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.

Download full size | View in the Article

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

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

Download full size | View in the Article

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

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

Download full size | View in the Article

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

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.

Download full size | View in the Article

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

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

Download full size | View in the Article

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

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.

Download full size | View in the Article

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

Download full size | View in the Article

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

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.

Download full size | View in the Article

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

View in the Article

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

View in the Article

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

View in the Article

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)

View in the Article

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