Macadam D L
Appl Opt. 1971 Jan 1;10(1):1-7. doi: 10.1364/AO.10.000001.
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 (sigma-units) between any two-terminal, equiluminous chromaticities is the straight line drawn between the points that represent those terminal colors. The total number of sigma-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 xi = 3751a(2) - 10a(4) - 520b(2) + 13295b(3) + 32327ab - 25491a(2)b - 41672ab(2) + 10a(3)b - 5227a((1/2)) + 2952(4)a((1/4)), where a = 10x/(2.4x + 34y + 1) and b = 10y/(2.4x + 34y + 1). The vertical coordinate in the geodesic chromaticity diagram is eta = 404b - 185b(2) + 52b(3) + 69a(1 - b(2)) - 3a(2)b + 30ab(2), where a = 10x/(4.2y - x + 1) and b = 10y/(4.2y - 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 xi,eta 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 xi,eta 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 xi,eta 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.
基于14名正常观察者的综合颜色匹配方差数据，推导出了CIE x,y色度坐标的非线性变换。在所得图表中，在任意两个端点等亮度色度之间，颜色匹配标准差（sigma单位）数量最少的等亮度色度系列是连接代表这些端点颜色的点所画的直线。这些端点颜色之间的sigma单位差异总数是这两点之间的欧几里得距离。根据薛定谔的假设，恒定色调的轨迹是从该图表中代表无色颜色的点辐射出的直线（测地线）。测地色度图中的横坐标为ξ = 3751a(2) - 10a(4) - 520b(2) + 13295b(3) + 32327ab - 25491a(2)b - 41672ab(2) + 10a(3)b - 5227a((1/2)) + 2952(4)a((1/4))，其中a = 10x/(2.4x + 34y + 1)且b = 10y/(2.4x + 34y + 1)。测地色度图中的纵坐标为η = 404b - 185b(2) + 52b(3) + 69a(1 - b(2)) - 3a(2)b + 30ab(2)，其中a = 10x/(4.2y - x + 1)且b = 10y/(4.2y - x + 1)。这些公式是通过使用两名观察者的数据平均值（其个人数据于1949年发表）以及12名年轻观察者的加权平均值（于1957年发表），再加上单个观察者PGN的数据（其数据于1942 - 1945年发表）得出的。基于对这些数据的广泛研究，在推导新的ξ,η图时，PGN数据被赋予30%的权重。1949年的数据被赋予44%的权重，即每位观察者22%，1957年的数据被赋予26%的权重，即每位观察者约2.2%。通过假设根据1949年数据的颜色匹配标准差的总体平均值是PGN标准差的1.2倍，以及12名观察者的加权平均标准差是PGN标准差的1.04倍，找到了最佳拟合。在此基础上进行调整后，三组观测值的方差椭圆半径在ξ,η图上拟合单位距离，均方误差为0.056。PGN数据的均方误差为0.052，可与仅从PGN数据推导的ξ,η图版本的0.02进行比较。1949年平均数据与单位距离的均方偏差为0.053，1955年加权平均椭圆的均方偏差为0.076。
