Abstract

A visually uniform color coordinate system, based upon simple mathematical formulas, is described. This system resembles the Adams chromatic-value system but replaces the quintic-parabola function with a cube-root function. For colors having reflectances greater than 0.5% the color spacing obtained agrees with Munsell spacing as closely as the modified Adams system. At lower reflectances an expanded color spacing over that of the Munsell system is provided. The cube-root equations can be solved directly for color coordinate differences in terms of simple functions of the difference in colorimeter readings or tristimulus values. The computation of color coordinates in this system is simpler and requires less computational precision than other visually uniform color coordinate systems. A simple slide rule for computing color differences in cube-root color coordinates is described. A modification of the cube-root color coordinate system which provides nearly perfect representation of the spacing of Munsell colors is described, and the appropriateness of the assumptions required to obtain this behavior is discussed.

© 1958 Optical Society of America

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References

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  1. E. Q. Adams, J. Opt. Soc. Am. 32, 168 (1942).
    [Crossref]
  2. D. Nickerson and K. F. Stultz, J. Opt. Soc. Am. 34, 550 (1944).
    [Crossref]
  3. D. B. Judd, Color in Business, Science, and Industry (John Wiley & Sons, Inc., New York, 1952).
  4. J. L. Saunderson and B. I. Milner, J. Opt. Soc. Am. 36, 36 (1946).
    [Crossref] [PubMed]
  5. L. G. Glasser and D. J. Troy, J. Opt. Soc. Am. 42, 652 (1952).
    [Crossref]
  6. J. R. L. Delboeuf, Etude Psychophysique, 51 (1873).
  7. S. S. Stevens, Phys. Today 8, 12 (October, 1955).
    [Crossref]
  8. R. W. G. Hunt, Nature 179, 1026 (May18, 1957).
    [Crossref]
  9. J. L. Saunderson and B. I. Milner, J. Opt. Soc. Am. 34, 169 (1944).
  10. J. H. Ladd and J. E. Pinney, Proc. Inst. Radio Engrs. 43, 1137 (1955).

1957 (1)

R. W. G. Hunt, Nature 179, 1026 (May18, 1957).
[Crossref]

1955 (2)

S. S. Stevens, Phys. Today 8, 12 (October, 1955).
[Crossref]

J. H. Ladd and J. E. Pinney, Proc. Inst. Radio Engrs. 43, 1137 (1955).

1952 (1)

1946 (1)

1944 (2)

J. L. Saunderson and B. I. Milner, J. Opt. Soc. Am. 34, 169 (1944).

D. Nickerson and K. F. Stultz, J. Opt. Soc. Am. 34, 550 (1944).
[Crossref]

1942 (1)

1873 (1)

J. R. L. Delboeuf, Etude Psychophysique, 51 (1873).

Adams, E. Q.

Delboeuf, J. R. L.

J. R. L. Delboeuf, Etude Psychophysique, 51 (1873).

Glasser, L. G.

Hunt, R. W. G.

R. W. G. Hunt, Nature 179, 1026 (May18, 1957).
[Crossref]

Judd, D. B.

D. B. Judd, Color in Business, Science, and Industry (John Wiley & Sons, Inc., New York, 1952).

Ladd, J. H.

J. H. Ladd and J. E. Pinney, Proc. Inst. Radio Engrs. 43, 1137 (1955).

Milner, B. I.

J. L. Saunderson and B. I. Milner, J. Opt. Soc. Am. 36, 36 (1946).
[Crossref] [PubMed]

J. L. Saunderson and B. I. Milner, J. Opt. Soc. Am. 34, 169 (1944).

Nickerson, D.

Pinney, J. E.

J. H. Ladd and J. E. Pinney, Proc. Inst. Radio Engrs. 43, 1137 (1955).

Saunderson, J. L.

J. L. Saunderson and B. I. Milner, J. Opt. Soc. Am. 36, 36 (1946).
[Crossref] [PubMed]

J. L. Saunderson and B. I. Milner, J. Opt. Soc. Am. 34, 169 (1944).

Stevens, S. S.

S. S. Stevens, Phys. Today 8, 12 (October, 1955).
[Crossref]

Stultz, K. F.

Troy, D. J.

Etude Psychophysique (1)

J. R. L. Delboeuf, Etude Psychophysique, 51 (1873).

J. Opt. Soc. Am. (5)

Nature (1)

R. W. G. Hunt, Nature 179, 1026 (May18, 1957).
[Crossref]

Phys. Today (1)

S. S. Stevens, Phys. Today 8, 12 (October, 1955).
[Crossref]

Proc. Inst. Radio Engrs. (1)

J. H. Ladd and J. E. Pinney, Proc. Inst. Radio Engrs. 43, 1137 (1955).

Other (1)

D. B. Judd, Color in Business, Science, and Industry (John Wiley & Sons, Inc., New York, 1952).

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Figures (5)

Fig. 1
Fig. 1

Comparison of cube-root and Munsell value functions.

Fig. 2
Fig. 2

Rate of change of reflectance with value.

Fig. 3
Fig. 3

Normalized cube-root chromaticity coordinates of Munsell colors at values 1 to 9.

Fig. 4
Fig. 4

Normalized cube-root coordinates of Munsell hues 5 PB and 5 Y.

Fig. 5
Fig. 5

Color difference slide rule for cube-root coordinates.

Tables (1)

Tables Icon

Table I Comparison of cube root and modified Adams color coordinates for assorted Munsell colors.

Equations (14)

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v ( Y ) = k Y 1 3 - m ,
Modified Adams system Cube-root system L = 10 V ( G ) L = 10 v ( G ) a = 41.86 [ V ( R ) - V ( G ) ] a = K a [ v ( R ) - v ( G ) ] b = 16.74 [ V ( G ) - V ( B ) ] b = K b [ v ( G ) - v ( B ) ] .
R = 1.02 X ,             G = Y ,             and             B = 0.847 Z .
log ( Δ r / Δ v ) = ( 2 3 ) log r - log k ,
Δ v = k r - 2 3 Δ r .
v = k r 1 3 + m ,
L = 25.29 G 1 3 - 18.38 a = 106.0 ( R 1 3 - G 1 3 ) b = 42.34 ( G 1 3 - B 1 3 ) .
Δ E = ( Δ L 2 + Δ a 2 + Δ b 2 ) 1 2 ,
Δ L = 25.29 Δ ( G 1 3 ) Δ a = 106.0 [ Δ ( R 1 3 ) - Δ ( G 1 3 ) ] Δ b = 42.34 [ Δ ( G 1 3 ) - Δ ( B 1 3 ) ] .
Δ ( G 1 3 ) ( G 1 + Δ G ) 1 3 - ( G 1 ) 1 3 ,
Δ L = 8.43 G 1 3 ( Δ G / G ) × [ 1 - 1 3 ( Δ G / G ) + 1 / 5.4 ( Δ G / G ) 2 - ] ,
for R > G : a = 125 ( R 1 3 - G 1 3 ) for G > B : b = 30.5 ( G 1 3 - B 1 3 ) ,
for R < G : a = 105 ( R 1 3 - G 1 3 ) for G < B : b = 53.6 ( G 1 3 - B 1 3 ) .
Δ L = K L × Δ G [ 0.5 ( G 1 + G 2 ) ] - 2 3 .