On relaxed averaged alternating reflections (RAAR) algorithm for phase retrieval with structured illumination

Ji, Li; Zhou, Tong

doi:10.1088/1361-6420/aa518e

Cited by 17 publications

(21 citation statements)

References 38 publications

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“…We follow the analysis approach initiated by Chen and Fannjiang [8] where they established a local linear convergence result for the DR algorithm. The mentioned result of [8] was later extended for the RAAR algorithm in [34]. We will show that DRAP also enjoys that kind of convergence result.…”

Section: A Results From Spectral Analysismentioning

confidence: 71%

Convex combination of alternating projection and Douglas–Rachford operators for phase retrieval

2021

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We present the convergence analysis of convex combination of the alternating projection and Douglas–Rachford operators for solving the phase retrieval problem. New convergence criteria for iterations generated by the algorithm are established by applying various schemes of numerical analysis and exploring both physical and mathematical characteristics of the phase retrieval problem. Numerical results demonstrate the advantages of the algorithm over the other widely known projection methods in practically relevant simulations.

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Section: A Results From Spectral Analysismentioning

confidence: 71%

Convex combination of alternating projection and Douglas–Rachford operators for phase retrieval

2021

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“…The key is to show the eigen structure of the isometric matrix B. The local convergence analysis for specific Douglas-Rachford algorithms for Fourier phase retrieval problem can be found in [6,14]. However, their convergence results require m ≥ 2n while our analysis is valid for arbitrary m and n. Here we also prove convergence of the newly proposed robust version.…”

Section: Technical Details For Proofmentioning

confidence: 78%

“…However, they do not apply in the case of non-isometric measurements with more general prior constraints, e.g., total variation regularization. Recently, RAAR has been adapted to nonisometric measurements [14], but it does not support prior information. Thus these existing algorithms for non-Gaussian measurements have different kinds of restrictions.…”

Section: Introductionmentioning

confidence: 99%

Solving phase retrieval via graph projection splitting

Zhao

2020

Inverse Problems

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Phase retrieval with prior information can be cast as a nonsmooth and nonconvex optimization problem. To decouple the signal and measurement variables, we introduce an auxiliary variable and reformulate it as an optimization with an equality constraint. We then solve the reformulated problem by graph projection splitting (GPS), where the two proximity subproblems and the graph projection step can be solved efficiently. With slight modification, we also propose a robust graph projection splitting (RGPS) method to stabilize the iteration for noisy measurements. Contrary to intuition, RGPS outperforms GPS with fewer iterations to locate a satisfying solution even for noiseless case. Based on the connection between GPS and Douglas-Rachford iteration, under mild conditions on the sampling vectors, we analyze the fixed point sets and provide the local convergence of GPS and RGPS applied to noiseless phase retrieval without prior information. For noisy case, we provide the error bound of the reconstruction. Compared to other existing methods, thanks for the splitting approach, GPS and RGPS can efficiently solve phase retrieval with prior information regularization for general sampling vectors which are not necessarily isometric. For Gaussian phase retrieval, compared to existing gradient flow approaches, numerical results show that GPS and RGPS are much less sensitive to the initialization. Thus they markedly improve the phase transition in noiseless case and reconstruction in the presence of noise respectively. GPS shows sharpest phase transition among existing methods including RGPS, while it needs more iterations than RGPS when the number of measurement is large enough. RGPS outperforms GPS in terms of stability for noisy measurements. When applying RGPS to more general non-Gaussian measurements with prior information, such as support, sparsity and TV minimization, RGPS either outperforms state-of-the-art solvers or can be combined with state-of-the-art solvers to improve their reconstruction quality. *

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“…which tends to 0 and ∞ as β tends to 1 and 1/2, respectively. Local geometric convergence of RAAR has been proved by Li and Zhou (2017). Moreover, like Theorem 4.6, RAAR possesses the desirable property that every RAAR sequence is explicitly bounded in terms of β as follows.…”

Section: Noise-agnostic Methodsmentioning

confidence: 98%

“…According to Li and Zhou (2017), the optimal β is usually between 0.8 and 0.9, corresponding to ρ = 0.125 and 0.333 according to (4.46). We set β = 0.9 in Figure 4.3.…”

Section: Optimal Parametermentioning

confidence: 99%

The numerics of phase retrieval

Fannjiang

Strohmer

2020

Acta Numerica

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Phase retrieval, i.e. the problem of recovering a function from the squared magnitude of its Fourier transform, arises in many applications, such as X-ray crystallography, diffraction imaging, optics, quantum mechanics and astronomy. This problem has confounded engineers, physicists, and mathematicians for many decades. Recently, phase retrieval has seen a resurgence in research activity, ignited by new imaging modalities and novel mathematical concepts. As our scientific experiments produce larger and larger datasets and we aim for faster and faster throughput, it is becoming increasingly important to study the involved numerical algorithms in a systematic and principled manner. Indeed, the past decade has witnessed a surge in the systematic study of computational algorithms for phase retrieval. In this paper we will review these recent advances from a numerical viewpoint.

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On relaxed averaged alternating reflections (RAAR) algorithm for phase retrieval with structured illumination

Cited by 17 publications

References 38 publications

Convex combination of alternating projection and Douglas–Rachford operators for phase retrieval

Convex combination of alternating projection and Douglas–Rachford operators for phase retrieval

Solving phase retrieval via graph projection splitting

The numerics of phase retrieval

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