Abstract

A fast and accurate model to compute optimal colors under a given illuminant is introduced. The model estimates the reflectance (transmittance) distributions of optimal colors, which is bandpass (Type 1) or bandstop (Type 2), at a given luminance factor with user-specified small tolerances for the bandwidth (e.g., 1010nm). The tristimulus values of the optimal colors are obtained by using trapezoidal integration of the product of color-matching functions and illuminant spectrums sampled with small wavelength steps instead of performing summation of the product values. Selecting the distribution type whether Type 1 or Type 2 is avoided in the algorithm to reduce computing cost. Some optimal color solids computed by a MATLAB program have been demonstrated.

© 2010 Optical Society of America

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References

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  1. E. Schrödinger, Ann. Phys. 62, 603 (1920).
    [CrossRef]
  2. S. Rösch, Kristallogr. Petrogr. 13, 143 (1929).
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    [CrossRef]
  6. CIE, “Recommended practice for tabulating spectral data for use in colour computations,” CIE 167 (Commission Internationale de l’Eclairage, 2005).

2007 (1)

1935 (2)

1929 (1)

S. Rösch, Kristallogr. Petrogr. 13, 143 (1929).

1920 (1)

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

Ann. Phys. (1)

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

J. Opt. Soc. Am. (2)

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

Kristallogr. Petrogr. (1)

S. Rösch, Kristallogr. Petrogr. 13, 143 (1929).

Other (1)

CIE, “Recommended practice for tabulating spectral data for use in colour computations,” CIE 167 (Commission Internationale de l’Eclairage, 2005).

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Figures (7)

Fig. 1
Fig. 1

Two types of spectral reflectances (transmittances) for optimal colors.

Fig. 2
Fig. 2

Smallest differences between given lightness values L const * and their closest L * for each sampled wavelength.

Fig. 3
Fig. 3

Extended reflectance distribution R ¯ and concatenated T ¯ .

Fig. 4
Fig. 4

Diagram of trapezoidal integration of R ¯ T ¯ .

Fig. 5
Fig. 5

Source code for computing optimal colors using MATLAB.

Fig. 6
Fig. 6

Optimal color loci at the constant luminance factors ( Y const = 0 , 5 , , 100 ) and central wavelengths ( λ n = 380 , 390 , , 780 nm ) in the CIE x y chromaticity diagram.

Fig. 7
Fig. 7

Optimal color loci at several lightness values ( L const * = 0 , 5 , , 100 ) and hue ( h * = 0 , 15 , , 345 ° ) in the CIELAB color space.

Equations (5)

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k = 1 N S ( k ) y ¯ ( k ) = k = 1 N T ( k ) = 100 ,
Y = k = 1 N R ( k ) T ( k ) .
R ¯ n ( l ) = { 1 , 0 , n h n l n + h n otherwise ,
Y n = 1 3 N R ¯ n ( l ) T ¯ ( l ) d l ,
Y n = ( T ¯ ( n h n 1 ) · { h n } + T ¯ ( n h n ) · ( 2 { h n } ) ) · { h n } / 2 + ( T ¯ ( n + h n + 1 ) · { h n } + T ¯ ( n + h n ) · ( 2 { h n } ) ) · { h n } / 2 + k = n h n n + h n T ¯ ( k ) T ¯ ( n h n ) / 2 T ¯ ( n + h n ) / 2 .

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