Sparse phase retrieval with near minimal measurements: A structured sampling based approach

The problem of sparse phase retrieval is considered where the goal is to recover a sparse complex valued vector (with s non zero elements) from the magnitudes of its linear measurements. Using a modified and partially randomized version of a newly proposed structured sampler, namely the Partial Nested Fourier Sampler (PNFS), it is shown to be possible to recover the unknown signal (up to a global phase ambiguity) from O(s log N) phaseless measurements where N is the dimension of the vector. The reconstruction is based on a novel idea of "decoupling" certain quadratic terms in the phaseless measurements acquired by the PNFS, leading to a simple l1-minimization-based recovery algorithm, without the need for "lifting" the unknown variable to a higher dimensional space. The proposed algorithm is also proved to be stable in presence of bounded noise.

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

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

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

[5]  Holger Rauhut,et al.  An Invitation to Compressive Sensing , 2013 .

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

[7]  Yonina C. Eldar,et al.  STFT Phase Retrieval: Uniqueness Guarantees and Recovery Algorithms , 2015, IEEE Journal of Selected Topics in Signal Processing.

[8]  Andrea J. Goldsmith,et al.  Exact and Stable Covariance Estimation From Quadratic Sampling via Convex Programming , 2013, IEEE Transactions on Information Theory.

[9]  Piya Pal,et al.  Sparse phase retrieval using partial nested fourier samplers , 2015, 2015 IEEE Global Conference on Signal and Information Processing (GlobalSIP).

[10]  Xiaodong Li,et al.  Phase Retrieval via Wirtinger Flow: Theory and Algorithms , 2014, IEEE Transactions on Information Theory.

[11]  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).

[12]  Allen Y. Yang,et al.  CPRL -- An Extension of Compressive Sensing to the Phase Retrieval Problem , 2012, NIPS.

[13]  Holger Boche,et al.  Fast compressive phase retrieval from Fourier measurements , 2015, 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[14]  Zhiqiang Xu,et al.  Phase Retrieval for Sparse Signals , 2013, ArXiv.

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

[16]  Justin K. Romberg,et al.  Efficient Compressive Phase Retrieval with Constrained Sensing Vectors , 2015, NIPS.

[17]  Peter Jung,et al.  Stable recovery from the magnitude of symmetrized fourier measurements , 2014, 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

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

[19]  慧 廣瀬.  A Mathematical Introduction to Compressive Sensing , 2015 .

[20]  Xiaodong Li,et al.  Solving Quadratic Equations via PhaseLift When There Are About as Many Equations as Unknowns , 2012, Found. Comput. Math..

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

[22]  Holger Rauhut,et al.  A Mathematical Introduction to Compressive Sensing , 2013, Applied and Numerical Harmonic Analysis.

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

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

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

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

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