Phase Retrieval With One or Two Diffraction Patterns by Alternating Projections of the Null Vector

Two versions of alternating projection (AP), the parallel alternating projection (PAP) and the serial alternating projection (SAP), are proposed to solve phase retrieval with at most two coded diffraction patterns. The proofs of geometric convergence are given with sharp bounds on the rates of convergence in terms of a spectral gap condition. To compensate for the local nature of convergence, the null vector method is proposed for initialization and proved to produce asymptotically accurate initialization for the Gaussian case. Extensive numerical experiments are performed to show that the null vector method produces more accurate initialization than the spectral vector method and that PAP/SAP converge faster to more accurate solutions than other iterative schemes for non-convex optimization such as the Wirtinger flow. Moreover, SAP converges still faster than PAP. In practice AP and the null vector method together produce globally convergent iterates to the true object.

[1]  W. Cheney,et al.  Proximity maps for convex sets , 1959 .

[2]  A. Goldstein.  Convex programming in Hilbert space , 1964 .

[3]  Boris Polyak,et al.  Constrained minimization methods , 1966 .

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

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

[6]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

[7]  Heinz H. Bauschke,et al.  On Projection Algorithms for Solving Convex Feasibility Problems , 1996, SIAM Rev..

[8]  J. Miao,et al.  Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects , 1998 .

[9]  J. Miao,et al.  Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens , 1999, Nature.

[10]  J. Hajdu,et al.  Potential for biomolecular imaging with femtosecond X-ray pulses , 2000, Nature.

[11]  F. Deutsch.  Best approximation in inner product spaces , 2001 .

[12]  Heinz H. Bauschke,et al.  Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization. , 2002, Journal of the Optical Society of America. A, Optics, image science, and vision.

[13]  Heinz H. Bauschke,et al.  Finding best approximation pairs relative to two closed convex sets in Hilbert spaces , 2004, J. Approx. Theory.

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

[15]  Adrian S. Lewis,et al.  Local Linear Convergence for Alternating and Averaged Nonconvex Projections , 2009, Found. Comput. Math..

[16]  Georg Weidenspointner,et al.  Femtosecond X-ray protein nanocrystallography , 2011, Nature.

[17]  G. Papanicolaou,et al.  Array imaging using intensity-only measurements , 2010 .

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

[19]  Garth J. Williams,et al.  Single mimivirus particles intercepted and imaged with an X-ray laser , 2011, Nature.

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

[21]  A. Fannjiang,et al.  Phase retrieval with random phase illumination. , 2012, Journal of the Optical Society of America. A, Optics, image science, and vision.

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

[23]  Chao Yang,et al.  Alternating direction methods for classical and ptychographic phase retrieval , 2012 .

[24]  Wenjing Liao,et al.  Fourier phasing with phase-uncertain mask , 2012, 1212.3858.

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

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

[27]  Yonina C. Eldar,et al.  Phase Retrieval via Matrix Completion , 2013, SIAM J. Imaging Sci..

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

[29]  Yuxin Chen,et al.  Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems , 2015, NIPS.

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

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