Abstract

A nonlinear transformation of the CIE x,y chromaticity coordinates has been derived from the combined color-matching-variance data of 14 normal observers. In the resulting diagram, the series of equiluminous chromaticities entailing the least number of standard deviations of color matching (σ-units) between any two-terminal, equiluminous chromaticities is the straight line drawn between the points that represent those terminal colors. The total number of σ-unit differences between those terminal colors is the euclidean distance between those two points. According to Schrödinger’s hypothesis, the loci of constant hue are the straight lines (geodesics) radiating from the point that represents hueless colors in this diagram. The horizontal coordinate in the geodesic chromaticity diagram is ξ=3751a2-10a4-520b2+13295b3+32327ab-25491a2b-41672ab2+10a3b-5227a12+29524a14, where a = 10x/(2.4x + 34y + 1) and b = 10y/(2.4x + 34y + 1). The vertical coordinate in the geodesic chromaticity diagram is η = 404b − 185b2 + 52b3 + 69a(1 − b2) − 3a2b + 30ab3, where a = 10x/(4.2yx + 1) and b = 10y/(42yx + 1). These formulas were obtained by use of averages of data for two observers whose individual data were published in 1949 and the weighted averages for 12 young observers, which were published in 1957, together with the data for the single observer, PGN, whose data were published in 1942–45. On the basis of extensive studies of these data, the PGN data were assigned 30% weight in the derivation of the new ξ,η diagram. The 1949 data were assigned 44% weight, or 22% per observer, and the 1957 data were assigned 26%, or about 2.2% per observer. The best fit was found by assuming that the over-all mean of the standard deviation of color matching according to the 1949 data was 1.2 times as much as the standard deviation for PGN, and that the weighted-mean standard deviation for the 12 observers was 1.04 times the standard deviation for PGN. When adjusted to this basis, the radii of the variance ellipses for the three sets of observations fit unit distance on the in diagram with a mean-square error of 0.056. The mean-square error for the PGN data is 0.052, which may be compared with 0.02 for a version of the ξ,η diagram that was derived from the PGN data alone. The mean-square deviation from unit distance of the 1949 average data is 0.053, and for the 1955 weighted-average ellipses it is 0.076.

© 1971 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. L. MacAdam, J. Opt. Soc. Amer. 32, 247 (1942).
    [CrossRef]
  2. D. L. MacAdam, J. Opt. Soc. Amer. 33, 18 (1943).
    [CrossRef]
  3. L. Silberstein, D. L. MacAdam, J. Opt. Soc. Amer. 35, 32 (1945).
    [CrossRef]
  4. D. L. MacAdam, Official Digest, Fed. Soc. Paint Technol. 37, 1487 (1965).
  5. D. L. MacAdam, Acta Chromatica 1, 147 (1965).
  6. D. L. MacAdam, J. Phot. Sci. 14, 229 (1966).
  7. D. L. MacAdam, Phys. Today 20, 27 (Jan.1967).
    [CrossRef]
  8. W. R. J. Brown, D. L. MacAdam, J. Opt. Soc. Amer. 39, 808 (1949).
    [CrossRef]
  9. W. R. J. Brown, J. Opt. Soc. Amer. 47, 137 (1957).
    [CrossRef]
  10. F. W. Billmeyer, Opt. Spectra 4, 63 (1970), especially p. 65.
  11. D. L. MacAdam, J. Opt. Soc. Amer. 54, 249 (1964).
    [CrossRef]
  12. D. L. MacAdam, J. Opt. Soc. Amer. 54, 1161 (1964).
    [CrossRef]
  13. D. L. MacAdam, J. Opt. Soc. Amer. 55, 91 (1965).
    [CrossRef]
  14. D. L. MacAdam, J. Opt. Soc. Amer. 56, 1784 (1966).
    [CrossRef]
  15. L. F. C. Friele, Die Farbe 10, 193 (1961).
  16. L. F. C. Friele, J. Opt. Soc. Amer. 55, 1314 (1965).
    [CrossRef]
  17. K. D. Chickering, J. Opt. Soc. Amer. 57, 537 (1967).
    [CrossRef]
  18. G. Wyszecki, J. Opt. Soc. Amer. 58, 290 (1967), especially p. 292, but K1 and K2 should be set identically equal to 1.0, to correspond to the results of Ref. 17 and to represent the data of Ref. 3.
  19. D. L. MacAdam, Die Farbe18, in press (1969).
  20. D. L. MacAdam, Die Farbe19, in press (1970).

1970

F. W. Billmeyer, Opt. Spectra 4, 63 (1970), especially p. 65.

1967

K. D. Chickering, J. Opt. Soc. Amer. 57, 537 (1967).
[CrossRef]

G. Wyszecki, J. Opt. Soc. Amer. 58, 290 (1967), especially p. 292, but K1 and K2 should be set identically equal to 1.0, to correspond to the results of Ref. 17 and to represent the data of Ref. 3.

D. L. MacAdam, Phys. Today 20, 27 (Jan.1967).
[CrossRef]

1966

D. L. MacAdam, J. Phot. Sci. 14, 229 (1966).

D. L. MacAdam, J. Opt. Soc. Amer. 56, 1784 (1966).
[CrossRef]

1965

D. L. MacAdam, J. Opt. Soc. Amer. 55, 91 (1965).
[CrossRef]

D. L. MacAdam, Official Digest, Fed. Soc. Paint Technol. 37, 1487 (1965).

D. L. MacAdam, Acta Chromatica 1, 147 (1965).

L. F. C. Friele, J. Opt. Soc. Amer. 55, 1314 (1965).
[CrossRef]

1964

D. L. MacAdam, J. Opt. Soc. Amer. 54, 249 (1964).
[CrossRef]

D. L. MacAdam, J. Opt. Soc. Amer. 54, 1161 (1964).
[CrossRef]

1961

L. F. C. Friele, Die Farbe 10, 193 (1961).

1957

W. R. J. Brown, J. Opt. Soc. Amer. 47, 137 (1957).
[CrossRef]

1949

W. R. J. Brown, D. L. MacAdam, J. Opt. Soc. Amer. 39, 808 (1949).
[CrossRef]

1945

L. Silberstein, D. L. MacAdam, J. Opt. Soc. Amer. 35, 32 (1945).
[CrossRef]

1943

D. L. MacAdam, J. Opt. Soc. Amer. 33, 18 (1943).
[CrossRef]

1942

D. L. MacAdam, J. Opt. Soc. Amer. 32, 247 (1942).
[CrossRef]

Billmeyer, F. W.

F. W. Billmeyer, Opt. Spectra 4, 63 (1970), especially p. 65.

Brown, W. R. J.

W. R. J. Brown, J. Opt. Soc. Amer. 47, 137 (1957).
[CrossRef]

W. R. J. Brown, D. L. MacAdam, J. Opt. Soc. Amer. 39, 808 (1949).
[CrossRef]

Chickering, K. D.

K. D. Chickering, J. Opt. Soc. Amer. 57, 537 (1967).
[CrossRef]

Friele, L. F. C.

L. F. C. Friele, J. Opt. Soc. Amer. 55, 1314 (1965).
[CrossRef]

L. F. C. Friele, Die Farbe 10, 193 (1961).

MacAdam, D. L.

D. L. MacAdam, Phys. Today 20, 27 (Jan.1967).
[CrossRef]

D. L. MacAdam, J. Opt. Soc. Amer. 56, 1784 (1966).
[CrossRef]

D. L. MacAdam, J. Phot. Sci. 14, 229 (1966).

D. L. MacAdam, J. Opt. Soc. Amer. 55, 91 (1965).
[CrossRef]

D. L. MacAdam, Acta Chromatica 1, 147 (1965).

D. L. MacAdam, Official Digest, Fed. Soc. Paint Technol. 37, 1487 (1965).

D. L. MacAdam, J. Opt. Soc. Amer. 54, 249 (1964).
[CrossRef]

D. L. MacAdam, J. Opt. Soc. Amer. 54, 1161 (1964).
[CrossRef]

W. R. J. Brown, D. L. MacAdam, J. Opt. Soc. Amer. 39, 808 (1949).
[CrossRef]

L. Silberstein, D. L. MacAdam, J. Opt. Soc. Amer. 35, 32 (1945).
[CrossRef]

D. L. MacAdam, J. Opt. Soc. Amer. 33, 18 (1943).
[CrossRef]

D. L. MacAdam, J. Opt. Soc. Amer. 32, 247 (1942).
[CrossRef]

D. L. MacAdam, Die Farbe18, in press (1969).

D. L. MacAdam, Die Farbe19, in press (1970).

Silberstein, L.

L. Silberstein, D. L. MacAdam, J. Opt. Soc. Amer. 35, 32 (1945).
[CrossRef]

Wyszecki, G.

G. Wyszecki, J. Opt. Soc. Amer. 58, 290 (1967), especially p. 292, but K1 and K2 should be set identically equal to 1.0, to correspond to the results of Ref. 17 and to represent the data of Ref. 3.

Acta Chromatica

D. L. MacAdam, Acta Chromatica 1, 147 (1965).

Die Farbe

L. F. C. Friele, Die Farbe 10, 193 (1961).

Fed. Soc. Paint Technol.

D. L. MacAdam, Official Digest, Fed. Soc. Paint Technol. 37, 1487 (1965).

J. Opt. Soc. Amer.

W. R. J. Brown, D. L. MacAdam, J. Opt. Soc. Amer. 39, 808 (1949).
[CrossRef]

W. R. J. Brown, J. Opt. Soc. Amer. 47, 137 (1957).
[CrossRef]

L. F. C. Friele, J. Opt. Soc. Amer. 55, 1314 (1965).
[CrossRef]

K. D. Chickering, J. Opt. Soc. Amer. 57, 537 (1967).
[CrossRef]

G. Wyszecki, J. Opt. Soc. Amer. 58, 290 (1967), especially p. 292, but K1 and K2 should be set identically equal to 1.0, to correspond to the results of Ref. 17 and to represent the data of Ref. 3.

D. L. MacAdam, J. Opt. Soc. Amer. 32, 247 (1942).
[CrossRef]

D. L. MacAdam, J. Opt. Soc. Amer. 33, 18 (1943).
[CrossRef]

L. Silberstein, D. L. MacAdam, J. Opt. Soc. Amer. 35, 32 (1945).
[CrossRef]

D. L. MacAdam, J. Opt. Soc. Amer. 54, 249 (1964).
[CrossRef]

D. L. MacAdam, J. Opt. Soc. Amer. 54, 1161 (1964).
[CrossRef]

D. L. MacAdam, J. Opt. Soc. Amer. 55, 91 (1965).
[CrossRef]

D. L. MacAdam, J. Opt. Soc. Amer. 56, 1784 (1966).
[CrossRef]

J. Phot. Sci.

D. L. MacAdam, J. Phot. Sci. 14, 229 (1966).

Opt. Spectra

F. W. Billmeyer, Opt. Spectra 4, 63 (1970), especially p. 65.

Phys. Today

D. L. MacAdam, Phys. Today 20, 27 (Jan.1967).
[CrossRef]

Other

D. L. MacAdam, Die Farbe18, in press (1969).

D. L. MacAdam, Die Farbe19, in press (1970).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Nonlinear transformation of CIE (x,y)-coordinate system, derived from variances of color matching by 14 normal observers. The averaged observed variances are represented by 1 unit on the ξ and η scales, with less than 24% rms error.

Fig. 2
Fig. 2

Constant-hue loci derived by transforming, back to the CIE (x,y) diagram, straight lines drawn outward from the point representing CIE source D65 (ξ = 861.2, η = 395.73) in the (ξ,η) diagram.

Fig. 3
Fig. 3

Variances of color matching by observer PGN, enlarged ten times on the ξ,η diagram. The rms variation from unit radius, of the observational ellipses, is 23%.

Fig. 4
Fig. 4

Variances of color matching by observer DLM, enlarged ten times on the ξ,η diagram.

Fig. 5
Fig. 5

Variances of color matching by observer WRJB (Ref. 8), enlarged ten times on the ξ,η diagram.

Fig. 6
Fig. 6

Variances of color matching by 12 young observers with normal color vision, enlarged ten times on the ξ,η diagram.

Fig. 7
Fig. 7

Averaged variances of color matching by observers DLM and WRJB, enlarged ten times on the ξ,η diagram. The rms variation from unit radius of the ellipses for the averaged data is 23%.

Fig. 8
Fig. 8

Weighted-average variances of color matching by 12 young normal observers (Ref. 9), enlarged ten times on the ξ,η diagram. The rms variation from unit radius of the ellipses for the averaged data is 28%.

Tables (2)

Tables Icon

Table I Mean-Square Errors of Fitting Radii of Experimental Color-Matching-Variance Ellipses for 14 Observers, Using 5 Different Types of Color-Difference Formulas

Tables Icon

Table II Coefficients of Projective Transformations, Optimized for 14 Observers

Equations (2)

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

ξ = 3751 a 2 - 10 a 4 - 520 b 2 + 13295 b 3 + 32327 a b - 25491 a 2 b - 41672 a b 2 + 10 a 3 b - 5227 a 1 2 + 2952 a 1 4 ,
η = 404 b - 185 b 2 + 52 b 3 + 69 a ( 1 - b 2 ) - 3 a 2 b + 30 a b 3 ,

Metrics