## Geodesic Chromaticity Diagram Based on Variances of Color Matching by 14 Normal Observers

Applied Optics, Vol. 10, Issue 1, pp. 1-7 (1971)

http://dx.doi.org/10.1364/AO.10.000001

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### 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
*a* = 10*x*/(2.4*x* + 34*y* + 1) and *b* = 10*y*/(2.4*x* + 34*y* + 1). The vertical coordinate in the geodesic chromaticity diagram is *η* = 404*b* − 185*b*^{2} + 52*b*^{3} + 69*a*(1 − *b*^{2}) − 3*a*^{2}*b* + 30*ab*^{3}, where *a* = 10*x*/(4.2*y* − *x* + 1) and *b* = 10*y*/(42*y* − *x* + 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

**History**

Original Manuscript: July 6, 1970

Published: January 1, 1971

**Citation**

D. L. MacAdam, "Geodesic Chromaticity Diagram Based on Variances of Color Matching by 14 Normal Observers," Appl. Opt. **10**, 1-7 (1971)

http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-10-1-1

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