Proximity operators for phase retrieval.

We present a new formulation of a family of proximity operators that generalize the projector step for phase retrieval. These proximity operators for noisy intensity measurements can replace the classical "noise-free" projection in any projection-based algorithm. They are derived from a maximum-likelihood formulation and admit closed form solutions for both the Gaussian and the Poisson cases. In addition, we extend these proximity operators to under-sampled intensity measurements. To assess their performance, these operators are exploited in a classical Gerchberg-Saxton algorithm. We present numerical experiments showing that the reconstructed complex amplitudes with these proximity operators always perform better than using the classical intensity projector, while their computational overhead is moderate.

[1]  James V. Burke,et al.  Optical Wavefront Reconstruction: Theory and Numerical Methods , 2002, SIAM Rev..

[2]  Aude Rondepierre,et al.  On Local Convergence of the Method of Alternating Projections , 2013, Foundations of Computational Mathematics.

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

[4]  V. Elser.  Solution of the crystallographic phase problem by iterated projections. , 2002, Acta crystallographica. Section A, Foundations of crystallography.

[5]  David Mary,et al.  PAINTER: a spatio-spectral image reconstruction algorithm for optical interferometry , 2014, Journal of the Optical Society of America. A, Optics, image science, and vision.

[6]  Andrew V. Martin,et al.  Noise-robust coherent diffractive imaging with a single diffraction pattern , 2012 .

[7]  Florent Krzakala,et al.  Phase recovery from a Bayesian point of view: The variational approach , 2014, 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[8]  J. Dorband,et al.  Hubble Space Telescope Faint Object Camera calculated point-spread functions. , 1997, Applied optics.

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

[10]  Yonina C. Eldar,et al.  Dictionary learning from phaseless measurements , 2016, 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[11]  Warren Hare,et al.  Computing proximal points of nonconvex functions , 2008, Math. Program..

[12]  Ian Robinson,et al.  Effectiveness of iterative algorithms in recovering phase in the presence of noise. , 2007, Acta crystallographica. Section A, Foundations of crystallography.

[13]  Alexandre d'Aspremont,et al.  Phase retrieval for imaging problems , 2013, Mathematical Programming Computation.

[14]  D. Russell Luke,et al.  Nonconvex Notions of Regularity and Convergence of Fundamental Algorithms for Feasibility Problems , 2012, SIAM J. Optim..

[15]  Jeffrey A. Fessler,et al.  Undersampled Phase Retrieval With Outliers , 2014, IEEE Transactions on Computational Imaging.

[16]  Patrick L. Combettes,et al.  Proximal Splitting Methods in Signal Processing , 2009, Fixed-Point Algorithms for Inverse Problems in Science and Engineering.

[17]  D. Russell Luke.  Local linear convergence of approximate projections onto regularized sets , 2012 .

[18]  Émilie Chouzenoux,et al.  A nonconvex regularized approach for phase retrieval , 2014, 2014 IEEE International Conference on Image Processing (ICIP).

[19]  D. L. Misell.  Comment onA method for the solution of the phase problem in electron microscopy , 1973 .

[20]  Tatiana Latychevskaia,et al.  Novel Fourier-domain constraint for fast phase retrieval in coherent diffraction imaging. , 2011, Optics express.

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

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

[23]  Dima Grigoriev,et al.  Subtraction-Free Complexity, Cluster Transformations, and Spanning Trees , 2013, Found. Comput. Math..

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

[25]  J. Moreau.  Proximité et dualité dans un espace hilbertien , 1965 .

[26]  Nelly Pustelnik,et al.  Parallel Proximal Algorithm for Image Restoration Using Hybrid Regularization , 2009, IEEE Transactions on Image Processing.

[27]  Xiaodong Li,et al.  Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flow , 2015, ArXiv.

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

[29]  Henry Stark,et al.  Image restoration by the method of generalized projections with application to restoration from magnitude , 1984, ICASSP.

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

[31]  Heinz H. Bauschke,et al.  Hybrid projection-reflection method for phase retrieval. , 2003, Journal of the Optical Society of America. A, Optics, image science, and vision.

[32]  D. R. Luke.  Relaxed averaged alternating reflections for diffraction imaging , 2004, math/0405208.

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

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

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

[36]  Keith A. Nugent,et al.  Coherent diffractive imaging: a new statistically regularized amplitude constraint , 2010 .

[37]  James R. Fienup,et al.  Iterative Method Applied To Image Reconstruction And To Computer-Generated Holograms , 1980 .