Phase retrieval of sparse signals using optimization transfer and ADMM

Weller, Daniel S.; Pnueli, Ayelet; Radzyner, Ori; Divon, Gilad; Eldar, Yonina C.; Fessler, Jeffrey A.

doi:10.1109/icip.2014.7025268

Cited by 7 publications

(16 citation statements)

References 31 publications

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“…In [44], β scales roughly linearly with the number of measurements for the proposed method without normalization. With normalization, the optimal β appears to remain fairly constant between 0.1 and 10 −0.9 .…”

Section: Experimental Setup and Resultsmentioning

confidence: 99%

“…The N = 512 × 512-pixel star of David phantom used in [44] is inspired by the real example image shown in [68]. The pattern in the image is constructed using 30 discs, each 21-pixels wide.…”

Section: Experimental Setup and Resultsmentioning

confidence: 99%

“…Otherwise, an iterative algorithm like FISTA [60] could be embedded within the ADMM method [44]. Instead, we use “preconditioned” ADMM (PADMM) [61], [62] accelerated 1 using Nesterov momentum [64], essentially using a single FISTA step as the x -update in (12):

x^{i} \leftarrow soft false(z^{i - 1} - 1 \frac{1}{c} A^{'} false(bold-italicA z^{i - 1} - u^{i - 1} + b^{i - 1} false); 1 \frac{β}{μ c} false) .

t^{i} \leftarrow false(1 + \sqrt{1 + 4 {false(t^{i - 1} false)}^{2}} false) / 2.

z^{i} \leftarrow x^{i} + 1 \frac{t^{i - 1} - 1}{t^{i}} false(x^{i} - x^{i - 1} false) .

…”

Section: Solving the Majorized Objective With Admmmentioning

confidence: 99%

“…Many existing methods either ignore measurement noise or use quadratic data fit terms. The proposed method, based on [44], employs a robust 1-norm data fit term, corresponding to the negative log-likelihood of a Laplace distribution, to improve robustness to outliers. This data fit term can also be found in some matrix lifting phase retrieval methods [40], [41], at the expense of much larger memory and computational resources.…”

Section: Introductionmentioning

confidence: 99%

“…More specifically, majorization yields a tight convex surrogate for the original nonconvex objective. Introducing an auxiliary variable enables efficient minimization of this majorizer via a more easily separable preconditioned variant of ADMM (ordinary ADMM was used in [44]).…”

Section: Introductionmentioning

confidence: 99%

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Undersampled Phase Retrieval With Outliers

Weller

Pnueli

Divon

et al. 2015

IEEE Trans. Comput. Imaging

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This paper proposes a general framework for reconstructing sparse images from undersampled (squared)-magnitude data corrupted with outliers and noise. This phase retrieval method uses a layered approach, combining repeated minimization of a convex majorizer (surrogate for a nonconvex objective function), and iterative optimization of that majorizer using a preconditioned variant of the alternating direction method of multipliers (ADMM). Since phase retrieval is nonconvex, this implementation uses multiple initial majorization vectors. The introduction of a robust 1-norm data fit term that is better adapted to outliers exploits the generality of this framework. The derivation also describes a normalization scheme for the regularization parameter and a known adaptive heuristic for the ADMM penalty parameter. Both 1D Monte Carlo tests and 2D image reconstruction simulations suggest the proposed framework, with the robust data fit term, reduces the reconstruction error for data corrupted with both outliers and additive noise, relative to competing algorithms having the same total computation.

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Section: Experimental Setup and Resultsmentioning

confidence: 99%

Section: Experimental Setup and Resultsmentioning

confidence: 99%

x^{i} \leftarrow soft false(z^{i - 1} - 1 \frac{1}{c} A^{'} false(bold-italicA z^{i - 1} - u^{i - 1} + b^{i - 1} false); 1 \frac{β}{μ c} false) .

t^{i} \leftarrow false(1 + \sqrt{1 + 4 {false(t^{i - 1} false)}^{2}} false) / 2.

z^{i} \leftarrow x^{i} + 1 \frac{t^{i - 1} - 1}{t^{i}} false(x^{i} - x^{i - 1} false) .

…”

Section: Solving the Majorized Objective With Admmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Undersampled Phase Retrieval With Outliers

Weller

Pnueli

Divon

et al. 2015

IEEE Trans. Comput. Imaging

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ADMM-TGV image restoration for scientific applications with unbiased parameter choice

Zietlow,

Lindner

2024

Numer Algor

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Image restoration via alternating direction method of multipliers (ADMM) has gained large interest within the last decade. Solving standard problems of Gaussian and Poisson noise, the set of “Total Variation” (TV)-based regularizers proved to be efficient and versatile. In the last few years, the “Total Generalized Variation” (TGV) approach combined TV regularizers of different orders adaptively to better suit local regions in the image. This improved the technique significantly. The approach solved the staircase problem inherent of the first-order TV while keeping the beneficial edge preservation. The iterative minimization for the augmented Lagrangian of TGV problems requires four important parameters: two penalty parameters $${\rho }$$ ρ and $${\eta }$$ η and two regularization parameters $${\lambda _{0}}$$ λ 0 and $${\lambda _{1}}$$ λ 1 . The choice of penalty parameters decides on the convergence speed, and the regularization parameters decide on the impact of the respective regularizer and are determined by the noise level in the image. For scientific applications of such algorithms, an automated and thus objective method to determine these parameters is essential to receive unbiased results independent of the user. Obviously, both sets of parameters are to be well chosen to achieve optimal results, too. In this paper, a method is proposed to adaptively choose optimal $${\rho }$$ ρ and $${\eta }$$ η values for the iteration to converge faster, based on the primal and dual residuals arising from the optimality conditions of the augmented Lagrangian. Further, we show how to choose $${\lambda _{0}}$$ λ 0 and $${\lambda _{1}}$$ λ 1 based on the inherent noise in the image.

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Inexact alternating optimization for phase retrieval with outliers

Qian

Sidiropoulos

2016

2016 24th European Signal Processing Conference (EUSIPCO)

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Phase retrieval of sparse signals using optimization transfer and ADMM

Cited by 7 publications

References 31 publications

Undersampled Phase Retrieval With Outliers

Undersampled Phase Retrieval With Outliers

ADMM-TGV image restoration for scientific applications with unbiased parameter choice

Inexact alternating optimization for phase retrieval with outliers

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