Fast and robust phase retrieval for masked coherent diffractive imaging

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

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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Table 1. Numerical Comparison between ADMM and Baseline Method for Pb Growth Experiment
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Table 1. Numerical Comparison between ADMM and Baseline Method for Pb Growth Experiment
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. PnP-ADMM for Masked CDI

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Table 1. PnP-ADMM for Masked CDI

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 2. Mean Total Variation Loss of Different Methods for Glioblastoma Experiment
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Table 2. Mean Total Variation Loss of Different Methods for Glioblastoma Experiment
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 3. Processing Time of Different Algorithms in Pb Growth Experiment (Unit: s)
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Table 3. Processing Time of Different Algorithms in Pb Growth Experiment (Unit: s)
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 4. Processing Time of Different Algorithms in Glioblastoma Experiment (Unit: s)
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Table 4. Processing Time of Different Algorithms in Glioblastoma Experiment (Unit: s)
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

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

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Paper Information

Category: Image Processing and Image Analysis

Received: Nov. 9, 2021

Accepted: Jan. 20, 2022

Published Online: Mar. 1, 2022

The Author Email: Edmund Y. Lam (elam@eee.hku.hk)

DOI:10.1364/PRJ.447862

Topics

laser devices and laser physics

Lasers and Laser Optics

Laser physics

laser manufacturing

Instrumentation, Measurement and Metrology

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

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

Table 1. PnP-ADMM for Masked CDI

Table 1. PnP-ADMM for Masked CDI

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

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

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

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

Table 4. Processing Time of Different Algorithms in Glioblastoma Experiment (Unit: s)

Table 4. Processing Time of Different Algorithms in Glioblastoma Experiment (Unit: s)