv.

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

[48]  Albert Fannjiang,et al.  Absolute uniqueness of phase retrieval with random illumination , 2011, ArXiv.

[49]  Peter G. Casazza,et al.  Phase retrieval , 2015, SPIE Optical Engineering + Applications.

[50]  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).

[51]  E. Loewen,et al.  Diffraction Gratings and Applications , 2018 .

[52]  J. Franklin,et al.  Ambiguities in the X‐ray analysis of crystal structures , 1974 .

[53]  Steven Skiena,et al.  Reconstructing sets from interpoint distances (extended abstract) , 1990, SCG '90.

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

[55]  J. Rodenburg.  Ptychography and Related Diffractive Imaging Methods , 2008 .

[56]  J. Rodenburg,et al.  Ptychographic electron microscopy using high-angle dark-field scattering for sub-nanometre resolution imaging , 2012, Nature Communications.

[57]  Babak Hassibi,et al.  Sparse Phase Retrieval: Uniqueness Guarantees and Recovery Algorithms , 2013, IEEE Transactions on Signal Processing.

[58]  Yonina C. Eldar,et al.  Phase Retrieval with Application to Optical Imaging: A contemporary overview , 2015, IEEE Signal Processing Magazine.

[59]  Prateek Jain,et al.  Phase Retrieval Using Alternating Minimization , 2013, IEEE Transactions on Signal Processing.

[60]  P. Seymour,et al.  The Structure of Homometric Sets , 1982 .

[61]  Anton van den Hengel,et al.  Semidefinite Programming , 2014, Computer Vision, A Reference Guide.

[62]  James R. Fienup,et al.  Reconstruction of the support of an object from the support of its autocorrelation , 1982 .

[63]  Martin Vetterli,et al.  Sparse spectral factorization: Unicity and reconstruction algorithms , 2011, 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

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

[65]  Pablo A. Parrilo,et al.  Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization , 2007, SIAM Rev..

[66]  Stephen P. Boyd,et al.  Enhancing Sparsity by Reweighted ℓ1 Minimization , 2007, 0711.1612.

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

[68]  K. Nugent,et al.  Phase retrieval in x-ray imaging based on using structured illumination , 2008 .

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

[70]  R. Balan,et al.  On signal reconstruction without phase , 2006 .

[71]  Michel Verhaegen,et al.  Quadratic Basis Pursuit , 2013, 1301.7002.

[72]  A.V. Oppenheim,et al.  Enhancement and bandwidth compression of noisy speech , 1979, Proceedings of the IEEE.

[73]  A. L. Patterson.  A Direct Method for the Determination of the Components of Interatomic Distances in Crystals , 1935 .

[74]  Emmanuel J. Candès,et al.  Exact Matrix Completion via Convex Optimization , 2008, Found. Comput. Math..

[75]  Babak Hassibi,et al.  Sparse phase retrieval: Convex algorithms and limitations , 2013, 2013 IEEE International Symposium on Information Theory.

[76]  A. L. Patterson.  A Fourier Series Method for the Determination of the Components of Interatomic Distances in Crystals , 1934 .

[77]  Felix Krahmer,et al.  Improved Recovery Guarantees for Phase Retrieval from Coded Diffraction Patterns , 2014, arXiv.org.

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

[79]  Yonina C. Eldar,et al.  Sparsity-based single-shot sub-wavelength coherent diffractive imaging , 2011, 2012 IEEE 27th Convention of Electrical and Electronics Engineers in Israel.