Abstract

The shape of the color-mixture curve for any primary is completely determined by the choice of the line in the chromaticity diagram representing the additive mixture of the other two primaries. Formulas for the linear combinations necessary to transform one set of color-mixture curves to another are given in terms of the slopes and axis-intercepts of the lines joining the new primaries, represented in the chromaticity diagram based on the original color-mixture functions.

Orthogonal color-mixture functions are defined and an example is given, in which one of the color-mixture functions is proportional to the luminosity function. Formulas for deriving all other orthogonal sets of color-mixture functions are given. The importance of the concept of orthogonality in the analysis of the propagation of errors in spectrophotometric colorimetry is discussed.

© 1953 Optical Society of America

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References

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  1. H. E. Ives, J. Franklin Inst. 180, 673–701 (1915); J. Franklin Inst. 195, 23–44 (1923).
    [Crossref]
  2. J. Guild, Trans. Opt. Soc. (London) 26, 95–108, 139–174 (1924–1925).
    [Crossref]
  3. D. B. Judd, Bur. Standards J. Research 4, 543–544 (1930).
  4. E. Schrödinger, Ann. Physik (4) 63, 397–456 (1920).
    [Crossref]
  5. D. L. MacAdam, J. Opt. Soc. Amer. 31, 456T (1941).
  6. D. L. MacAdam, J. Franklin Inst. 238, 195–210 (1944).
    [Crossref]

1944 (1)

D. L. MacAdam, J. Franklin Inst. 238, 195–210 (1944).
[Crossref]

1941 (1)

D. L. MacAdam, J. Opt. Soc. Amer. 31, 456T (1941).

1930 (1)

D. B. Judd, Bur. Standards J. Research 4, 543–544 (1930).

1920 (1)

E. Schrödinger, Ann. Physik (4) 63, 397–456 (1920).
[Crossref]

1915 (1)

H. E. Ives, J. Franklin Inst. 180, 673–701 (1915); J. Franklin Inst. 195, 23–44 (1923).
[Crossref]

Guild, J.

J. Guild, Trans. Opt. Soc. (London) 26, 95–108, 139–174 (1924–1925).
[Crossref]

Ives, H. E.

H. E. Ives, J. Franklin Inst. 180, 673–701 (1915); J. Franklin Inst. 195, 23–44 (1923).
[Crossref]

Judd, D. B.

D. B. Judd, Bur. Standards J. Research 4, 543–544 (1930).

MacAdam, D. L.

D. L. MacAdam, J. Franklin Inst. 238, 195–210 (1944).
[Crossref]

D. L. MacAdam, J. Opt. Soc. Amer. 31, 456T (1941).

Schrödinger, E.

E. Schrödinger, Ann. Physik (4) 63, 397–456 (1920).
[Crossref]

Ann. Physik (4) (1)

E. Schrödinger, Ann. Physik (4) 63, 397–456 (1920).
[Crossref]

Bur. Standards J. Research (1)

D. B. Judd, Bur. Standards J. Research 4, 543–544 (1930).

J. Franklin Inst. (2)

D. L. MacAdam, J. Franklin Inst. 238, 195–210 (1944).
[Crossref]

H. E. Ives, J. Franklin Inst. 180, 673–701 (1915); J. Franklin Inst. 195, 23–44 (1923).
[Crossref]

J. Opt. Soc. Amer. (1)

D. L. MacAdam, J. Opt. Soc. Amer. 31, 456T (1941).

Trans. Opt. Soc. (London) (1)

J. Guild, Trans. Opt. Soc. (London) 26, 95–108, 139–174 (1924–1925).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1

CIE chromaticity diagram, showing relation between location of line connecting two primaries and coefficients of linear combination which specifies color-mixture function for third primary.

Fig. 2
Fig. 2

A set of normal and orthogonal color-mixture functions, one of which is the luminosity function, and another yields zero total for an equal-energy source.

Fig. 3
Fig. 3

CIE chromaticity diagram, showing loci of primaries orthogonal to pairs of orthogonal primaries which determine lines having the indicated slopes, m.

Fig. 4
Fig. 4

CIE chromaticity diagram, showing loci of primaries orthogonal to pairs of orthogonal primaries which determine lines having the indicated y-axis intercepts, b. The corresponding values (1−b−1) of the coefficient of y ¯ in Formula (12) are also shown.

Fig. 5
Fig. 5

CIE chromaticity diagram, showing loci of primaries orthogonal to pairs of orthogonal primaries which determine lines having the indicated x-axis intercepts, B. The corresponding values (1−B−1) of the coefficient of x ¯ in Formula (12) are also shown.

Fig. 6
Fig. 6

Chromaticity diagram based on the set of normal and orthogonal color-mixture functions specified by Eq. (8) and shown in Fig. 2. The x- and y-axes of the 1931 CIE chromaticity diagram are shown, as are also the spectrum locus and the loci (10 times enlarged, dotted ellipses) of standard deviations of visual color matches. The solid circles show (10 times enlarged) the root mean square errors of location of the point representing any color, if the root mean square error of measurement of spectral reflectance is 0.01.

Equations (27)

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U = a 11 X + a 12 Y + a 13 Z , V = a 21 X + a 22 Y + a 23 Z , W = a 31 X + a 32 Y + a 33 Z ,
x u = X u / S u ,             y u = Y u / S u ,
U = S [ ( a 11 - a 13 ) x + ( a 12 - a 13 ) y + a 13 ] , V = S [ ( a 21 - a 23 ) x + ( a 22 - a 23 ) y + a 23 ] , W = S [ ( a 31 - a 33 ) x + ( a 32 - a 33 ) y + a 33 ] .
or a 12 = a 11 ( b 1 - 1 ) / ( b 1 + m 1 ) , a 13 = a 11 b 1 / ( b 1 + m 1 ) , a 12 = a 11 ( B 1 + M 1 ) / ( B 1 - 1 ) , a 13 = a 11 B 1 / ( B 1 - 1 ) .
m 2 x - y + b 2 = 0 ;
a 21 = a 22 ( b 2 + m 2 ) / ( b 2 - 1 ) , a 23 = a 22 b 2 / ( b 2 - 1 ) .
a 31 = a 33 ( b 3 + m 3 ) / b 3 , a 32 = a 33 ( b 3 - 1 ) / b 3 .
X = A 1 [ ( b 2 - b 3 ) U + ( b 3 - b 1 ) V + ( b 1 - b 2 ) W ] , Y = A 2 [ ( m 3 b 2 - m 2 b 3 ) U + ( m 1 b 3 - m 3 b 1 ) V + ( m 1 b 2 - m 2 b 1 ) W ] , Z = A 3 [ ( m 3 - m 2 ) U + ( m 1 - m 3 ) V + ( m 2 - m 1 ) W - X / A 1 - Y / A 2 ] .
U V F d λ = V W F d λ = U W F d λ = 0 ,
U = 0.5657 x ¯ - 0.3965 y ¯ - 0.1691 z ¯ , V =                                         0.3598 y ¯ , W = 0.2032 x ¯ - 0.1738 y ¯ + 0.2239 z ¯ .
Σ U 2 = Σ V 2 = Σ W 2 = 1.
U = λ 1 U + μ 1 V + ν 1 W , V = λ 2 U + μ 2 V + ν 2 W , W = λ 3 U + μ 3 V + ν 3 W ,
λ 1 λ 2 + μ 1 μ 2 + ν 1 ν 2 = λ 2 λ 3 + μ 2 μ 3 + ν 2 ν 3 = λ 1 λ 3 + μ 1 μ 3 + ν 1 ν 3 = 0.
λ 1 2 + μ 1 2 + ν 1 2 = λ 2 2 + μ 2 2 + ν 2 2 = λ 3 2 + μ 3 2 + ν 3 2 = 1 ,
x u = 0.2383 + 0.0889 / ( 0.2383 - x w ) .
m = ( 3.3945 x - y - 0.8091 ) / ( x - 2.1343 y + 0.3182 ) , b = ( - 0.8091 x - 0.3182 y + 0.4930 ) / ( x - 2.1343 y + 0.3182 ) , B = ( 0.8091 x + 0.3182 y - 0.4930 ) / ( 3.3945 x - y - 0.8091 ) ,
U = C [ ( 1 - B - 1 ) x ¯ + ( 1 - b - 1 ) y ¯ + z ¯ ] .
U = C [ ( x u - 0.51 y u - 0.1224 ) x ¯ - ( 0.7 x u - 0.7025 y u - 0.0676 ) y ¯ - ( 0.313 x u + 0.1231 y u - 0.1906 ) z ¯ ] .
3.3945 x - y - 0.8091 = 0.
x - 2.1343 y + 0.3182 = 0.
- 0.8091 x - 0.3182 y + 0.4930 = 0.
Σ x ¯ 2 = 14.39396 , Σ y ¯ 2 = 15.44122 , Σ z ¯ 2 = 28.07834 , Σ x ¯ y ¯ = 11.33290 , Σ y ¯ z ¯ = 1.70088 , Σ x ¯ z ¯ = 5.12472.
σ x 2 = ( x X ) 2 σ X 2 + ( x Y ) 2 σ Y 2 + ( x Z ) 2 σ Z 2 σ y 2 = ( y X ) 2 σ X 2 + ( y Y ) 2 σ Y 2 + ( y Z ) 2 σ Z 2 .
δ x λ = ( x X x ¯ + x Y y ¯ + x Z z ¯ ) δ R λ .
σ x 2 = ( x X ) 2 ( x ¯ 2 ) σ 2 + 2 x X x Y ( x ¯ y ¯ ) σ 2 + ( x Y ) 2 ( y ¯ 2 ) σ 2 + 2 x Y x Z ( y ¯ z ¯ ) σ 2 + ( x Z ) 2 ( z ¯ 2 ) σ 2 + 2 x Z x X ( z ¯ x ¯ ) σ 2 .
σ x 2 = ( x X ) 2 σ X 2 + ( x Y ) 2 σ Y + ( x Z ) 2 σ Z 2 + 2 σ 2 [ x X x Y ( x ¯ y ¯ ) + x Y x Z ( y ¯ z ¯ ) + x Z x X ( z ¯ x ¯ ) ] .
σ u = [ ( u U ) 2 + ( u V ) 2 + ( u W ) 2 ] 1 2 σ             and             σ v = [ ( v U ) 2 + ( v V ) 2 + ( v W ) 2 ] 1 2 σ .