David R. Williams, David H. Brainard, Matthew J. McMahon, and Rafael Navarro, "Double-pass and interferometric measures of the optical quality of the eye," J. Opt. Soc. Am. A 11, 3123-3135 (1994)
We compare two methods for measuring the modulation transfer function (MTF) of the human eye: an interferometric method similar to that of Campbell and Green [
J. Physiol. (London) 181,
576 (
1965)] and a double-pass procedure similar to that of Santamaria et al. [
J. Opt. Soc. Am. A 4,
1109 (
1987)]. We implemented various improvements in both techniques to reduce error in the estimates of the MTF. We used the same observers, refractive state, pupil size (3 mm), and wavelength (632.8 nm) for both methods. In the double-pass method we found close agreement between the plane of subjective best focus for the observer and the plane of objective best focus, suggesting that much of the reflected light is confined within individual cones throughout its double pass through the receptor layer. The double-pass method produced MTF’s that were similar to but slightly lower than those of the interferometric method. This additional loss in modulation transfer is probably attributable to light reflected from the choroid, because green light, which reduces the contribution of the choroid to the fundus reflection, produces somewhat higher MTF’s that are consistent with the interferometric results. When either method is used, the MTF’s lie well below those obtained with the aberroscope method [
Vision Res. 28,
659 (
1988)]. On the basis of the interferometric method, we propose a new estimate of the monochromatic MTF of the eye.
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Tabulated Values of the Interferometric MTF Averaged across Three Observers and Standard Deviation Based on the Variability among Thema
Spatial Frequency
Average MTF
Standard Deviation
Observer DHB
Observer DRW
Observer RNB
MTF
SEM
MTF
SEM
MTF
SEM
10
0.458
0.034
0.482
0.050
0.472
0.011
0.419
0.043
20
0.291
0.055
0.317
0.054
0.228
0.025
0.327
0.027
30
0.178
0.037
0.220
0.015
0.164
0.029
0.150
0.027
40
0.147
0.037
0.185
0.013
0.112
0.018
0.145
0.021
50
0.119
0.052
0.178
0.025
0.080
0.014
0.099
0.024
Individual MTF’s are also tabulated for each observer, along with the standard error of the mean, SEM, based on the variability between estimates of the MTF from three experimental sessions.
Table 2
Tabulated Values of the PSF and LSF Estimated from the Mean Interferometric MTFa
Distance (arcmin)
PSF
LSF
Distance (arcmin)
PSF (× 104)
LSF
0.00
6.98 × 106
1.95 × 103
2.60
7.69
1.27 × 102
0.10
6.69 × 106
1.87 × 103
2.70
7.22
1.19 × 102
0.20
5.87 × 106
1.68 × 103
2.80
6.47
1.11 × 102
0.30
4.71 × 106
1.40 × 103
2.90
5.62
1.04 × 102
0.40
3.43 × 106
1.10 × 103
3.00
4.89
9.79 × 10
0.50
2.27 × 106
8.35 × 102
3.10
4.45
9.35 × 10
0.60
1.37 × 106
6.36 × 102
3.20
4.26
8.99 × 10
0.70
7.90 × 105
5.13 × 102
3.30
4.17
8.63 × 10
0.80
5.02 × 105
4.52 × 102
3.40
4.01
8.22 × 10
0.90
4.12 × 105
4.26 × 102
3.50
3.70
7.77 × 10
1.00
4.14 ×105
4.10 × 102
3.60
3.29
7.33 × 10
1.10
4.26 × 105
3.89 × 102
3.70
2.91
6.97 × 10
1.20
4.06 × 105
3.58 × 102
3.80
2.65
6.70 × 10
1.30
3.52 × 105
3.21 × 102
3.90
2.54
6.48 × 10
1.40
2.84 × 105
2.87 × 102
4.00
2.50
6.28 × 10
1.50
2.25 × 105
2.60 × 102
4.10
2.44
6.04 × 10
1.60
1.89 × 105
2.41 × 102
4.20
2.31
5.77 × 10
1.70
1.74 ×105
2.28 × 102
4.30
2.10
5.50 × 10
1.80
1.68 × 105
2.16 × 102
4.40
1.88
5.26 × 10
1.90
1.61 × 105
2.03 × 102
4.50
1.72
5.08 × 10
2.00
1.47 × 105
1.87 × 102
4.60
1.64
4.94 ×10
2.10
1.27 × 105
1.71 × 102
4.70
1.62
4.81 ×10
2.20
1.06 × 105
1.58 × 102
4.80
1.61
4.67 × 10
2.30
9.10 × 104
1.47 × 102
4.90
1.54
4.50 × 10
2.40
8.28 ×104
1.40 × 102
5.00
1.43
4.32 × 10
2.50
7.93 × 104
1.34 × 102
We made two important assumptions to compute these data: (1) that the 2-D MTF is circularly symmetric, although our measurements were restricted to one dimension, and (2) that the PSF is an even function (i.e., that the phase transfer function is zero), although our measurements provide no information about phase. See text for details. To compute the PSF, we used the analytic fit to the measured MTF to generate a circularly symmetric MTF on a 512 × 512 pixel grid. We used the fast Fourier transform to compute a raw PSF from the MTF. To generate the table, we extracted a radial slice of the raw PSF and interpolated with a piecewise polynomial. The tabulated PSF is normalized so that it represents the fraction of incident light scattered per steradian. To normalize, we divided the PSF values by the volume under the entire 2-D PSF. The tabulated values may be converted to units of fraction scattered per square degree by multiplication by a factor of (π/180)2. To compute the LSF, we used the analytic fit to the MTF to generate a 1-D MTF on a 512-pixel line. We used the fast Fourier transform to compute a raw LSF. To generate the table, we interpolated the raw LSF with a piecewise polynomial. The tabulated LSF is normalized so that it represents the fraction of incident light scattered per radian. To normalize, we divided the LSF values by the area under the entire 1-D LSF. The tabulated values may be converted to units of fraction scattered per degree by multiplication by a factor of (π/180).
Table 3
Mean Objective Minus Mean Subjective Focus in Diopters for Various Observers at Various Retinal Locations
Retinal Location
Observer
Mean Objective Minus Mean Subjective Focus (D)
Standard Error of the Mean Difference
Fovea
DRW
−0.01
0.08
RNB
0.00
0.17
PA
0.07
0.04
Extrafovea
DRW
4-deg nasal
0.08
0.09
2-deg inferior
DRW
8-deg nasal
−0.14
0.10
RBN
4.4-deg nasal
−0.05
0.33
Mean
−0.01
Standard error
0.03
Tables (3)
Table 1
Tabulated Values of the Interferometric MTF Averaged across Three Observers and Standard Deviation Based on the Variability among Thema
Spatial Frequency
Average MTF
Standard Deviation
Observer DHB
Observer DRW
Observer RNB
MTF
SEM
MTF
SEM
MTF
SEM
10
0.458
0.034
0.482
0.050
0.472
0.011
0.419
0.043
20
0.291
0.055
0.317
0.054
0.228
0.025
0.327
0.027
30
0.178
0.037
0.220
0.015
0.164
0.029
0.150
0.027
40
0.147
0.037
0.185
0.013
0.112
0.018
0.145
0.021
50
0.119
0.052
0.178
0.025
0.080
0.014
0.099
0.024
Individual MTF’s are also tabulated for each observer, along with the standard error of the mean, SEM, based on the variability between estimates of the MTF from three experimental sessions.
Table 2
Tabulated Values of the PSF and LSF Estimated from the Mean Interferometric MTFa
Distance (arcmin)
PSF
LSF
Distance (arcmin)
PSF (× 104)
LSF
0.00
6.98 × 106
1.95 × 103
2.60
7.69
1.27 × 102
0.10
6.69 × 106
1.87 × 103
2.70
7.22
1.19 × 102
0.20
5.87 × 106
1.68 × 103
2.80
6.47
1.11 × 102
0.30
4.71 × 106
1.40 × 103
2.90
5.62
1.04 × 102
0.40
3.43 × 106
1.10 × 103
3.00
4.89
9.79 × 10
0.50
2.27 × 106
8.35 × 102
3.10
4.45
9.35 × 10
0.60
1.37 × 106
6.36 × 102
3.20
4.26
8.99 × 10
0.70
7.90 × 105
5.13 × 102
3.30
4.17
8.63 × 10
0.80
5.02 × 105
4.52 × 102
3.40
4.01
8.22 × 10
0.90
4.12 × 105
4.26 × 102
3.50
3.70
7.77 × 10
1.00
4.14 ×105
4.10 × 102
3.60
3.29
7.33 × 10
1.10
4.26 × 105
3.89 × 102
3.70
2.91
6.97 × 10
1.20
4.06 × 105
3.58 × 102
3.80
2.65
6.70 × 10
1.30
3.52 × 105
3.21 × 102
3.90
2.54
6.48 × 10
1.40
2.84 × 105
2.87 × 102
4.00
2.50
6.28 × 10
1.50
2.25 × 105
2.60 × 102
4.10
2.44
6.04 × 10
1.60
1.89 × 105
2.41 × 102
4.20
2.31
5.77 × 10
1.70
1.74 ×105
2.28 × 102
4.30
2.10
5.50 × 10
1.80
1.68 × 105
2.16 × 102
4.40
1.88
5.26 × 10
1.90
1.61 × 105
2.03 × 102
4.50
1.72
5.08 × 10
2.00
1.47 × 105
1.87 × 102
4.60
1.64
4.94 ×10
2.10
1.27 × 105
1.71 × 102
4.70
1.62
4.81 ×10
2.20
1.06 × 105
1.58 × 102
4.80
1.61
4.67 × 10
2.30
9.10 × 104
1.47 × 102
4.90
1.54
4.50 × 10
2.40
8.28 ×104
1.40 × 102
5.00
1.43
4.32 × 10
2.50
7.93 × 104
1.34 × 102
We made two important assumptions to compute these data: (1) that the 2-D MTF is circularly symmetric, although our measurements were restricted to one dimension, and (2) that the PSF is an even function (i.e., that the phase transfer function is zero), although our measurements provide no information about phase. See text for details. To compute the PSF, we used the analytic fit to the measured MTF to generate a circularly symmetric MTF on a 512 × 512 pixel grid. We used the fast Fourier transform to compute a raw PSF from the MTF. To generate the table, we extracted a radial slice of the raw PSF and interpolated with a piecewise polynomial. The tabulated PSF is normalized so that it represents the fraction of incident light scattered per steradian. To normalize, we divided the PSF values by the volume under the entire 2-D PSF. The tabulated values may be converted to units of fraction scattered per square degree by multiplication by a factor of (π/180)2. To compute the LSF, we used the analytic fit to the MTF to generate a 1-D MTF on a 512-pixel line. We used the fast Fourier transform to compute a raw LSF. To generate the table, we interpolated the raw LSF with a piecewise polynomial. The tabulated LSF is normalized so that it represents the fraction of incident light scattered per radian. To normalize, we divided the LSF values by the area under the entire 1-D LSF. The tabulated values may be converted to units of fraction scattered per degree by multiplication by a factor of (π/180).
Table 3
Mean Objective Minus Mean Subjective Focus in Diopters for Various Observers at Various Retinal Locations