Abstract

Experimental data concerning the accuracy of visual chromaticity matching (automatically constant luminance) which were previously summarized graphically, are compared with predictions based on the assumption of a normal frequency distribution in two dimensions. The standard deviations and correlation coefficient of the two conventional coordinates vary from point to point in the chromaticity diagram and values computed from the experimental data are presented. The major and minor axes, and the angle of inclination of the major axis of the ellipse representing the standard deviation of matchings in any specified direction from any central color, are computed from the parameters of the distribution function, as are the coefficients of the quadratic differential form representing noticeability of chromaticity difference. The conclusions previously published, based on graphical analysis of the data, are confirmed and strengthened by these results, eliminating the former dependence upon personal judgment in construction of ellipses through scattered points.

© 1945 Optical Society of America

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References

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  1. L. Silberstein, J. Opt. Soc. Am. 33, 1–10 (1943).
    [Crossref]
  2. D. L. MacAdam, J. Opt. Soc. Am. 33, 18–26 (1943).
    [Crossref]
  3. D. L. MacAdam, J. Opt. Soc. Am. 32, 247–274 (1942).
    [Crossref]
  4. Reference 3, Figs. 23 to 47.
  5. J. Guild, Phil. Trans. Roy. Soc. A230, 149–187 (1931).

1943 (2)

1942 (1)

1931 (1)

J. Guild, Phil. Trans. Roy. Soc. A230, 149–187 (1931).

J. Opt. Soc. Am. (3)

Phil. Trans. Roy. Soc. (1)

J. Guild, Phil. Trans. Roy. Soc. A230, 149–187 (1931).

Other (1)

Reference 3, Figs. 23 to 47.

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Tables (6)

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Table I Distributions of errors along radii of Fig. 26 (reference 3), center 0.150, 0.680.

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Table II Distributions of errors along radii of Fig. 44 (reference 3).

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Table III Summary of analysis of deviations greater and less than “probable error.”

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Table IV Computation of constants of normal distribution law.

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Table V Comparison of calculated and experimental values of σ.

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Table VI Computed values of constants of ellipses.

Equations (45)

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x 0 = 1 n x i = x ¯ ,             y 0 = 1 n y i = y ¯ .
f ( x , y ) = C exp [ - { u 2 σ 1 2 - 2 ρ σ 1 σ 2 u v + v 2 σ 2 2 } / 2 ( 1 - ρ 2 ) ] ,
C = 1 2 π σ 1 σ 2 ( 1 - ρ 2 ) 1 2 .
σ 1 2 = u 2 Av ,             σ 2 2 = v 2 Av ,             ρ = u v Av σ 1 σ 2 ,
f ( r , ω ˜ ) = C r exp [ - φ 2 ( ω ˜ ) r 2 ] ,
φ 2 ( ω ˜ ) = { cos 2 ω ˜ σ 1 2 - ρ sin 2 ω ˜ σ 1 σ 2 + sin 2 ω ˜ σ 2 2 } / 2 ( 1 - ρ 2 ) .
f ( r ) = c exp ( - φ 2 r 2 ) ,
f ( r ) = φ π exp ( - φ 2 r 2 ) ,
σ 2 = 2 0 r 2 f ( r ) d r = 2 φ 2 π 0 u 2 exp ( - u 2 ) d u = 1 2 φ 2 .
f ( r ) = 1 σ ( 2 π ) 1 2 exp ( - r 2 / 2 σ 2 ) ,
1 σ 2 = 1 1 - ρ 2 { cos 2 ω ˜ σ 1 2 - ρ sin 2 ω ˜ σ 1 σ 2 + sin 2 ω ˜ σ 2 2 } .
1 σ 1 2 U 2 + 1 σ 2 2 V 2 - 2 ρ σ 1 σ 2 U V = 1 - ρ 2 ,
p ( r ) = 2 0 r f ( r ) d r = 2 π 0 r / σ 2 exp ( - u 2 ) d u = Φ ( r σ 2 ) ,
p = Φ ( 0.477 ) = 1 2 .
σ 1 2 = 2 C ω ˜ = 0 π r = 0 r 2 cos 2 ω ˜ exp ( - r 2 / 2 σ 2 ) r d r d ω ˜ = 4 C 0 π U 2 σ 2 d ω ˜ ,
U = σ cos ω ˜ .
2 C 0 π 0 exp ( - r 2 / 2 σ 2 ) r d r d ω ˜ = 1 ,
C = 1 / 2 π σ 2 Av .
σ 1 2 = 2 U 2 σ 2 Av / σ 2 Av .
σ 2 2 = 2 V 2 σ 2 Av / σ 2 Av ,
V = σ sin ω ˜ .
u v Av = 2 U V σ 2 Av / σ 2 Av ,
ρ = u v Av / σ 1 σ 2 .
σ 2 Av = 1 360 1 n σ k 2 · Δ ω ˜ k ,
Δ ω ˜ k = ω ˜ k + 1 - ω ˜ k - 1 ,             ω ˜ 0 = ω ˜ n - 180 °
ω ˜ n + 1 = ω ˜ 1 + 180 ° .
U 2 σ 2 Av = 1 360 1 n σ k 4 cos 2 ω ˜ k · Δ ω ˜ k ,            
V 2 σ 2 Av = 1 360 1 n σ k 4 sin 2 ω ˜ k · Δ ω ˜ k ,            
U V σ 2 Av = 1 360 1 n σ k 4 sin ω ˜ k cos ω ˜ k · Δ ω ˜ k .            
( 1 + t 2 ) / σ 2 = [ 1 σ 1 2 - 2 ρ t σ 1 σ 2 + t 2 σ 2 2 ] / ( 1 - ρ 2 ) ,
1 σ 2 = ( A t 2 + B t + C ) / ( 1 + t 2 ) .
( A t 2 + B t + C ) t / ( 1 + t 2 ) = A t + B / 2
t 2 + 2 ( C - A ) t / B = 1 ,
t m = [ A - C ± { ( A - C ) 2 + B 2 } 1 2 ] / B ,
1 σ 2 = A + B / 2 t m ,
A , B , C = ( 1 σ 2 2 , - 2 ρ σ 1 σ 2 , 1 σ 1 2 ) / ( 1 - ρ 2 ) .
σ 2 = ( 1 + t 2 ) / ( 0.896 t 2 - 1.016 t + 0.473 ) .
a = 2.73 ,             b = 0.90 ,             θ = 33.7 ° .
d s = PQ / σ             or             d s 2 = ( d x 2 + d y 2 ) / σ 2 ,
d s 2 = ( d x 2 σ 1 2 - 2 ρ σ 1 σ 2 d x d y + d y 2 σ 2 2 ) / ( 1 - ρ 2 ) ,
d s 2 = g 11 d x 2 + 2 g 12 d x d y + g 22 d y 2 ,
g 11 , g 12 , g 22 = ( 1 σ 1 2 , - ρ σ 1 σ 2 , 1 σ 2 2 ) / ( 1 - ρ 2 ) .
2 log f 0 f = g 11 ( x - x 0 ) 2 + 2 g 12 ( x - x 0 ) ( y - y 0 ) + g 22 ( y - y 0 ) 2 ;
d s 2 = g 11 d X 2 + 2 g 12 d X d Y + + g 33 d Z 2 ,
d s 2 = 2 log f 0 / f .