Reconstructing a signal from squared linear (rank-1 quadratic) measurements is a challenging problem with important applications in optics and imaging, where it is known as phase retrieval. This paper proposes two new phase retrieval algorithms based on nonconvex quadratically constrained quadratic programming) formulations, and a recently proposed approximation technique dubbed feasible point pursuit (FPP). The first is designed for uniformly distributed bounded measurement errors, such as those arising from high-rate quantization (B-FPP). The second is designed for Gaussian measurement errors, using a least-squares criterion (LS-FPP). Their performance is measured against state-of-the-art algorithms and the Cramér-Rao bound (CRB), which is also derived here. Simulations show that LS-FPP outperforms the existing schemes and operates close to the CRB. Compact CRB expressions, properties, and insights are obtained by explicitly computing the CRB in various special cases - including when the signal of interest admits a sparse parametrization, using harmonic retrieval as an example.
Bibliographical noteFunding Information:
The work of N. Sidiropoulos was supported by NSF CIF-1525194. K. Huang was supported by a UMII dissertation fellowship. C. Qian is on leave from the Department of Electronics and Information Engineering, Harbin Institute of Technology, China, supported in part by the Natural Science Foundation of China (NSFC) under Grant No. 61171187 and the Chinese Scholarship Council.
© 2016 IEEE.
- Cramér-Rao bound (CRB)
- feasible point pursuit (FPP)
- phase retrieval
- quadratically constrained quadratic programming (QCQP)
- semidefinite programming (SDP)