Abstract
We address the problem of signal reconstruction from intensity measurements with respect to a measurement frame. This nonconvex inverse problem is known as phase retrieval. The case considered in this paper concerns phaseless measurements taken with respect to a Gabor frame. It arises naturally in many practical applications, such as diffraction imaging and speech recognition. We present a reconstruction algorithm that uses a nearly optimal number of phaseless time-frequency structured measurements and discuss its robustness in the case when the measurements are corrupted by noise. We show how geometric properties of the measurement frame are related to the robustness of the phaseless reconstruction.
Original language | English |
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Pages (from-to) | 736-761 |
Number of pages | 26 |
Journal | SIAM Journal on Imaging Sciences |
Volume | 12 |
Issue number | 2 |
DOIs | |
Publication status | Published - 16 Apr 2019 |
Externally published | Yes |
Funding
\ast Received by the editors August 6, 2018; accepted for publication (in revised form) February 7, 2019; published electronically April 16, 2019. http://www.siam.org/journals/siims/12-2/M120552.html Funding: The work of the second author was supported by a Research Grant for Doctoral Candidates and Young Academics and Scientists of German Academic Exchange Service (DAAD). \dagger Department of Scientific Computing, Catholic University of Eichst\a"tt-Ingolstadt, 85072 Germany (pfander@ku. de). \ddagger Department of Mathematics, University of California, Los Angeles, CA 90095-1555 ([email protected]).
Keywords
- Angular synchronization
- Expander graphs
- Gabor frames
- Order statistics of frame coefficients
- Phase retrieval
- Spectral clustering