Abstract

We evaluate how well three different parametric shapes, ellipsoids, rectangles, and parallelograms, serve as models of three-dimensional detection contours. We describe how the procedures for deriving the best-fitting shapes constrain inferences about the theoretical visual detection mechanisms. The ellipsoidal shape, commonly assumed by vector-length theories, is related to a class of visual mechanisms that are unique only up to orthogonal transformations. The rectangle shape is related to a unique set of visual mechanisms, but since the rectangle is not invariant with respect to linear transformations the estimated visual mechanisms are dependent on the stimulus coordinate frame. The parallelogram is related to a unique set of visual mechanisms and can be derived by methods that are independent of the stimulus coordinate frame. We evaluate how well these shapes approximate detection contours, using 2-deg test fields with a long (1-sec) Gaussian time course. Two statistical tests suggest that the parallelogram model is too strong. First, we find that the ellipsoid and rectangle shapes fit the data with the same precision as the variance in repeated threshold measurements. The parallelogram model, which has more free parameters, fits the data with more precision than the variance in repeated threshold measurements. Second, although the parallelogram model provides a slightly better fit of our data than the other two shapes, it does not serve as a better guide than the ellipsoidal model for interpolating from the measurements to thresholds in novel color directions.

© 1990 Optical Society of America

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References

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  1. W. S. Stiles, “The directional sensitivity of the retina and the spectral sensitivities of the rods and cones,” Proc. R. Soc. London Ser. B 127, 64–105 (1939).
    [CrossRef]
  2. W. S. Stiles, “Color vision: the approach through increment-threshold sensitivity,” Proc. Natl. Acad. Sci. USA 45, 100–114 (1959).
    [CrossRef]
  3. W. S. Stiles, in Mechanisms of Colour Vision, (Academic, London, 1978).
  4. D. L. MacAdam, “Visual sensitivities to color differences in daylight,” J. Opt. Soc. Am. 32, 247–274 (1942).
    [CrossRef]
  5. W. R. J. Brown, D. L. MacAdam, “Visual sensitivities to combined chromaticity and luminance differences,” J. Opt. Soc. Am. 39, 808–834 (1949).
    [CrossRef] [PubMed]
  6. W. R. J. Brown, “Color discrimination of twelve observers,” J. Opt. Soc. Am. 47, 137–143 (1957).
    [CrossRef] [PubMed]
  7. A. B. Watson, “Probability summation over time,” Vision Res. 19, 515–522 (1979).
    [CrossRef] [PubMed]
  8. C. F. Stromeyer, G. R. Cole, R. E. Kronauer, “Secondsite adaptation in the red-green chromatic pathways,” Vision Res. 25, 219–237 (1985).
    [CrossRef]
  9. K. Kranda, P. E. King-Smith, “Detection of coloured stimuli by independent linear systems,” Vision Res. 19, 733–746 (1979).
    [CrossRef] [PubMed]
  10. B. A. Wandell, “Color measurement and discrimination,” J. Opt. Soc. Am. A 2, 62–71 (1985).
    [CrossRef] [PubMed]
  11. J. P. Chandler, stepit, Quantum Chemistry Program Exchange, Department of Chemistry, Indiana University, Bloomington, Ind. (1965).
  12. S. L. Guth, “Non-additivity and inhibition among chromatic luminances at threshold,” Vision Res. 7, 319–328 (1967).
    [CrossRef] [PubMed]
  13. S. L. Guth, H. R. Lodge, “Heterochromatic additivity, foveal spectral sensitivity, and a new color model,” J. Opt. Soc, Am. 63, 450–462 (1973).
    [CrossRef]
  14. S. L. Guth, R. W. Massof, T. Benzschawel, “Vector model for normal and dichromatic color vision,” J. Opt. Soc. Am. 70, 197–212 (1980).
    [CrossRef] [PubMed]
  15. C. R. Ingling, B. H-P Tsou, “Orthogonal combination of the three visual channels,” Vision Res. 17, 1075–1082 (1977).
    [CrossRef] [PubMed]
  16. D. B. Judd; “Report of the U.S. Secretariat Committee on Colorimetry and Artificial Daylight,” in CIE Proceedings (Bureau de la CIE, Paris, 1951), Vol. 1, Part 7, p. 11.
  17. H. G. Sperling, R. S. Harwerth, “Red-green cone interactions in the increment-threshold spectral sensitivity of primates,” Science 172, 180–184 (1971).
    [CrossRef] [PubMed]

1985 (2)

C. F. Stromeyer, G. R. Cole, R. E. Kronauer, “Secondsite adaptation in the red-green chromatic pathways,” Vision Res. 25, 219–237 (1985).
[CrossRef]

B. A. Wandell, “Color measurement and discrimination,” J. Opt. Soc. Am. A 2, 62–71 (1985).
[CrossRef] [PubMed]

1980 (1)

1979 (2)

K. Kranda, P. E. King-Smith, “Detection of coloured stimuli by independent linear systems,” Vision Res. 19, 733–746 (1979).
[CrossRef] [PubMed]

A. B. Watson, “Probability summation over time,” Vision Res. 19, 515–522 (1979).
[CrossRef] [PubMed]

1977 (1)

C. R. Ingling, B. H-P Tsou, “Orthogonal combination of the three visual channels,” Vision Res. 17, 1075–1082 (1977).
[CrossRef] [PubMed]

1973 (1)

S. L. Guth, H. R. Lodge, “Heterochromatic additivity, foveal spectral sensitivity, and a new color model,” J. Opt. Soc, Am. 63, 450–462 (1973).
[CrossRef]

1971 (1)

H. G. Sperling, R. S. Harwerth, “Red-green cone interactions in the increment-threshold spectral sensitivity of primates,” Science 172, 180–184 (1971).
[CrossRef] [PubMed]

1967 (1)

S. L. Guth, “Non-additivity and inhibition among chromatic luminances at threshold,” Vision Res. 7, 319–328 (1967).
[CrossRef] [PubMed]

1959 (1)

W. S. Stiles, “Color vision: the approach through increment-threshold sensitivity,” Proc. Natl. Acad. Sci. USA 45, 100–114 (1959).
[CrossRef]

1957 (1)

1949 (1)

W. R. J. Brown, D. L. MacAdam, “Visual sensitivities to combined chromaticity and luminance differences,” J. Opt. Soc. Am. 39, 808–834 (1949).
[CrossRef] [PubMed]

1942 (1)

D. L. MacAdam, “Visual sensitivities to color differences in daylight,” J. Opt. Soc. Am. 32, 247–274 (1942).
[CrossRef]

1939 (1)

W. S. Stiles, “The directional sensitivity of the retina and the spectral sensitivities of the rods and cones,” Proc. R. Soc. London Ser. B 127, 64–105 (1939).
[CrossRef]

Benzschawel, T.

Brown, W. R. J.

W. R. J. Brown, “Color discrimination of twelve observers,” J. Opt. Soc. Am. 47, 137–143 (1957).
[CrossRef] [PubMed]

W. R. J. Brown, D. L. MacAdam, “Visual sensitivities to combined chromaticity and luminance differences,” J. Opt. Soc. Am. 39, 808–834 (1949).
[CrossRef] [PubMed]

Chandler, J. P.

J. P. Chandler, stepit, Quantum Chemistry Program Exchange, Department of Chemistry, Indiana University, Bloomington, Ind. (1965).

Cole, G. R.

C. F. Stromeyer, G. R. Cole, R. E. Kronauer, “Secondsite adaptation in the red-green chromatic pathways,” Vision Res. 25, 219–237 (1985).
[CrossRef]

Guth, S. L.

S. L. Guth, R. W. Massof, T. Benzschawel, “Vector model for normal and dichromatic color vision,” J. Opt. Soc. Am. 70, 197–212 (1980).
[CrossRef] [PubMed]

S. L. Guth, H. R. Lodge, “Heterochromatic additivity, foveal spectral sensitivity, and a new color model,” J. Opt. Soc, Am. 63, 450–462 (1973).
[CrossRef]

S. L. Guth, “Non-additivity and inhibition among chromatic luminances at threshold,” Vision Res. 7, 319–328 (1967).
[CrossRef] [PubMed]

Harwerth, R. S.

H. G. Sperling, R. S. Harwerth, “Red-green cone interactions in the increment-threshold spectral sensitivity of primates,” Science 172, 180–184 (1971).
[CrossRef] [PubMed]

Ingling, C. R.

C. R. Ingling, B. H-P Tsou, “Orthogonal combination of the three visual channels,” Vision Res. 17, 1075–1082 (1977).
[CrossRef] [PubMed]

Judd, D. B.

D. B. Judd; “Report of the U.S. Secretariat Committee on Colorimetry and Artificial Daylight,” in CIE Proceedings (Bureau de la CIE, Paris, 1951), Vol. 1, Part 7, p. 11.

King-Smith, P. E.

K. Kranda, P. E. King-Smith, “Detection of coloured stimuli by independent linear systems,” Vision Res. 19, 733–746 (1979).
[CrossRef] [PubMed]

Kranda, K.

K. Kranda, P. E. King-Smith, “Detection of coloured stimuli by independent linear systems,” Vision Res. 19, 733–746 (1979).
[CrossRef] [PubMed]

Kronauer, R. E.

C. F. Stromeyer, G. R. Cole, R. E. Kronauer, “Secondsite adaptation in the red-green chromatic pathways,” Vision Res. 25, 219–237 (1985).
[CrossRef]

Lodge, H. R.

S. L. Guth, H. R. Lodge, “Heterochromatic additivity, foveal spectral sensitivity, and a new color model,” J. Opt. Soc, Am. 63, 450–462 (1973).
[CrossRef]

MacAdam, D. L.

W. R. J. Brown, D. L. MacAdam, “Visual sensitivities to combined chromaticity and luminance differences,” J. Opt. Soc. Am. 39, 808–834 (1949).
[CrossRef] [PubMed]

D. L. MacAdam, “Visual sensitivities to color differences in daylight,” J. Opt. Soc. Am. 32, 247–274 (1942).
[CrossRef]

Massof, R. W.

Sperling, H. G.

H. G. Sperling, R. S. Harwerth, “Red-green cone interactions in the increment-threshold spectral sensitivity of primates,” Science 172, 180–184 (1971).
[CrossRef] [PubMed]

Stiles, W. S.

W. S. Stiles, “Color vision: the approach through increment-threshold sensitivity,” Proc. Natl. Acad. Sci. USA 45, 100–114 (1959).
[CrossRef]

W. S. Stiles, “The directional sensitivity of the retina and the spectral sensitivities of the rods and cones,” Proc. R. Soc. London Ser. B 127, 64–105 (1939).
[CrossRef]

W. S. Stiles, in Mechanisms of Colour Vision, (Academic, London, 1978).

Stromeyer, C. F.

C. F. Stromeyer, G. R. Cole, R. E. Kronauer, “Secondsite adaptation in the red-green chromatic pathways,” Vision Res. 25, 219–237 (1985).
[CrossRef]

Tsou, B. H-P

C. R. Ingling, B. H-P Tsou, “Orthogonal combination of the three visual channels,” Vision Res. 17, 1075–1082 (1977).
[CrossRef] [PubMed]

Wandell, B. A.

Watson, A. B.

A. B. Watson, “Probability summation over time,” Vision Res. 19, 515–522 (1979).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

W. R. J. Brown, D. L. MacAdam, “Visual sensitivities to combined chromaticity and luminance differences,” J. Opt. Soc. Am. 39, 808–834 (1949).
[CrossRef] [PubMed]

J. Opt. Soc, Am. (1)

S. L. Guth, H. R. Lodge, “Heterochromatic additivity, foveal spectral sensitivity, and a new color model,” J. Opt. Soc, Am. 63, 450–462 (1973).
[CrossRef]

J. Opt. Soc. Am. (1)

D. L. MacAdam, “Visual sensitivities to color differences in daylight,” J. Opt. Soc. Am. 32, 247–274 (1942).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Proc. Natl. Acad. Sci. USA (1)

W. S. Stiles, “Color vision: the approach through increment-threshold sensitivity,” Proc. Natl. Acad. Sci. USA 45, 100–114 (1959).
[CrossRef]

Proc. R. Soc. London Ser. B (1)

W. S. Stiles, “The directional sensitivity of the retina and the spectral sensitivities of the rods and cones,” Proc. R. Soc. London Ser. B 127, 64–105 (1939).
[CrossRef]

Science (1)

H. G. Sperling, R. S. Harwerth, “Red-green cone interactions in the increment-threshold spectral sensitivity of primates,” Science 172, 180–184 (1971).
[CrossRef] [PubMed]

Vision Res. (2)

K. Kranda, P. E. King-Smith, “Detection of coloured stimuli by independent linear systems,” Vision Res. 19, 733–746 (1979).
[CrossRef] [PubMed]

S. L. Guth, “Non-additivity and inhibition among chromatic luminances at threshold,” Vision Res. 7, 319–328 (1967).
[CrossRef] [PubMed]

Vision Res. (3)

C. R. Ingling, B. H-P Tsou, “Orthogonal combination of the three visual channels,” Vision Res. 17, 1075–1082 (1977).
[CrossRef] [PubMed]

A. B. Watson, “Probability summation over time,” Vision Res. 19, 515–522 (1979).
[CrossRef] [PubMed]

C. F. Stromeyer, G. R. Cole, R. E. Kronauer, “Secondsite adaptation in the red-green chromatic pathways,” Vision Res. 25, 219–237 (1985).
[CrossRef]

Other (3)

W. S. Stiles, in Mechanisms of Colour Vision, (Academic, London, 1978).

D. B. Judd; “Report of the U.S. Secretariat Committee on Colorimetry and Artificial Daylight,” in CIE Proceedings (Bureau de la CIE, Paris, 1951), Vol. 1, Part 7, p. 11.

J. P. Chandler, stepit, Quantum Chemistry Program Exchange, Department of Chemistry, Indiana University, Bloomington, Ind. (1965).

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Figures (5)

Fig. 1
Fig. 1

Thresholds (filled symbols) and cross sections of fitted surfaces (curves) in three color planes. The axes are the percent contrast modulation at detection threshold for the test stimulus primaries. Thresholds were measured on a white, 2-deg background field, surrounded by a 490-nm annulus with an outer diameter of 4 deg. The white disk was formed by the mixture of 480-, 650-, and 540-nm lights at 9.250,9.609, and 9.103 log quanta sec−1 deg−2, respectively. The annulus intensity was 9.708 log quanta sec−1 deg−2. The cross sections are from parallelogram (solid lines), rectangle (dashed lines), and ellipsoid (dotted curves) shapes.

Fig. 2
Fig. 2

Minkowski length frequency distributions for the three different shapes: ellipsoidal (crosses), rectangular (filled diamonds), and parallelogram (filled boxes). If any of the shapes fitted perfectly, then the corresponding Minkowski length distribution would be a delta function at 1.

Fig. 3
Fig. 3

Cumulative frequency distribution of the normalized thresholds (open circles) and cumulative distributions of the Minkowski lengths of the ellipsoidal (crosses), rectangular (filled squares) and parallelogram (filled diamonds) shapes.

Fig. 4
Fig. 4

Spectral sensitivities derived from observer PC’s ellipsoids (crosses) compared with spectral sensitivity measurements from two observers (filled symbols) measured on 4-log Td, 10-deg achromatic backgrounds, as described by Sperling and Harwerth.17 Observer PC viewed a 2-deg achromatic background field formed by the mixture of 480-, 650-, and 540-nm lights at 8.755, 9.544, and 9.000 log quanta sec−1 deg−2, respectively. The test field was also 2 deg, centered upon the background. To facilitate comparison of the curves, the data have been shifted to coincide at 500 nm. Since the test field seen by PC is larger, longer in duration, and more gradual in its onset and offset than the one used by Sperling and Harwerth, we expect his sensitivity to be relatively greater in the short-wave-length region.

Fig. 5
Fig. 5

Observer PC’s spectral sensitivity curves derived from the ellipsoidal approximation to data collected at three intensities of the achromatic background field. If we call the intensity of the achromatic test field 1.0 in the first condition, then the intensities in the second and third conditions are 2.0 and 6.0, respectively. The spectral sensitivity curve derived in condition one is the top curve (filled diamonds), in condition two is the middle curve (crosses), and in condition three is the bottom curve (filled squares). The log quanta sec−1 deg−2 values for the 480-, 650- and 540-nm components were, respectively, 8.434, 9.243, and 8.699 for condition one, 8.755, 9.544, and 9.000 for condition two, and 9.232, 10.021, and 9.477 for condition three.

Equations (5)

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( j = 1 , 3 m n j γ ) 1 / γ = 1.0 .
n [ ( j = 1 , 3 m n j γ ) 1 / γ 1.0 ] 2 .
t n t Q t n = 1.0
Q 11 t n 1 2 + 2 Q 12 t n 1 t n 2 + 2 Q 13 t n 1 t n 3 + Q 22 t n 2 2 + 2 Q 23 t n 2 t n 3 + Q 33 t n 3 2 = 1.0 ,
MAX j ( m n j ) = 1.0 .

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