Article Contents

2024, Volume 18, Issue 1: 286-310. Doi: 10.3934/ipi.2023033 open access

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Finding robust minimizer for non-convex phase retrieval

1.
School of Science, Nanjing University of Posts and Telecommunications, Nanjing, China
2.
Department of Mathematics, The Chinese University of Hong Kong, Shatin, N. T., Hong Kong, China

^*Corresponding author: Tieyong Zeng
^*Corresponding author: Tieyong Zeng

Received: March 2023

Revised: June 2023

Early access: August 2023

Published: February 2024

This work was supported in part by the National Key R & D Program of China under Grant 2021YFE0203700, Grant NSFC/RGC N_CUHK 415/19, Grant ITF MHP/038/20, Grant RGC 14300219, 14302920, 14301121, and CUHK Direct Grant for Research; and in part by the Natural Science Foundation of China (Grant No. 61971234, 12126340, 12126304, 11501301), the "QingLan" Project for Colleges and Universities of Jiangsu Province

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Abstract

The phase retrieval task has received considerable attention in recent years. Here, phase retrieval refers to the problem of recovering the clear image from magnitude-only data of its Fourier transform, or other linear transforms. As the observed magnitude signal is usually corrupted by heavy noise, retrieving the original object is rather difficult. In this paper, we investigate a regularization-based framelet method to recover the phase information and alleviate the influence of noise. Indeed, since phase retrieval is usually modeled by a non-convex model, how to find the solution efficiently is an intricate challenge. For this reason, we first reformulate the objective function as the difference between two convex functions. By introducing the boosted difference of convex algorithm (BDCA), the proposed scheme has good performance in handling the phase retrieval problem. Theoretically, we also present the convergence analysis of the numerical algorithm. To exhibit the effectiveness of our approach, we consider two classical regularizers with the proposed framework. Besides, two different measurement models are carefully studied to illustrate the robustness of our scheme. Numerical experiments demonstrate clearly that the proposed framelet method is effective and robust in tackling the non-convex phase retrieval task.

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Mathematics Subject Classification: Primary: 68U10, 94A08k, 90C47; Secondary: 65K10.

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Figure 1. The sensitive analysis of initial values. The proposed model (10) of the image 'cameraman' with the degradation of the $ 3n $ Fourier mask ratio 50% and $ \mathrm{SNR} = 30 $. (a) SNR(dB) results with 100 random values of $ \Phi(u) = \Vert\nabla u\Vert_1 $; (c) SNR(dB) results with 100 random values of $ \Phi(u) = \Vert\mathcal{W} u\Vert_1 $; the average visual results with (b) $ \mathrm{SNR} = 22.67 $ and (d) $ \mathrm{SNR} = 24.47 $

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Figure 2. Comparisons between our BDCA and DC algorithm of Image 'cameraman' with CDP mask type $ J = 2 $. Here the red, green, and blue lines are with $ \sigma = 0 $, $ \sigma = 10 $, and $ \sigma = 30 $ respectively

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Figure 3. Sensitive analysis with parameters $ \lambda $, $ \rho $. Image 'cameraman' with the degradation of the $ 3n $ Fourier mask ratio 50% and $ \mathrm{SNR} = 30 $

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Figure 4. Sensitive analysis with parameters $ r_1 $, $ r_2 $, and $ r_3 $. Image 'cameraman' with the degradation of the $ 3n $ Fourier mask ratio 50% and $ \mathrm{SNR} = 30 $. The first row displays the results of the proposed model with $ \Phi(u) = \Vert\nabla u\Vert_1 $ and the second row is the results of the proposed model with $ \Phi(u) = \Vert\mathcal{W}u\Vert_1 $

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Figure 5. The convergence behavior of the proposed model (10) of the image 'cameraman' with the degradation of the $ 3n $ Fourier mask ratio 50% and $ \mathrm{SNR} = 30 $

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Figure 6. The comparison results (SNR/SSIM) of Img10 with $ 3n $ Fourier mask ratio $ 50% $ and $ \sigma = 0 $

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Figure 7. The comparison results (SNR/SSIM) of Img08 with $ 3n $ Fourier mask ratio $ 50% $ and $ \sigma = 10 $

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Figure 8. The comparison results (SNR/SSIM) of (2D projection slice of molecule) with CDP mask type $ J = 2 $ and $ \sigma = 1 $

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Figure 9. The comparison results (SNR/SSIM) of Img13 with CDP mask type $ J = 2 $ and $ \sigma = 10 $

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Figure 10. The comparison results (SNR/SSIM) of Img03 with CDP mask type $ J = 2 $ and $ \sigma = 30 $

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Figure 11. The comparison results (SNR/SSIM) with random initial value of (2D projection slice of molecule) with CDP mask type $ J = 2 $ and $ \sigma = 10 $. (a) the original image; (b) The random value image with Matlab command 'u0 = rand(m, n)'; (c) RAF [48]; (d) TAF [47]; (e) TWF [15]; (f) Our TV with parameter $ \lambda = 1e30 $; (g) Our TF with parameter $ \lambda = 1e30 $; (h) Our TV; (i) Our TF

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Table 1. Average results of Set18 with SNR/SSIM/running time in second of phase retrieval (3n Fourier measurements)

mask $ 50% $	TVB [11]	Our TV	Our TF
$ \sigma=0 $	27.83/0.7411/24.12	27.99/0.7755/21.80	28.97/0.7952/60.33
$ \sigma=10 $	23.14/0.5472/24.37	24.00/0.5616/19.70	24.59/0.6148/57.28
$ \sigma=30 $	19.40/0.4117/24.70	20.08/0.4365/19.93	21.27/0.4677/60.17

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Table 2. Average results of Set18 with SNR/SSIM of phase retrieval (CDP measurement with $ J = 2 $)

CDP mask	ER [21]	RAAR [30]	RAF [48]	RWF [52]	TAF [47]	CDA [54]	HIO [19]	TWF [15]	WF [9]	Our TV	Our TF
$ \sigma=30 $	22.55/0.3406	22.94/0.3432	22.55/0.3408	22.55/0.3406	22.16/0.3418	21.72/0.3404	12.59/0.1212	22.03/0.3394	22.55/0.3406	27.02/0.6292	27.11/0.6406
$ \sigma=10 $	30.16/0.7323	30.56/0.7490	30.16/0.7324	30.16/0.7323	30.21/0.7345	30.06/0.7315	20.63/0.3424	30.20/0.7042	29.04/0.7402	34.53/0.8902	35.29/0.9002

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Table 3. Average running time in second of Set18 of phase retrieval (CDP measurement with $ J = 2 $)

CDP mask	ER [21]	RAAR [30]	RAF [48]	RWF [52]	TAF [47]	CDA [54]	HIO [19]	TWF [15]	WF [9]	Our TV	Our TF
$ \sigma=30 $	0.67	3.20	2.83	14.39	2.91	1.55	3.42	3.06	154.91	62.77	75.09
$ \sigma=10 $	0.66	3.06	2.75	10.61	2.81	1.27	2.92	3.01	136.06	61.21	73.85

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