Fast and robust phase retrieval for masked coherent diffractive imaging

Li Song; Edmund Y. Lam

doi:10.1364/PRJ.447862

Journals >Photonics Research >Volume 10 >Issue 3 >Page 758 > Article

Photonics Research
Vol. 10, Issue 3, 758 (2022)

Fast and robust phase retrieval for masked coherent diffractive imaging

Li Song and Edmund Y. Lam^*

Author Affiliations

Department of Electrical and Electronic Engineering, The University of Hong Kong, Pokfulam, Hong Kong, China

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DOI: 10.1364/PRJ.447862 Cite this Article Set citation alerts

Li Song, Edmund Y. Lam. Fast and robust phase retrieval for masked coherent diffractive imaging[J]. Photonics Research, 2022, 10(3): 758 Copy Citation Text

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Fig. 1. Typical masked CDI: in situ CDI setup [31].

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$Imaging results of the Pb growth experiment (amplitude reconstructions). (a)–(f), (m)–(r) ADMM phase retrieval results for No. 1–12 diffraction patterns; (g)–(l), (s)–(x) phase retrieval results for No. 1–12 diffraction patterns using the baseline method, respectively.$

Fig. 2. Imaging results of the Pb growth experiment (amplitude reconstructions). (a)–(f), (m)–(r) ADMM phase retrieval results for No. 1–12 diffraction patterns; (g)–(l), (s)–(x) phase retrieval results for No. 1–12 diffraction patterns using the baseline method, respectively.

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$ADMM iterations for the Pb growth experiment (different colors for different diffraction patterns). Both converge quickly. (a) Error. (b) R-factor.$

Fig. 3. ADMM iterations for the Pb growth experiment (different colors for different diffraction patterns). Both converge quickly. (a) Error. (b) R-factor.

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$Reconstructed object planes for No. 1 diffraction pattern. The main difference is the area outside the support. (a) ADMM. (b) Baseline.$

Fig. 4. Reconstructed object planes for No. 1 diffraction pattern. The main difference is the area outside the support. (a) ADMM. (b) Baseline.

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$Imaging results of No. 1 diffraction pattern in the Pb growth experiment (amplitude reconstructions) using different phase retrieval methods. (a) Baseline. (b) ADMM. (c) WF. (d) TWF.$

Fig. 5. Imaging results of No. 1 diffraction pattern in the Pb growth experiment (amplitude reconstructions) using different phase retrieval methods. (a) Baseline. (b) ADMM. (c) WF. (d) TWF.

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$ADMM iterations for the glioblastoma experiment (different colors for different diffraction patterns). (a) Error. (b) R-factor.$

Fig. 6. ADMM iterations for the glioblastoma experiment (different colors for different diffraction patterns). (a) Error. (b) R-factor.

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$Imaging results of the glioblastoma experiment (phase reconstructions). (a)–(f), (m)–(r) ADMM phase retrieval results for No. 1–12 diffraction patterns; (g)–(l), (s)–(x) phase retrieval results for No. 1–12 diffraction patterns using the baseline method, respectively.$

Fig. 7. Imaging results of the glioblastoma experiment (phase reconstructions). (a)–(f), (m)–(r) ADMM phase retrieval results for No. 1–12 diffraction patterns; (g)–(l), (s)–(x) phase retrieval results for No. 1–12 diffraction patterns using the baseline method, respectively.

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Fig. 8. Processing time in different experiments (unit: s). (a) Pb growth experiment. (b) Glioblastoma experiment.

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$Phase retrieval results with single diffraction pattern. (a), (b) Baseline method and ADMM method for No. 10 diffraction pattern in the Pb growth experiment; (c), (d) baseline method and ADMM method for No. 7 diffraction pattern in the glioblastoma experiment, respectively.$

Fig. 9. Phase retrieval results with single diffraction pattern. (a), (b) Baseline method and ADMM method for No. 10 diffraction pattern in the Pb growth experiment; (c), (d) baseline method and ADMM method for No. 7 diffraction pattern in the glioblastoma experiment, respectively.

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No.	1	2	3	4	5	6
PSNR (dB)	31.0	32.5	33.5	34.7	34.7	34.1
SSIM	0.88	0.85	0.84	0.86	0.86	0.86
No.	7	8	9	10	11	12
PSNR (dB)	34.0	33.4	33.7	34.3	33.6	33.1
SSIM	0.85	0.84	0.86	0.86	0.86	0.86

Table 1. Numerical Comparison between ADMM and Baseline Method for Pb Growth Experiment

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Input:

o

: captured diffraction pattern;

L

: probe;

W_{o}, W_{b}

: dynamic and static masks;

r

: known static pattern;

R

: regularization function;

{\hat{u}}^{(0)}, ϕ^{(0)}, u^{(0)}, λ^{(0)}, μ^{(0)}

: initial values;

ρ

τ

: penalty parameters;

ϵ_{tol}

: error tolerance.

Output: optimal reconstructed image

u^{*}

1: repeat

{\hat{u}}^{(k + 1)} = {(ρ L^{H} F^{H} F L + τ W_{o}^{H} W_{o})}^{- 1} [ρ {(F L)}^{H} (o \circ ϕ^{(k)} - λ^{(k)})

+ τ W_{o}^{H} (u^{(k)} + W_{o} W_{b} r + μ^{(k)})]

;

ϕ^{(k + 1)} = P_{o} (λ^{(k)} + F L {\hat{u}}^{(k + 1)})

;

u^{(k + 1)} = {prox}_{\frac{R}{τ}} [W_{o} ({\hat{u}}^{(k + 1)} - W_{b} r - μ^{(k)})]

;

λ^{(k + 1)} = λ^{(k)} + F L {\hat{u}}^{(k + 1)} - o \circ ϕ^{(k + 1)}

;

μ^{(k + 1)} = μ^{(k)} + u^{(k + 1)} - W_{o} ({\hat{u}}^{(k + 1)} - W_{b} r)

8: until

(\frac{‖ u^{(k + 1)} - u^{(k)} ‖_{2}}{‖ u^{(k + 1)} ‖_{2}} < ϵ_{tol})

Table 1. PnP-ADMM for Masked CDI

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No.	1	2	3	4	5	6
ADMM	0.599	0.601	0.595	0.642	0.816	0.820
Baseline	0.846	0.822	0.869	0.940	1.372	1.527
No.	7	8	9	10	11	12
ADMM	0.818	0.811	0.779	0.722	0.705	0.648
Baseline	1.485	1.445	1.397	1.316	1.365	1.378

Table 2. Mean Total Variation Loss of Different Methods for Glioblastoma Experiment

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No.	1	2	3	4	5	6	7
ADMM	0.39	0.38	0.37	0.37	0.38	0.37	0.37
Baseline	39.41	40.74	38.80	39.10	39.29	38.96	39.23
No.	8	9	10	11	12	Avg	Var
ADMM	0.37	0.36	0.37	0.36	0.37	0.37	$7 \times 10^{- 5}$
Baseline	39.24	39.12	39.24	38.88	39.45	39.29	0.25

Table 3. Processing Time of Different Algorithms in Pb Growth Experiment (Unit: s)

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No.	1	2	3	4	5	6	7
ADMM	0.35	0.34	0.34	0.33	0.33	0.34	0.33
Baseline	40.25	39.97	39.93	40.64	40.56	40.14	40.15
No.	8	9	10	11	12	Avg	Var
ADMM	0.34	0.34	0.34	0.34	0.34	0.34	$2 \times 10^{- 5}$
Baseline	41.09	39.76	40.56	40.00	41.50	40.38	0.27