Abstract

We have developed a method for determining the parameters of color-difference-perceptibility ellipses for surface colors and have evaluated it by use of new visual data. A new theory was devised, based on the assumption of a normal distribution of color differences in local areas of CIE 1931 chromaticity space. A likelihood function that combines the observer characteristic with the probability of a certain chromaticity point being seen as a match to a color center provides a mathematical description of color-difference perception in <i>x</i>, <i>y</i> space. Best estimates of the color-difference-perception-ellipse parameters and the value of a false-alarm-rate parameter are found by maximizing this likelihood function. Use of this function leads to predicted data that are not significantly different from collected visual data. The unit color-difference-perception ellipse thus calculated describes the locus of chromaticities that have approximately a 60% chance of being seen as matches to the color represented by the center of the ellipse.

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  1. D. L. MacAdam, J. Opt. Soc. Am. 33, 18 (1943).
  2. W. R. J. Brown and D. L. MacAdam, J. Opt. Soc. Am. 39, 808 (1949).
  3. G. Wyszecki and G. H. Fielder, J. Opt. Soc. Am. 61, 1135 (1971).
  4. W. R. J. Brown, W. G. Howe, J. E. Jackson, and R. M. Morris, J. Opt. Soc. Am. 46, 46 (1956).
  5. Kollmorgen Corp., Newburgh, N. Y. 12550.
  6. W. H. Foster, Jr., R. Gans, E. I. Stearns, and R. E. Stearns, Color Eng. 8, No. 3, 35 (1970).
  7. D. I. Morley, R. Munn, and F. W. Billmeyer, Jr., J. Soc. Dyers and Colourists 91, 000 (1975).
  8. Diano Corp., Mansfield, Mass. 02048.
  9. M. J. D. Powell, Computer J. 7, 155 (1964).
  10. The error of the estimates of the ellipse parameters and of the false-alarm-rate parameter may be found from the negative inverse of the matrix of second partial derivatives of the function that was minimized. On occasion, these values did not provide the positive definite matrix that would be expected. This situation was probably due to the ill-conditioned state of the function. The minimization algorithms work best when the values of all of the first partial derivatives are of the same order of magnitude. This was not the case for the function given here. Where it was possible to calculate the error, it appeared to be of the order of ± 10% of the value of the ellipse parameter and somewhat less for the false-alarm-rate parameter.
  11. K. McLaren, J. Soc. Dyers and Colourists 88, 389 (1970).

Billmeyer, Jr., F. W.

D. I. Morley, R. Munn, and F. W. Billmeyer, Jr., J. Soc. Dyers and Colourists 91, 000 (1975).

Brown, W. R. J.

W. R. J. Brown, W. G. Howe, J. E. Jackson, and R. M. Morris, J. Opt. Soc. Am. 46, 46 (1956).

W. R. J. Brown and D. L. MacAdam, J. Opt. Soc. Am. 39, 808 (1949).

Fielder, G. H.

G. Wyszecki and G. H. Fielder, J. Opt. Soc. Am. 61, 1135 (1971).

Foster, Jr., W. H.

W. H. Foster, Jr., R. Gans, E. I. Stearns, and R. E. Stearns, Color Eng. 8, No. 3, 35 (1970).

Gans, R.

W. H. Foster, Jr., R. Gans, E. I. Stearns, and R. E. Stearns, Color Eng. 8, No. 3, 35 (1970).

Howe, W. G.

W. R. J. Brown, W. G. Howe, J. E. Jackson, and R. M. Morris, J. Opt. Soc. Am. 46, 46 (1956).

Jackson, J. E.

W. R. J. Brown, W. G. Howe, J. E. Jackson, and R. M. Morris, J. Opt. Soc. Am. 46, 46 (1956).

MacAdam, D. L.

D. L. MacAdam, J. Opt. Soc. Am. 33, 18 (1943).

W. R. J. Brown and D. L. MacAdam, J. Opt. Soc. Am. 39, 808 (1949).

McLaren, K.

K. McLaren, J. Soc. Dyers and Colourists 88, 389 (1970).

Morley, D. I.

D. I. Morley, R. Munn, and F. W. Billmeyer, Jr., J. Soc. Dyers and Colourists 91, 000 (1975).

Morris, R. M.

W. R. J. Brown, W. G. Howe, J. E. Jackson, and R. M. Morris, J. Opt. Soc. Am. 46, 46 (1956).

Munn, R.

D. I. Morley, R. Munn, and F. W. Billmeyer, Jr., J. Soc. Dyers and Colourists 91, 000 (1975).

Powell, M. J. D.

M. J. D. Powell, Computer J. 7, 155 (1964).

Stearns, E. I.

W. H. Foster, Jr., R. Gans, E. I. Stearns, and R. E. Stearns, Color Eng. 8, No. 3, 35 (1970).

Stearns, R. E.

W. H. Foster, Jr., R. Gans, E. I. Stearns, and R. E. Stearns, Color Eng. 8, No. 3, 35 (1970).

Wyszecki, G.

G. Wyszecki and G. H. Fielder, J. Opt. Soc. Am. 61, 1135 (1971).

Other

D. L. MacAdam, J. Opt. Soc. Am. 33, 18 (1943).

W. R. J. Brown and D. L. MacAdam, J. Opt. Soc. Am. 39, 808 (1949).

G. Wyszecki and G. H. Fielder, J. Opt. Soc. Am. 61, 1135 (1971).

W. R. J. Brown, W. G. Howe, J. E. Jackson, and R. M. Morris, J. Opt. Soc. Am. 46, 46 (1956).

Kollmorgen Corp., Newburgh, N. Y. 12550.

W. H. Foster, Jr., R. Gans, E. I. Stearns, and R. E. Stearns, Color Eng. 8, No. 3, 35 (1970).

D. I. Morley, R. Munn, and F. W. Billmeyer, Jr., J. Soc. Dyers and Colourists 91, 000 (1975).

Diano Corp., Mansfield, Mass. 02048.

M. J. D. Powell, Computer J. 7, 155 (1964).

The error of the estimates of the ellipse parameters and of the false-alarm-rate parameter may be found from the negative inverse of the matrix of second partial derivatives of the function that was minimized. On occasion, these values did not provide the positive definite matrix that would be expected. This situation was probably due to the ill-conditioned state of the function. The minimization algorithms work best when the values of all of the first partial derivatives are of the same order of magnitude. This was not the case for the function given here. Where it was possible to calculate the error, it appeared to be of the order of ± 10% of the value of the ellipse parameter and somewhat less for the false-alarm-rate parameter.

K. McLaren, J. Soc. Dyers and Colourists 88, 389 (1970).

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