Abstract

Metamerism, with reference to an arbitrary pattern and any ternary combination of colorants suitable for matching its color, is generally not antisymmetrical with respect to any pair of illuminants. For the exceptional case of antisymmetry, the prerequisite, necessary, and sufficient condition is formulated. There are, therefore, two measures of metamerism, and not one. These two measures are vectors that differ both in sign and generally in absolute value of all three components. <i>A priori</i>, there is no reason to prefer any one of the two measures of metamerism—but there may be technical reasons for doing so. Formulas are deduced for the practical computation of both of these vectors. These are applicable even when the colors of the samples differ in both of two illuminants.

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  1. I. Nimeroff, Color Eng. 9 (2), 13 (1971).
  2. In terms of Nimeroff's terminology (Ref. 1), this is a "source-specimen-variable match." In this paper, the term metamerism is used in the customary sense. Two samples are involved, the reflectance spectra of which are not identical. Two illuminants are also involved, which are defined by different spectral energy distributions. Finally, the observer remains the same; this is specified in Eq. (5) by the matrix T.
  3. E. Allen, J. Opt. Soc. Am. 56, 1256 (1966).
  4. G. Wyszecki and W. S. Stiles, Color Science (Wiley, New York, 1967), p. 435ff.
  5. A. Brockes, Color Eng. 6 (3), 48 (1968).
  6. D. L. MacAdam, Vision Res. 1, 9 (1961).
  7. It is doubtful whether chromatic adaptation (Refs. 4 and 5) is sufficiently described by a linear transformation. MacAdam (Ref. 6) has proposed a nonlinear hypothesis.
  8. The adaptation matrix CA, where C refers to illuminant C, is usually defined as the unity matrix, and may, therefore, be omitted.
  9. D. B. Judd and G. Wyszecki, Color in Business, Science and Industry (Wiley, New York, 1963), p. 153.
  10. Judd and Wyszecki (Ref. 9) define this quantity by D = ΔEEm. The scale-normalizing factor 1/ ΔEm is set equal to 1 here and throughout this paper.
  11. U. Gugerli, Chimia 15, 39 (1961).
  12. S. M. Newhall, D. Nickerson, and D. B. Judd, J. Opt. Soc. Am. 33, 417 (1943).
  13. R. H. Park and E. I. Stearns, J. Opt. Soc. Am. 34, 112 (1944).
  14. Park and Stearns (Ref. 13) have proposed (for the first time, to the best of the author’s knowledge) the use essentially of the matrix P-1 to correct approximate, precalculated matches.
  15. With the exception of the special case defined by Eq. (15).
  16. This deduction has been restricted to ternary colorant combinations, but this is not necessary. It would be possible, for example, to show that Eq. (9′) is also valid for quaternary combinations. However, the deduction would then become much more complicated without changing anything of fundamental significance.
  17. By this, the technical suitability is first envisaged. Obviously, only such colorants can be considered that enable a given substrate to be colored serviceably. In addition, it must be assumed that matching the pattern in both illuminants is possible using the colorants given. As is well known, this may be decided by numerical computation on the basis of the absorptivity matrix Φ.

Allen, E.

E. Allen, J. Opt. Soc. Am. 56, 1256 (1966).

Brockes, A.

A. Brockes, Color Eng. 6 (3), 48 (1968).

Gugerli, U.

U. Gugerli, Chimia 15, 39 (1961).

Judd, D. B.

S. M. Newhall, D. Nickerson, and D. B. Judd, J. Opt. Soc. Am. 33, 417 (1943).

D. B. Judd and G. Wyszecki, Color in Business, Science and Industry (Wiley, New York, 1963), p. 153.

MacAdam, D. L.

D. L. MacAdam, Vision Res. 1, 9 (1961).

Newhall, S. M.

S. M. Newhall, D. Nickerson, and D. B. Judd, J. Opt. Soc. Am. 33, 417 (1943).

Nickerson, D.

S. M. Newhall, D. Nickerson, and D. B. Judd, J. Opt. Soc. Am. 33, 417 (1943).

Nimeroff, I.

I. Nimeroff, Color Eng. 9 (2), 13 (1971).

Park, R. H.

R. H. Park and E. I. Stearns, J. Opt. Soc. Am. 34, 112 (1944).

Stearns, E. I.

R. H. Park and E. I. Stearns, J. Opt. Soc. Am. 34, 112 (1944).

Stiles, W. S.

G. Wyszecki and W. S. Stiles, Color Science (Wiley, New York, 1967), p. 435ff.

Wyszecki, G.

G. Wyszecki and W. S. Stiles, Color Science (Wiley, New York, 1967), p. 435ff.

D. B. Judd and G. Wyszecki, Color in Business, Science and Industry (Wiley, New York, 1963), p. 153.

Other (17)

I. Nimeroff, Color Eng. 9 (2), 13 (1971).

In terms of Nimeroff's terminology (Ref. 1), this is a "source-specimen-variable match." In this paper, the term metamerism is used in the customary sense. Two samples are involved, the reflectance spectra of which are not identical. Two illuminants are also involved, which are defined by different spectral energy distributions. Finally, the observer remains the same; this is specified in Eq. (5) by the matrix T.

E. Allen, J. Opt. Soc. Am. 56, 1256 (1966).

G. Wyszecki and W. S. Stiles, Color Science (Wiley, New York, 1967), p. 435ff.

A. Brockes, Color Eng. 6 (3), 48 (1968).

D. L. MacAdam, Vision Res. 1, 9 (1961).

It is doubtful whether chromatic adaptation (Refs. 4 and 5) is sufficiently described by a linear transformation. MacAdam (Ref. 6) has proposed a nonlinear hypothesis.

The adaptation matrix CA, where C refers to illuminant C, is usually defined as the unity matrix, and may, therefore, be omitted.

D. B. Judd and G. Wyszecki, Color in Business, Science and Industry (Wiley, New York, 1963), p. 153.

Judd and Wyszecki (Ref. 9) define this quantity by D = ΔEEm. The scale-normalizing factor 1/ ΔEm is set equal to 1 here and throughout this paper.

U. Gugerli, Chimia 15, 39 (1961).

S. M. Newhall, D. Nickerson, and D. B. Judd, J. Opt. Soc. Am. 33, 417 (1943).

R. H. Park and E. I. Stearns, J. Opt. Soc. Am. 34, 112 (1944).

Park and Stearns (Ref. 13) have proposed (for the first time, to the best of the author’s knowledge) the use essentially of the matrix P-1 to correct approximate, precalculated matches.

With the exception of the special case defined by Eq. (15).

This deduction has been restricted to ternary colorant combinations, but this is not necessary. It would be possible, for example, to show that Eq. (9′) is also valid for quaternary combinations. However, the deduction would then become much more complicated without changing anything of fundamental significance.

By this, the technical suitability is first envisaged. Obviously, only such colorants can be considered that enable a given substrate to be colored serviceably. In addition, it must be assumed that matching the pattern in both illuminants is possible using the colorants given. As is well known, this may be decided by numerical computation on the basis of the absorptivity matrix Φ.

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