Abstract

From larger–smaller judgments of color differences, compared visually two at a time, the perceived sizes may be evaluated on an interval scale. Given numbers B so evaluated, and such that B is linearly connected to some power p, of the physical measure D (such as distance on any chromaticity diagram) of the differences, the additive constant Kbr such that the numbers B+Kbr are expressed on a ratio scale may be found from judgments of the ratio of sizes of pairs of differences. To evaluate p, it is sufficient to observe the three differences, 12, 23, and 13 between the pairs of three colors, 1, 2, 3, forming a geodesic series, and chosen so that B12 is not much different from B23. The scale formed by the numbers (B+Kbr)1/p is additive if the D scale is additive. If the largest of the color differences judged exceeds the smallest by a factor not greater than 3, a close approximation to the (B+Kbr)1/p scale may be found without evaluating Kbr by ratio judgments. This approximation is based on the empirical discovery that scales based on the additivity condition: (B12+Kbd)1/p+(B13+Kbd)1/p=(B13+Kbd)1/p, though it implies that Kbd depends strongly on p, are essentially identical regardless of the choice of p between 1 and 13. It is sufficient therefore, to derive the additive scale by setting p=1, and computing Kbd as B13B12B23.

© 1967 Optical Society of America

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References

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  1. G. Wyszecki, J. Opt. Soc. Am. 44, 725 (1954); D. B. Judd, J. Opt. Soc. Am. 45, 673 (1955).
    [CrossRef]
  2. H. A. David, The Method of Paired Comparisons (Hafner Publishing Co., New York, 1963), Sec. 1.3 and Ch. 4, especially p. 55.
  3. J. E. Jackson and M. Fleckenstein, Biometrics 13, 51 (1957).
    [CrossRef]
  4. W. S. Torgerson, Theory and Methods of Scaling (John Wiley & Sons, New York, 1958), p. 271.
  5. D. L. MacAdam, J. Opt. Soc. Am. 53, 754 (1963).
    [CrossRef]
  6. G. L. Howett, Report of Eighth Meeting of OSA Committee on Uniform Color Scales, 20November1964, p. 4, unpublished.

1963 (1)

1957 (1)

J. E. Jackson and M. Fleckenstein, Biometrics 13, 51 (1957).
[CrossRef]

1954 (1)

David, H. A.

H. A. David, The Method of Paired Comparisons (Hafner Publishing Co., New York, 1963), Sec. 1.3 and Ch. 4, especially p. 55.

Fleckenstein, M.

J. E. Jackson and M. Fleckenstein, Biometrics 13, 51 (1957).
[CrossRef]

Howett, G. L.

G. L. Howett, Report of Eighth Meeting of OSA Committee on Uniform Color Scales, 20November1964, p. 4, unpublished.

Jackson, J. E.

J. E. Jackson and M. Fleckenstein, Biometrics 13, 51 (1957).
[CrossRef]

MacAdam, D. L.

Torgerson, W. S.

W. S. Torgerson, Theory and Methods of Scaling (John Wiley & Sons, New York, 1958), p. 271.

Wyszecki, G.

Biometrics (1)

J. E. Jackson and M. Fleckenstein, Biometrics 13, 51 (1957).
[CrossRef]

J. Opt. Soc. Am. (2)

Other (3)

W. S. Torgerson, Theory and Methods of Scaling (John Wiley & Sons, New York, 1958), p. 271.

G. L. Howett, Report of Eighth Meeting of OSA Committee on Uniform Color Scales, 20November1964, p. 4, unpublished.

H. A. David, The Method of Paired Comparisons (Hafner Publishing Co., New York, 1963), Sec. 1.3 and Ch. 4, especially p. 55.

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Figures (1)

Fig. 1
Fig. 1

Scales of 2[(B+Kbd)/Kbd]1/p for p = 1 3 , 1 2 , 3 4, and 1 plotted against the scale for p=1. Note that within scale values varying by a factor of 3 (1.5 to 4.5) it makes little difference which of these scales is chosen.

Tables (2)

Tables Icon

Table I Demonstration of the degree to which [(B13+Kbd)/(B12+Kbd)]1/p approaches 2 for Kbd approximated from Eq. (11), B12 taken equal to B23=−0.5, and B13=0.7.

Tables Icon

Table II Dependence of the values of 2[(B+Kbd)/Kbd]1/p on the value of p over the range p=1 to p = 1 3.

Equations (28)

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B = m b r R - K b r .
B = m b r R .
[ B i - B j - F ( f i - f j ) ] 2 = a minimum
( B i + K b r ) / ( B j + K b r ) = r i j ,
K b r = ( r i j B j - B i ) / ( 1 - r i j ) .
D = [ ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 ] 1 2
B 12 + B 23 = B 13 .
K b d = B 13 - B 12 - B 23 .
( B 12 + K b d ) / m b d + ( B 23 + K b d ) / m b d = ( B 13 + K b d ) / m b d ,
B + K b d = m b d D .
B 12 + K b d + B 23 + K b d = B 13 + K b d ,
m b r R 12 - K b r + K b d + m b r R 23 - K b r + K b d = m b r R 13 - K b r + K b d ,
B + K b d = m b d D .
B + K b r = m b d D p .
( B 13 + K b r ) / ( B 12 + K b r ) = 2 p ,
p = log [ B 13 + K b r B 12 + K b r ] / [ log 2 = 0.301 ] .
R 12 = R 23 = R 34 = R n - 1 , n R 1 n R 12 = B 1 n + K b r B 12 + K b r = ( n - 1 ) p ,
p = log [ B 1 n + K b r B 12 + K b r ] / log ( n - 1 ) .
B + K b r = m b d D p             or             R 1 n / R 12 = ( D 1 n / D 12 ) p .
( B 12 + K b d ) 1 / p + ( B 23 + K b d ) 1 / p = ( B 13 + K b d ) 1 / p .
K b d = B 13 - B 12 - B 23 + [ 2 ( B 13 - B 12 ) ( B 13 - B 23 ) ] 1 2 .
K b d B 13 - B 12 - B 23 - 2 1 2 ( 1 - 1 / p ) × [ ( B 13 - B 12 ) ( B 13 - B 23 ) ] 1 2 .
( B 13 + K b d ) / ( B 12 + K b d ) = [ 2 - ( 1 - 1 / p ) 2 1 2 ] / [ 1 - ( 1 - 1 / p ) 2 1 2 ] , B 12 = B 23 .
( B i + K b r ) / ( B j + K b r ) = r i j
K b r = ( B i - r i j B j ) / ( r i j - 1 ) .
B + K b r = m b d D p .
p = ( 1 / 0.301 ) log [ ( B 13 + K b r ) / ( B 12 + K b r ) ] .
K b d = B 13 - B 12 - B 23 ,