Abstract

All of the spectrally different colors that match according to an observer form a metamer ensemble. We present a new method to generate the metamer ensembles for both direct light and reflecting-surface-color problems. The method is based on the properties of particular metameric functions (the simple elements) and is particularly appropriate for treating problems of theoretical limits of metamerism. The method is illustrated with several practical examples and is compared with previously known methods.

© 1976 Optical Society of America

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References

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  1. R. H. Morris, J. Opt. Soc. Am. 37, 669 (1947).
    [CrossRef]
  2. W. L. Brewer and F. R. Holly, J. Opt. Soc. Am. 38, 858 (1948).
    [CrossRef] [PubMed]
  3. I. Hennicke, Die Farbe 7, 93 (1958).
  4. M. Richter, Die Farbe 7, 163 (1958).
    [CrossRef]
  5. G. Wyszecki, J. Opt. Soc. Am. 48, 451 (1958).
    [CrossRef]
  6. K. Takahama and Y. Nayatani, J. Opt. Soc. Am. 62, 1516 (1972).
    [CrossRef]
  7. G. Wyszecki, J. Opt. Soc. Am. 49, 811 (1959).
    [CrossRef]
  8. W. S. Stiles and G. Wyszecki, J. Opt. Soc. Am. 52, 313 (1962).
    [CrossRef]
  9. G. Wyszecki and W. S. Stiles, Color Science (Wiley, New York, 1967).
  10. E. Allen, J. Opt. Soc. Am. 56, 559 (1966).
    [CrossRef]
  11. E. Allen, Color Eng. 7, 35 (1969).
  12. N. Ohta and G. Wyszecki, J. Opt. Soc. Am. 65, 327 (1975).
    [CrossRef]
  13. N. Ohta and G. Wyszecki, J. Opt. Soc. Am. 65, 1152 (1975).
    [CrossRef]
  14. N. Ohta and G. Wyszecki, J. Opt. Soc. Am. 65, 834 (1975).
    [CrossRef]
  15. Demonstrations presented in the thesis to be submitted by the author at the University P. et M, Curie, Paris 75005, France.
  16. E. Schrödinger, Ann. Phys. (N.Y.) 62, 603 (1920).
  17. D. L. MacAdam, J. Opt. Soc. Am. 25, 249 (1935).
    [CrossRef]

1975 (3)

1972 (1)

1969 (1)

E. Allen, Color Eng. 7, 35 (1969).

1966 (1)

E. Allen, J. Opt. Soc. Am. 56, 559 (1966).
[CrossRef]

1962 (1)

1959 (1)

1958 (3)

I. Hennicke, Die Farbe 7, 93 (1958).

M. Richter, Die Farbe 7, 163 (1958).
[CrossRef]

G. Wyszecki, J. Opt. Soc. Am. 48, 451 (1958).
[CrossRef]

1948 (1)

1947 (1)

1935 (1)

1920 (1)

E. Schrödinger, Ann. Phys. (N.Y.) 62, 603 (1920).

Allen, E.

E. Allen, Color Eng. 7, 35 (1969).

E. Allen, J. Opt. Soc. Am. 56, 559 (1966).
[CrossRef]

Brewer, W. L.

Hennicke, I.

I. Hennicke, Die Farbe 7, 93 (1958).

Holly, F. R.

MacAdam, D. L.

Morris, R. H.

Nayatani, Y.

Ohta, N.

Richter, M.

M. Richter, Die Farbe 7, 163 (1958).
[CrossRef]

Schrödinger, E.

E. Schrödinger, Ann. Phys. (N.Y.) 62, 603 (1920).

Stiles, W. S.

W. S. Stiles and G. Wyszecki, J. Opt. Soc. Am. 52, 313 (1962).
[CrossRef]

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

Takahama, K.

Wyszecki, G.

Ann. Phys. (N.Y.) (1)

E. Schrödinger, Ann. Phys. (N.Y.) 62, 603 (1920).

Color Eng. (1)

E. Allen, Color Eng. 7, 35 (1969).

Die Farbe (2)

I. Hennicke, Die Farbe 7, 93 (1958).

M. Richter, Die Farbe 7, 163 (1958).
[CrossRef]

J. Opt. Soc. Am. (11)

Other (2)

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

Demonstrations presented in the thesis to be submitted by the author at the University P. et M, Curie, Paris 75005, France.

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Figures (13)

FIG. 1
FIG. 1

Example of the spectral-reflectance function that corresponds to a simple element in a metamer ensemble R.

FIG. 2
FIG. 2

Dispersion in the CIE 1964 chromaticity diagram of simple elements of 20 metamer ensembles F that correspond to the CIE 1931 standard observer, defined in Table I.

FIG. 3
FIG. 3

Limits in the CIE 1964 chromaticity diagram of the color-mismatch volumes of the 20 metamer ensembles F defined in Table I. The geometric barycenter (xg, yg, Yg) of every simple-element set is noted by a star and is connected with the point of CIE 1931 chromaticity coordinates (x, y), plotted on the same axes; that point defines the corresponding metamer ensemble (see Table I).

FIG. 4
FIG. 4

Limits in the CIE 1931 chromaticity diagram of the color-mismatch volume of the 20 metamer ensembles F that correspond to the CIE 1964 standard observer and the 20 geometric barycenters (xg, yg, Yg) of Table I.

FIG. 5
FIG. 5

Projections in the CIE 1964 (x10, y10, Y10) color space of the simple elements of the metamer ensemble F that corresponds to the equienergy white W with Y = 100, for the CIE 1931 standard observer. Calculations made in the wavelength range from 400 to 700 nm with 10 nm steps.

FIG. 6
FIG. 6

Same as Fig. 5, but with calculations in the range from 380 to 770 nm with 10 nm steps.

FIG. 7
FIG. 7

Same as Figs. 5 and 6, but with calculations in the range 400 to 700 nm with 5 nm steps.

FIG. 8
FIG. 8

Color-mismatch-volume limits in the CIE 1964 (x10, y10, Y10) color space, corresponding to the metamer ensemble F of the equienergy white W (Y = 100), with respect to the CIE 1931 standard observer. Calculation was performed in the 400–700 nm wavelength range with 5 nm steps. The standard-deviation ellipsoid computed with the distribution of the simple-element projections is shown. Dashed lines represent the boundaries of the same volume for a calculation made with 10 nm steps.

FIG. 9
FIG. 9

Color-mismatch-volume limits and standard-deviation ellipsoid in the CIE 1931 (x, y, Y) color space for metameric grays with Y = 10%. Reference illuminant is A, test illuminant is D65.

FIG. 10
FIG. 10

Color-mismatch-volume boundaries and standard-deviation ellipsoid for metameric grays with Y = 10%. Reference illuminant is D65 and test illuminant is A.

FIG. 11
FIG. 11

Standard-deviation ellipsoid and limits of the same color-mismatch volume as defined in Fig. 10 but for a computation with 20 nm steps. Limits for 10 nm steps are indicated with dashed lines.

FIG. 12
FIG. 12

Projections of the simple elements obtained for 20 nm step calculation as in Fig. 11.

FIG. 13
FIG. 13

Color-mismatch-volume limits and standard-deviation ellipsoid in the CIE 1964 (x10, y10, Y10) color space for grays that are metameric for the CIE 1931 standard observer with Y = 10%. Reference illuminant is A. Dashed lines show the color-mismatch volume that corresponds to the same change of observer for the metamer ensemble F defined at the chromaticity coordinates of A.

Tables (1)

Tables Icon

TABLE I Examples of 20 metamer ensembles F(x, y, 100) for the CIE 1931 standard colorimetric observer. For every ensemble, the computation gives N simple elements which, when projected on the CIE 1964 (x10, y10, Y10) color space, have a geometric barycenter (xg, yg, Yg). Inversely for the metamer ensemble F(xg, yg, Yg) of the CIE 1964 standard observer, the N′ corresponding simple elements have the geometric barycenter ( x g , y g , Y g ) of their projections in the CIE 1931 (x, y, Y) color space.

Equations (14)

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X = k λ f ( λ ) x ¯ ( λ ) d λ , Y = k λ f ( λ ) y ¯ ( λ ) d λ , Z = k λ f ( λ ) z ¯ ( λ ) d λ ,
k = 100 / λ S ( λ ) y ¯ ( λ ) d ( λ ) .
X 1 = X ,             X 2 = Y ,             X 3 = Z , x ¯ 1 ( λ ) = x ¯ ( λ ) ,             x ¯ 2 ( λ ) = y ¯ ( λ ) ,             x ¯ 3 ( λ ) = z ¯ ( λ ) .
X a = k i = 1 M Δ λ i f ( λ ) x ¯ a ( λ ) d ( λ ) ,             ( a = 1 , 2 , 3 ) .
X a = k i = 1 M f i x ¯ a , i Δ λ i ,
X a = 100 i = 1 M r i ( S x ¯ a ) i Δ λ i / i = 1 M ( S x ¯ a ) i Δ λ i ,
f i 0 ,             i = 1 - M
b = α f + α f , α 0 ,             α 0 ,             α + α = 1.
( M 3 ) ;
f = j = 1 α j E j ,             with j = 1 α j = 1.
0 r i 1             ( i = 1 - M ) .
X a = k i = 1 M f i x ¯ a , i Δ λ i             ( a = 1 , 2 , 3 ) .
X a = 100 i = 1 M r i ( S x ¯ a ) i Δ λ i / i = 1 M ( S x ¯ a ) i Δ λ i ,
2 M - 3 ( M 3 ) ,