Phase retrieval using iterative Fourier transform and convex optimization algorithm

2015 ◽  
Author(s):  
Fen Zhang ◽  
Hong Cheng ◽  
Quanbing Zhang ◽  
Sui Wei
Keyword(s):  
Fourier Transform ◽  
Convex Optimization ◽  
Optimization Algorithm ◽  
Phase Retrieval

Translation uniqueness of phase retrieval and magnitude retrieval of band-limited signals

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Lung-Hui Chen
Keyword(s):  
Fourier Transform ◽  
Entire Function ◽  
Convex Hull ◽  
Scattering Theory ◽  
Phase Retrieval ◽  
Indicator Function ◽  
Band Limited ◽  
The Fourier Transform ◽  
Complex Valued ◽  
Spectral Invariant

Abstract In this paper, we discuss how to partially determine the Fourier transform F ⁢ ( z ) = ∫ - 1 1 f ⁢ ( t ) ⁢ e i ⁢ z ⁢ t ⁢ 𝑑 t , z ∈ ℂ , F(z)=\int_{-1}^{1}f(t)e^{izt}\,dt,\quad z\in\mathbb{C}, given the data | F ⁢ ( z ) | {\lvert F(z)\rvert} or arg ⁡ F ⁢ ( z ) {\arg F(z)} for z ∈ ℝ {z\in\mathbb{R}} . Initially, we assume [ - 1 , 1 ] {[-1,1]} to be the convex hull of the support of the signal f. We start with reviewing the computation of the indicator function and indicator diagram of a finite-typed complex-valued entire function, and then connect to the spectral invariant of F ⁢ ( z ) {F(z)} . Then we focus to derive the unimodular part of the entire function up to certain non-uniqueness. We elaborate on the translation of the signal including the non-uniqueness associates of the Fourier transform. We show that the phase retrieval and magnitude retrieval are conjugate problems in the scattering theory of waves.

scholarly journals Beam Performance Optimization of Multibeam Imaging Sonar Based on the Hybrid Algorithm of Binary Particle Swarm Optimization and Convex Optimization

2016 ◽  
Vol 2016 ◽  
pp. 1-6 ◽  
Author(s):  
Weijie Xia ◽  
Xue Jin ◽  
Fawang Dou
Keyword(s):  
Particle Swarm Optimization ◽  
Convex Optimization ◽  
Optimization Algorithm ◽  
Performance Optimization ◽  
Hybrid Algorithm ◽  
Particle Swarm ◽  
Binary Particle Swarm Optimization ◽  
Swarm Optimization ◽  
Two Factors ◽  
Imaging Sonar

It should be noted that the peak sidelobe level (PSLL) significantly influences the performance of the multibeam imaging sonar. Although a great amount of work has been done to suppress the PSLL of the array, one can verify that these methods do not provide optimal results when applied to the case of multiple patterns. In order to suppress the PSLL for multibeam imaging sonar array, a hybrid algorithm of binary particle swarm optimization (BPSO) and convex optimization is proposed in this paper. In this algorithm, the PSLL of multiple patterns is taken as the optimization objective. BPSO is considered as a global optimization algorithm to determine best common elements’ positions and convex optimization is considered as a local optimization algorithm to optimize elements’ weights, which guarantees the complete match of the two factors. At last, simulations are carried out to illustrate the effectiveness of the proposed algorithm in this paper. Results show that, for a sparse semicircular array with multiple patterns, the hybrid algorithm can obtain a lower PSLL compared with existing methods and it consumes less calculation time in comparison with other hybrid algorithms.

Generalized Phase-Shifting Phase Retrieval Approach Based on Time-Domain Fourier Transform

2015 ◽  
Vol 42 (9) ◽  
pp. 0908004
Author(s):  
张望平 Zhang Wangping ◽  
吕晓旭 Lü Xiaoxu ◽  
刘胜德 Liu Shengde ◽  
赵晖 Zhao Hui ◽  
钟丽云 Zhong Liyun
Keyword(s):  
Fourier Transform ◽  
Time Domain ◽  
Phase Retrieval ◽  
Phase Shifting

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

2018 ◽  
Vol 8 (10) ◽  
pp. 1797 ◽  
Author(s):  
Zhuolei Xiao ◽  
Yerong Zhang ◽  
Kaixuan Zhang ◽  
Dongxu Zhao ◽  
Guan Gui
Keyword(s):  
Fourier Transform ◽  
Local Search ◽  
Phase Retrieval ◽  
Greedy Algorithms ◽  
Optimal Estimation ◽  
Retrieval Algorithm ◽  
Marquardt Algorithm ◽  
Levenberg Marquardt ◽  
Local Search Procedure ◽  
Unknown Signal

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.

scholarly journals Inverting spectrogram measurements via aliased Wigner distribution deconvolution and angular synchronization

2020 ◽  
Author(s):  
Michael Perlmutter ◽  
Sami Merhi ◽  
Aditya Viswanathan ◽  
Mark Iwen
Keyword(s):  
Fourier Transform ◽  
Phase Retrieval ◽  
Wigner Distribution ◽  
Short Time Fourier Transform ◽  
Fourier Inversion ◽  
New Class ◽  
Bandlimited Signals ◽  
Rank One ◽  
Noise Robust ◽  
Short Time

Abstract We propose a two-step approach for reconstructing a signal $\textbf x\in \mathbb{C}^d$ from subsampled discrete short-time Fourier transform magnitude (spectogram) measurements: first, we use an aliased Wigner distribution deconvolution approach to solve for a portion of the rank-one matrix $\widehat{\textbf{x}}\widehat{\textbf{x}}^{*}.$ Secondly, we use angular synchronization to solve for $\widehat{\textbf{x}}$ (and then for $\textbf{x}$ by Fourier inversion). Using this method, we produce two new efficient phase retrieval algorithms that perform well numerically in comparison to standard approaches and also prove two theorems; one which guarantees the recovery of discrete, bandlimited signals $\textbf{x}\in \mathbb{C}^{d}$ from fewer than $d$ short-time Fourier transform magnitude measurements and another which establishes a new class of deterministic coded diffraction pattern measurements which are guaranteed to allow efficient and noise robust recovery.

scholarly journals Binary Sparse Phase Retrieval via Simulated Annealing

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Wei Peng ◽  
Hongxia Wang
Keyword(s):  
Fourier Transform ◽  
Simulated Annealing ◽  
Recovery Rate ◽  
Phase Retrieval ◽  
High Recovery ◽  
Phase Recovery ◽  
Greedy Strategy ◽  
High Recovery Rate ◽  
Binary Model ◽  
The Fourier Transform

This paper presents the Simulated Annealing Sparse PhAse Recovery (SASPAR) algorithm for reconstructing sparse binary signals from their phaseless magnitudes of the Fourier transform. The greedy strategy version is also proposed for a comparison, which is a parameter-free algorithm. Sufficient numeric simulations indicate that our method is quite effective and suggest the binary model is robust. The SASPAR algorithm seems competitive to the existing methods for its efficiency and high recovery rate even with fewer Fourier measurements.

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