CPRL -- An Extension of Compressive Sensing to the Phase Retrieval Problem

While compressive sensing (CS) has been one of the most vibrant research fields in the past few years, most development only applies to linear models. This limits its application in many areas where CS could make a difference. This paper presents a novel extension of CS to the phase retrieval problem, where intensity measurements of a linear system are used to recover a complex sparse signal. We propose a novel solution using a lifting technique - CPRL, which relaxes the NP-hard problem to a nonsmooth semidefinite program. Our analysis shows that CPRL inherits many desirable properties from CS, such as guarantees for exact recovery. We further provide scalable numerical solvers to accelerate its implementation.

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

[2]  L. Mandel,et al.  Source reconstruction from the modulus of the correlation function: a practical approach to the phase problem of optical coherence theory* , 1973 .

[3]  John N. Tsitsiklis,et al.  Parallel and distributed computation , 1989 .

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

[5]  Stéphane Mallat,et al.  Matching pursuits with time-frequency dictionaries , 1993, IEEE Trans. Signal Process..

[6]  Michael A. Saunders,et al.  Atomic Decomposition by Basis Pursuit , 1998, SIAM J. Sci. Comput..

[7]  H. Konno,et al.  A cutting plane algorithm for semi-definite programming problems with applications to failure discriminant analysis , 2002 .

[8]  R. J.,et al.  Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint , 2002 .

[9]  Michael Elad,et al.  Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ1 minimization , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[10]  Emmanuel J. Candès,et al.  Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information , 2004, IEEE Transactions on Information Theory.

[11]  E.J. Candes.  Compressive Sampling , 2022 .

[12]  Michael I. Jordan,et al.  A Direct Formulation for Sparse Pca Using Semidefinite Programming , 2004, SIAM Rev..

[13]  Stefano Marchesini,et al.  Phase retrieval and saddle-point optimization. , 2006, Journal of the Optical Society of America. A, Optics, image science, and vision.

[14]  Piotr Indyk,et al.  Combining geometry and combinatorics: A unified approach to sparse signal recovery , 2008, 2008 46th Annual Allerton Conference on Communication, Control, and Computing.

[15]  E. Candès.  The restricted isometry property and its implications for compressed sensing , 2008 .

[16]  Michael Elad,et al.  From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images , 2009, SIAM Rev..

[17]  Wotao Yin,et al.  Alternating direction augmented Lagrangian methods for semidefinite programming , 2010, Math. Program. Comput..

[18]  Lei Tian,et al.  Compressive Phase Retrieval , 2011 .

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

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

[21]  Stephen P. Boyd,et al.  Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers , 2011, Found. Trends Mach. Learn..

[22]  Yonina C. Eldar,et al.  Sparsity Based Sub-wavelength Imaging with Partially Incoherent Light via Quadratic Compressed Sensing References and Links , 2022 .

[23]  Yi Ma,et al.  Robust principal component analysis? , 2009, JACM.

[24]  S. Sastry,et al.  Compressive Phase Retrieval From Squared Output Measurements Via Semidefinite Programming , 2011, 1111.6323.

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

[26]  Yonina C. Eldar,et al.  Sparsity-based single-shot subwavelength coherent diffractive imaging , 2012, 2012 Conference on Lasers and Electro-Optics (CLEO).

[27]  Yonina C. Eldar,et al.  Sparsity Constrained Nonlinear Optimization: Optimality Conditions and Algorithms , 2012, SIAM J. Optim..

[28]  K. Schittkowski,et al.  NONLINEAR PROGRAMMING , 2022 .