Phase retrieval from low rate samples

The paper considers the phase retrieval problem in N-dimensional complex vector spaces. It provides two sets of deterministic measurement vectors which guarantee signal recovery for all signals, excluding only a specific subspace and a union of subspaces, respectively. A stable analytic reconstruction procedure of low complexity is given. Additionally it is proven that signal recovery from these measurements can be solved exactly via a semidefinite program. A practical implementation with 4 deterministic diffraction patterns is provided and some numerical experiments with noisy measurements complement the analytic approach.

[1]  M. Fiddy,et al.  The phase problem in scattering phenomena: the zeros of entire functions and their significance , 1978, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[2]  A. Oppenheim,et al.  Signal reconstruction from phase or magnitude , 1980 .

[3]  R. Young,et al.  An introduction to nonharmonic Fourier series , 1980 .

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

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

[6]  J. H. Seldin,et al.  Hubble Space Telescope characterized by using phase-retrieval algorithms. , 1993, Applied optics.

[7]  Kim-Chuan Toh,et al.  SDPT3 -- A Matlab Software Package for Semidefinite Programming , 1996 .

[8]  Yurii Lyubarskii,et al.  Lectures on entire functions , 1996 .

[9]  Vladimir I. Levenshtein,et al.  On designs in compact metric spaces and a universal bound on their size , 1998, Discret. Math..

[10]  O. Christensen.  An introduction to frames and Riesz bases , 2002 .

[11]  J. Finkelstein.  Pure-state informationally complete and "really" complete measurements (3 pages) , 2004 .

[12]  C. Caves,et al.  Minimal Informationally Complete Measurements for Pure States , 2004, quant-ph/0404137.

[13]  Robert M. Gray,et al.  Toeplitz and Circulant Matrices: A Review , 2005, Found. Trends Commun. Inf. Theory.

[14]  Q. Shen,et al.  Wave propagation and phase retrieval in Fresnel diffraction by a distorted-object approach , 2005 .

[15]  Robert M. Gray,et al.  Toeplitz And Circulant Matrices: A Review (Foundations and Trends(R) in Communications and Information Theory) , 2006 .

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

[17]  G. Pedrini,et al.  Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation , 2007 .

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

[19]  R. Balan,et al.  Painless Reconstruction from Magnitudes of Frame Coefficients , 2009 .

[20]  C. Falldorf,et al.  Phase retrieval by means of a spatial light modulator in the Fourier domain of an imaging system. , 2010, Applied optics.

[21]  P. Jaming,et al.  The phase retrieval problem for the Radar Ambiguity Function and vice versa , 2010, 2010 IEEE Radar Conference.

[22]  Emmanuel J. Candès,et al.  The Power of Convex Relaxation: Near-Optimal Matrix Completion , 2009, IEEE Transactions on Information Theory.

[23]  P. Jaming,et al.  Uniqueness results for the phase retrieval problem of fractional Fourier transforms of variable order , 2010, 1009.3418.

[24]  Emmanuel J. Candès,et al.  Templates for convex cone problems with applications to sparse signal recovery , 2010, Math. Program. Comput..

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

[26]  G. Zauner,et al.  QUANTUM DESIGNS: FOUNDATIONS OF A NONCOMMUTATIVE DESIGN THEORY , 2011 .

[27]  G. Thakur.  Reconstruction of Bandlimited Functions from Unsigned Samples , 2010, 1007.0198.

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

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

[30]  L. Demanet,et al.  Stable Optimizationless Recovery from Phaseless Linear Measurements , 2012, Journal of Fourier Analysis and Applications.

[31]  Holger Boche,et al.  Phase Retrieval via Structured Modulations in Paley-Wiener Spaces , 2013, ArXiv.

[32]  Dan Edidin,et al.  An algebraic characterization of injectivity in phase retrieval , 2013, ArXiv.

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

[34]  Felix Krahmer,et al.  A Partial Derandomization of PhaseLift Using Spherical Designs , 2013, Journal of Fourier Analysis and Applications.

[35]  Fan Yang.  Signal Reconstruction from Magnitude Measurements in Infinite Dimensional Spaces , 2013 .

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

[37]  Holger Boche,et al.  Phaseless Signal Recovery in Infinite Dimensional Spaces Using Structured Modulations , 2013, ArXiv.

[38]  Yang Wang,et al.  Phase retrieval from very few measurements , 2013, ArXiv.

[39]  T. Heinosaari,et al.  Quantum Tomography under Prior Information , 2011, 1109.5478.

[40]  Dustin G. Mixon,et al.  Saving phase: Injectivity and stability for phase retrieval , 2013, 1302.4618.

[41]  Holger Boche,et al.  A phase retrieval method for signals in modulation-invariant spaces , 2014, 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

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

[43]  Dustin G. Mixon,et al.  Phase Retrieval with Polarization , 2012, SIAM J. Imaging Sci..

[44]  P. Jaming,et al.  Uniqueness results in an extension of Pauli's phase retrieval problem , 2014 .

[45]  Bernhard G. Bodmann,et al.  Stable phase retrieval with low-redundancy frames , 2013, Adv. Comput. Math..

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

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

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