Iterative and one-step reconstruction from nonuniform samples by convex projections

Shu-jen Yeh; Henry Stark

doi:10.1364/JOSAA.7.000491

Journal of the Optical Society of America A
Vol. 7,
Issue 3,
pp. 491-499
(1990)
•https://doi.org/10.1364/JOSAA.7.000491

Iterative and one-step reconstruction from nonuniform samples by convex projections

Shu-jen Yeh and Henry Stark

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Shu-jen Yeh and Henry Stark, "Iterative and one-step reconstruction from nonuniform samples by convex projections," J. Opt. Soc. Am. A 7, 491-499 (1990)

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Abstract

We apply the method of projections onto convex sets (POCS) to recover a signal or image from nonuniform samples and prior knowledge. Both iterative and one-step algorithms to implement POCS are furnished. We discuss our results in relation to the recent work by Sauer and Allebach [ IEEE Trans. Circuits Syst. CAS-34, 1497 ( 1987)], who considered the same problem from a different point of view.

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Data Sets	One-Step Projection Method: Mean-Square Error	Iterative Method		Iterative Method with Magnitude Constraint

		Mean Square Error	Number of Iterations	Mean Squared Error	Number of Iterations
S1	4.42 × 10⁻²	7.54 × 10⁻²	149	6.48 × 10⁻²	144
S2	1.75 × 10⁻³	.275	82	.228	63
S3	4.73 × 10⁻²	8.89 × 10⁻²	81	7.86 × 10⁻²	75
S4	0.48	.196	75	.149	216
S5	0.24	.2097	107	.247	118
Average	6.55 × 10⁻²	1.36 × 10⁻¹	88.2	1.39 × 10⁻¹	89.9

Data Sets	Without Ma gnitude Constraint		With Magnitude Constratint

	Mean-Squared Error	Number of Iterations	Mean-Squared Error	Number of Iterations
S1	0.271	106	0.242	96
S2	0.141	130	0.114	90
S3	0.340	89	0.345	81
S4	0.491	96	0.367	85
S5	0.571	129	0.473	64

Data Sets	One-Step Method: S/E (dB)	POCS Method Iterative		Smeared Iterative Method

		S/E (dB)	Number of Iterations	S/E (dB)	Number of Iterations
S1	12.56	12.37	200	11.24	200
S2	14.30	14.28	95	13.47	200
S3	6.75	6.57	200	7.89	200
S4	7.10	7.05	200	8.46	200
S5	12.60	12.60	54	15.47	126
S6	13.47	13.74	37	17.03	184
S7	11.64	11.64	57	18.26	150
S8	15.59	15.59	37	<11.20	162
Average	11.79	11.73	110	12.92	177.8

Data Sets	One-Step Projection Method: Mean-Square Error	Iterative Method		Iterative Method with Magnitude Constraint

		Mean Square Error	Number of Iterations	Mean Squared Error	Number of Iterations
S1	4.42 × 10⁻²	7.54 × 10⁻²	149	6.48 × 10⁻²	144
S2	1.75 × 10⁻³	.275	82	.228	63
S3	4.73 × 10⁻²	8.89 × 10⁻²	81	7.86 × 10⁻²	75
S4	0.48	.196	75	.149	216
S5	0.24	.2097	107	.247	118
Average	6.55 × 10⁻²	1.36 × 10⁻¹	88.2	1.39 × 10⁻¹	89.9

Data Sets	Without Ma gnitude Constraint		With Magnitude Constratint

	Mean-Squared Error	Number of Iterations	Mean-Squared Error	Number of Iterations
S1	0.271	106	0.242	96
S2	0.141	130	0.114	90
S3	0.340	89	0.345	81
S4	0.491	96	0.367	85
S5	0.571	129	0.473	64

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Journal of the Optical Society of America A