Abstract

A method is presented for obtaining spectral distributions having colorimetric characteristics which have been pre-specified in terms of a set of tristimulus values or trichromatic coefficients. The distributions may be chosen to correspond to energy radiated from light sources, or to the transmittances or reflectances of materials acting on such energy. Tables and figures are included to facilitate a rapid solution to any given problem. Several examples are given which illustrate the method and indicate the kinds of applications possible.

© 1948 Optical Society of America

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References

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  1. T. Smith and J. Guild, “The C.I.E. colorimetric standards and their use,” Trans. Opt. Soc. 33, 73–130 (1931–32).
    [Crossref]
  2. Massachusetts Institute of Technology, the Color Measurement Laboratory, Handbook of Colorimetry (The Technology Press, Cambridge, Massachusetts, 1936), 87 pp.
  3. Committee on Colorimetry, “Quantitative data and methods for colorimetry,” J. Opt. Soc. Am. 34, 633–688 (1944).
  4. W. D. Wright, The Measurement of Colour (Adam Hilger, Ltd., London, 1944), 223 pp.
  5. Ronald H. Bingham and Herman Hoerlin, “Optimum spectral sensitivities for a reversible color film process,” J. Opt. Soc. Am. 37, 199–210 (1947).
    [Crossref] [PubMed]
  6. Robert H. Morris, “Metameric formulation,” J. Opt. Soc. Am. 37, 669 (1947).
    [Crossref]
  7. R. H. Park and E. I. Stearns, “Spectrophotometric formulation,” J. Opt. Soc. Am. 34, 112–113 (1944).
    [Crossref]
  8. Committee on Colorimetry, “The psychophysics of color,” J. Opt. Soc. Am. 34, 245–266 (1944), p. 252.
  9. This curve was taken from G. E. Inman, “Characteristics of fluorescent lamps,” Trans. Illum. Eng. Soc. 34, 65–86 (1939), p. 72.
  10. Taken from the Handbook of Colorimetry (reference 2), p. 2.
  11. The spectral transmittances for this filter is given in Eastman Kodak Company, Wratten Light Filters (Eastman Kodak Company, Rochester, New York, 1945), 17th edition, p. 72.
  12. David L. MacAdam, “Colorimetric specifications of Wratten light filters,” J. Opt. Soc. Am. 35, 670–675 (1945), p. 671.
    [Crossref]

1947 (2)

1945 (1)

1944 (3)

1939 (1)

This curve was taken from G. E. Inman, “Characteristics of fluorescent lamps,” Trans. Illum. Eng. Soc. 34, 65–86 (1939), p. 72.

Bingham, Ronald H.

Guild, J.

T. Smith and J. Guild, “The C.I.E. colorimetric standards and their use,” Trans. Opt. Soc. 33, 73–130 (1931–32).
[Crossref]

Hoerlin, Herman

Inman, G. E.

This curve was taken from G. E. Inman, “Characteristics of fluorescent lamps,” Trans. Illum. Eng. Soc. 34, 65–86 (1939), p. 72.

MacAdam, David L.

Morris, Robert H.

Park, R. H.

Smith, T.

T. Smith and J. Guild, “The C.I.E. colorimetric standards and their use,” Trans. Opt. Soc. 33, 73–130 (1931–32).
[Crossref]

Stearns, E. I.

Wright, W. D.

W. D. Wright, The Measurement of Colour (Adam Hilger, Ltd., London, 1944), 223 pp.

J. Opt. Soc. Am. (6)

Trans. Illum. Eng. Soc. (1)

This curve was taken from G. E. Inman, “Characteristics of fluorescent lamps,” Trans. Illum. Eng. Soc. 34, 65–86 (1939), p. 72.

Trans. Opt. Soc. (1)

T. Smith and J. Guild, “The C.I.E. colorimetric standards and their use,” Trans. Opt. Soc. 33, 73–130 (1931–32).
[Crossref]

Other (4)

Massachusetts Institute of Technology, the Color Measurement Laboratory, Handbook of Colorimetry (The Technology Press, Cambridge, Massachusetts, 1936), 87 pp.

W. D. Wright, The Measurement of Colour (Adam Hilger, Ltd., London, 1944), 223 pp.

Taken from the Handbook of Colorimetry (reference 2), p. 2.

The spectral transmittances for this filter is given in Eastman Kodak Company, Wratten Light Filters (Eastman Kodak Company, Rochester, New York, 1945), 17th edition, p. 72.

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Figures (13)

Fig. 1
Fig. 1

Graphical representations illustrating the combining of three f’s to produce a distribution with a given set of trichromatic coefficients.

Fig. 2
Fig. 2

Families of curves of the f’s.

Fig. 3
Fig. 3

Chromaticity loci of the f’s used as energy distributions.

Fig. 4
Fig. 4

Spectral-energy distribution of the white fluorescent light indicated in Fig. 5 and three calculated distributions which yield the same tristimulus values.

Fig. 5
Fig. 5

Chromaticity plots of white fluorescent light and of f’s used as energy distributions which can be combined to match it.

Fig. 6
Fig. 6

Portion of Fig. 5 enlarged to illustrate graphical solution.

Fig. 7
Fig. 7

Chromaticity loci of the f’s used as transmittance or reflectance distributions with Illuminant C.

Fig. 8
Fig. 8

Spectral-reflectance distributions of the green paint of Fig. 9 and the calculated distributions which yield matches with it when all are used with Illuminant C.

Fig. 9
Fig. 9

Chromaticity plot of a green paint under Illuminant C and plots of f’s used as reflectance distributions with the same illuminant which can be combined to match it.

Fig. 10
Fig. 10

Spectral-transmittance distributions which, with Illuminant C, are of the same chromaticities and which have transmittances of 0.60 at 630 mμ.

Fig. 11
Fig. 11

Spectral-transmittance distributions which, with Illuminant C, have the same tristimulus values and which have transmittances of 0.25 at 530 mμ.

Fig. 12
Fig. 12

Spectral-transmittance distributions which, with Illuminant C, have the same tristimulus values and which have maxima at 550 mμ.

Fig. 13
Fig. 13

Spectral-transmittance distributions which, with Illuminant C, have the same chromaticities, and which have maxima of 0.50 at 550 mμ.

Tables (4)

Tables Icon

Table I List of f’s.

Tables Icon

Table II Numerical values of the f’s as functions of (ii0).

Tables Icon

Table III Values of the x’s, y’s, and m’s for the f’s taken as energy distributions.

Tables Icon

Table IV Values of the x’s, y’s, and m’s for the f’s taken as transmittance or reflectance distributions with I.C.I. Illuminant C.

Equations (39)

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X = K m 0 x ¯ P d λ ,             Y = K m 0 y ¯ P d λ , Z = K m 0 z ¯ P d λ ,
P = a 1 f 1 + a 2 f 2 + a 3 f 3 ,
X 1 = K m 0 x ¯ f 1 d λ ,             Y 1 = K m 0 y ¯ f 1 d λ , Z 1 = K m 0 z ¯ f 1 d λ ,
X = a 1 X 1 + a 2 X 2 + a 3 X 3 , Y = a 1 Y 1 + a 2 Y 2 + a 3 Y 3 , Z = a 1 Z 1 + a 2 Z 2 + a 3 Z 3 .
m = X + Y + Z ;             m i = X i + Y i + Z i ;
x = X / m , y = Y / m , z = Z / m ; x i = X i / m i , y i = Y i / m i , z i = Z i / m i ;
α i = a i m i / m ;
x = α 1 x 1 + α 2 x 2 + α 3 x 3 , y = α 1 y 1 + α 2 y 2 + α 3 y 3 , z = α 1 z 1 + α 2 z 2 + α 3 z 3 ,
x + y + z = 1 ,             x i + y i + z i = 1 ,             α 1 + α 2 + α 3 = 1.
α 1 = x ( y 2 - y 3 ) + x 2 ( y 3 - y ) + x 3 ( y - y 2 ) x 1 ( y 2 - y 3 ) + x 2 ( y 3 - y 1 ) + x 3 ( y 1 - y 2 ) , α 2 = x 1 ( y - y 3 ) + x ( y 3 - y 1 ) + x 3 ( y 1 - y ) x 1 ( y 2 - y 3 ) + x 2 ( y 3 - y 1 ) + x 3 ( y 1 - y 2 ) , α 3 = x 1 ( y 2 - y ) + x 2 ( y - y 1 ) + x ( y 1 - y 2 ) x 1 ( y 2 - y 3 ) + x 2 ( y 3 - y 1 ) + x 3 ( y 1 - y 2 ) .
α 1 = A 1 / A ,             α 2 = A 2 / A ,             α 3 = A 3 / A .
i = ( λ - 400 ) / 10 ,
f 11 = ( sin i - 8 30 π ) 2 ,             f 12 = 1 + ( sin i - 20 30 π ) 3 , f 13 = 1 + ( sin i - 56 30 π ) 3 .
x 11 = 0.51158 , x 12 = 0.47876 , x 13 = 0.29761 , y 11 = 0.39694 , y 12 = 0.47322 , y 13 = 0.35483.
α 11 = 0.55688 ,             α 12 = 0.18346 ,             α 13 = 0.25966.
a 11 = 9.6666 ,             α 12 = 2.4641 ,             a 13 = 1.3547.
P 1 = 9.6666 ( sin i - 8 30 π ) 2 + 2.4641 [ 1 + ( sin i - 20 30 π ) 3 ] + 1.3547 [ 1 + ( sin i - 56 30 π ) 3 ] .
P 2 = 4.5145 [ 1 + ( sin i - 16 15 π ) 3 ] + 4.9675 [ 1 + ( sin i - 20 30 π ) 3 ] + 0.7121 , P 3 = 1.8819 + 0.20170 i + 0.015516 i 2 .
A 1 = 23.3 ,             A 2 = 7.6 ,             A 3 = 11.2 ,             A = 42.1.
α 1 = 23.3 42.1 = 0.55 ,             α 2 = 7.6 42.1 = 0.18 , α 3 = 11.2 42.1 = 0.26 ,
X = 0 x ¯ T P d λ / 0 y ¯ P d λ , Y = 0 y ¯ T P d λ / 0 y ¯ P d λ , Z = 0 z ¯ T P d λ / 0 y ¯ P d λ ,
T = a 1 f 1 + a 2 f 2 + a 3 f 3 .
X = a 1 X 1 + a 2 X 2 + a 3 X 3 , Y = a 1 Y 1 + a 2 Y 2 + a 3 Y 3 , Z = a 1 Z 1 + a 2 Z 2 + a 3 Z 3 ,
X i = 0 x ¯ f i P d λ / 0 y ¯ P d λ , Y i = 0 y ¯ f i P d λ / 0 y ¯ P d λ , Z i = 0 z ¯ f i P d λ / 0 y ¯ P d λ .
R 1 = 0.08800 [ 1 + sin i - 7 10 π ] + 0.07073 [ 1 + sin i - 5 10 π ] + 0.03923 [ 1 + sin i 30 π ] , R 2 = 0.16819 [ 1 + sin i - 6 10 π ] + 0.003069 i + 0.001308 ( 30 - i ) , R 3 = 0.19593 [ e - ( i - 11 ) 2 50 ] + 0.28548 [ e - ( i - 12 ) 2 50 ] - 0.15784 [ e - ( i - 11 ) 2 18 ] .
X = m x = a 1 X 1 + a 2 X 2 + a 3 X 3 , Y = m y = a 1 Y 1 + a 2 Y 2 + a 3 Y 3 , Z = m z = a 1 Z 1 + a 2 Z 2 + a 3 Z 3 ,
P = θ P 1 + η P 2 ,
θ + η = 1 ,
T 1 = 0.09523 e - ( i - 4 ) 2 50 + 0.40229 e - ( i - 14 ) 2 8 + 0.44264 e - ( i - 26 ) 2 50 , T 2 = 0.47755 e - ( i - 16 ) 2 8 + 0.10179 e - ( i - 4 ) 2 50 + 0.24356 e - ( i - 26 ) 2 50 , T 3 = 0.08831 + 0.15258 e - ( i - 16 ) 2 18 + 0.19046 e - ( i - 18 ) 2 18 , T 4 = 0.03460 + 0.0012576 ( i - 7 ) 2 + 0.27928 e - ( i - 12 ) 2 18 .
T 1 = 0.38894 , T 2 = 0.18347 , T 3 = 0.22837 , T 4 = 0.34405.
T a = θ a T 1 + η a T 2 ,             T b = θ b T 1 + η b T 3 , T c = θ c T 3 + η c T 4 .
T a = 0.32379 T 1 + 0.67621 T 2 ; T b = 0.13473 T 1 + 0.86527 T 3 ; T c = 0.81300 T 3 + 0.18700 T 4 .
T = θ T + η T ;
d T / d i = θ ( d T / d i ) + η ( d T / d i ) .
( d T / d i ) i = 15 = 0
θ ( d T / d i ) i = 15 + η ( d T / d i ) i = 15 = 0.
T a = 0.59605 T 1 + 0.40395 T 2 , T b = 0.42051 T 1 + 0.57949 T 3 , T c = 0.39987 T 3 + 0.60013 T 4 ,
d 2 T / d i 2 = θ ( d 2 T / d i 2 ) + η ( d 2 T / d i 2 ) < 0.
T a = ( 0.50 / 0.42276 ) T a T b = ( 0.50 / 0.37116 ) T b = 1.1827 T a = 1.3471 T b , T c = ( 0.50 / 0.30994 ) T c = 1.6132 T c .