Phase Retrieval with One or Two Diffraction Patterns by Alternating Projections with the Null Initialization

Chen, Pengwen; Fannjiang, Albert; Liu, Gi Ren

doi:10.1007/s00041-017-9536-8

Cited by 46 publications

(64 citation statements)

References 52 publications

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“…Second, Gaussian-DRS with ρ = 1 produces the optimal convergence rate for the range ρ ≥ 1 (Corollary (6.2) and Remark 6.3) which is smaller than that of CDR (Corollary 6.4) but the same as AP [10].…”

Section: Optimal Step Sizementioning

confidence: 96%

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Fixed Point Analysis of Douglas--Rachford Splitting for Ptychography and Phase Retrieval

Fannjiang¹,

Zhang²

2020

SIAM J. Imaging Sci.

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Douglas-Rachford Splitting (DRS) methods based on the proximal point algorithms for the Poisson and Gaussian log-likelihood functions are proposed for ptychography and phase retrieval.Fixed point analysis shows that the DRS iterated sequences are always bounded explicitly in terms of the step size and that the fixed points are linearly stable if and only if the fixed points are regular solutions. This alleviates two major drawbacks of the classical Douglas-Rachford (CDR) algorithm: slow convergence when the feasibility problem is consistent and divergent behavior when the feasibility problem is inconsistent.Moreover, the fixed point analysis decisively leads to the selection of an optimal step size which in turn renders the Gaussian DRS method in a particularly simple form with no tuning parameter (Averaged Projection-Reflection). When applied to the challenging problem of blind ptychography, which seeks to recover both the object and the probe simultaneously, the DRS methods converge geometrically and globally when properly initialized.

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Section: Optimal Step Sizementioning

confidence: 96%

“…We see that the Gaussian-DRS outperforms the Poisson-DRS, especially when the Poisson RE becomes unstable for NSR ≥ 35%. As noted in [10,31,50] fast convergence (with the Poisson log-likelihood function) may introduce noisy artifacts and reduce reconstruction quality.…”

Section: Test Objectsmentioning

confidence: 99%

Fixed Point Analysis of Douglas--Rachford Splitting for Ptychography and Phase Retrieval

Fannjiang¹,

Zhang²

2020

SIAM J. Imaging Sci.

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“…7 Corollary 2.4. Consider the multi-part ptychography (4)- (5). Suppose that the assumptions of Proposition 2.2 hold and that for every…”

Section: Uniqueness Of Ptychographic Solutionmentioning

confidence: 99%

“…The following result identifies any AP limit point with an AP fixed point. The proof of Proposition 3.2 can be adapted from [5] verbatim and is omitted.…”

Section: Fixed Point Algorithmsmentioning

confidence: 99%

Coded aperture ptychography: uniqueness and reconstruction

Chen

Fannjiang

2018

Inverse Problems

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Abstract. Uniqueness of solution is proved for any ptychographic scheme with a random masks under a minimum overlap condition and local geometric convergence analysis is given for the alternating projection (AP) and Douglas-Rachford (DR) algorithms. DR is shown to possess a unique fixed point in the object domain and for AP a simple criterion for distinguishing the true solution among possibly many fixed points is given.A minimalist scheme is proposed where the adjacent masks overlap 50% of area and each pixel of the object is illuminated by exactly four times during the whole measurement process. Such a scheme is conveniently parametrized by the number q of shifted masks in each direction. The lower bound 1 − C/q 2 is proved for the geometric convergence rate of the minimalist scheme, predicting a poor performance with large q which is confirmed by numerical experiments.Extensive numerical experiments are performed to explore what the general features of a wellperforming mask are like, what the best-performing values of q for a given mask are, how robust the minimalist scheme is with respect to measurement noise and what the significant factors affecting the noise stability are.

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“…In this setting, even the ER algorithm behaves well for nonnegative image. And recently the local geometrical convergence to a solution for HIO is proved [24,25]. An important difference between the optimization approaches in [20][21][22] and the standard iterative projection methods is that their coded diffraction patterns are not oversampled.…”

Section: Introductionmentioning

confidence: 99%

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

Zhou

2017

Inverse Problems

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In this paper, as opposed to the random phase masks, the structured illuminations with a pixeldependent deterministic phase shift are considered to derandomize the model setup. The RAAR algorithm is modified to adapt to two or more diffraction patterns, and the modified RAAR algorithm operates in Fourier domain rather than space domain. The local convergence of the RAAR algorithm is proved by some eigenvalue analysis. Numerical simulations is presented to demonstrate the effectiveness and stability of the algorithm compared to the HIO (Hybrid Input-Output) method. The numerical performances show the global convergence of the RAAR in our tests.

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Phase Retrieval with One or Two Diffraction Patterns by Alternating Projections with the Null Initialization

Cited by 46 publications

References 52 publications

Fixed Point Analysis of Douglas--Rachford Splitting for Ptychography and Phase Retrieval

Fixed Point Analysis of Douglas--Rachford Splitting for Ptychography and Phase Retrieval

Coded aperture ptychography: uniqueness and reconstruction

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

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