Abstract

By assuming that color matches are normally distributed in XYZ space, we present a rigorous statistical technique to obtain regions of equally noticeable chromaticity differences. The probability density function of color matches in (x, y, Y) space is calculated according to standard techniques in probability theory. The geometry of the chromaticity thresholds is computed for various confidence levels α. Because of the asymmetry of the probability density function in (x, y, Y) space, the chromaticity thresholds are not symmetrical around the color center. The asymmetries depend on the color center, and they increase when high confidence levels (α < 0.32) are considered. It is our opinion that the technique proposed here can provide a useful tool for checking and evaluating deviations from the elliptic geometry of the chromaticity thresholds. It is formally demonstrated that regions of equally noticeable chromaticity differences are not ellipses when the normality hypothesis is assumed in XYZ space.

© 1999 Optical Society of America

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References

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  1. D. L. MacAdam, “Visual sensitivities to color differences in daylight,” J. Opt. Soc. Am. 32, 247–274 (1942).
    [CrossRef]
  2. J. M. Zoido, F. Carreño, E. Bernabeu, “On MacAdam’s ellipses,” Opt. Pura Apl. 29, 81–108 (1996).
  3. J. M. Zoido, F. Carreño, E. Bernabeu, “Some considerations on MacAdam’s ellipses,” Die Farbe 43, 1–16 (1997).
  4. Y. Legrand, “Théories sur la métrique de léspace des couleurs,” in Proceedings of the International Color Congress, Color 69, Stockholm (Göttingen, Musterschmidt, 1970), pp. 257–265.
  5. B. H. Crawford, “Just perceptible chromaticity shifts,” in Proceedings of the International Color Congress, Color 69, Stockholm (Göttingen, Musterschmidt, 1970), pp. 302–311.
  6. F. Parra, “Seuils différentiels de couleur,” in Proceedings of the International Color Congress, Color 69, Stockholm (Göttingen, Musterschmidt, 1970), pp. 727–733.
  7. G. Wyszecky, G. H. Fielder, “New color-matching ellipses,” J. Opt. Soc. Am. 61, 1135–1152 (1971).
    [CrossRef]
  8. E. Hita, E. Jiménez, “Differential thresholds of color from isomeric matching,” Atti Fond. “Georgio Ronchi,” 35, 511–526 (1980).
  9. E. Hita, F. González-Caballero, A. Cervantes, “On the geometric structure of color discrimination thresholds,” Die Farbe 30, 55–64 (1982).
  10. M. R. Luo, B. Rigg, “Chromaticity-discrimination ellipses for surface colors,” Color Res. Appl. 11, 25–42 (1986).
    [CrossRef]
  11. A. R. Robertson, “Comparison of chromaticity discrimination ellipses,” J. Opt. Soc. Am. A 4, P119 (1987).
  12. A. L. Nagy, R. R. Sánchez, “Critical color differences determined with a visual search task,” J. Opt. Soc. Am. A 7, 1209–1217 (1990).
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  13. G. R. Cole, T. Hine, W. MeIlhagga, “Detection mechanism in L-, M-, and S-cone contrast space,” J. Opt. Soc. Am. A 10, 38–51 (1993).
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  15. C. Oleari, “Uniform-scale chromaticity diagram with angular coordinates in zero-curvature space,” J. Opt. Soc. Am. A 8, 415–421 (1991).
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    [CrossRef]
  17. J. A. García, J. Romero, L. Jiménez del Barco, E. Hita, “Improved formula for evaluating color-differential thresholds,” Appl. Opt. 31, 6292–6298 (1992).
    [CrossRef] [PubMed]
  18. T. Indow, A. R. Robertson, M. von Grunau, G. H. Fielder, “Discrimination ellipsoids of aperture and simulated colors by matching and paired comparison,” Color Res. Appl. 17, 6–23 (1992).
    [CrossRef]
  19. J. Romero, J. A. García, L. Jiménez del Barco, E. Hita, “Evaluation of color-discrimination ellipsoids in two color spaces,” J. Opt. Soc. Am. A 10, 827–837 (1993).
    [CrossRef]
  20. M. Mahy, L. Van Eycken, A. Oosterlinck, “Evaluation of uniform color spaces developed after the adoption of CIELAB and CIELUV,” Color Res. Appl. 19, 105–120 (1994).
  21. G. Wyszecki, G. H. Fielder, “Color-difference matches,” J. Opt. Soc. Am. 61, 1501–1513 (1971).
    [CrossRef] [PubMed]
  22. L. Silberstein, D. L. MacAdam, “The distribution of color matchings around a color center,” J. Opt. Soc. Am. 35, 32–39 (1945).
    [CrossRef]
  23. 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]
  24. W. R. J. Brown, “Statistics of color-matching data,” J. Opt. Soc. Am. 42, 252–256 (1952).
    [CrossRef]
  25. W. R. J. Brown, W. G. Howe, J. E. Jackson, R. H. Morris, “Multivariate normality of the color-matching process,” J. Opt. Soc. Am. 46, 46–49 (1956).
    [CrossRef]
  26. E. Hita, E. Jimenez, M. Alvarez-Claro, “Color discrimination from isomeric matching and different types of primaries fusion,” Atti Fond. Georgio Ronchi 34, 718–727 (1979).
  27. K. Witt, G. Döring, “Parametric variations in a threshold color-difference ellipsoid for green painted samples,” Color Res. Appl. 8, 153–163 (1983).
    [CrossRef]
  28. M. Cheung, B. Rigg, “Color-difference ellipsoids for five CIE color centers,” Color Res. Appl. 11, 185–195 (1986).
    [CrossRef]
  29. W. R. J. Brown, “The effect of field size and chromatic surroundings on color discrimination,” J. Opt. Soc. Am. 42, 837–844 (1952).
    [CrossRef] [PubMed]
  30. W. R. J. Brown, “Color discrimination of twelve observers,” J. Opt. Soc. Am. 47, 137–143 (1957).
    [CrossRef] [PubMed]
  31. L. T. Sharpe, G. Wyszecki, “Proximity factor in color-difference evaluations,” J. Opt. Soc. Am. 66, 40–49 (1976).
    [CrossRef] [PubMed]
  32. G. Wyszecki, “Propagation of errors in colorimetric transformation,” J. Opt. Soc. Am. 49, 389–393 (1959).
    [CrossRef]
  33. A. L. Vingrys, L. E. Mahon, “Color and luminance detection and discrimination asymmetries and interactions,” Vision Res. 38, 1085–1095 (1998).
    [CrossRef] [PubMed]
  34. A. Chaparro, C. F. Stromeyer, R. E. Kronauer, R. T. Eskew, “Separable red-green and luminance detectors for small flashes,” Vision Res. 34, 751–762 (1994).
    [CrossRef] [PubMed]
  35. H. R. Davidson, E. Friede, “The size of acceptable color differences,” J. Opt. Soc. Am. 43, 581–589 (1953).
    [CrossRef] [PubMed]
  36. R. M. Rich, F. W. Billmeyer, “Method for deriving color-difference-perceptibility ellipses for surface-color samples,” J. Opt. Soc. Am. 65, 956–959 (1975).
    [CrossRef]
  37. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, Singapore, 1984).
  38. J. M. Zoido, “Distancia estadística generalizada. La métrica del color,” Ph.D. dissertation (Universidad Complutense de Madrid, Madrid, 1997).
  39. K. Witt, “Three-dimensional threshold of color-difference perceptibility in painted samples: variability of observers in four CIE color regions,” Color Res. Appl. 12, 128–134 (1987).
    [CrossRef]
  40. A. R. Robertson, “CIE guidelines for coordinated research on color-difference evaluation,” Color Res. Appl. 3, 149–151 (1978).
  41. J. Fox, Linear Statistical Models and Related Methods (Wiley, New York, 1981).

1998 (1)

A. L. Vingrys, L. E. Mahon, “Color and luminance detection and discrimination asymmetries and interactions,” Vision Res. 38, 1085–1095 (1998).
[CrossRef] [PubMed]

1997 (1)

J. M. Zoido, F. Carreño, E. Bernabeu, “Some considerations on MacAdam’s ellipses,” Die Farbe 43, 1–16 (1997).

1996 (1)

J. M. Zoido, F. Carreño, E. Bernabeu, “On MacAdam’s ellipses,” Opt. Pura Apl. 29, 81–108 (1996).

1994 (2)

M. Mahy, L. Van Eycken, A. Oosterlinck, “Evaluation of uniform color spaces developed after the adoption of CIELAB and CIELUV,” Color Res. Appl. 19, 105–120 (1994).

A. Chaparro, C. F. Stromeyer, R. E. Kronauer, R. T. Eskew, “Separable red-green and luminance detectors for small flashes,” Vision Res. 34, 751–762 (1994).
[CrossRef] [PubMed]

1993 (2)

1992 (3)

J. M. Troost, C. M. M. de Weert, “Techniques for simulating object color under changing illuminant conditions on electronic displays,” Color Res. Appl. 17, 316–327 (1992).
[CrossRef]

J. A. García, J. Romero, L. Jiménez del Barco, E. Hita, “Improved formula for evaluating color-differential thresholds,” Appl. Opt. 31, 6292–6298 (1992).
[CrossRef] [PubMed]

T. Indow, A. R. Robertson, M. von Grunau, G. H. Fielder, “Discrimination ellipsoids of aperture and simulated colors by matching and paired comparison,” Color Res. Appl. 17, 6–23 (1992).
[CrossRef]

1991 (1)

1990 (1)

1987 (2)

A. R. Robertson, “Comparison of chromaticity discrimination ellipses,” J. Opt. Soc. Am. A 4, P119 (1987).

K. Witt, “Three-dimensional threshold of color-difference perceptibility in painted samples: variability of observers in four CIE color regions,” Color Res. Appl. 12, 128–134 (1987).
[CrossRef]

1986 (2)

M. Cheung, B. Rigg, “Color-difference ellipsoids for five CIE color centers,” Color Res. Appl. 11, 185–195 (1986).
[CrossRef]

M. R. Luo, B. Rigg, “Chromaticity-discrimination ellipses for surface colors,” Color Res. Appl. 11, 25–42 (1986).
[CrossRef]

1985 (1)

B. A. Wandell, “Color constancy and color discrimination,” J. Opt. Soc. Am. A 2, P11 (1985).
[CrossRef]

1983 (1)

K. Witt, G. Döring, “Parametric variations in a threshold color-difference ellipsoid for green painted samples,” Color Res. Appl. 8, 153–163 (1983).
[CrossRef]

1982 (1)

E. Hita, F. González-Caballero, A. Cervantes, “On the geometric structure of color discrimination thresholds,” Die Farbe 30, 55–64 (1982).

1980 (1)

E. Hita, E. Jiménez, “Differential thresholds of color from isomeric matching,” Atti Fond. “Georgio Ronchi,” 35, 511–526 (1980).

1979 (1)

E. Hita, E. Jimenez, M. Alvarez-Claro, “Color discrimination from isomeric matching and different types of primaries fusion,” Atti Fond. Georgio Ronchi 34, 718–727 (1979).

1978 (1)

A. R. Robertson, “CIE guidelines for coordinated research on color-difference evaluation,” Color Res. Appl. 3, 149–151 (1978).

1976 (1)

1975 (1)

1971 (2)

1959 (1)

1957 (1)

1956 (1)

1953 (1)

1952 (2)

1949 (1)

1945 (1)

1942 (1)

Alvarez-Claro, M.

E. Hita, E. Jimenez, M. Alvarez-Claro, “Color discrimination from isomeric matching and different types of primaries fusion,” Atti Fond. Georgio Ronchi 34, 718–727 (1979).

Bernabeu, E.

J. M. Zoido, F. Carreño, E. Bernabeu, “Some considerations on MacAdam’s ellipses,” Die Farbe 43, 1–16 (1997).

J. M. Zoido, F. Carreño, E. Bernabeu, “On MacAdam’s ellipses,” Opt. Pura Apl. 29, 81–108 (1996).

Billmeyer, F. W.

Brown, W. R. J.

Carreño, F.

J. M. Zoido, F. Carreño, E. Bernabeu, “Some considerations on MacAdam’s ellipses,” Die Farbe 43, 1–16 (1997).

J. M. Zoido, F. Carreño, E. Bernabeu, “On MacAdam’s ellipses,” Opt. Pura Apl. 29, 81–108 (1996).

Cervantes, A.

E. Hita, F. González-Caballero, A. Cervantes, “On the geometric structure of color discrimination thresholds,” Die Farbe 30, 55–64 (1982).

Chaparro, A.

A. Chaparro, C. F. Stromeyer, R. E. Kronauer, R. T. Eskew, “Separable red-green and luminance detectors for small flashes,” Vision Res. 34, 751–762 (1994).
[CrossRef] [PubMed]

Cheung, M.

M. Cheung, B. Rigg, “Color-difference ellipsoids for five CIE color centers,” Color Res. Appl. 11, 185–195 (1986).
[CrossRef]

Cole, G. R.

Crawford, B. H.

B. H. Crawford, “Just perceptible chromaticity shifts,” in Proceedings of the International Color Congress, Color 69, Stockholm (Göttingen, Musterschmidt, 1970), pp. 302–311.

Davidson, H. R.

de Weert, C. M. M.

J. M. Troost, C. M. M. de Weert, “Techniques for simulating object color under changing illuminant conditions on electronic displays,” Color Res. Appl. 17, 316–327 (1992).
[CrossRef]

Döring, G.

K. Witt, G. Döring, “Parametric variations in a threshold color-difference ellipsoid for green painted samples,” Color Res. Appl. 8, 153–163 (1983).
[CrossRef]

Eskew, R. T.

A. Chaparro, C. F. Stromeyer, R. E. Kronauer, R. T. Eskew, “Separable red-green and luminance detectors for small flashes,” Vision Res. 34, 751–762 (1994).
[CrossRef] [PubMed]

Fielder, G. H.

T. Indow, A. R. Robertson, M. von Grunau, G. H. Fielder, “Discrimination ellipsoids of aperture and simulated colors by matching and paired comparison,” Color Res. Appl. 17, 6–23 (1992).
[CrossRef]

G. Wyszecki, G. H. Fielder, “Color-difference matches,” J. Opt. Soc. Am. 61, 1501–1513 (1971).
[CrossRef] [PubMed]

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

Fox, J.

J. Fox, Linear Statistical Models and Related Methods (Wiley, New York, 1981).

Friede, E.

García, J. A.

González-Caballero, F.

E. Hita, F. González-Caballero, A. Cervantes, “On the geometric structure of color discrimination thresholds,” Die Farbe 30, 55–64 (1982).

Hine, T.

Hita, E.

J. Romero, J. A. García, L. Jiménez del Barco, E. Hita, “Evaluation of color-discrimination ellipsoids in two color spaces,” J. Opt. Soc. Am. A 10, 827–837 (1993).
[CrossRef]

J. A. García, J. Romero, L. Jiménez del Barco, E. Hita, “Improved formula for evaluating color-differential thresholds,” Appl. Opt. 31, 6292–6298 (1992).
[CrossRef] [PubMed]

E. Hita, F. González-Caballero, A. Cervantes, “On the geometric structure of color discrimination thresholds,” Die Farbe 30, 55–64 (1982).

E. Hita, E. Jiménez, “Differential thresholds of color from isomeric matching,” Atti Fond. “Georgio Ronchi,” 35, 511–526 (1980).

E. Hita, E. Jimenez, M. Alvarez-Claro, “Color discrimination from isomeric matching and different types of primaries fusion,” Atti Fond. Georgio Ronchi 34, 718–727 (1979).

Howe, W. G.

Indow, T.

T. Indow, A. R. Robertson, M. von Grunau, G. H. Fielder, “Discrimination ellipsoids of aperture and simulated colors by matching and paired comparison,” Color Res. Appl. 17, 6–23 (1992).
[CrossRef]

Jackson, J. E.

Jimenez, E.

E. Hita, E. Jimenez, M. Alvarez-Claro, “Color discrimination from isomeric matching and different types of primaries fusion,” Atti Fond. Georgio Ronchi 34, 718–727 (1979).

Jiménez, E.

E. Hita, E. Jiménez, “Differential thresholds of color from isomeric matching,” Atti Fond. “Georgio Ronchi,” 35, 511–526 (1980).

Jiménez del Barco, L.

Kronauer, R. E.

A. Chaparro, C. F. Stromeyer, R. E. Kronauer, R. T. Eskew, “Separable red-green and luminance detectors for small flashes,” Vision Res. 34, 751–762 (1994).
[CrossRef] [PubMed]

Legrand, Y.

Y. Legrand, “Théories sur la métrique de léspace des couleurs,” in Proceedings of the International Color Congress, Color 69, Stockholm (Göttingen, Musterschmidt, 1970), pp. 257–265.

Luo, M. R.

M. R. Luo, B. Rigg, “Chromaticity-discrimination ellipses for surface colors,” Color Res. Appl. 11, 25–42 (1986).
[CrossRef]

MacAdam, D. L.

Mahon, L. E.

A. L. Vingrys, L. E. Mahon, “Color and luminance detection and discrimination asymmetries and interactions,” Vision Res. 38, 1085–1095 (1998).
[CrossRef] [PubMed]

Mahy, M.

M. Mahy, L. Van Eycken, A. Oosterlinck, “Evaluation of uniform color spaces developed after the adoption of CIELAB and CIELUV,” Color Res. Appl. 19, 105–120 (1994).

MeIlhagga, W.

Morris, R. H.

Nagy, A. L.

Oleari, C.

Oosterlinck, A.

M. Mahy, L. Van Eycken, A. Oosterlinck, “Evaluation of uniform color spaces developed after the adoption of CIELAB and CIELUV,” Color Res. Appl. 19, 105–120 (1994).

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, Singapore, 1984).

Parra, F.

F. Parra, “Seuils différentiels de couleur,” in Proceedings of the International Color Congress, Color 69, Stockholm (Göttingen, Musterschmidt, 1970), pp. 727–733.

Rich, R. M.

Rigg, B.

M. R. Luo, B. Rigg, “Chromaticity-discrimination ellipses for surface colors,” Color Res. Appl. 11, 25–42 (1986).
[CrossRef]

M. Cheung, B. Rigg, “Color-difference ellipsoids for five CIE color centers,” Color Res. Appl. 11, 185–195 (1986).
[CrossRef]

Robertson, A. R.

T. Indow, A. R. Robertson, M. von Grunau, G. H. Fielder, “Discrimination ellipsoids of aperture and simulated colors by matching and paired comparison,” Color Res. Appl. 17, 6–23 (1992).
[CrossRef]

A. R. Robertson, “Comparison of chromaticity discrimination ellipses,” J. Opt. Soc. Am. A 4, P119 (1987).

A. R. Robertson, “CIE guidelines for coordinated research on color-difference evaluation,” Color Res. Appl. 3, 149–151 (1978).

Romero, J.

Sánchez, R. R.

Sharpe, L. T.

Silberstein, L.

Stromeyer, C. F.

A. Chaparro, C. F. Stromeyer, R. E. Kronauer, R. T. Eskew, “Separable red-green and luminance detectors for small flashes,” Vision Res. 34, 751–762 (1994).
[CrossRef] [PubMed]

Troost, J. M.

J. M. Troost, C. M. M. de Weert, “Techniques for simulating object color under changing illuminant conditions on electronic displays,” Color Res. Appl. 17, 316–327 (1992).
[CrossRef]

Van Eycken, L.

M. Mahy, L. Van Eycken, A. Oosterlinck, “Evaluation of uniform color spaces developed after the adoption of CIELAB and CIELUV,” Color Res. Appl. 19, 105–120 (1994).

Vingrys, A. L.

A. L. Vingrys, L. E. Mahon, “Color and luminance detection and discrimination asymmetries and interactions,” Vision Res. 38, 1085–1095 (1998).
[CrossRef] [PubMed]

von Grunau, M.

T. Indow, A. R. Robertson, M. von Grunau, G. H. Fielder, “Discrimination ellipsoids of aperture and simulated colors by matching and paired comparison,” Color Res. Appl. 17, 6–23 (1992).
[CrossRef]

Wandell, B. A.

B. A. Wandell, “Color constancy and color discrimination,” J. Opt. Soc. Am. A 2, P11 (1985).
[CrossRef]

Witt, K.

K. Witt, “Three-dimensional threshold of color-difference perceptibility in painted samples: variability of observers in four CIE color regions,” Color Res. Appl. 12, 128–134 (1987).
[CrossRef]

K. Witt, G. Döring, “Parametric variations in a threshold color-difference ellipsoid for green painted samples,” Color Res. Appl. 8, 153–163 (1983).
[CrossRef]

Wyszecki, G.

Wyszecky, G.

Zoido, J. M.

J. M. Zoido, F. Carreño, E. Bernabeu, “Some considerations on MacAdam’s ellipses,” Die Farbe 43, 1–16 (1997).

J. M. Zoido, F. Carreño, E. Bernabeu, “On MacAdam’s ellipses,” Opt. Pura Apl. 29, 81–108 (1996).

J. M. Zoido, “Distancia estadística generalizada. La métrica del color,” Ph.D. dissertation (Universidad Complutense de Madrid, Madrid, 1997).

Appl. Opt. (1)

Atti Fond. “Georgio Ronchi,” (1)

E. Hita, E. Jiménez, “Differential thresholds of color from isomeric matching,” Atti Fond. “Georgio Ronchi,” 35, 511–526 (1980).

Atti Fond. Georgio Ronchi (1)

E. Hita, E. Jimenez, M. Alvarez-Claro, “Color discrimination from isomeric matching and different types of primaries fusion,” Atti Fond. Georgio Ronchi 34, 718–727 (1979).

Color Res. Appl. (8)

K. Witt, G. Döring, “Parametric variations in a threshold color-difference ellipsoid for green painted samples,” Color Res. Appl. 8, 153–163 (1983).
[CrossRef]

M. Cheung, B. Rigg, “Color-difference ellipsoids for five CIE color centers,” Color Res. Appl. 11, 185–195 (1986).
[CrossRef]

M. Mahy, L. Van Eycken, A. Oosterlinck, “Evaluation of uniform color spaces developed after the adoption of CIELAB and CIELUV,” Color Res. Appl. 19, 105–120 (1994).

T. Indow, A. R. Robertson, M. von Grunau, G. H. Fielder, “Discrimination ellipsoids of aperture and simulated colors by matching and paired comparison,” Color Res. Appl. 17, 6–23 (1992).
[CrossRef]

J. M. Troost, C. M. M. de Weert, “Techniques for simulating object color under changing illuminant conditions on electronic displays,” Color Res. Appl. 17, 316–327 (1992).
[CrossRef]

M. R. Luo, B. Rigg, “Chromaticity-discrimination ellipses for surface colors,” Color Res. Appl. 11, 25–42 (1986).
[CrossRef]

K. Witt, “Three-dimensional threshold of color-difference perceptibility in painted samples: variability of observers in four CIE color regions,” Color Res. Appl. 12, 128–134 (1987).
[CrossRef]

A. R. Robertson, “CIE guidelines for coordinated research on color-difference evaluation,” Color Res. Appl. 3, 149–151 (1978).

Die Farbe (2)

E. Hita, F. González-Caballero, A. Cervantes, “On the geometric structure of color discrimination thresholds,” Die Farbe 30, 55–64 (1982).

J. M. Zoido, F. Carreño, E. Bernabeu, “Some considerations on MacAdam’s ellipses,” Die Farbe 43, 1–16 (1997).

J. Opt. Soc. Am. (13)

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

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

G. Wyszecki, G. H. Fielder, “Color-difference matches,” J. Opt. Soc. Am. 61, 1501–1513 (1971).
[CrossRef] [PubMed]

L. Silberstein, D. L. MacAdam, “The distribution of color matchings around a color center,” J. Opt. Soc. Am. 35, 32–39 (1945).
[CrossRef]

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]

W. R. J. Brown, “Statistics of color-matching data,” J. Opt. Soc. Am. 42, 252–256 (1952).
[CrossRef]

W. R. J. Brown, W. G. Howe, J. E. Jackson, R. H. Morris, “Multivariate normality of the color-matching process,” J. Opt. Soc. Am. 46, 46–49 (1956).
[CrossRef]

H. R. Davidson, E. Friede, “The size of acceptable color differences,” J. Opt. Soc. Am. 43, 581–589 (1953).
[CrossRef] [PubMed]

R. M. Rich, F. W. Billmeyer, “Method for deriving color-difference-perceptibility ellipses for surface-color samples,” J. Opt. Soc. Am. 65, 956–959 (1975).
[CrossRef]

W. R. J. Brown, “The effect of field size and chromatic surroundings on color discrimination,” J. Opt. Soc. Am. 42, 837–844 (1952).
[CrossRef] [PubMed]

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

L. T. Sharpe, G. Wyszecki, “Proximity factor in color-difference evaluations,” J. Opt. Soc. Am. 66, 40–49 (1976).
[CrossRef] [PubMed]

G. Wyszecki, “Propagation of errors in colorimetric transformation,” J. Opt. Soc. Am. 49, 389–393 (1959).
[CrossRef]

J. Opt. Soc. Am. A (6)

Opt. Pura Apl. (1)

J. M. Zoido, F. Carreño, E. Bernabeu, “On MacAdam’s ellipses,” Opt. Pura Apl. 29, 81–108 (1996).

Vision Res. (2)

A. L. Vingrys, L. E. Mahon, “Color and luminance detection and discrimination asymmetries and interactions,” Vision Res. 38, 1085–1095 (1998).
[CrossRef] [PubMed]

A. Chaparro, C. F. Stromeyer, R. E. Kronauer, R. T. Eskew, “Separable red-green and luminance detectors for small flashes,” Vision Res. 34, 751–762 (1994).
[CrossRef] [PubMed]

Other (6)

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, Singapore, 1984).

J. M. Zoido, “Distancia estadística generalizada. La métrica del color,” Ph.D. dissertation (Universidad Complutense de Madrid, Madrid, 1997).

Y. Legrand, “Théories sur la métrique de léspace des couleurs,” in Proceedings of the International Color Congress, Color 69, Stockholm (Göttingen, Musterschmidt, 1970), pp. 257–265.

B. H. Crawford, “Just perceptible chromaticity shifts,” in Proceedings of the International Color Congress, Color 69, Stockholm (Göttingen, Musterschmidt, 1970), pp. 302–311.

F. Parra, “Seuils différentiels de couleur,” in Proceedings of the International Color Congress, Color 69, Stockholm (Göttingen, Musterschmidt, 1970), pp. 727–733.

J. Fox, Linear Statistical Models and Related Methods (Wiley, New York, 1981).

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Figures (4)

Fig. 1
Fig. 1

Contours levels of f R a : (a) stimulus 36 and (b) stimulus 38. The dots are the coordinates of the reference stimulus. The data are for observer WRJB and were taken from Ref. 23.

Fig. 2
Fig. 2

Procedure for determining the region of chromatic indistinguishability when the color matchings are guided. For a given color center x0 and a previously selected direction θ, we determined the intersections of the straight line with the chromaticity diagram, x(1) and x(2). For a certain confidence level α, we performed the numerical integrations indicated in Eqs. (23) and (24), thus determining points x s and x i , which are the color-difference thresholds for that direction in the chromaticity diagram.

Fig. 3
Fig. 3

Regions of chromatic indistinguishability computed from Eqs. (23) and (24) for α = 0.32 (*), 0.16 (+), 0.05 (○), observer WRJB, and color centers 36 and 38 of Ref. 23. The continuous curve is that of the ellipse obtained by Brown and MacAdam.23

Fig. 4
Fig. 4

Local estimation of the asymmetry by means of the magnitude q a for color center 33 in different directions for α = 0.32 (*), 0.16 (+), 0.05 (○), and observer WRJB.

Tables (2)

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Table 1 Percentage of Deviation δcij x for Observer WRJB

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Table 2 Asymmetry Coefficients and Kurtosis for the Color Centers Studied for WRJB

Equations (29)

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Qi0=j=1nQijn,  i=1, 2, 3.
σiQ=j=1nΔQij2/n1/2,  i=1, 2, 3,
μijQ=1nk=1n ΔQikΔQjk.
fQ=AQ exp-12i=13j=13 cijQΔQiΔQj,
Xi=fiQ1, Q2, Q3,  i=1, 2, 3.
Qi=hiX1, X2, X3,  i=1, 2, 3.
fX=JXfQ,
X=AQ.
fX=AX exp-12i=13j=13 cijXΔXiΔXj,
X1=x1x2 x3,  X2=x3,  X3=1-x1-x2x2 x3.
fRax=AYx32x23exp-12i=13j=13 cijxΔxiΔxj,
Δx1=x1x3x2-x10x30x20, Δx2=x3-x30, Δx3=x31-x1-x2x2-x301-x10-x20x20.
AY=Dxdx1dx20dx3x32x23×exp-12i=13j=13 cijxΔxiΔxj-1,
Ix fRaxdx1dx2dx3=1-α,
fdiax1, x2=Axx302x23exp-12 qx1, x2,
qx1, x2=c11Xx1x30x2-x10x30x202+2c13Xx1x30x2-x10x30x20×x301-x1-x2x2-x301-x10-x20x20+c33Xx301-x1-x2x2-x301-x10-x20x202,
Ax=x302 Dx1,x21x23exp-12 qx1, x2dx1dx2-1.
xi=Dx1,x2 xifdiax1, x2dx1dx2,  i=1, 2
μijx=xi-xixj-xji, j=1, 2.
Asi=xi-xi3xi-xi23/2,  i=1, 2,  Ki=xi-xi4xi-xi22,  i=1, 2.
Dx1, x2 fdiax1, x2dx1dx2=1-α
x2x1, θ=x1-x10tanθ+x20.
fθx1=Aθx302x1-x10tanθ+x203×exp-12 qx1, x2x1, θ.
Aθ=x11x12dx1x302x1-x10tanθ+x203×exp-12 qx1, x2x1, θ-1.
x10x1s fθx1dx1=1-α2,
x1ix10 fθx1dx1=1-α2.
δcijx=100 |gij-cijx|cijx,  i, j=1, 2,
dsθ=x10-x1s2+y10-y1s21/2, diθ=x10-x1i2+y10-y1i21/2,
qaθ=|ds-di|maxds, di 100,

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