Abstract

A new numerical method is developed for producing metameric spectral-reflectance functions of given object colors. The computational procedure is simpler than those found in earlier studies. For a given illuminant, observer, and set of tristimulus values, spectral-reflectance functions of object colors are derived, which are metameric with respect to the given conditions. Smoothed metameric spectral-reflectance functions of object colors can also be produced, which resemble those found in actual paint and dye products. The method can be extended to generate spectral-reflectance functions of object colors that are metameric with respect to several conditions of illuminant-observer combinations taken simultaneously. For two spectral-reflectance functions that give an incomplete color match with respect to the given illuminant and observer, the present method can be used to make an appropriate adjustment to one of the spectral-reflectance functions so that a metameric match results.

© 1972 Optical Society of America

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References

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  1. G. Wyszecki, Farbe 2, 39 (1953).
  2. G. Wyszecki, J. Opt. Soc. Am. 48, 451 (1958).
    [CrossRef]
  3. M. Richter, Farbe 9, 163 (1958).
  4. G. Wyszecki, Acta Chromatica 1, 1 (1962).
  5. G. Wyszecki and W. S. Stiles, Color Science (Wiley, New York, 1967), pp. 344–357, where most of the earlier work has been summarized.
  6. V. I. Smirnov, A Course of Higher Mathematics, Vol. IV (Pergamon, New York, 1964).
  7. W. S. Stiles and G. Wyszecki, J. Opt. Soc. Am. 52, 313 (1962).
    [CrossRef]

1962 (2)

1958 (2)

1953 (1)

G. Wyszecki, Farbe 2, 39 (1953).

Richter, M.

M. Richter, Farbe 9, 163 (1958).

Smirnov, V. I.

V. I. Smirnov, A Course of Higher Mathematics, Vol. IV (Pergamon, New York, 1964).

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), pp. 344–357, where most of the earlier work has been summarized.

Wyszecki, G.

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

G. Wyszecki, Acta Chromatica 1, 1 (1962).

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

G. Wyszecki, Farbe 2, 39 (1953).

G. Wyszecki and W. S. Stiles, Color Science (Wiley, New York, 1967), pp. 344–357, where most of the earlier work has been summarized.

Acta Chromatica (1)

G. Wyszecki, Acta Chromatica 1, 1 (1962).

Farbe (2)

G. Wyszecki, Farbe 2, 39 (1953).

M. Richter, Farbe 9, 163 (1958).

J. Opt. Soc. Am. (2)

Other (2)

G. Wyszecki and W. S. Stiles, Color Science (Wiley, New York, 1967), pp. 344–357, where most of the earlier work has been summarized.

V. I. Smirnov, A Course of Higher Mathematics, Vol. IV (Pergamon, New York, 1964).

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Figures (2)

F. 1
F. 1

Spectral-reflectance curves of objects that yield metameric colors when illuminated with CIE standard illuminant A and viewed by the CIE 1964 supplementary observer. The full line with dots corresponds to ρ*(λ) in Table I, and the dotted line with crosses to ρ*(λ) in Table II.

F. 2
F. 2

Two spectral-reflectance curves ρ1(λ) and ρ*2(λ) of objects that yield metameric colors when illuminated with CIE standard illuminant D65 and viewed by the CIE 1931 standard observer, and also with CIE standard illuminant A and the same observer.

Tables (4)

Tables Icon

Table I Spectral reflectances ρ(λ) and ρ*(λ) derived by means of random-numbers procedure used at 10-nm intervals.

Tables Icon

Table II Spectral reflectances ρ(λ) and ρ*(λ) derived by means of random-numbers procedure used at 30-nm intervals with subsequent linear interpolations.

Tables Icon

Table III Spectral reflectances of 12 gray objects that yield metameric colors when illuminated with CIE standard illuminant D65 and viewed by the CIE 1931 standard observer.

Tables Icon

Table IV Spectral reflectances ρ1(λ), ρ2(λ), ρ*2(λ) with smooth distributions.

Equations (23)

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X 1 = X , X 2 = Y , X 3 = Z , x 1 ( λ ) = J ( λ ) x ¯ ( λ ) , x 2 ( λ ) = J ( λ ) y ¯ ( λ ) , x 3 ( λ ) = J ( λ ) z ¯ ( λ ) , ( f · g ) = f ( λ ) g ( λ ) d λ .
X i = ( ρ · x i ) ( i = 1 , 2 , 3 ) ,
X 0 i = ( ρ * · x i ) ( i = 1 , 2 , 3 ) .
X 0 i = ( ρ * · x i ) + ( · x i ) ,
Δ X i = ( · x i ) ,
Δ X i = X 0 i X i .
( · ) = { ρ * ( λ ) ρ ( λ ) } 2 d λ
[ { ( λ ) } 2 2 j = 1 3 μ j ( λ ) x j ( λ ) ] d λ ,
( λ ) [ { ( λ ) } 2 2 j = 1 3 μ j ( λ ) x j ( λ ) ] = 0
( λ ) = j = 1 3 μ j x j ( λ ) .
Δ X i = j = 1 3 A i j μ j ,
A i j = A j i = ( x i · x j ) .
μ j = i = 1 3 Δ X i B i j ,
( λ ) = i = 1 3 j = 1 3 Δ X i B i j x j ( λ ) .
ρ * ( λ ) = ρ ( λ ) + i = 1 3 j = 1 3 Δ X i B i j x j ( λ ) .
0 ρ * α ( λ ) 1
X 0 , 3 n 2 = X 0 ( n ) , X 0 , 3 n 1 = Y 0 ( n ) , X 0 , 3 n = Z 0 ( n ) , x 3 n 2 ( λ ) = J ( n ) ( λ ) x ¯ ( n ) ( λ ) , x 3 n 1 ( λ ) = J ( n ) ( λ ) y ¯ ( n ) ( λ ) , x 3 n ( λ ) = J ( n ) ( λ ) z ¯ ( n ) ( λ ) ( n = 1 , 2 , , N ) .
ρ * ( λ ) = ρ ( λ ) + i = 1 3 N j = 1 3 N Δ X i B i j x j ( λ ) ,
Δ X i = X 0 i ( ρ · x i ) ( i = 1 , 2 , , 3 N ) ,
A i j = ( x i · x j ) , ( i , j = 1 , 2 , , 3 N ) .
3 N M ,
X 1 ( D 65 ) = 33.80 , Y 1 ( D 65 ) = 30.54 , Z 1 ( D 65 ) = 25.15
X 1 ( A ) = 43.34 , Y 1 ( A ) = 34.54 , Z 1 ( A ) = 8.18