The Matrix Completion Method for Phase Retrieval from Fractional Fourier Transform Magnitudes

Qi Luo,Hongxia Wang

doi:10.1155/2016/4617327

Abstract

Inspired by the implementation of the fractional Fourier transform (FRFT) and its applications in optics, we address the problem of reconstructing a signal from its several FRFT magnitudes (or intensities). The matrix completion method is adopted here. Through numerical tests, the matrix completion method is proven effective in both noisy and noise-free situations. We also compare our method with the Gerchberg-Saxton (GS) algorithm based on FRFT. Numerical tests show that the matrix completion method gains a certain advantage in recovering uniqueness and convergence over the GS algorithm in the noise-free case. Furthermore, in terms of noisy signals, the matrix completion method performs robustly and adding more measurements can generally increase accuracy of recovered signals.

Highlights

Phase retrieval has a long history since the first application to the optics and X-ray crystallography [1]
Inspired by the implementation of the fractional Fourier transform (FRFT) and its applications in optics, we address the problem of reconstructing a signal from its several FRFT magnitudes
In the last few decades phase retrieval has arisen in various fields including diffraction imaging [2], image encryption [3], and quantum mechanics [4]

Summary

Introduction

Phase retrieval has a long history since the first application to the optics and X-ray crystallography [1]. Multiple FRFT magnitudes can provide more information of the signal, which may lead to the correct recovery without extra constraints of the original signal This is a notable advantage because one of most stringent limitations for most algorithms is the need for isolated objects (the support constraint) [9]. Most studies deal with magnitudes of two FRFTs, which need some additional assumptions about the signal, like support of the signal and nonnegativity Without these assumptions algorithms dealing with two FRFTs cannot ensure recovering correctness in many cases [16, 19]. At least 4 FRFTs are chosen to reconstruct the original signal to ensure probability of successful recovery without further assumption upon signals Numerical experiments show that the matrix completion method gains certain advantages in stability and convergency

Matrix Completion Method for FRFT Phase Retrieval

Numerical Experiments

Findings

Conclusions