Abstract

Spectrally different object colors that match for one illuminant generally do not remain matched when the illuminant is replaced by another. However, all possible colors that match under one illuminant are, with any other illuminant, confined to a region of color space that has a closed boundary. This boundary, called the theoretical chromaticity-mismatch limits of metamers due to a change of illuminant, can be computed by means of a linear-programming technique. The size and shape of the theoretical limits in color space depend on the two illuminants involved and on the color of the metamers. A number of examples are given, and some practical implications of the method are discussed.

© 1975 Optical Society of America

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References

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  1. International Commission on Illumination, Colorimetry, Publ. CIE No. 15 (E-1. 3.1) (Paris, 1971).
  2. E. Allen, J. Opt. Soc. Am. 56, 559 (1966).
    [CrossRef]
  3. E. Allen, Color Eng. 7, 35 (1967).
  4. N. Ohta, Phot. Sci. Eng. 16, 203 (1972).
  5. N. Ohta, J. Phot. Sci. 20, 149 (1972).
  6. G. Wyszecki and W. S. Stiles, Color Science (Wiley, New York, 1967).
  7. G. S. G. Beveridge and R. S. Schechter, Optimization: Theory and Practice (McGraw-Hill, New York, 1970).
  8. P. Rabinowitz, SIAM Review 10, 121 (1968).
    [CrossRef]
  9. N. Ohta and H. Urabe, Appl. Opt. 11, 2551 (1972).
    [CrossRef] [PubMed]
  10. D. L. MacAdam, J. Opt. Soc. Am. 25, 249 (1935).
    [CrossRef]
  11. D. L. MacAdam, J. Opt. Soc. Am. 25, 361 (1935).
    [CrossRef]
  12. E. Schrödinger, Ann. Phys. 62, 603 (1920).
    [CrossRef]
  13. International Commission on Illumination, Method of Measuring and Specifying Color-Rendering Properties of Light Sources, Publ. CIE No. 13 (E-1. 3.2) (Paris, 1965).
  14. G. Wyszecki, Die Farbe 19, 43 (1970).
  15. Y. Nayatani, Y. Kurioka, and H. Sobagaki, J. Ill. Eng. Inst. Jpn. 54, 461 (1970).
    [CrossRef]

1972 (3)

N. Ohta, Phot. Sci. Eng. 16, 203 (1972).

N. Ohta, J. Phot. Sci. 20, 149 (1972).

N. Ohta and H. Urabe, Appl. Opt. 11, 2551 (1972).
[CrossRef] [PubMed]

1970 (2)

G. Wyszecki, Die Farbe 19, 43 (1970).

Y. Nayatani, Y. Kurioka, and H. Sobagaki, J. Ill. Eng. Inst. Jpn. 54, 461 (1970).
[CrossRef]

1968 (1)

P. Rabinowitz, SIAM Review 10, 121 (1968).
[CrossRef]

1967 (1)

E. Allen, Color Eng. 7, 35 (1967).

1966 (1)

E. Allen, J. Opt. Soc. Am. 56, 559 (1966).
[CrossRef]

1935 (2)

1920 (1)

E. Schrödinger, Ann. Phys. 62, 603 (1920).
[CrossRef]

Allen, E.

E. Allen, Color Eng. 7, 35 (1967).

E. Allen, J. Opt. Soc. Am. 56, 559 (1966).
[CrossRef]

Beveridge, G. S. G.

G. S. G. Beveridge and R. S. Schechter, Optimization: Theory and Practice (McGraw-Hill, New York, 1970).

Kurioka, Y.

Y. Nayatani, Y. Kurioka, and H. Sobagaki, J. Ill. Eng. Inst. Jpn. 54, 461 (1970).
[CrossRef]

MacAdam, D. L.

Nayatani, Y.

Y. Nayatani, Y. Kurioka, and H. Sobagaki, J. Ill. Eng. Inst. Jpn. 54, 461 (1970).
[CrossRef]

Ohta, N.

N. Ohta and H. Urabe, Appl. Opt. 11, 2551 (1972).
[CrossRef] [PubMed]

N. Ohta, Phot. Sci. Eng. 16, 203 (1972).

N. Ohta, J. Phot. Sci. 20, 149 (1972).

Rabinowitz, P.

P. Rabinowitz, SIAM Review 10, 121 (1968).
[CrossRef]

Schechter, R. S.

G. S. G. Beveridge and R. S. Schechter, Optimization: Theory and Practice (McGraw-Hill, New York, 1970).

Schrödinger, E.

E. Schrödinger, Ann. Phys. 62, 603 (1920).
[CrossRef]

Sobagaki, H.

Y. Nayatani, Y. Kurioka, and H. Sobagaki, J. Ill. Eng. Inst. Jpn. 54, 461 (1970).
[CrossRef]

Stiles, W. S.

G. Wyszecki and W. S. Stiles, Color Science (Wiley, New York, 1967).

Urabe, H.

Wyszecki, G.

G. Wyszecki, Die Farbe 19, 43 (1970).

G. Wyszecki and W. S. Stiles, Color Science (Wiley, New York, 1967).

Ann. Phys. (1)

E. Schrödinger, Ann. Phys. 62, 603 (1920).
[CrossRef]

Appl. Opt. (1)

Color Eng. (1)

E. Allen, Color Eng. 7, 35 (1967).

Die Farbe (1)

G. Wyszecki, Die Farbe 19, 43 (1970).

J. Ill. Eng. Inst. Jpn. (1)

Y. Nayatani, Y. Kurioka, and H. Sobagaki, J. Ill. Eng. Inst. Jpn. 54, 461 (1970).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Phot. Sci. (1)

N. Ohta, J. Phot. Sci. 20, 149 (1972).

Phot. Sci. Eng. (1)

N. Ohta, Phot. Sci. Eng. 16, 203 (1972).

SIAM Review (1)

P. Rabinowitz, SIAM Review 10, 121 (1968).
[CrossRef]

Other (4)

International Commission on Illumination, Colorimetry, Publ. CIE No. 15 (E-1. 3.1) (Paris, 1971).

G. Wyszecki and W. S. Stiles, Color Science (Wiley, New York, 1967).

G. S. G. Beveridge and R. S. Schechter, Optimization: Theory and Practice (McGraw-Hill, New York, 1970).

International Commission on Illumination, Method of Measuring and Specifying Color-Rendering Properties of Light Sources, Publ. CIE No. 13 (E-1. 3.2) (Paris, 1965).

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Figures (12)

FIG. 1
FIG. 1

Theoretical limits of metamerism for grays of Y = 10. Left: Reference illuminant is A and test illuminant is D65. Right: Reference illuminant is D65 and test illuminant is A.

FIG. 2
FIG. 2

Extreme spectral reflectance functions to give points M and m in Fig. 1.

FIG. 3
FIG. 3

Extreme spectral reflectance functions (m) calculated for different wavelength intervals, Δλ = 1, 5, and 10 nm.

FIG. 4
FIG. 4

Expected extreme spectral reflectance function for Δλ → 0.

FIG. 5
FIG. 5

Theoretical limits of metamerism for grays of Y = 10. Reference illuminant is A and test illuminants are fluorescent lamps F(3000), F(4500), and F(6500). The chromaticity coordinates of F(3000), F(4500), and F(6500) are shown by 1, 2, and 3.

FIG. 6
FIG. 6

Approximation of theoretical color gamut by the smallest rectangular box.

FIG. 7
FIG. 7

Upper part: Variation of the logarithm of the volume VB with X for Y = 10, 50, 90 at Z = 50. Solid dots indicate the logarithm of volume VE (right-hand-side scale), where VE is volume of actual gamut of metamers. Lower part: Loci of optimal colors for Y = 10, 50, 90.

FIG. 8
FIG. 8

Theoretical limits of metamerism for grays of Y = 10. Reference illuminants are A and D65, and test illuminant is fluorescent lamp F (4500). Cross sections for Yg = 9, 10, and 11 are shown.

FIG. 9
FIG. 9

Theoretical limits of metamerism for grays of Y = 10. Reference illuminants are A and D65, and test illuminants are fluorescent lamps F(3000), F(4500), and F(6500). Only cross sections for Yg = 10 are shown.

FIG. 10
FIG. 10

Extreme spectral reflectance functions to give points M and m in Fig. 9.

FIG. 11
FIG. 11

Relative spectral-power distributions of filtered xenon arc (Xe), filtered tungsten lamp (T), fluorescent lamp (Fl), and D65. All have approximately the same chromaticity.

FIG. 12
FIG. 12

Theoretical limits of metamerism for grays of Y = 10. Reference illuminant is D65 and test illuminants are filtered xenon arc (Xe), filtered tungsten lamp (T), and fluorescent lamp (Fl). The three test illuminants have nearly the same CIE color-rendering index. The chromaticity of D65 is shown by a cross. The chromaticities of Xe, T, and Fl are nearly the same as that of D65.

Equations (12)

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X = 100 λ ρ ( λ ) S ( λ ) x ¯ ( λ ) Δ λ / λ S ( λ ) y ¯ ( λ ) Δ λ , Y = 100 λ ρ ( λ ) S ( λ ) y ¯ ( λ ) Δ λ / λ S ( λ ) y ¯ ( λ ) Δ λ , Z = 100 λ ρ ( λ ) S ( λ ) z ¯ ( λ ) Δ λ / λ S ( λ ) y ¯ ( λ ) Δ λ ,
λ ρ ( λ ) S ( λ ) x ¯ ( λ ) Δ λ = λ ρ ( λ ) S ( λ ) x ¯ ( λ ) Δ λ , λ ρ ( λ ) S ( λ ) y ¯ ( λ ) Δ λ = λ ρ ( λ ) S ( λ ) y ¯ ( λ ) Δ λ , λ ρ ( λ ) S ( λ ) z ¯ ( λ ) Δ λ = λ ρ ( λ ) S ( λ ) z ¯ ( λ ) Δ λ ,
100 i = 1 n ρ i j S i ( 1 ) x ¯ i / i = 1 n S i ( 1 ) y ¯ i = X ( 1 ) , 100 i = 1 n ρ i j S i ( 1 ) y ¯ i / i = 1 n S i ( 1 ) y ¯ i = Y ( 1 ) , 100 i = 1 n ρ i j S i ( 1 ) z ¯ i / i = 1 n S i ( 1 ) y ¯ i = Z ( 1 ) ,
100 i = 1 n ρ i j S i ( 2 ) y ¯ i / i = 1 n S i ( 2 ) y ¯ i = Y g , 100 i = 1 n ρ i j S i ( 2 ) [ ( 1 x g 1 ) x ¯ i z ¯ i ] / i = 1 n S i ( 2 ) y ¯ i = Y g .
0 ρ j i 1 , for all j and i = 1 to n .
y = ( 1 x g ) Y g / ( Y g + 100 i = 1 n ρ i j S i ( 2 ) z ¯ i / i = 1 n S i ( 2 ) y ¯ i ) .
Z = i = 1 n ρ i j S ( 2 ) z ¯ i .
y = ( 1 x g ) Y g / ( Y g + 100 i = 1 n ρ i j S i ( 3 ) z ¯ i / i = 1 n S i ( 3 ) y ¯ i ) ,
100 i = 1 n ρ i j S i ( 1 ) x ¯ i / i = 1 n S i ( 1 ) y ¯ i = X ( 1 ) ,
100 i = 1 n ρ i j S i ( 3 ) y ¯ i / i = 1 n S i ( 3 ) y ¯ i = Y g , 100 i = 1 n ρ i j S i ( 3 ) [ ( 1 x g 1 ) x ¯ i z ¯ i ] / i = 1 n S i ( 3 ) y ¯ i = Y g ,
X ( 2 ) = 100 i = 1 n ρ i j S i ( 2 ) x ¯ i / i = 1 n S i ( 2 ) y ¯ i ,
R a ( Xe ) = 95.4 , R a ( T ) = 94.3 , R a ( F 1 ) = 93.1 .