Phase Retrieval Algorithm via Nonconvex Minimization Using a Smoothing Function

Phase retrieval is an inverse problem which consists in recovering an unknown signal from a set of absolute squared projections. Recently, gradient descent algorithms have been developed to solve this problem. However, their optimization cost functions are non-convex and non-smooth. To address the n...

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Veröffentlicht in:IEEE transactions on signal processing Jg. 66; H. 17; S. 4574 - 4584
Hauptverfasser: Pinilla, Samuel, Bacca, Jorge, Arguello, Henry
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York IEEE 01.09.2018
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:
Algorithms
Computational geometry
Computer simulation
Conjugate gradient method
Conjugates
Convergence
Convexity
Cost function
Crystallography
Inverse problems
Mathematical analysis
Non-convex problem
non-smooth function
Optimization
Parameters
Phase measurement
Phase retrieval
Signal processing algorithms
Smoothing
smoothing function
Smoothing methods
Smoothness
X-ray imaging
ISSN:1053-587X, 1941-0476
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Zusammenfassung:Phase retrieval is an inverse problem which consists in recovering an unknown signal from a set of absolute squared projections. Recently, gradient descent algorithms have been developed to solve this problem. However, their optimization cost functions are non-convex and non-smooth. To address the non-smoothness of the cost function, some of these methods use truncation thresholds to calculate a truncated step gradient direction. But, the truncation requires designing parameters to obtain a desired performance in the phase recovery, which drastically modifies the search direction update, increasing the sampling complexity. Therefore, this paper develops the Phase Retrieval Smoothing Conjugate Gradient method (PR-SCG) which uses a smoothing function to retrieve the signal. PR-SCG is based on the smooth-ing projected gradient method which is useful for non-convex optimization problems. PR-SCG uses a nonlinear conjugate gradient of the smoothing function as the search direction to accelerate the convergence. Furthermore, the incremental Stochastic Smoothing Phase Retrieval algorithm (SSPR) is developed. SSPR involves a single equation per iteration which results in a simple, scalable, and fast approach useful when the size of the signal is large. Also, it is shown that SSPR converges linearly to the true signal, up to a global unimodular constant. Additionally, the proposed methods do not require truncation parameters. Simulation results are provided to validate the efficiency of PR-SCG and SSPR compared to existing phase retrieval algorithms. It is shown that PR-SCG and SSPR are able to reduce the number of measurements and iterations to recover the phase, compared with recently developed algorithms.
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ISSN:1053-587X
1941-0476
DOI:10.1109/TSP.2018.2855667