Abstract

We present an application of linear-programming techniques to generate metameric spectral radiant power distributions. Conditions concerning physical color properties, dominant wavelength, and excitation purity lead us to propose criteria that provide distributions associated with color stimuli with high-excitation purity values of any brightness. An unlimited number of metameric distributions can be obtained from the degrees of freedom introduced by the proposed criteria.

© 1995 Optical Society of America

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References

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  1. G. Wyszecki, “Evaluation of metameric colors,” J. Opt. Soc. Am. 48, 451–454 (1958).
    [CrossRef]
  2. K. Takahama, Y. Nayatani, “New method for generating metameric stimuli of object colors,” J. Opt. Soc. Am. 62, 1516–1520 (1972).
    [CrossRef]
  3. N. Ohta, G. Wyszecki, “Location of the nodes of metameric color stimuli,” Color Res. Appl. 2, 183–186 (1977).
    [CrossRef]
  4. N. Ohta, “Intersections of spectral curves of metameric colors,” Color Res. Appl. 12, 85–87 (1987).
    [CrossRef]
  5. G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982).
  6. N. Ohta, “Generating metameric object colors,” J. Opt. Soc. Am. 65, 1081–1082 (1975).
    [CrossRef]
  7. N. Ohta, G. Wyszecki, “Designing illuminants that render given objects in prescriberd colors,” J. Opt. Soc. Am. 66, 269–275 (1976).
    [CrossRef]
  8. J. A. Worthey, “Calculations of metameric reflectances,” Color Res. Appl. 13, 76–84 (1988).
    [CrossRef]
  9. S. Suzuki, T. Kusunoki, M. Mori, “Color characteristic design for color scanners,” Appl. Opt. 29, 5187–5192 (1990).
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  13. J. H. Wilkinson, C. Reinsch, Linear Algebra, Vol. 2 of Handbook for Automatic Computation (Springer, New York, 1971).
  14. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1988).
  15. D. L. MacAdam, Color Measurement, 2nd ed. (Springer-Verlag, Berlin, 1985), Chap. 4.
  16. D. L. MacAdam, “Maximum attainable luminous efficiency of various chromaticities,” J. Opt. Soc. Am. 40, 120 (1950).
    [CrossRef]

1994 (1)

1993 (1)

1992 (1)

1990 (1)

1988 (1)

J. A. Worthey, “Calculations of metameric reflectances,” Color Res. Appl. 13, 76–84 (1988).
[CrossRef]

1987 (1)

N. Ohta, “Intersections of spectral curves of metameric colors,” Color Res. Appl. 12, 85–87 (1987).
[CrossRef]

1977 (1)

N. Ohta, G. Wyszecki, “Location of the nodes of metameric color stimuli,” Color Res. Appl. 2, 183–186 (1977).
[CrossRef]

1976 (1)

1975 (1)

1972 (1)

1958 (1)

1950 (1)

Alonso, J.

Bernabeu, E.

Engelhardt, K.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1988).

Kusunoki, T.

MacAdam, D. L.

Marimont, D. H.

Mori, M.

Nayatani, Y.

Ohta, N.

N. Ohta, “Intersections of spectral curves of metameric colors,” Color Res. Appl. 12, 85–87 (1987).
[CrossRef]

N. Ohta, G. Wyszecki, “Location of the nodes of metameric color stimuli,” Color Res. Appl. 2, 183–186 (1977).
[CrossRef]

N. Ohta, G. Wyszecki, “Designing illuminants that render given objects in prescriberd colors,” J. Opt. Soc. Am. 66, 269–275 (1976).
[CrossRef]

N. Ohta, “Generating metameric object colors,” J. Opt. Soc. Am. 65, 1081–1082 (1975).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1988).

Quiroga, J. A.

Reinsch, C.

J. H. Wilkinson, C. Reinsch, Linear Algebra, Vol. 2 of Handbook for Automatic Computation (Springer, New York, 1971).

Seitz, P.

Stiles, W. S.

G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982).

Suzuki, S.

Takahama, K.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1988).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1988).

Wandell, B. W.

Wilkinson, J. H.

J. H. Wilkinson, C. Reinsch, Linear Algebra, Vol. 2 of Handbook for Automatic Computation (Springer, New York, 1971).

Worthey, J. A.

J. A. Worthey, “Calculations of metameric reflectances,” Color Res. Appl. 13, 76–84 (1988).
[CrossRef]

Wyszecki, G.

N. Ohta, G. Wyszecki, “Location of the nodes of metameric color stimuli,” Color Res. Appl. 2, 183–186 (1977).
[CrossRef]

N. Ohta, G. Wyszecki, “Designing illuminants that render given objects in prescriberd colors,” J. Opt. Soc. Am. 66, 269–275 (1976).
[CrossRef]

G. Wyszecki, “Evaluation of metameric colors,” J. Opt. Soc. Am. 48, 451–454 (1958).
[CrossRef]

G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982).

Zoido, J.

Appl. Opt. (3)

Color Res. Appl. (3)

J. A. Worthey, “Calculations of metameric reflectances,” Color Res. Appl. 13, 76–84 (1988).
[CrossRef]

N. Ohta, G. Wyszecki, “Location of the nodes of metameric color stimuli,” Color Res. Appl. 2, 183–186 (1977).
[CrossRef]

N. Ohta, “Intersections of spectral curves of metameric colors,” Color Res. Appl. 12, 85–87 (1987).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (1)

Other (4)

G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982).

J. H. Wilkinson, C. Reinsch, Linear Algebra, Vol. 2 of Handbook for Automatic Computation (Springer, New York, 1971).

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1988).

D. L. MacAdam, Color Measurement, 2nd ed. (Springer-Verlag, Berlin, 1985), Chap. 4.

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Figures (4)

Fig. 1
Fig. 1

(a) Spectral radiant power distribution p(λ) generated for a color stimulus with tristimulus values X = 25.5623, Y = 48.3473, and 26.0904. The excitation purity of the color stimulus is 0.3. (b) Spectral radiant power distribution p(λ) generated for the same color stimulus taking into account the softness criterion of Eq. (14) with var = 20% and Vmax = 10.

Fig. 2
Fig. 2

Spectral radiant power distribution p(λ) generated maximizing Eq. (15) for a color stimulus with tristimulus values X = 10.0203, Y = 78.3753, and 11.6043. The excitation purity of the color stimulus is 0.9. The value of σ = 110 nm, var = 15%, and Vmax = 10.

Fig. 3
Fig. 3

Spectral radiant power distribution p(λ) generated maximizing Eq. (25) for a color stimulus with tristimulus values X = 68.6724, Y = 27.9942, and 3.3333. The excitation purity of the color stimulus is 0.9. The value of σ = 100 nm, var = 20%, and Vmax = 10.

Fig. 4
Fig. 4

(a) Spectral radiant power distribution p(λ) generated for a color stimulus with tristimulus values X = 11.5494, Y = 15.2763, and 73.1743 and λd = 480 nm. The excitation purity of the color stimulus is 0.9. The bounds are generated by a Gaussian function with σc = 42 nm and Vmax = 10. (b) Loci in the chromaticity diagram of the color stimuli with excitation purities 0.5 and 0.9.

Equations (27)

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X i = k p ( λ ) x ^ i ( λ ) d λ             ( i = 1 , 2 , 3 ) ,
p ( 1 ) ( λ ) x ^ i ( λ ) d λ = p ( 2 ) ( λ ) x ^ i ( λ ) d λ             ( i = 1 , 2 , 3 ) .
p ( λ ) = ρ ( λ ) E ( λ ) ,
k = 100 x ^ 2 E ( λ ) d λ .
0 ρ ( λ ) 1.
0 p ( λ ) V ( λ ) ,
0 p j V j ,
X 1 = p 1 x 11 + p 2 x 12 + + p p x 1 p , X 2 = p 1 x 21 + p 2 x 22 + + p p x 2 p , X 3 = p 1 x 31 + p 2 x 32 + + p p x 3 p , V 1 = p 1 + p p + 1 , V 2 = p 2 + p p + 2 , V p = p p + p 2 p ,
z = c 1 p 1 + c 2 p 2 + + c 2 p p 2 p
z = c · p ,
A · p = b ,
z = ( p 1 + p 2 + p p ) Δ λ ,
c = ( Δ λ , Δ λ , Δ λ ( p ) , 0 , , 0 ( p ) ) .
var = V j V j - V j + 1 100             ( j = 2 , , p - 1 ) .
z = λ d - σ λ d + σ p ( λ ) d λ .
p ( λ ) = V max p n ( λ ) ,
0 p n ( λ ) 1.
X i = k V max p n ( λ ) x ^ i ( λ ) d λ k V max x ^ i ( λ ) d λ             ( i = 1 , 2 , 3 ) ,
V max X i k x ^ i ( λ ) d λ .
V max max [ X 1 k x ^ 1 ( λ ) d λ , X 2 k x ^ 2 ( λ ) d λ , X 3 k x ^ 3 ( λ ) d λ ] .
i = 1 i = 3 X i = k V max i = 1 i = 3 p n ( λ ) x ^ i ( λ ) d λ k V max i = 1 i = 3 x ^ i ( λ ) d λ ,
V max i = 1 i = 3 X i k i = 1 i = 3 x ^ i ( λ ) d λ .
E v ( X 1 , X 2 , X 3 ) = k p ( λ ) x ^ 2 ( λ ) d λ p ( λ ) d λ
k p ( λ ) x ^ 2 ( λ ) d λ p ( λ ) d λ E v max .
z L = [ k - x ^ 2 ( λ ) ] p ( λ ) d λ ,
k ( X 1 , X 2 , X 3 ) = E v max ( X 1 , X 2 , X 3 ) / k .
V j = V max exp [ - ( λ j - λ d ) 2 / σ c 2 ] ,

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