Abstract

A common colorimetric transformation is a homogeneous linear transformation of tristimulus values from one primary system to another. The transformation coefficients are obtained by spectrophotornetry and/or visual color matching and consequently are subject to certain errors. These errors, together with those associated with the given tristimulus values, propagate into the tristimulus values to be computed by means of the transformation equations. The theory of the propagation of such errors is outlined and illustrated by an example.

© 1959 Optical Society of America

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References

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  1. International Committee on Illumination, Proceedings of the 8th Session, Cambridge, 1931 (The Cambridge University Press, London, 1932), p. 19.
  2. I. Nimferoff, J. Opt. Soc. Am. 43, 531 (1953).
    [Crossref]
  3. I. Nimeroff, J. Opt. Soc. Am. 47, 697 (1957).
    [Crossref]
  4. L. Silberstein and D. L. MacAdam, J. Opt. Soc. Am. 35, 32 (1945).
    [Crossref]
  5. W. R. J. Brown, J. Opt. Soc. Am. 42, 252 (1952).
    [Crossref]
  6. W. S. Stiles, “The average color matching functions for a large matching field,” Symposium on Visual Color Problems, Teddington, England, September, 1957.

1957 (1)

1953 (1)

1952 (1)

1945 (1)

Brown, W. R. J.

MacAdam, D. L.

Nimeroff, I.

Nimferoff, I.

Silberstein, L.

Stiles, W. S.

W. S. Stiles, “The average color matching functions for a large matching field,” Symposium on Visual Color Problems, Teddington, England, September, 1957.

J. Opt. Soc. Am. (4)

Other (2)

W. S. Stiles, “The average color matching functions for a large matching field,” Symposium on Visual Color Problems, Teddington, England, September, 1957.

International Committee on Illumination, Proceedings of the 8th Session, Cambridge, 1931 (The Cambridge University Press, London, 1932), p. 19.

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Figures (2)

Fig. 1
Fig. 1

Portion of the CIE (x, y)-chromaticity diagram with contour lines of three 95% confidence ellipsoids and two sets of chromaticity points obtained from color matching.

Fig. 2
Fig. 2

Portion of the CIE (x, y)-chromaticity diagram with contour lines at different luminance levels for three 95% confidence ellipsoids already described in Fig. 1. The various levels are given as follows:

Tables (4)

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Table I Numerical values of color matching experiments.

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Table II Transformed experimental data.

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Table III Numerical values of experimental data transformed into the x ¯ , y ¯ , Y ¯ color space.

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Table IV Equations of ellipsoids derived from values of Table III.

Equations (16)

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U = a 11 R + a 12 G + a 13 B V = a 21 R + a 22 G + a 23 B W = a 31 R + a 32 G + a 33 B .
Var ( U k ) = i ( U k v i ) 2 Var ( v i ) + 2 i , j ( U k v i ) ( U k v j ) Cov ( v i v j )
Cov ( U k , U l ) = i ( U k v i ) ( U l v i ) Var ( v i ) + i , j [ ( U k v i ) ( U l v j ) + ( U k v j ) ( U l v i ) ] Cov ( v i v j ) i , j = 1 , 2 , 3 , ( i j ) k , l = 1 , 2 , 3 , ( k l ) .
Var ( U ) = R 2 Var ( a 11 ) + G 2 Var ( a 12 ) + B 2 Var ( a 13 ) + a 11 2 Var ( R ) + a 12 2 Var ( G ) + a 13 2 Var ( B ) + 2 [ R G Cov ( a 11 a 12 ) + R B Cov ( a 11 a 13 ) + G B Cov ( a 12 a 13 ) + a 11 a 12 Cov ( R G ) + a 11 a 13 Cov ( R B ) + a 12 a 13 Cov ( G B ) + a 11 R Cov ( R a 11 ) + a 11 G Cov ( R a 12 ) + a 11 B Cov ( R a 13 ) + a 12 R Cov ( G a 11 ) + a 12 G Cov ( G a 12 ) + a 12 B Cov ( G a 13 ) + a 13 R Cov ( B a 11 ) + a 13 G Cov ( B a 12 ) + a 13 B Cov ( B a 13 ) ] .
Cov ( U V ) = a 11 a 21 Var ( R ) + a 12 a 22 Var ( G ) + a 13 a 23 Var ( B ) + ( a 11 a 22 + a 12 a 21 ) Cov ( R G ) + ( a 11 a 23 + a 13 a 21 ) Cov ( R B ) + ( a 12 a 23 + a 13 a 22 ) Cov ( G B ) + R a 21 Cov ( a 11 R ) + R a 22 Cov ( a 11 G ) + R a 23 Cov ( a 11 B ) + G a 21 Cov ( a 12 R ) + G a 22 Cov ( a 12 G ) + G a 23 Cov ( a 12 B ) + B a 21 Cov ( a 13 R ) + B a 22 Cov ( a 13 G ) + B a 23 Cov ( a 13 B ) + R 2 Cov ( a 11 a 21 ) + R G Cov ( a 11 a 22 ) + R B Cov ( a 11 a 23 ) + G R Cov ( a 12 a 21 ) + G 2 Cov ( a 12 a 22 ) + G B Cov ( a 12 a 23 ) + B R Cov ( a 13 a 21 ) + B G Cov ( a 13 a 22 ) + B 2 Cov ( a 13 a 23 ) + a 11 R Cov ( R a 21 ) + a 11 G Cov ( R a 22 ) + a 11 B Cov ( R a 23 ) + a 12 R Cov ( G a 21 ) + a 12 G Cov ( G a 22 ) + a 12 B Cov ( G a 23 ) + a 13 R Cov ( B a 21 ) + a 13 G Cov ( B a 22 ) + a 13 B Cov ( B a 23 ) .
Var ( v i ) = m ( v i , m - v ¯ i ) 2 / ( n - 1 ) Cov ( v i v j ) = m ( v i , m - v ¯ i ) ( v j , m - v ¯ j ) / ( n - 1 ) . ( i j )
v ¯ i = m = 1 n v i , m / n             v ¯ j = m = 1 n v j , m / n .
M = ( Var ( U ) Cov ( U V ) Cov ( U W ) Cov ( V U ) Var ( V ) Cov ( V W ) Cov ( W U ) Cov ( W V ) Var ( W ) ) ,
Q = N M - 1 N *
Q - 7.81 = 0.
M 11 ( - 1 ) ( U - Ū ) 2 + M 22 ( - 1 ) ( V - V ¯ ) 2 + M 33 ( - 1 ) ( W - W ¯ ) 2 + 2 M 12 ( - 1 ) ( U - Ū ) ( V - V ¯ ) + 2 M 23 ( - 1 ) ( V - V ¯ ) ( W ¯ - W ¯ ) + 2 M 31 ( - 1 ) ( W - W ¯ ) ( U - Ū )             - 7.81 = 0.
x = 1.6890 U + 1.4853 V + 1.2625 W 2.3175 U + 6.1502 V + 2.3557 W y = 0.6285 U + 4.3937 V + 0.2490 W 2.3175 U + 6.1502 V + 2.3557 W Y = 0.6285 U + 4.3937 V + 0.2490 W .
E 1 : ( Y - Y ¯ ) = 1.0 , 2.0 , 3.0 , 4.0 , 5.0 E 2 : ( Y - Y ¯ ) = 1.0 , 2.0 , 3.0 , 4.0 E 3 : ( Y - Y ¯ ) = 1.0 , 2.0 , 3.0 , 3.9.
M i j ( - 1 ) = C j i / D ,
M 21 ( - 1 ) = - [ Cov ( V U ) Var ( W ) - Cov ( V W ) Cov ( W U ) ] / D .
E 1 : ( Y - Y ¯ ) = 0.0 , 1.0 , 2.0 , 3.0 , 4.0 , 5.0 E 2 : ( Y - Y ¯ ) = 0.0 , 1.0 , 2.0 , 3.0 , 4.0 E 3 : ( Y - Y ¯ ) = 0.0 , 1.0 , 2.0 , 3.0 , 3.9.