Abstract

Many problems regarding color opponency are still unsolved. In this study the system of the uniform color scales of the Optical Society of America (OSA-UCS) is analyzed with the aim of obtaining answers to very general questions on color opponency. The perceptual color opponencies in the OSA-UCS system, represented by its coordinates (j, g), appear to work in a mutually interacting way. On the hypothesis that such an interaction is due to a linear mixing of a pair of independent opponent mechanisms with scales satisfying a proper Weber fraction, three chromatic opponency functions are derived, whose sum is equal to zero. These functions are the logarithms of the ratios of two tristimulus values in a proper reference frame (called the “main reference frame”) and therefore are antisymmetric and zero-degree homogeneous functions of these tristimulus values. Any pair of these three functions is a set of two independent functions. A new formula for color opponency in the OSA-UCS system is derived in which the perceptual color opponencies (j, g) are written as products of the lightness by a proper linear mixing of any pair of the three chromatic opponency functions. All this is possible because the lattice of the OSA-UCS system is composed of geodesic lines.

© 2004 Optical Society of America

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References

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  1. L. M. Hurvich, “Two decades of opponent processes,” in Color 77, F. W. Billmayer, G. Wyszecki, eds., Proceedings of the International Color Association (AIC) (Adam Hilger, Bristol, UK, 1978), pp. 33–61.
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  4. E. J. Chichilinsky, B. A. Wandell, “Trichromatic opponent color classification,” Vision Res. 39, 3444–3458 (1999).
    [CrossRef]
  5. P. K. Kaiser, R. M. Boynton, Human Color Vision (Optical Society of America, Washington D.C., 1966), Chaps. 5–8.
  6. M. D. Fairchild, Color Appearance Models (Addison-Wesley Longman, Inc., Reading, Mass., 1997), Chaps. 11–13.
  7. D. L. MacAdam, “Uniform color scales,” J. Opt. Soc. Am. 64, 1691–1702 (1974).
    [CrossRef] [PubMed]
  8. D. L. MacAdam, “Colorimetric data for samples of OSA uniform color scales,” J. Opt. Soc. Am. 68, 121–130 (1978).
    [CrossRef]
  9. D. L. MacAdam, Color Measurement (Springer-Verlag, Berlin, 1985), pp. 165–177.
  10. D. Nickerson, “OSA uniform color samples: a unique set,” Color Res. Appl. 6, 7–33 (1981).
    [CrossRef]
  11. C. Oleari, “Color opponency and scale uniformity in the OSA-UCS system: the geometrical structure,” AIC 2001, The 9th Congress of the International Colour Association, R. Chung, A. Rodrigues, eds., Proc. SPIE4421, 852–855 (2002).
    [CrossRef]
  12. C. Oleari, “Inter-observer comparison of color-matching functions,” Color Res. Appl. 24, 177–184 (1999).
    [CrossRef]
  13. V. Smith, J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
    [CrossRef] [PubMed]
  14. E. Schrödinger, “Grundlinien einer Theorie der Farbenmetrik im Tagessehen (III),” Ann. Phys. (Leipzig) 62(IV), 481–520 (1920).
    [CrossRef]

1999 (2)

E. J. Chichilinsky, B. A. Wandell, “Trichromatic opponent color classification,” Vision Res. 39, 3444–3458 (1999).
[CrossRef]

C. Oleari, “Inter-observer comparison of color-matching functions,” Color Res. Appl. 24, 177–184 (1999).
[CrossRef]

1993 (1)

R. L. De Valois, K. K. De Valois, “A multi-stage color model,” Vision Res. 33, 1053–1065 (1993).
[CrossRef] [PubMed]

1991 (1)

1981 (1)

D. Nickerson, “OSA uniform color samples: a unique set,” Color Res. Appl. 6, 7–33 (1981).
[CrossRef]

1978 (1)

1975 (1)

V. Smith, J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
[CrossRef] [PubMed]

1974 (1)

1920 (1)

E. Schrödinger, “Grundlinien einer Theorie der Farbenmetrik im Tagessehen (III),” Ann. Phys. (Leipzig) 62(IV), 481–520 (1920).
[CrossRef]

Boynton, R. M.

P. K. Kaiser, R. M. Boynton, Human Color Vision (Optical Society of America, Washington D.C., 1966), Chaps. 5–8.

Chichilinsky, E. J.

E. J. Chichilinsky, B. A. Wandell, “Trichromatic opponent color classification,” Vision Res. 39, 3444–3458 (1999).
[CrossRef]

De Valois, K. K.

R. L. De Valois, K. K. De Valois, “A multi-stage color model,” Vision Res. 33, 1053–1065 (1993).
[CrossRef] [PubMed]

De Valois, R. L.

R. L. De Valois, K. K. De Valois, “A multi-stage color model,” Vision Res. 33, 1053–1065 (1993).
[CrossRef] [PubMed]

Fairchild, M. D.

M. D. Fairchild, Color Appearance Models (Addison-Wesley Longman, Inc., Reading, Mass., 1997), Chaps. 11–13.

Guth, S. L.

Hurvich, L. M.

L. M. Hurvich, “Two decades of opponent processes,” in Color 77, F. W. Billmayer, G. Wyszecki, eds., Proceedings of the International Color Association (AIC) (Adam Hilger, Bristol, UK, 1978), pp. 33–61.

Kaiser, P. K.

P. K. Kaiser, R. M. Boynton, Human Color Vision (Optical Society of America, Washington D.C., 1966), Chaps. 5–8.

MacAdam, D. L.

Nickerson, D.

D. Nickerson, “OSA uniform color samples: a unique set,” Color Res. Appl. 6, 7–33 (1981).
[CrossRef]

Oleari, C.

C. Oleari, “Inter-observer comparison of color-matching functions,” Color Res. Appl. 24, 177–184 (1999).
[CrossRef]

C. Oleari, “Color opponency and scale uniformity in the OSA-UCS system: the geometrical structure,” AIC 2001, The 9th Congress of the International Colour Association, R. Chung, A. Rodrigues, eds., Proc. SPIE4421, 852–855 (2002).
[CrossRef]

Pokorny, J.

V. Smith, J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
[CrossRef] [PubMed]

Schrödinger, E.

E. Schrödinger, “Grundlinien einer Theorie der Farbenmetrik im Tagessehen (III),” Ann. Phys. (Leipzig) 62(IV), 481–520 (1920).
[CrossRef]

Smith, V.

V. Smith, J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
[CrossRef] [PubMed]

Wandell, B. A.

E. J. Chichilinsky, B. A. Wandell, “Trichromatic opponent color classification,” Vision Res. 39, 3444–3458 (1999).
[CrossRef]

Ann. Phys. (Leipzig) (1)

E. Schrödinger, “Grundlinien einer Theorie der Farbenmetrik im Tagessehen (III),” Ann. Phys. (Leipzig) 62(IV), 481–520 (1920).
[CrossRef]

Color Res. Appl. (2)

D. Nickerson, “OSA uniform color samples: a unique set,” Color Res. Appl. 6, 7–33 (1981).
[CrossRef]

C. Oleari, “Inter-observer comparison of color-matching functions,” Color Res. Appl. 24, 177–184 (1999).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Vision Res. (1)

V. Smith, J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
[CrossRef] [PubMed]

Vision Res. (2)

R. L. De Valois, K. K. De Valois, “A multi-stage color model,” Vision Res. 33, 1053–1065 (1993).
[CrossRef] [PubMed]

E. J. Chichilinsky, B. A. Wandell, “Trichromatic opponent color classification,” Vision Res. 39, 3444–3458 (1999).
[CrossRef]

Other (5)

P. K. Kaiser, R. M. Boynton, Human Color Vision (Optical Society of America, Washington D.C., 1966), Chaps. 5–8.

M. D. Fairchild, Color Appearance Models (Addison-Wesley Longman, Inc., Reading, Mass., 1997), Chaps. 11–13.

D. L. MacAdam, Color Measurement (Springer-Verlag, Berlin, 1985), pp. 165–177.

C. Oleari, “Color opponency and scale uniformity in the OSA-UCS system: the geometrical structure,” AIC 2001, The 9th Congress of the International Colour Association, R. Chung, A. Rodrigues, eds., Proc. SPIE4421, 852–855 (2002).
[CrossRef]

L. M. Hurvich, “Two decades of opponent processes,” in Color 77, F. W. Billmayer, G. Wyszecki, eds., Proceedings of the International Color Association (AIC) (Adam Hilger, Bristol, UK, 1978), pp. 33–61.

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Figures (7)

Fig. 1
Fig. 1

CIE 64 chromaticity diagram with zero-lightness OSA-UCS lattice; thick lines G and J, which subdivide the plane into four parts, and thin straight lines that, radiating from the points D ̲ R , D ̲ G , T ̲ B , and T ̲ Y , are fitting the points of the lattice and define the grid. The points are D ̲ R = ( x 10 = 1.0000 ,   y 10 = 0.1157 ) , D ̲ G = ( x 10 = 0.9278 ,   y 10 = 0.0910 ) , T ̲ Y = ( x 10 = 0.0538 ,   y 10 = 0.0976 ) , and T ̲ B = ( x 10 = 0.0934 ,   y 10 = 0.0399 ) . Point is coincident with T ̲ B .

Fig. 2
Fig. 2

CIE 1964 chromaticity diagram referred to the main reference primaries A, B, and C. Any point Q = ( A ,   B ,   C ) can be defined by the crossing of a pair of the straight lines radiating from the vertices A, B, and C or by a pair of the angles CÂQ, AB̂Q, BĈQ, whose cotangents are linearly related to the ratios of main tristimulus values.

Fig. 3
Fig. 3

Lines associated with the Weber fractions [Eq. (5)], Δ ( A / B ) = k AB ( A / B ) , Δ ( B / A ) = k BA ( B / A ) , Δ ( B / C ) = k BC ( B / C ) , Δ ( C / B ) = k CB ( C / B ) , Δ ( C / A ) = k CA ( C / A ) , and Δ ( A / C ) = k AC ( A / C ) , and fitting of the points obtained from the OSA-UCS system at lightness L OSA = 0 . The increments Δ ( B / C ) , Δ ( C / B ) , Δ ( C / A ) , and Δ ( A / C ) are evaluated on the lines at constant g value of the OSA-UCS lattice and the increments Δ ( B / A ) and Δ ( A / B ) on the lines at constant j. All the increments correspond to 1 jnd.

Fig. 4
Fig. 4

(a) (b) (c) CIE 64 chromaticity diagram and the OSA-UCS lattice at L OSA = 0 on coordinates [ ln ( A / B ) ,   ln ( B / C ) ] , on [ ln ( A / C ) ,   ln ( B / C ) ] , and on [ ln ( A / B ) ,   ln ( A / C ) ] , respectively.

Fig. 5
Fig. 5

Scale factors S G , i and S J , i as functions of the lightness. ( i = 1 , 2, 3 pertain to the pair of main chromatic opponency functions considered: i = 1 is for [ ln ( A / B ) ,   ln ( B / C ) ] , i = 2 for [ ln ( B / C ) ,   ln ( C / A ) ] , and i = 3 for [ ln ( C / A ) ,   ln ( A / B ) ] and the corresponding fitting lines represented by the elements of the diagonal matrix of Eqs. (9) and (10). These lines are mutually crossing at a point with distance 0.10 ± 0.22 from the abscissa axis at lightness L OSA = - 12.24 ± 0.12 . This point defines the absolute zero of the lightness.

Fig. 6
Fig. 6

Plane L OSA = 0 spanned by the coordinates ( J ,   G ) and the OSA-UCS lattice. This diagram is obtained in three different ways by Eq. (10) from the three diagrams of Fig. 4. The unit of distance represented by the grid is equal to 1 OSA-UCS unit and corresponds to 10 jnd.

Fig. 7
Fig. 7

Constant-hue lines (a) according to Schrödinger definition represented in the ABC diagram and (b) compared with the original Schrödinger “theoretical” representation in the fundamental reference frame.

Equations (16)

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Δ ( B ̲ / Y ̲ ) ( B ̲ / Y ̲ ) = - Δ ( Y ̲ / B _ ) ( Y ̲ / B _ ) = k BY ,
Δ ( A i / B i ) ( A i / B i ) = - Δ ( B i / A i ) ( B i / A i ) = k A i B i
ln A i B i ( i = 1 ,   2 ) ,
A = ( x 10 = 0.9057 ,   y 10 = 0.2391 ) ,
B = ( x 10 = 1.1134 ,   y 10 = - 1.3384 ) ,
C = ( x 10 = 0.1604 ,   y 10 = - 0.0258 ) ,
Δ ( A / B ) ( A / B ) = - Δ ( B / A ) ( B / A )   = k AB ,
Δ ( B / C ) ( B / C ) = - Δ ( C / B ) ( C / B ) = k BC ,
Δ ( C / A ) ( C / A ) = - Δ ( A / C ) ( A / C ) = k CA ,
ln ( A / B ) , ln ( B / C ) , ln ( C / A )
ln ( A / B ) + ln ( B / C ) + ln ( C / A ) = 0 .
A B C = 1.6817 - 0.7091 0.0274 0.2734   0.7063 0.0203 0.2785   0.1735 0.5480   L M S =   0.65973 0.44916 - 0.10889 - 0.30528 1.21255   0.09273 - 0.03740 0.47951   0.55789   X 10 Y 10 Z 10 ,
J G = S J 0 0 S G   - sin   α cos   α sin   β - cos   β   ln A / B A n / B n ln B / C B n / C n ,
J G = 2 ( 0.5735 L OSA + 7.0892 ) 0 0 - 2 ( 0.7640 L OSA + 9.2521 )   0.1792   0.9837 0.9482 - 0.3175   ln A / B 0.9366 ln B / C 0.9807 ,
= 2 ( 0.4728 L OSA + 5.8437 ) 0 0 - 2 ( 1.2054 L OSA + 14.6251 )   0.2176   0.9759 0.5996 - 0.8003   ln A / C 0.9185 ln B / C 0.9807 ,
= 2 ( 0.7290 L OSA + 9.0095 ) 0 0 - 2 ( 0.9892 L OSA + 12.0545 )   - 0.6330   0.7741   0.9699 - 0.2432   ln A / B 0.9366 ln A / C 0.9185 ,

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