Convergence of the Schulz-Snyder phase retrieval algorithm to local minima

Kerkil Choi; Aaron D. Lanterman; Raviv Raich

doi:10.1364/JOSAA.23.001835

Journal of the Optical Society of America A
Vol. 23,
Issue 8,
pp. 1835-1845
(2006)
•https://doi.org/10.1364/JOSAA.23.001835

Convergence of the Schulz–Snyder phase retrieval algorithm to local minima

Kerkil Choi, Aaron D. Lanterman, and Raviv Raich

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Kerkil Choi, Aaron D. Lanterman, and Raviv Raich, "Convergence of the Schulz-Snyder phase retrieval algorithm to local minima," J. Opt. Soc. Am. A 23, 1835-1845 (2006)

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Abstract

The Schulz–Snyder iterative algorithm for phase retrieval attempts to recover a nonnegative function from its autocorrelation by minimizing the I-divergence between a measured autocorrelation and the autocorrelation of the estimated image. We illustrate that the Schulz–Snyder algorithm can become trapped in a local minimum of the I-divergence surface. To show that the estimates found are indeed local minima, sufficient conditions involving the gradient and the Hessian matrix of the I-divergence are given. Then we build a brief proof showing how an estimate that satisfies these conditions is a local minimum. The conditions are used to perform numerical tests determining local minimality of estimates. Along with the tests, related numerical issues are examined, and some interesting phenomena are discussed.

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Experiment	$mx \nabla f_{0}$	$mn \nabla f_{0}$	$∣ A ∣$	$mx \nabla f (S_{1})$	$mn \nabla f (S_{1})$	$mx f (S_{2})$	$\nabla {mx f (S_{2})}$
1	58.66	$- 39.81$	1	$1.2 \times 10^{- 12}$	$5.1 \times 10^{- 12}$	$1.7 \times 10^{- 9}$	$5.2 \times 10^{- 5}$
2	86.33	$- 17.62$	2	$4.0 \times 10^{- 13}$	$9.3 \times 10^{- 13}$	$6.3 \times 10^{- 12}$	$1.8 \times 10^{- 4}$
3	305.99	254.64	0	$9.3 \times 10^{- 8}$	$9.7 \times 10^{- 8}$	$4.5 \times 10^{- 149}$	$8.8 \times 10^{- 3}$
4	27.76	$- 27.14$	1	$1.6 \times 10^{- 7}$	$8.0 \times 10^{- 7}$	$4.7 \times 10^{- 166}$	$6.5 \times 10^{- 3}$
5	35.51	$- 32.58$	2	$2.0 \times 10^{- 7}$	$3.4 \times 10^{- 7}$	$6.0 \times 10^{- 167}$	$2.5 \times 10^{- 2}$
6	36.55	$- 28.74$	1	$3.8 \times 10^{- 8}$	$2.8 \times 10^{- 6}$	$9.0 \times 10^{- 9}$	$1.2 \times 10^{- 4}$
7	31.52	$- 20.75$	2	$8.8 \times 10^{- 8}$	$1.0 \times 10^{- 7}$	$4.4 \times 10^{- 323}$	$5.2 \times 10^{- 2}$
8	34.08	$- 20.79$	2	$3.2 \times 10^{- 6}$	$3.5 \times 10^{- 9}$	$1.1 \times 10^{- 7}$	$3.3 \times 10^{- 3}$

Quantity	Exp. 1	Exp. 2	Exp. 3	Exp. 4

$D [S, S]$	$6.0 \times 10^{- 13}$	$7.5 \times 10^{- 13}$	$8.9 \times 10^{- 12}$	$6.2 \times 10^{- 13}$
$D [S, R_{f 0}]$	3220.71	716.34	18021.14	1581.95
$D [S, R_{f_{k}}]$	4.5947	6.3621	45.0530	0.1572
k	308,320	118,113	38,210	57,720
${Max}_{i} ∣ f_{k} (x_{i}) - f_{k - 1} (x_{i}) ∣$	$9.9 \times 10^{- 14}$	$9.9 \times 10^{- 14}$	$9.9 \times 10^{- 8}$	$9.9 \times 10^{- 8}$
$Size (H)$	$120 \times 120$	$122 \times 122$	$307 \times 307$	$111 \times 111$
$Max {eigenvalues (H)}$	484.8513	493.0640	1236.4256	469.1720
$Min {eigenvalues (H)}$	0.02472	0.0558	0.0327	0.0574

Quantity	Exp. 5	Exp. 6	Exp. 7	Exp. 8

$D [S, S]$	$5.9 \times 10^{- 13}$	$5.0 \times 10^{- 13}$	$8.5 \times 10^{- 13}$	$5.7 \times 10^{- 13}$
$D [S, R_{f_{0}}]$	1645.03	1982.54	3716.34	2685.69
$D [S, R_{f_{k}}]$	1.0190	1.2542	0.0807	0.0292
k	15,108	101,158	94,240	257,827
${Max}_{i} ∣ f_{k} (x_{i}) - f_{k - 1} (x_{i}) ∣$	$9.9 \times 10^{- 8}$	$9.9 \times 10^{- 9}$	$9.9 \times 10^{- 8}$	$9.9 \times 10^{- 9}$
$Size (H)$	$86 \times 86$	$110 \times 110$	$110 \times 110$	$122 \times 122$
$Max {eigenvalues (H)}$	353.1094	454.0213	474.7070	522.9897
$Min {eigenvalues (H)}$	0.1541	0.0543	0.0032	0.0022

Experiment	$mx \nabla f_{0}$	$mn \nabla f_{0}$	$∣ A ∣$	$mx \nabla f (S_{1})$	$mn \nabla f (S_{1})$	$mx f (S_{2})$	$\nabla {mx f (S_{2})}$
1	58.66	$- 39.81$	1	$1.2 \times 10^{- 12}$	$5.1 \times 10^{- 12}$	$1.7 \times 10^{- 9}$	$5.2 \times 10^{- 5}$
2	86.33	$- 17.62$	2	$4.0 \times 10^{- 13}$	$9.3 \times 10^{- 13}$	$6.3 \times 10^{- 12}$	$1.8 \times 10^{- 4}$
3	305.99	254.64	0	$9.3 \times 10^{- 8}$	$9.7 \times 10^{- 8}$	$4.5 \times 10^{- 149}$	$8.8 \times 10^{- 3}$
4	27.76	$- 27.14$	1	$1.6 \times 10^{- 7}$	$8.0 \times 10^{- 7}$	$4.7 \times 10^{- 166}$	$6.5 \times 10^{- 3}$
5	35.51	$- 32.58$	2	$2.0 \times 10^{- 7}$	$3.4 \times 10^{- 7}$	$6.0 \times 10^{- 167}$	$2.5 \times 10^{- 2}$
6	36.55	$- 28.74$	1	$3.8 \times 10^{- 8}$	$2.8 \times 10^{- 6}$	$9.0 \times 10^{- 9}$	$1.2 \times 10^{- 4}$
7	31.52	$- 20.75$	2	$8.8 \times 10^{- 8}$	$1.0 \times 10^{- 7}$	$4.4 \times 10^{- 323}$	$5.2 \times 10^{- 2}$
8	34.08	$- 20.79$	2	$3.2 \times 10^{- 6}$	$3.5 \times 10^{- 9}$	$1.1 \times 10^{- 7}$	$3.3 \times 10^{- 3}$

Quantity	Exp. 1	Exp. 2	Exp. 3	Exp. 4

$D [S, S]$	$6.0 \times 10^{- 13}$	$7.5 \times 10^{- 13}$	$8.9 \times 10^{- 12}$	$6.2 \times 10^{- 13}$
$D [S, R_{f 0}]$	3220.71	716.34	18021.14	1581.95
$D [S, R_{f_{k}}]$	4.5947	6.3621	45.0530	0.1572
k	308,320	118,113	38,210	57,720
${Max}_{i} ∣ f_{k} (x_{i}) - f_{k - 1} (x_{i}) ∣$	$9.9 \times 10^{- 14}$	$9.9 \times 10^{- 14}$	$9.9 \times 10^{- 8}$	$9.9 \times 10^{- 8}$
$Size (H)$	$120 \times 120$	$122 \times 122$	$307 \times 307$	$111 \times 111$
$Max {eigenvalues (H)}$	484.8513	493.0640	1236.4256	469.1720
$Min {eigenvalues (H)}$	0.02472	0.0558	0.0327	0.0574

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Figures (21)

Tables (3)

Equations (20)

Journal of the Optical Society of America A