Abstract

We have studied the ability of observers to discriminate between suprathreshold vertical sinusoidal spatial-frequency gratings on the basis of spatial frequency. The results show that spatial-frequency discrimination is not a smooth function of spatial frequency but instead appears regularly segmented. Similar results were also obtained in an experiment in which observers discriminated between pairs of narrow vertical lines on the basis of their separation. Angular resolutions achieved for both discrimination tasks were less than the spacing between photoreceptors, requiring some type of neural interpolation. The similarity between the two sets of data indicates that discrimination between spatial-frequency gratings is probably based on the separation between two features exactly one cycle apart. We suggest that the segmentation reflects the existence of neural-image representations with discrete levels of spatial accuracy.

© 1982 Optical Society of America

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  1. H. B. Barlow, “Reconstructing the visual image in space and time,” Nature,  279, 189–190 (1979).
    [Crossref] [PubMed]
  2. G. Westheimer, “The spatial sense of the eye,” Invest. Ophthalmol. Visual Sci. 18, 893–912 (1979).
  3. M. Fahle and T. Poggio, “Visual hyperacuity: spatiotemporal interpolation in human vision,” Proc. R. Soc. Lond. B 213, 451–477 (1981).
    [Crossref]
  4. It was found that the contrast-matching experiments were necessary for medium and high spatial frequencies because test gratings of different frequencies and equal contrasts had perceptibly different apparent contrasts that were due to the slope of the modulation transfer function. The contrast-matching experiments were similar to the frequency-discrimination experiment described here. The reference grating was presented at 30% contrast, and one test-grating frequency was randomly presented at seven different contrast levels. On each trial the observer indicated whether the contrast of the test grating was higher or lower than the contrast of the reference grating. A contrast level that appeared equivalent to the reference grating at 30% contrast was determined from the psychometric function.
  5. G. Westheimer and S. P. McKee, “Spatial configurations for visual hyperacuity,” Vision Res. 17, 941–947 (1977).
    [Crossref] [PubMed]
  6. G. Westheimer and S. P. McKee, “Integration regions for visual hyperacuity,” Vision Res. 17, 89–93 (1977).
    [Crossref] [PubMed]
  7. F. W. Campbell, J. Nachmias, and J. Jukes, “Spatial-frequency discrimination in human vision,” J. Opt. Soc. Am. 60, 555–559 (1970).
    [Crossref] [PubMed]
  8. Each neural unit receives input from M photoreceptors, where M must be greater than 2. If M were equal to 2, only linear interpolation would be possible, and it would not be possible to reconstruct a peak falling between two photoreceptors. If M were equal to 3, quadratic interpolation would be possible. Since sinusoids and line-spread functions are quadratic near a peak, this might be sufficient. If the interpolation were done in two dimensions, quadratic interpolation would require input from at least six photoreceptors. The figure shows the case of M= 2 for clarity.
  9. If N were not an integer, all photoreceptors would not project to the same number of neural units, and the neural map would not be coherent with the photoreceptor lattice, thus resulting in a chaotic structure. Strictly speaking, coherence requires only that the ratio of the photoreceptor spacing to the neural-array spacing be the ratio of two integers. However, the period of the resulting structure is more than one photoreceptor, and many more weights would be required.
  10. Spatial frequencies below 2 c/deg are somewhat problematic. If we say that they should fall on the N= 1 line, Δf/f would become too small, violating the assumption that the whole mechanism is intended to keep Δf/f roughly constant. Thus the model does not make predictions below 2 c/deg.
  11. The fitting procedure determined the values of ξ and f1 (the first transition frequency), which minimized the chi squared for the agreement between the model and the data. The procedure was not entirely straightforward since close inspection of the data shows that the transition frequencies are not exactly evenly spaced, and there are also points that fall in the transition regions between segments. Good chi-squared values (confidence greater than 0.05) were obtained by permitting a small error (roughly 0.2 c/deg) in the transition frequencies and also by omitting a small number of points from the fit (1 of 26 points above 2 c/deg for JH, 3 of 25 points for BA, 2 of 25 points for MM, and no points for the JH line-separation data). These omitted points fell in transition regions between segments, except those for observer MM, for whom points at 6 and 6.5 c/deg seem to be outliers. However, the fitted value of ξ was insensitive to these points. The best f1 values for the three frequency-discrimination observers are 3.75, 4.25, and 4.38, and for the line separation observer it is 3.75. Since chi squared is not a continuous function of f1 (and we permit small deviations from equally spaced transitions), the errors on f1 are difficult to define.
  12. G. Westheimer and F. W. Campbell, “Light distribution in the image formed by the living human eye,” J. Opt. Soc. Am. 52, 1040–1045 (1962).
    [Crossref] [PubMed]
  13. M. B. Sachs, J. Nachmias, and J. G. Robson, “Spatial frequency channels in human vision,” J. Opt. Soc. Am. 61, 1176–1186 (1971).
    [Crossref] [PubMed]
  14. N. Graham, “Psychophysics of spatial-frequency channels,” in Perceptual Organization, M. Kubovy and J. Pomerantz, eds. (Lawrence Erlbaum, Hillsdale, N.J., 1981), pp. 1–25.
  15. H. R. Wilson and J. R. Bergen, “A four mechanism model for threshold spatial vision,” Vision Res. 19, 19–31 (1978).
    [Crossref]

1981 (1)

M. Fahle and T. Poggio, “Visual hyperacuity: spatiotemporal interpolation in human vision,” Proc. R. Soc. Lond. B 213, 451–477 (1981).
[Crossref]

1979 (2)

H. B. Barlow, “Reconstructing the visual image in space and time,” Nature,  279, 189–190 (1979).
[Crossref] [PubMed]

G. Westheimer, “The spatial sense of the eye,” Invest. Ophthalmol. Visual Sci. 18, 893–912 (1979).

1978 (1)

H. R. Wilson and J. R. Bergen, “A four mechanism model for threshold spatial vision,” Vision Res. 19, 19–31 (1978).
[Crossref]

1977 (2)

G. Westheimer and S. P. McKee, “Spatial configurations for visual hyperacuity,” Vision Res. 17, 941–947 (1977).
[Crossref] [PubMed]

G. Westheimer and S. P. McKee, “Integration regions for visual hyperacuity,” Vision Res. 17, 89–93 (1977).
[Crossref] [PubMed]

1971 (1)

1970 (1)

1962 (1)

Barlow, H. B.

H. B. Barlow, “Reconstructing the visual image in space and time,” Nature,  279, 189–190 (1979).
[Crossref] [PubMed]

Bergen, J. R.

H. R. Wilson and J. R. Bergen, “A four mechanism model for threshold spatial vision,” Vision Res. 19, 19–31 (1978).
[Crossref]

Campbell, F. W.

Fahle, M.

M. Fahle and T. Poggio, “Visual hyperacuity: spatiotemporal interpolation in human vision,” Proc. R. Soc. Lond. B 213, 451–477 (1981).
[Crossref]

Graham, N.

N. Graham, “Psychophysics of spatial-frequency channels,” in Perceptual Organization, M. Kubovy and J. Pomerantz, eds. (Lawrence Erlbaum, Hillsdale, N.J., 1981), pp. 1–25.

Jukes, J.

McKee, S. P.

G. Westheimer and S. P. McKee, “Spatial configurations for visual hyperacuity,” Vision Res. 17, 941–947 (1977).
[Crossref] [PubMed]

G. Westheimer and S. P. McKee, “Integration regions for visual hyperacuity,” Vision Res. 17, 89–93 (1977).
[Crossref] [PubMed]

Nachmias, J.

Poggio, T.

M. Fahle and T. Poggio, “Visual hyperacuity: spatiotemporal interpolation in human vision,” Proc. R. Soc. Lond. B 213, 451–477 (1981).
[Crossref]

Robson, J. G.

Sachs, M. B.

Westheimer, G.

G. Westheimer, “The spatial sense of the eye,” Invest. Ophthalmol. Visual Sci. 18, 893–912 (1979).

G. Westheimer and S. P. McKee, “Spatial configurations for visual hyperacuity,” Vision Res. 17, 941–947 (1977).
[Crossref] [PubMed]

G. Westheimer and S. P. McKee, “Integration regions for visual hyperacuity,” Vision Res. 17, 89–93 (1977).
[Crossref] [PubMed]

G. Westheimer and F. W. Campbell, “Light distribution in the image formed by the living human eye,” J. Opt. Soc. Am. 52, 1040–1045 (1962).
[Crossref] [PubMed]

Wilson, H. R.

H. R. Wilson and J. R. Bergen, “A four mechanism model for threshold spatial vision,” Vision Res. 19, 19–31 (1978).
[Crossref]

Invest. Ophthalmol. Visual Sci. (1)

G. Westheimer, “The spatial sense of the eye,” Invest. Ophthalmol. Visual Sci. 18, 893–912 (1979).

J. Opt. Soc. Am. (3)

Nature (1)

H. B. Barlow, “Reconstructing the visual image in space and time,” Nature,  279, 189–190 (1979).
[Crossref] [PubMed]

Proc. R. Soc. Lond. B (1)

M. Fahle and T. Poggio, “Visual hyperacuity: spatiotemporal interpolation in human vision,” Proc. R. Soc. Lond. B 213, 451–477 (1981).
[Crossref]

Vision Res. (3)

G. Westheimer and S. P. McKee, “Spatial configurations for visual hyperacuity,” Vision Res. 17, 941–947 (1977).
[Crossref] [PubMed]

G. Westheimer and S. P. McKee, “Integration regions for visual hyperacuity,” Vision Res. 17, 89–93 (1977).
[Crossref] [PubMed]

H. R. Wilson and J. R. Bergen, “A four mechanism model for threshold spatial vision,” Vision Res. 19, 19–31 (1978).
[Crossref]

Other (6)

N. Graham, “Psychophysics of spatial-frequency channels,” in Perceptual Organization, M. Kubovy and J. Pomerantz, eds. (Lawrence Erlbaum, Hillsdale, N.J., 1981), pp. 1–25.

It was found that the contrast-matching experiments were necessary for medium and high spatial frequencies because test gratings of different frequencies and equal contrasts had perceptibly different apparent contrasts that were due to the slope of the modulation transfer function. The contrast-matching experiments were similar to the frequency-discrimination experiment described here. The reference grating was presented at 30% contrast, and one test-grating frequency was randomly presented at seven different contrast levels. On each trial the observer indicated whether the contrast of the test grating was higher or lower than the contrast of the reference grating. A contrast level that appeared equivalent to the reference grating at 30% contrast was determined from the psychometric function.

Each neural unit receives input from M photoreceptors, where M must be greater than 2. If M were equal to 2, only linear interpolation would be possible, and it would not be possible to reconstruct a peak falling between two photoreceptors. If M were equal to 3, quadratic interpolation would be possible. Since sinusoids and line-spread functions are quadratic near a peak, this might be sufficient. If the interpolation were done in two dimensions, quadratic interpolation would require input from at least six photoreceptors. The figure shows the case of M= 2 for clarity.

If N were not an integer, all photoreceptors would not project to the same number of neural units, and the neural map would not be coherent with the photoreceptor lattice, thus resulting in a chaotic structure. Strictly speaking, coherence requires only that the ratio of the photoreceptor spacing to the neural-array spacing be the ratio of two integers. However, the period of the resulting structure is more than one photoreceptor, and many more weights would be required.

Spatial frequencies below 2 c/deg are somewhat problematic. If we say that they should fall on the N= 1 line, Δf/f would become too small, violating the assumption that the whole mechanism is intended to keep Δf/f roughly constant. Thus the model does not make predictions below 2 c/deg.

The fitting procedure determined the values of ξ and f1 (the first transition frequency), which minimized the chi squared for the agreement between the model and the data. The procedure was not entirely straightforward since close inspection of the data shows that the transition frequencies are not exactly evenly spaced, and there are also points that fall in the transition regions between segments. Good chi-squared values (confidence greater than 0.05) were obtained by permitting a small error (roughly 0.2 c/deg) in the transition frequencies and also by omitting a small number of points from the fit (1 of 26 points above 2 c/deg for JH, 3 of 25 points for BA, 2 of 25 points for MM, and no points for the JH line-separation data). These omitted points fell in transition regions between segments, except those for observer MM, for whom points at 6 and 6.5 c/deg seem to be outliers. However, the fitted value of ξ was insensitive to these points. The best f1 values for the three frequency-discrimination observers are 3.75, 4.25, and 4.38, and for the line separation observer it is 3.75. Since chi squared is not a continuous function of f1 (and we permit small deviations from equally spaced transitions), the errors on f1 are difficult to define.

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Figures (5)

Fig. 1
Fig. 1

Fractional jnd in spatial frequency (Δf/f) as a function of spatial frequency of the reference grating (cycles/deg) for observers JH, BA, and MM. The straight lines passing through the origin represent regions of constant angular jnd Δs. Determination of the lines is discussed in the text.

Fig. 2
Fig. 2

Fractional jnd in spatial frequency (Δf/f) as a function of field size (number of cycles in the reference grating) for observer BA at 4, 8, and 12 c/deg.

Fig. 3
Fig. 3

Fractional jnd in separation (Δs/s) as a function of 1/reference separation (deg−1). The straight lines passing through the origin represent the regions of constant angular jnd Δs as shown in Fig. 1.

Fig. 4
Fig. 4

Comparison of (a) spatial-frequency discrimination and (b) line-pair-separation discrimination for observer JH. The data from Figs. 1(a) and 3 are replotted [(a) and (b), respectively] to show the similarity between the two experiments as discussed in the text. The dashed lines drawn through the data in both cases are the same as the lines drawn through the origin of Figs. 1 and 3. The data illustrate the steplike characteristic of As for both experiments. Open circles show similar line-pair-separation results previously reported by Westheimer.2

Fig. 5
Fig. 5

Schematic model for neural interpolation. The upper curve represents the luminance profile of the image. Triangles represent photoreceptors, and circles represent neural units on which the image is reconstructed. Lines indicate connections from photoreceptors to neural units. Each neural unit performs an interpolation between photoreceptors. For clarity, each neural unit has been drawn with only two inputs.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

s = 1 / f ,
Δ s = Δ f / f 2 .
Δ f / f = Δ s / s .
Δ f / f = Δ s * f
α max = Δ s / s min .
( ξ / N ) / s N = α max .
f N = N α max / ξ .