Abstract

On the assumption of a color space of constant negative curvature, von Schelling has given simple formulas for the color metric coefficients. An attempt has been made to fit such formulas to newly available color-discrimination data for 12 normal observers. A least-squares analysis indicates that Y should be replaced with the quantity, Y′=Y−0.085X+0.011Z, in the special role assigned to luminance by von Schelling. Corresponding to this change, coordinates x′=X/(X+Y′+Z), y′=Y′/(X+Y′+Z) yield the simplest formulas for the coefficients of the expression for color difference:

ΔS=(g11Δx2+2g12ΔxΔy+g22Δy2+g33ΔY2/Y2)12.

Least-squares fitting of the experimental data to von Schelling’s formulas yielded:

g11=g33/(10x2-6.886x+2.083+2.5y-16.66xy+19.6y2).g12=g11(0.3443-x)/y.g22=g11[0.09+(0.3443-x)2]/y2.

These formulas are the best approximations obtainable on the assumption of a color space of constant negative or zero curvature. However, they give such poor approximations to the experimental data that it is concluded that the assumption of a color space inherently more complicated than a space of constant curvature would be required to fit the data to a closeness commensurate with the experimental reproducibility.

© 1957 Optical Society of America

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References

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  1. D. L. MacAdam, J. Opt. Soc. Am. 32, 247–274 (1942).
    [Crossref]
  2. W. R. J. Brown and D. L. MacAdam, J. Opt. Soc. Am. 39, 808–834 (1949).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  5. H. von Schelling, J. Opt. Soc. Am. 45, 1072–1079 (1955).
    [Crossref]
  6. H. von Schelling, J. Opt. Soc. Am. 46, 309–315 (1956).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  8. D. L. MacAdam, Die Farbe 4, 165 (1955).
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  10. D. L. MacAdam, Doc. Ophthalmol. 3, 214–223 (1947).
    [Crossref]
  11. L. Silberstein, J. Opt. Soc. Am. 33, 1–10 (1943).
    [Crossref]

1957 (1)

1956 (2)

1955 (2)

1949 (1)

1947 (2)

D. L. MacAdam, Rev. opt. 28, 161–173 (1947).

D. L. MacAdam, Doc. Ophthalmol. 3, 214–223 (1947).
[Crossref]

1943 (2)

1942 (1)

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Figures (8)

Fig. 1
Fig. 1

Two perspective views of a rectangular Cartesian coordinate system, in which various wavelengths of the equal-energy spectrum are represented according to their tristimulus values, X, Y, Z. The base plane, Y=0, represents zero luminance. Straight lines from the origin to points representing various colors intersect the 1,1,1 plane in points representing their chromaticities. The usual chromaticity diagram represents the same fractional coordinates, x and y, on a rectangular rather than this equilateral triangle.

Fig. 2
Fig. 2

Loci of equal values of g13/g33 according to Eq. (1), when F=0.011 and U=−0.096.

Fig. 3
Fig. 3

Loci of equal values of g23/g13 according to Eq. (1), when F/U=−0.115.

Fig. 4
Fig. 4

Loci of constant values of g11″=g11′/g33, according to Eq. (9a). These are concentric ellipses, all having the same shape and orientation. Their common center is at xc′=0.45, yc′=0.13, where g11″=1.41.

Fig. 5
Fig. 5

Graphical method for determining ω, which is constant along each vertical line. Locally uniform chromaticity-scale diagram, with angle, ω, between axes, is shown at upper right. Note that H=cosω and g 12 = ( g 11 g 22 ) 1 2 cos ω.

Fig. 6
Fig. 6

Loci of constant values of ratio, J, of length of equal coordinate increments along horizontal and oblique axes of locally uniform chromaticity-scale diagram (shown at upper right). Note that g22′=g11′/J2.

Fig. 7
Fig. 7

Solid ellipses show experimental results in the x′,y′ diagram. Dotted ellipses are based on the metric coefficients computed from Eqs. (9). All ellipses are enlarged 10 times.

Fig. 8
Fig. 8

Solid ellipses show experimental results in the standard CIE diagram. Dotted ellipses are based on the metric coefficients computed from Eqs. (9), after transformation to the standard (x,y) diagram. All ellipses are enlarged 10 times.

Tables (1)

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Table I Individual values of coefficients.

Equations (32)

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Δ S = ( g 11 Δ x 2 + 2 g 12 Δ x Δ y + g 22 Δ y 2 + g 33 Δ Y 2 / Y 2 ) 1 2 .
g 11 = g 33 / ( 10 x 2 - 6.886 x + 2.083 + 2.5 y - 16.66 x y + 19.6 y 2 ) . g 12 = g 11 ( 0.3443 - x ) / y . g 22 = g 11 [ 0.09 + ( 0.3443 - x ) 2 ] / y 2 .
g 13 / g 33 = 0.087 U / [ U x + ( 1 - F ) y + F ] g 23 / g 33 = 0.087 ( U x + F ) / [ U x + ( 1 - F ) y + F ] .
d s 2 = g 11 d x 2 + 2 g 12 d x d y + g 22 d y 2 + g 33 d l 2
x = X / ( X + Y + Z ) ,             y = Y / ( X + Y + Z ) ,
x = x / q ,             y = ( y + q - 1 ) / q
g j l = ( δ x i / δ x j ) g i k ( δ x k / δ x l ) ,
g = T g T ,
T j i = ( δ x i / δ x j ) = { ( U x + 1 ) q - U ( 1 - y ) q - 0.087 U q / y - F x q ( 1 + F - F y ) q 0.087 ( F q + U x ) q / ( q + y - 1 ) y 0 0 1 } ,
g 12 / ( g 11 g 22 ) 1 2 H .
H ± [ 1 + 1 / ( M - K x ) 2 ] - 1 2 ,
K = [ ( P C - Q 2 ) / P 2 ] - 1 2 ,             M = - K Q / P .
( g 11 / g 22 ) 1 2 J K y ( 1 - H 2 ) 1 2 .
L [ x 2 - 2 E x + α + N y + R x y + S y 2 ] = g 33 / g 11 ,
E = - M / K             and             α = ( 1 + M 2 ) / K 2 .
d s 2 = g 33 [ d l 2 + g 11 ( d x 2 + H d x d y / J + d y 2 / J 2 ) ] .
x 0 = ( α R + E N ) / ( N + E R ) y 0 = - 2 ( x 0 - E ) / R P = y 0 2 / Y 0 2 ( S y 0 2 - x 0 2 + 2 E x 0 - α ) ,
C = α P Q = E P B = C - Q A = P - C + 2 B ( P C - Q 2 ) / P 2 = 1 / K 2 .
M = - 1.148 K = 3.333 E = 0.3443 α = 0.2083.
L = 10 N = 0.25 R = - 1.666 S = 1.96.
Y 0 = 6 ,
x 0 = 0.807 y 0 = 0.552 P = 0.029 C = 0.006 Q = 0.010 A = 0.015 B = - 0.004.
x c = 0.45 y c = 0.13.
g 11 = g 33 / ( 10 x 2 - 6.886 x + 2.083 + 2.5 y - 16.66 x y + 19.6 y 2 )
g 12 = g 11 ( 0.3443 - x ) / y
g 22 = g 11 [ 0.09 + ( 0.3443 - x ) 2 ] / y 2 .
g i k = T - 1 g i k ( T - 1 ) ,
- F ( g 13 / g 33 ) ( 1 - y ) / y + U [ 0.087 - ( g 13 / g 33 ) x ] / y = g 13 / g 33 - F [ 0.087 + ( g 23 / g 33 ) ( 1 - y ) y ] / y 2 + U [ 0.087 + ( g 23 / g 33 ) y ] / y 2 = g 23 / g 33 .
v = ( a 3 x + a 4 y + a 5 ) / ( a 1 x + a 2 y + 1 ) ,
x = ( a 4 - a 2 a 5 ) u / [ ( a 2 a 3 - a 1 a 4 ) u - a 2 v + a 4 ] , y = [ ( a 1 a 5 - a 3 ) u + v - a 5 ] / [ ( a 2 a 3 - a 1 a 4 ) u - a 2 v + a 4 ] .
x 1 = x 1 / ( - U x 1 + F x 2 + 1 - F ) , x 2 = [ - U x 1 + ( 1 + F ) x 2 - F ] / ( - U x 1 + F x 2 + 1 - F ) ,
x 3 = x 3 + 0.087 ln [ - U x 1 + ( 1 + F ) x 2 - F ] / x 2 ,