Effect of point-spread functions on precision edge measurement

Raymond G. White; Robert A. Schowengerdt

doi:10.1364/JOSAA.11.002593

Journal of the Optical Society of America A
Vol. 11,
Issue 10,
pp. 2593-2603
(1994)
•https://doi.org/10.1364/JOSAA.11.002593

Effect of point-spread functions on precision edge measurement

Raymond G. White and Robert A. Schowengerdt

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Raymond G. White and Robert A. Schowengerdt, "Effect of point-spread functions on precision edge measurement," J. Opt. Soc. Am. A 11, 2593-2603 (1994)

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Abstract

We explore the effect that an image-formation system’s point-spread function (PSF) has on measurement of the orientation and position of step edges. The theory of locales is used for numerical computation of bounds on precision as a function of PSF size and shape. It is shown that in the noise-free case the PSF should be 0.5–0.9 pixel in radius for optimal precision to be achieved. PSF shape is shown to have a much lesser effect on precision. A one-dimensional noisy-edge model is analyzed, and results similar to the no-noise case are obtained.

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PSF Shape	Parameter	β_ρ (pixels)	${\bar{β}}_{ρ}$ (pixels)	β_θ (deg)	${\bar{β}}_{θ}$ (deg)
Size (N): 3
Rectangle	S	0.54	0.62	0.66	0.64
	Precision	0.131	0.037	9.018	2.409
Defocus	S	0.55	0.64	0.69	0.64
	Precision	0.123	0.037	9.018	2.440
Gaussian	S	0.66	0.77	0.83	0.75
	Precision	0.123	0.037	9.018	2.439
Airy	S	0.67	0.78	0.86	0.78
	Precision	0.123	0.037	9.018	2.439
Size (N): 5
Rectangle	S	0.54	0.59	0.66	0.58
	Precision	0.131	0.027	4.554	1.044
Defocus	S	0.55	0.64	0.71	0.57
	Precision	0.123	0.028	4.554	1.059
Gaussian	S	0.66	0.77	0.86	0.68
	Precision	0.123	0.028	4.554	1.058
Airy	S	0.67	0.78	0.89	0.69
	Precision	0.123	0.028	4.554	1.059
Size (N): 7
Rectangle	S	0.54	0.62	0.67	0.73
	Precision	0.131	0.021	3.036	0.599
Defocus	S	0.55	0.65	0.70	0.74
	Precision	0.123	0.022	3.036	0.606
Gaussian	S	0.66	0.79	0.84	0.89
	Precision	0.123	0.022	3.036	0.606
Airy	S	0.67	0.81	0.86	0.91
	Precision	0.123	0.022	3.036	0.606
Size (N): 9
Rectangle	S	0.54	0.54	0.64	0.55
	Precision	0.131	0.018	2.232	0.384
Defocus	S	0.55	0.66	0.71	0.64
	Precision	0.123	0.018	2.232	0.390
Gaussian	S	0.66	0.79	0.79	0.77
	Precision	0.123	0.018	2.232	0.390
Airy	S	0.67	0.81	0.86	0.78
	Precision	0.123	0.018	2.232	0.390
Size (N): 11
Rectangle	S	0.54	0.63	0.63	0.54
	Precision	0.131	0.016	1.696	0.271
Defocus	S	0.55	0.66	0.68	0.48
	Precision	0.123	0.016	1.786	0.275
Gaussian	S	0.66	0.81	0.79	0.59
	Precision	0.123	0.016	1.786	0.275
Airy	S	0.67	0.81	0.83	0.59
	Precision	0.123	0.016	1.786	0.275

PSF Shape	Parameter	β_ρ (pixels)	${\bar{β}}_{ρ}$ (pixels)	β_θ (deg)	${\bar{β}}_{θ}$ (deg)
Size (N): 3
Rectangle	S	0.52	0.56	0.58	0.55
	Precision	0.063	0.017	3.929	1.104
Defocus	S	0.51	0.55	0.57	0.56
	Precision	0.065	0.017	4.464	1.113
Gaussian	S	0.56	0.63	0.81	0.63
	Precision	0.071	0.017	4.732	1.132
Airy	S	0.54	0.62	0.81	0.62
	Precision	0.075	0.017	4.554	1.147
Size (N): 5
Rectangle	S	0.52	0.56	0.58	0.53
	Precision	0.063	0.012	1.964	0.470
Defocus	S	0.52	0.56	0.58	0.52
	Precision	0.065	0.012	2.143	0.475
Gaussian	S	0.56	0.65	0.83	0.59
	Precision	0.071	0.012	2.143	0.478
Airy	S	0.54	0.62	0.83	0.59
	Precision	0.075	0.012	2.143	0.480
Size (N): 7
Rectangle	S	0.52	0.56	0.58	0.53
	Precision	0.063	0.010	1.250	0.264
Defocus	S	0.52	0.56	0.59	0.55
	Precision	0.065	0.010	1.339	0.267
Gaussian	S	0.56	0.61	0.81	0.63
	Precision	0.071	0.010	1.429	0.268
Airy	S	0.54	0.62	0.83	0.62
	Precision	0.075	0.010	1.339	0.267
Size (N): 9
Rectangle	S	0.52	0.56	0.57	0.53
	Precision	0.063	0.008	0.893	0.167
Defocus	S	0.52	0.57	0.58	0.55
	Precision	0.065	0.008	0.982	0.170
Gaussian	S	0.56	0.61	0.74	0.61
	Precision	0.071	0.008	0.982	0.169
Airy	S	0.54	0.62	0.83	0.59
	Precision	0.075	0.008	0.982	0.168
Size (N): 11
Rectangle	S	0.52	0.56	0.55	0.53
	Precision	0.063	0.007	0.714	0.115
Defocus	S	0.52	0.57	0.58	0.48
	Precision	0.065	0.007	0.714	0.116
Gaussian	S	0.56	0.65	0.68	0.61
	Precision	0.071	0.007	0.714	0.115
Airy	S	0.54	0.62	0.65	0.62
	Precision	0.075	0.007	0.714	0.115

PSF Shape	Parameter (pixels)	β_ρ	${\bar{β}}_{ρ}$ (pixels)	β_θ	${\bar{β}}_{θ}$ (deg)
Rectangle Δ_x = 0.4 ⊗ defocus	R	0.50	0.65	0.75	0.80
Rectangle Δ_x = 0.4 ⊗ 0 Airy	B⁻¹	0.30	0.45	0.55	0.10
Rectangle Δ_x = 0.5 ⊗ defocus	R	0.30	0.60	0.70	0.75
Rectangle Δ_x = 0.5 ⊗ Airy	B⁻¹	0.10	0.35	0.40	0.50

PSF Shape	Parameter (pixels)	β_ρ	${\bar{β}}_{ρ}$ (pixels)	β_θ	${\bar{β}}_{θ}$ (deg)
Rectangle Δ_x = 0.4 ⊗ defocus	R	0.35	0.45	0.50	0.45
Rectangle Δ_x = 0.4 ⊗ Airy	B⁻¹	0.20	0.25	0.30	0.25
Rectangle Δ_x = 0.5 ⊗ defocus	R	0.20	0.30	0.35	0.20
Rectangle Δ_x = 0.5 ⊗ Airy	B⁻¹	0.05	0.10	0.15	0.10

PSF Shape	β_ρ (pixels)	${\bar{β}}_{ρ}$ (pixels)	β_θ (deg)	${\bar{β}}_{θ}$ (deg)
Rectangle Δ_x = 0.4 ⊗ defocus	0.128	0.022	3.060	0.612
Rectangle Δ_x = 0.4 ⊗ Airy	0.126	0.022	3.060	0.610
Rectangle Δ_x = 0.5 ⊗ defocus	0.124	0.022	3.060	0.611
Rectangle Δ_x = 0.5 ⊗ Airy	0.130	0.022	2.970	0.610

PSF Shape	β_ρ (pixels)	${\bar{β}}_{ρ}$ (pixels)	β_θ (deg)	${\bar{β}}_{θ}$ (deg)
Rectangle Δ_x = 0.4 ⊗ defocus	0.072	0.010	1.440	0.271
Rectangle Δ_x = 0.4 ⊗ Airy	0.068	0.010	1.440	0.269
Rectangle Δ_x = 0.5 ⊗ defocus	0.062	0.010	1.440	0.269
Rectangle Δ_x = 0.5 ⊗ Airy	0.064	0.010	1.440	0.267

Abstract

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Journal of the Optical Society of America A