Perturbed Amplitude Flow for Phase Retrieval

In this paper, we propose a new non-convex algorithm for solving the phase retrieval problem, i.e., the reconstruction of a signal <inline-formula><tex-math notation="LaTeX">$ {\mathbf x}\in {\mathbb {H}}^n$</tex-math></inline-formula> (<inline-formula><tex-math notation="LaTeX">${\mathbb {H}}={\mathbb {R}}$</tex-math></inline-formula> or <inline-formula><tex-math notation="LaTeX">${\mathbb {C}}$</tex-math></inline-formula>) from phaseless samples <inline-formula><tex-math notation="LaTeX">$ b_j=\vert \langle {\mathbf a}_j, {\mathbf x}\rangle \vert $</tex-math></inline-formula>, <inline-formula><tex-math notation="LaTeX">$ j=1,\ldots,m$</tex-math></inline-formula>. The proposed algorithm solves a new proposed model, perturbed amplitude-based model, for phase retrieval, and is correspondingly named as <italic>Perturbed Amplitude Flow</italic> (PAF). We prove that PAF can recover <inline-formula><tex-math notation="LaTeX">$c{\mathbf x}$</tex-math></inline-formula> (<inline-formula><tex-math notation="LaTeX">$\vert c\vert = 1$</tex-math></inline-formula>) under <inline-formula><tex-math notation="LaTeX">$\mathcal {O}(n)$</tex-math></inline-formula> Gaussian random measurements (optimal order of measurements). Starting with a designed initial point, our PAF algorithm iteratively converges to the true solution at a linear rate for both real, and complex signals. Besides, PAF algorithm needn’t any truncation or re-weighted procedure, so it enjoys simplicity for implementation. The effectiveness, and benefit of the proposed method are validated by both the simulation studies, and the experiment of recovering natural images.

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

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

[3]  Yue M. Lu,et al.  Fundamental limits of phasemax for phase retrieval: A replica analysis , 2017, 2017 IEEE 7th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP).

[4]  Vladislav Voroninski,et al.  An Elementary Proof of Convex Phase Retrieval in the Natural Parameter Space via the Linear Program PhaseMax , 2016, ArXiv.

[5]  Zhiqiang Xu,et al.  Generalized phase retrieval : measurement number, matrix recovery and beyond , 2016, Applied and Computational Harmonic Analysis.

[6]  Yingbin Liang,et al.  Reshaped Wirtinger Flow for Solving Quadratic System of Equations , 2016, NIPS.

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

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

[9]  J. Miao,et al.  Extending X-ray crystallography to allow the imaging of noncrystalline materials, cells, and single protein complexes. , 2008, Annual review of physical chemistry.

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

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

[12]  Bing Gao,et al.  Phaseless Recovery Using the Gauss–Newton Method , 2016, IEEE Transactions on Signal Processing.

[13]  Yonina C. Eldar,et al.  Phase Retrieval from 1D Fourier Measurements: Convexity, Uniqueness, and Algorithms , 2016, IEEE Transactions on Signal Processing.

[14]  Yuxin Chen,et al.  Implicit Regularization in Nonconvex Statistical Estimation: Gradient Descent Converges Linearly for Phase Retrieval, Matrix Completion, and Blind Deconvolution , 2017, Found. Comput. Math..

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

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

[17]  Meng Huang,et al.  The Estimation Performance of Nonlinear Least Squares for Phase Retrieval , 2020, IEEE Transactions on Information Theory.

[18]  Feng Ruan,et al.  Solving (most) of a set of quadratic equalities: Composite optimization for robust phase retrieval , 2017, Information and Inference: A Journal of the IMA.

[19]  Veit Elser,et al.  Benchmark problems for phase retrieval , 2017, SIAM J. Imaging Sci..

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

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

[22]  Laura Waller,et al.  Experimental robustness of Fourier Ptychography phase retrieval algorithms , 2015, Optics express.

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

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

[25]  John Wright,et al.  A Geometric Analysis of Phase Retrieval , 2016, 2016 IEEE International Symposium on Information Theory (ISIT).

[26]  Irène Waldspurger,et al.  Phase Retrieval With Random Gaussian Sensing Vectors by Alternating Projections , 2016, IEEE Transactions on Information Theory.

[27]  Yonina C. Eldar,et al.  Solving Systems of Random Quadratic Equations via Truncated Amplitude Flow , 2016, IEEE Transactions on Information Theory.

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

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

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

[31]  Tom Goldstein,et al.  PhaseMax: Convex Phase Retrieval via Basis Pursuit , 2016, IEEE Transactions on Information Theory.

[32]  Yonina C. Eldar,et al.  Non-Convex Phase Retrieval From STFT Measurements , 2016, IEEE Transactions on Information Theory.

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

[34]  Gang Wang,et al.  Phase Retrieval via Reweighted Amplitude Flow , 2018, IEEE Transactions on Signal Processing.

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

[36]  Yingbin Liang,et al.  A Nonconvex Approach for Phase Retrieval: Reshaped Wirtinger Flow and Incremental Algorithms , 2017, J. Mach. Learn. Res..