Abstract

Color-discrimination ellipses derived from the variability of color-matching data of six observers are analyzed in a normalized constant-luminance cone-excitation space. The analysis shows that the ellipses do not vary significantly in shape with chromaticity, observer, or experimental conditions. The discrimination contours are predictable from the thresholds on the two cardinal axes of this space; these are used to normalize the data at each chromaticity for each observer. Thresholds on these two axes vary with chromaticities, individuals, and experimental conditions in accordance with simple and familiar laws.

© 1987 Optical Society of America

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References

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  1. D. I. A. MacLeod, R. M. Boynton, “Chromaticity diagram showing cone excitation by stimuli of equal luminance,”J. Opt. Soc. Am. 69, 1183–1186 (1979).
    [CrossRef] [PubMed]
  2. V. C. Smith, J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
    [CrossRef] [PubMed]
  3. J. Krauskopf, D. R. Williams, D. W. Heeley, “Cardinal directions of color space,” Vision Res. 22, 1123–1131 (1982).
    [CrossRef] [PubMed]
  4. A. Eisner, D. I. A. MacLeod, “Blue-sensitive cones do not contribute to luminance,”J. Opt. Soc. Am. 70, 121–123 (1980).
    [CrossRef] [PubMed]
  5. R. M. Boynton, N. Kambe, “Chromatic difference steps of moderate size measured along theoretically critical axes,” Color Res. Appl. 5, 13–23 (1980); R. M. Boynton, “A system of colorimetry and photometry based on cone excitations,” Color Res. Appl. (to be published).
    [CrossRef]
  6. Y. LeGrand, “Les seuils différentiels de couleurs dans la théorie de Young,” Rev. Opt. Theor. Instrum. 28, 261–278 (1949).
  7. R. W. Rodieck, The Vertebrate Retina (Freeman, San Francisco, Calif., 1973).
  8. D. L. MacAdam, “Visual sensitivities to color differences in daylight,”J. Opt. Soc. Am. 32, 247–273 (1942).
    [CrossRef]
  9. R. M. Boynton, A. L. Nagy, C. X. Olson, “A flaw in color difference equations,” Color Res. Appl. 8, 69–74 (1983).
    [CrossRef]
  10. R. M. Boynton, A. L. Nagy, R. T. Eskew, “Chromatic discrimination in the R-B constant luminance chromaticity plot,” Perception (to be published).
  11. C. F. Stromeyer, G. R. Cole, R. E. Kronauer, “Second-site adaptation in the red-green chromatic pathways,” Vision Res. 25, 219–237 (1985).
    [CrossRef] [PubMed]
  12. K. Kranda, P. E. King-Smith, “Detection of coloured stimuli by independent linear systems,” Vision Res. 19, 733–745 (1979).
    [CrossRef] [PubMed]
  13. W. B. Cowan, G. Wyszecki, H. Yaguchi, “Probability summation among color channels,” J. Opt. Soc. Am. A 1, 1307 (A) (1984).
  14. L. Silberstein, “Investigations on the intrinsic properties of the color domain II,”J. Opt. Soc. Am. 33, 1–10 (1943).
    [CrossRef]
  15. G. Wyszecki, W. S. Stiles, Color Science (Wiley, New York, 1985), pp. 306–312.
  16. G. Wyszecki, G. H. Fielder, “New color matching ellipses,”J. Opt. Soc. Am. 61, 1135–1152 (1971).
    [CrossRef] [PubMed]
  17. 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]
  18. In order to parallel more closely the way in which the data were reported, we have specified the transformed, normalized ellipses by their major axis length and angle of orientation. For many purposes, however, it would be preferable to report semiaxis length at one orientation (either 45 or 135 deg); the other length is determined by Eq. (8). This procedure would prevent the fact that the major axis length is bounded below by 1.0 from distorting statistics calculated on it. The results in this paper were actually calculated both ways to be certain that the lower bound on major axis length did not alter the conclusions.
  19. One of the anonymous reviewers wondered how the pooled ellipse in Fig. 7 could have a larger a/b ratio than the individual constituents. Interestingly, the second reviewer provided an answer when he stated that this phenomenon “… is a consequence of normalizing noisy initial data which are elongated along the S axis on one day and elongated along the L− 2M axis on another day.” This reviewer went on to say that, because the within-subject variability is comparable with the between-subject variability after normalization, this “strengthens support for the validity of their general approach, but probably weakens support for the validity of the precise axes they have chosen. The authors could easily qualify some of the claims for the precise form of their model and minimize the possibility of readers forming an interpretation stronger than the data can support.” This reviewer points out that we have not attempted to determine whether the axes that we have chosen are the best possible set. Our reaction to this is simply that the axes that we have chosen, together with the normalization procedure, have enabled us to reconcile and interpret a previously disparate collection of data whose meaning has previously been obscure. As the second reviewer also states, “It is surprising how well such a simple normalization scheme works.”
  20. D. Jameson, L. M. Hurvich, “Opponent response functions related to measured cone photopigments,”J. Opt. Soc. Am. 58, 429–430 (1968); B. R. Wooten, J. Werner, “Shortwave cone input to the red–green opponent channel,” Vision Res. 19, 1053–1054 (1979); C. R. Ingling, P. W. Russell, M. S. Rea, B. H. Tsou, “Red–green opponent spectral sensitivity: disparity between cancellation and direct matching methods,” Science 201, 1221–1223 (1978).
    [CrossRef] [PubMed]

1985 (1)

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

1984 (1)

W. B. Cowan, G. Wyszecki, H. Yaguchi, “Probability summation among color channels,” J. Opt. Soc. Am. A 1, 1307 (A) (1984).

1983 (1)

R. M. Boynton, A. L. Nagy, C. X. Olson, “A flaw in color difference equations,” Color Res. Appl. 8, 69–74 (1983).
[CrossRef]

1982 (1)

J. Krauskopf, D. R. Williams, D. W. Heeley, “Cardinal directions of color space,” Vision Res. 22, 1123–1131 (1982).
[CrossRef] [PubMed]

1980 (2)

R. M. Boynton, N. Kambe, “Chromatic difference steps of moderate size measured along theoretically critical axes,” Color Res. Appl. 5, 13–23 (1980); R. M. Boynton, “A system of colorimetry and photometry based on cone excitations,” Color Res. Appl. (to be published).
[CrossRef]

A. Eisner, D. I. A. MacLeod, “Blue-sensitive cones do not contribute to luminance,”J. Opt. Soc. Am. 70, 121–123 (1980).
[CrossRef] [PubMed]

1979 (2)

D. I. A. MacLeod, R. M. Boynton, “Chromaticity diagram showing cone excitation by stimuli of equal luminance,”J. Opt. Soc. Am. 69, 1183–1186 (1979).
[CrossRef] [PubMed]

K. Kranda, P. E. King-Smith, “Detection of coloured stimuli by independent linear systems,” Vision Res. 19, 733–745 (1979).
[CrossRef] [PubMed]

1975 (1)

V. C. Smith, J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
[CrossRef] [PubMed]

1971 (1)

1968 (1)

1949 (2)

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]

Y. LeGrand, “Les seuils différentiels de couleurs dans la théorie de Young,” Rev. Opt. Theor. Instrum. 28, 261–278 (1949).

1943 (1)

1942 (1)

Boynton, R. M.

R. M. Boynton, A. L. Nagy, C. X. Olson, “A flaw in color difference equations,” Color Res. Appl. 8, 69–74 (1983).
[CrossRef]

R. M. Boynton, N. Kambe, “Chromatic difference steps of moderate size measured along theoretically critical axes,” Color Res. Appl. 5, 13–23 (1980); R. M. Boynton, “A system of colorimetry and photometry based on cone excitations,” Color Res. Appl. (to be published).
[CrossRef]

D. I. A. MacLeod, R. M. Boynton, “Chromaticity diagram showing cone excitation by stimuli of equal luminance,”J. Opt. Soc. Am. 69, 1183–1186 (1979).
[CrossRef] [PubMed]

R. M. Boynton, A. L. Nagy, R. T. Eskew, “Chromatic discrimination in the R-B constant luminance chromaticity plot,” Perception (to be published).

Brown, W. R. J.

Cole, G. R.

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

Cowan, W. B.

W. B. Cowan, G. Wyszecki, H. Yaguchi, “Probability summation among color channels,” J. Opt. Soc. Am. A 1, 1307 (A) (1984).

Eisner, A.

Eskew, R. T.

R. M. Boynton, A. L. Nagy, R. T. Eskew, “Chromatic discrimination in the R-B constant luminance chromaticity plot,” Perception (to be published).

Fielder, G. H.

Heeley, D. W.

J. Krauskopf, D. R. Williams, D. W. Heeley, “Cardinal directions of color space,” Vision Res. 22, 1123–1131 (1982).
[CrossRef] [PubMed]

Hurvich, L. M.

Jameson, D.

Kambe, N.

R. M. Boynton, N. Kambe, “Chromatic difference steps of moderate size measured along theoretically critical axes,” Color Res. Appl. 5, 13–23 (1980); R. M. Boynton, “A system of colorimetry and photometry based on cone excitations,” Color Res. Appl. (to be published).
[CrossRef]

King-Smith, P. E.

K. Kranda, P. E. King-Smith, “Detection of coloured stimuli by independent linear systems,” Vision Res. 19, 733–745 (1979).
[CrossRef] [PubMed]

Kranda, K.

K. Kranda, P. E. King-Smith, “Detection of coloured stimuli by independent linear systems,” Vision Res. 19, 733–745 (1979).
[CrossRef] [PubMed]

Krauskopf, J.

J. Krauskopf, D. R. Williams, D. W. Heeley, “Cardinal directions of color space,” Vision Res. 22, 1123–1131 (1982).
[CrossRef] [PubMed]

Kronauer, R. E.

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

LeGrand, Y.

Y. LeGrand, “Les seuils différentiels de couleurs dans la théorie de Young,” Rev. Opt. Theor. Instrum. 28, 261–278 (1949).

MacAdam, D. L.

MacLeod, D. I. A.

Nagy, A. L.

R. M. Boynton, A. L. Nagy, C. X. Olson, “A flaw in color difference equations,” Color Res. Appl. 8, 69–74 (1983).
[CrossRef]

R. M. Boynton, A. L. Nagy, R. T. Eskew, “Chromatic discrimination in the R-B constant luminance chromaticity plot,” Perception (to be published).

Olson, C. X.

R. M. Boynton, A. L. Nagy, C. X. Olson, “A flaw in color difference equations,” Color Res. Appl. 8, 69–74 (1983).
[CrossRef]

Pokorny, J.

V. C. Smith, J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
[CrossRef] [PubMed]

Rodieck, R. W.

R. W. Rodieck, The Vertebrate Retina (Freeman, San Francisco, Calif., 1973).

Silberstein, L.

Smith, V. C.

V. C. Smith, J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
[CrossRef] [PubMed]

Stiles, W. S.

G. Wyszecki, W. S. Stiles, Color Science (Wiley, New York, 1985), pp. 306–312.

Stromeyer, C. F.

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

Williams, D. R.

J. Krauskopf, D. R. Williams, D. W. Heeley, “Cardinal directions of color space,” Vision Res. 22, 1123–1131 (1982).
[CrossRef] [PubMed]

Wyszecki, G.

W. B. Cowan, G. Wyszecki, H. Yaguchi, “Probability summation among color channels,” J. Opt. Soc. Am. A 1, 1307 (A) (1984).

G. Wyszecki, G. H. Fielder, “New color matching ellipses,”J. Opt. Soc. Am. 61, 1135–1152 (1971).
[CrossRef] [PubMed]

G. Wyszecki, W. S. Stiles, Color Science (Wiley, New York, 1985), pp. 306–312.

Yaguchi, H.

W. B. Cowan, G. Wyszecki, H. Yaguchi, “Probability summation among color channels,” J. Opt. Soc. Am. A 1, 1307 (A) (1984).

Color Res. Appl. (2)

R. M. Boynton, N. Kambe, “Chromatic difference steps of moderate size measured along theoretically critical axes,” Color Res. Appl. 5, 13–23 (1980); R. M. Boynton, “A system of colorimetry and photometry based on cone excitations,” Color Res. Appl. (to be published).
[CrossRef]

R. M. Boynton, A. L. Nagy, C. X. Olson, “A flaw in color difference equations,” Color Res. Appl. 8, 69–74 (1983).
[CrossRef]

J. Opt. Soc. Am. (7)

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

W. B. Cowan, G. Wyszecki, H. Yaguchi, “Probability summation among color channels,” J. Opt. Soc. Am. A 1, 1307 (A) (1984).

Rev. Opt. Theor. Instrum. (1)

Y. LeGrand, “Les seuils différentiels de couleurs dans la théorie de Young,” Rev. Opt. Theor. Instrum. 28, 261–278 (1949).

Vision Res. (4)

V. C. Smith, J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
[CrossRef] [PubMed]

J. Krauskopf, D. R. Williams, D. W. Heeley, “Cardinal directions of color space,” Vision Res. 22, 1123–1131 (1982).
[CrossRef] [PubMed]

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

K. Kranda, P. E. King-Smith, “Detection of coloured stimuli by independent linear systems,” Vision Res. 19, 733–745 (1979).
[CrossRef] [PubMed]

Other (5)

G. Wyszecki, W. S. Stiles, Color Science (Wiley, New York, 1985), pp. 306–312.

In order to parallel more closely the way in which the data were reported, we have specified the transformed, normalized ellipses by their major axis length and angle of orientation. For many purposes, however, it would be preferable to report semiaxis length at one orientation (either 45 or 135 deg); the other length is determined by Eq. (8). This procedure would prevent the fact that the major axis length is bounded below by 1.0 from distorting statistics calculated on it. The results in this paper were actually calculated both ways to be certain that the lower bound on major axis length did not alter the conclusions.

One of the anonymous reviewers wondered how the pooled ellipse in Fig. 7 could have a larger a/b ratio than the individual constituents. Interestingly, the second reviewer provided an answer when he stated that this phenomenon “… is a consequence of normalizing noisy initial data which are elongated along the S axis on one day and elongated along the L− 2M axis on another day.” This reviewer went on to say that, because the within-subject variability is comparable with the between-subject variability after normalization, this “strengthens support for the validity of their general approach, but probably weakens support for the validity of the precise axes they have chosen. The authors could easily qualify some of the claims for the precise form of their model and minimize the possibility of readers forming an interpretation stronger than the data can support.” This reviewer points out that we have not attempted to determine whether the axes that we have chosen are the best possible set. Our reaction to this is simply that the axes that we have chosen, together with the normalization procedure, have enabled us to reconcile and interpret a previously disparate collection of data whose meaning has previously been obscure. As the second reviewer also states, “It is surprising how well such a simple normalization scheme works.”

R. W. Rodieck, The Vertebrate Retina (Freeman, San Francisco, Calif., 1973).

R. M. Boynton, A. L. Nagy, R. T. Eskew, “Chromatic discrimination in the R-B constant luminance chromaticity plot,” Perception (to be published).

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Figures (12)

Fig. 1
Fig. 1

Mean ellipses for each of the six observers. Major axis lengths were averaged across chromaticities. The lower panel shows the mean and the standard deviation for each observer, and the upper panel shows the largest and smallest mean ellipses. Unlabeled axes are scaled in threshold units in this figure and in Figs. 37.

Fig. 2
Fig. 2

Arbitrary division of the chromaticity diagram into six regions of different color. Numbers indicate the number of discrimination ellipses in each region.

Fig. 3
Fig. 3

Mean ellipses for each of the six regions of chromaticity. The major axis lengths of all ellipses within each region were averaged. The mean and the standard deviation are shown in the lower panel, and the largest and smallest mean ellipses are shown in the upper panel.

Fig. 4
Fig. 4

Major axis length plotted as a function of S excitation level. No systematic relationship appears to exist.

Fig. 5
Fig. 5

Major axis length plotted as a function of illuminance level. No systematic relationship appears to exist.

Fig. 6
Fig. 6

Major axis length plotted as a function of L − 2M excitation level. No systematic relationship appears to exist.

Fig. 7
Fig. 7

Ellipses obtained at the same chromaticity from the same observer on different days are indicated by solid lines. The dashed line indicates the ellipse fit to the pooled data.

Fig. 8
Fig. 8

Location of ellipses oriented at 45 deg. Open symbols indicate ellipses that may be oriented at 45 deg because of measurement error. Filled symbols indicate remaining ellipses oriented at 45 deg.

Fig. 9
Fig. 9

ΔS thresholds as a function of S excitation level for each of the six observers. Axes are scaled in S trolands (STd). The solid curves indicate the fits of Eq. (9).

Fig. 10
Fig. 10

ΔL thresholds as a function of illuminance level for WRJB and DLM. Axes are scaled in trolands. The solid lines indicate the fit of Eq. (11).

Fig. 11
Fig. 11

ΔL thresholds as a function of L − 2M excitation level for the four observers, measured at a constant illuminance level. The abscissa is normalized for illuminance level, and the ordinate is scaled in L trolands with 2M trolands varying oppositely. The solid lines indicate the fit of Eq. (11).

Fig. 12
Fig. 12

ΔL thresholds as a function of S excitation level for the four observers, measured at a constant illuminance level. Axes are again scaled in trolands. No systematic relationship appears to exist.

Tables (3)

Tables Icon

Table 1 Values of Parameters for Fits of Eq. (9) to the AS Thresholds for All Six Observersa

Tables Icon

Table 2 Values of Parameters for Fits of Eq. (11) to the ΔL Thresholda

Tables Icon

Table 3 Parameters for 184 Ellipses

Equations (14)

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

L = 0.15514 x ¯ + 0.54321 y ¯ - 0.03286 z ¯ , M = - 0.11514 x ¯ + 0.45684 y ¯ + 0.03286 z ¯ , S = z ¯ .
Δ E 2 = g 11 Δ x 2 + 2 g 12 Δ x Δ y + g 22 Δ y 2 ,
Δ x = ( x - x c ) ,             Δ y = ( y - y c ) ,
Δ E 2 = h 11 Δ L 2 + 2 h 12 Δ L Δ S + h 22 Δ S 2 .
Δ E 2 = k 11 Δ l 2 + 2 k 12 Δ l Δ s + k 22 Δ s 2 ,
Δ l = Δ L / h 11 ,             Δ s = Δ S h 22 .
k 11 = k 22 = 1.0
k 12 = h 12 / h 11 h 22 .
θ = 45 deg             if k 12 < 0 ,
θ = 135 deg             if k 12 > 0 ,
1 a 2 + 1 b 2 = 2.
Δ S = C ( S + k S 0 ) ,
Δ S = A { S + b [ S 0 + d ( L + M ) ] } ,
Δ L = W [ ( L + M ) + a ( L - 2 M ) + j ( S ) ] .

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