Abstract

An important part of the basis of instrumental colorimetry, namely, the uncertainty of spectrophotometric data, has been neglected in the investigations of color and color specification. The uncertainty of measurement of spectral data results from recurrent and random fluctuations such as slit-width and stray-energy effects and photometric-scale and wavelength-scale effects. By means of the theory of propagation of errors in a computed result the variances in each of the CIE chromaticity coordinates are determined. As the chromaticity coordinates are correlated, the covariance between x and y is also determined. Uncertainty ellipses are formed by assuming that the variates have normal distributions about their mean values with the determined covariance matrix. These ellipses have been computed for a representative series of colors for the special case of constant spectral uncertainty. These ellipses are compared with the perceptibility ellipses of MacAdam. As in visual colorimetry, an instrumental measurement of chromaticity coordinates does not establish a single point on the chromaticity diagram, but rather indicates that the true chromaticity point probably lies within an elliptical area surrounding this point.

© 1953 Optical Society of America

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References

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  1. Commission Internationale de l’Éclairage, Proceedings of the eighth session, Cambridge, England, 19–29 (September, 1931).
  2. A. G. Worthing and J. Geffner, Treatment of Experimental Data (John Wiley and Sons, Inc., New York, 1943).
  3. I. Nimeroff, J. Opt. Soc. Am. 42, 881, (1952).
  4. S. S. Wilks, Mathematical Statistics (Princeton University Press, Princeton, 1950), pp. 59 and 104.
  5. D. L. MacAdam, J. Opt. Soc. Am. 32, 247 (1942).
    [Crossref]

1952 (1)

I. Nimeroff, J. Opt. Soc. Am. 42, 881, (1952).

1942 (1)

Geffner, J.

A. G. Worthing and J. Geffner, Treatment of Experimental Data (John Wiley and Sons, Inc., New York, 1943).

MacAdam, D. L.

Nimeroff, I.

I. Nimeroff, J. Opt. Soc. Am. 42, 881, (1952).

Wilks, S. S.

S. S. Wilks, Mathematical Statistics (Princeton University Press, Princeton, 1950), pp. 59 and 104.

Worthing, A. G.

A. G. Worthing and J. Geffner, Treatment of Experimental Data (John Wiley and Sons, Inc., New York, 1943).

J. Opt. Soc. Am. (2)

I. Nimeroff, J. Opt. Soc. Am. 42, 881, (1952).

D. L. MacAdam, J. Opt. Soc. Am. 32, 247 (1942).
[Crossref]

Other (3)

S. S. Wilks, Mathematical Statistics (Princeton University Press, Princeton, 1950), pp. 59 and 104.

Commission Internationale de l’Éclairage, Proceedings of the eighth session, Cambridge, England, 19–29 (September, 1931).

A. G. Worthing and J. Geffner, Treatment of Experimental Data (John Wiley and Sons, Inc., New York, 1943).

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Figures (1)

Fig. 1
Fig. 1

Uncertainty ellipses compared with MacAdam perceptibility ellipses. Correlated uncertainty, solid line ellipses; MacAdam perceptibility, dotted line ellipses. (All ellipses are plotted on a times ten scale.)

Equations (12)

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X λ R λ A λ , Y λ R λ B λ , Z λ R λ C λ .             }
x X / S , y Y / S , z Z / S ,             }
x = λ ( R λ A λ ) / λ [ R λ ( A λ + B λ + C λ ) ] , y = λ ( R λ B λ ) / λ [ R λ ( A λ + B λ + C λ ) ] , z = λ ( R λ C λ ) / λ [ R λ ( A λ + B λ + C λ ) ] . }
σ U = [ ( U L ) 2 σ L 2 + ( U M ) 2 σ M 2 + ( U N ) 2 σ N 2 + ] 1 2
σ x 2 = λ ( S A λ - X s λ ) 2 σ R λ 2 / S 4 , σ y 2 = λ ( S B λ - Y s λ ) 2 σ R λ 2 / S 4 , σ z 2 = λ ( S C λ - Z s λ ) 2 σ R λ 2 / S 4 ,             }
s λ = ( A λ + B λ + C λ ) .
Σ = | σ x 2 σ x y 2 σ x y 2 σ y 2 | ,
σ x y 2 = λ ( S A λ - X s λ ) ( S B λ - Y s λ ) σ R λ 2 / S 4 .
N [ x n , y n ( x , y ) Σ ] = 1 2 π Σ 1 2 e - Q n / 2 d x n d y n ,
Q n = ( x n - x ) 2 σ y 2 - 2 ( x n - x ) ( y n - y ) σ x y 2 + ( y n - y ) 2 σ x 2 ( σ x 2 σ y 2 - σ x y 4 ) .
Q n = ( x - x n ) 2 σ y 2 - 2 ( x - x n ) ( y - y n ) σ x y 2 + ( y - y n ) 2 σ x 2 ( σ x 2 σ y 2 - σ x y 4 ) = T .
σ x 2 = [ λ ( S A λ - X s λ ) 2 ] σ R 2 / S 4 , σ y 2 = [ λ ( S B λ - Y s λ ) 2 ] σ R 2 / S 4 , and the covariance to σ x y 2 = [ λ ( S A λ - X s λ ) ( S B λ - Y s λ ) ] σ R 2 / S 4 . }