GARLM: Greedy Autocorrelation Retrieval Levenberg–Marquardt Algorithm for Improving Sparse Phase Retrieval

Zhuolei Xiao,Kaixuan Zhang,Dongxu Zhao,Guan Gui,Yerong Zhang

doi:10.3390/app8101797

Abstract

The goal of phase retrieval is to recover an unknown signal from the random measurements consisting of the magnitude of its Fourier transform. Due to the loss of the phase information, phase retrieval is considered as an ill-posed problem. Conventional greedy algorithms, e.g., greedy spare phase retrieval (GESPAR), were developed to solve this problem by using prior knowledge of the unknown signal. However, due to the defect of the Gauss–Newton method in the local convergence problem, especially when the residual is large, it is very difficult to use this method in GESPAR to efficiently solve the non-convex optimization problem. In order to improve the performance of the greedy algorithm, we propose an improved phase retrieval algorithm, which is called the greedy autocorrelation retrieval Levenberg–Marquardt (GARLM) algorithm. Specifically, the proposed GARLM algorithm is a local search iterative algorithm to recover the sparse signal from its Fourier transform magnitude. The proposed algorithm is preferred to existing greedy methods of phase retrieval, since at each iteration the problem of minimizing the objective function over a given support is solved by using the improved Levenberg–Marquardt (ILM) method and matrix transform. A local search procedure such as the 2-opt method is then invoked to get the optimal estimation. Simulation results are given to show that the proposed algorithm performs better than the conventional GESPAR algorithm.

Highlights

In recent years, sampling theory has made significant progress [1], which has a great impact on signal processing and communication
We proposed an improved and efficient local alternative search algorithm for phase retrieval named greedy autocorrelation retrieval Levenberg–Marquardt (GARLM)
The proposed GARLM algorithm is a local search iterative algorithm to recover the sparse signal from its Fourier transform magnitude

Summary

Introduction

In recent years, sampling theory has made significant progress [1], which has a great impact on signal processing and communication. Fourier analysis and wavelet analysis are essential for these signal processing techniques, and the mathematical tools involving Fourier and wavelet analysis for sampling and reconstruction are very important [2,3]. Reference [2] used Riesz bases and frames to study stable reconstructions and generalize the concept of oblique double frames to the infinite-dimensional framework. The authors of Reference [3] developed a redundant sampling procedure that can be used to reduce quantization errors when quantizing measurements before reconstruction. Concerning reconstruction techniques taking advantage of sparsity in general, the basic work of processing noise measurements and non-ideal sampling is an important part of recovering unknown signals from random measurements [4,5]. The problem of linear reconstruction based on Fourier

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Conclusion