Abstract

If we are given the spectrophotometric curves of a color and three colorants to be used in matching it, the computation of the concentrations of the three colorants required for a tristimulus match is a complicated nonlinear problem. However, with the help of an approximating assumption, a linear solution may be obtained by a matrix inversion technique. Although this is an approximate solution, it is better the less meta-meric the match. With this rough solution as a starting point, iteration may be used to approach an exact match to any desired degree of accuracy. The inverted matrix used for the iterative computation is identical to that used for the rough solution.

© 1966 Optical Society of America

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References

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  1. F. W. Billmeyer, J. K. Beasley, and J. A. Sheldon, J. Opt. Soc. Am. 50, 70 (1960).
    [Crossref]
  2. J. V. Alderson, E. Atherton, and A. N. Derbyshire, J. Soc. Dyers Colourists 77, 657 (1961).
    [Crossref]
  3. R. H. Park and E. I. Stearns, J. Soc. Opt Am. 34, 112 (1944).
    [Crossref]
  4. H. R. Davidson, H. Hemmendinger, and J. L. R. Landry, J. Soc. Dyers Colourists 79, 577 (1963).
    [Crossref]
  5. Manufactured by Davidson and Hemmendinger, Easton, Pa.
  6. Matrix notation as applied to colorimetric problems is explained by E. Allen, Color Eng. 4, No. 4, p. 24 (1966).
  7. The Ciba Q Method [E. Ganz, Textil-Rundschau 20, 255 (1965)] is a graphical method for just this. The Q values of the dyes are the tristimulus-integrated Kubelka–Munk K/S values, previously computed and tabulated. It is understandable that the dR/df(R) values cannot be incorporated into the Q values as they should, because they vary with the sample to be matched. Park and Stearns’s Eqs. (5) also call for summation of the f(R) values at “suitably selected ordinates.”

1966 (1)

Matrix notation as applied to colorimetric problems is explained by E. Allen, Color Eng. 4, No. 4, p. 24 (1966).

1965 (1)

The Ciba Q Method [E. Ganz, Textil-Rundschau 20, 255 (1965)] is a graphical method for just this. The Q values of the dyes are the tristimulus-integrated Kubelka–Munk K/S values, previously computed and tabulated. It is understandable that the dR/df(R) values cannot be incorporated into the Q values as they should, because they vary with the sample to be matched. Park and Stearns’s Eqs. (5) also call for summation of the f(R) values at “suitably selected ordinates.”

1963 (1)

H. R. Davidson, H. Hemmendinger, and J. L. R. Landry, J. Soc. Dyers Colourists 79, 577 (1963).
[Crossref]

1961 (1)

J. V. Alderson, E. Atherton, and A. N. Derbyshire, J. Soc. Dyers Colourists 77, 657 (1961).
[Crossref]

1960 (1)

1944 (1)

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

Alderson, J. V.

J. V. Alderson, E. Atherton, and A. N. Derbyshire, J. Soc. Dyers Colourists 77, 657 (1961).
[Crossref]

Allen, E.

Matrix notation as applied to colorimetric problems is explained by E. Allen, Color Eng. 4, No. 4, p. 24 (1966).

Atherton, E.

J. V. Alderson, E. Atherton, and A. N. Derbyshire, J. Soc. Dyers Colourists 77, 657 (1961).
[Crossref]

Beasley, J. K.

Billmeyer, F. W.

Davidson, H. R.

H. R. Davidson, H. Hemmendinger, and J. L. R. Landry, J. Soc. Dyers Colourists 79, 577 (1963).
[Crossref]

Derbyshire, A. N.

J. V. Alderson, E. Atherton, and A. N. Derbyshire, J. Soc. Dyers Colourists 77, 657 (1961).
[Crossref]

Ganz, E.

The Ciba Q Method [E. Ganz, Textil-Rundschau 20, 255 (1965)] is a graphical method for just this. The Q values of the dyes are the tristimulus-integrated Kubelka–Munk K/S values, previously computed and tabulated. It is understandable that the dR/df(R) values cannot be incorporated into the Q values as they should, because they vary with the sample to be matched. Park and Stearns’s Eqs. (5) also call for summation of the f(R) values at “suitably selected ordinates.”

Hemmendinger, H.

H. R. Davidson, H. Hemmendinger, and J. L. R. Landry, J. Soc. Dyers Colourists 79, 577 (1963).
[Crossref]

Landry, J. L. R.

H. R. Davidson, H. Hemmendinger, and J. L. R. Landry, J. Soc. Dyers Colourists 79, 577 (1963).
[Crossref]

Park, R. H.

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

Sheldon, J. A.

Stearns, E. I.

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

Color Eng. (1)

Matrix notation as applied to colorimetric problems is explained by E. Allen, Color Eng. 4, No. 4, p. 24 (1966).

J. Opt. Soc. Am. (1)

J. Soc. Dyers Colourists (2)

J. V. Alderson, E. Atherton, and A. N. Derbyshire, J. Soc. Dyers Colourists 77, 657 (1961).
[Crossref]

H. R. Davidson, H. Hemmendinger, and J. L. R. Landry, J. Soc. Dyers Colourists 79, 577 (1963).
[Crossref]

J. Soc. Opt Am. (1)

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

Textil-Rundschau (1)

The Ciba Q Method [E. Ganz, Textil-Rundschau 20, 255 (1965)] is a graphical method for just this. The Q values of the dyes are the tristimulus-integrated Kubelka–Munk K/S values, previously computed and tabulated. It is understandable that the dR/df(R) values cannot be incorporated into the Q values as they should, because they vary with the sample to be matched. Park and Stearns’s Eqs. (5) also call for summation of the f(R) values at “suitably selected ordinates.”

Other (1)

Manufactured by Davidson and Hemmendinger, Easton, Pa.

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Figures (1)

F. 1
F. 1

Reflectance curve of Munsell 7.5B 4/8, taken here as a sample to be matched. The match is desired under source C (curve Ec), but if the d factors are omitted from the equations the match will really be calculated for source O (curve Eo).

Equations (37)

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F 1 ( C 1 , C 2 , C 3 ) = X , F 2 ( C 1 , C 2 , C 3 ) = Y , F 3 ( C 1 , C 2 , C 3 ) = Z ,
t = [ X Y Z ] .
T = [ x ¯ 400 x ¯ 420 x ¯ 700 y ¯ 400 y ¯ 420 y ¯ 700 z ¯ 400 z ¯ 420 z ¯ 700 ] ,
E = [ E 400 0 0 0 E 420 0 0 0 E 700 ] ,
r ( s ) = [ R 400 ( s ) R 420 ( s ) R 700 ( s ) ] , r ( m ) = [ R 400 ( m ) R 420 ( m ) R 700 ( m ) ] ,
t = TE r ( s ) = TE r ( m ) ,
TE [ r ( s ) r ( m ) ] = 0 .
R i ( s ) R i ( m ) = Δ R i = [ d R / d f ( R ) ] i Δ f ( R ) i = [ d R / d f ( R ) ] i [ f ( R ) i ( s ) f ( R ) i ( m ) ] ,
f ( s ) = [ f ( R ) 400 ( s ) f ( R ) 420 ( s ) f ( R ) 700 ( s ) ] , f ( m ) = [ f ( R ) 400 ( m ) f ( R ) 420 ( m ) f ( R ) 700 ( m ) ] ,
D = [ d 400 0 0 0 d 420 0 0 0 d 700 ] ,
r ( s ) r ( m ) = D [ f ( s ) f ( m ) ] .
TED f ( s ) = TED f ( m ) .
c = [ C 1 C 2 C 3 ] , and Φ = [ ϕ 400 , 1 ϕ 400 , 2 ϕ 400 , 3 ϕ 420 , 1 ϕ 420 , 2 ϕ 420 , 3 ϕ 700 , 1 ϕ 700 , 2 ϕ 700 , 3 ] ,
f ( m ) = f ( t ) + Φ c ,
TED Φ c = TED [ f ( s ) f ( t ) ] ,
c = ( TED Φ ) 1 TED [ f ( s ) f ( t ) ] .
Δ T = [ Δ X Δ Y Δ Z ] , and Δ c = [ Δ C 1 Δ C 2 Δ C 3 ] ,
Δ t = B Δ c ,
B = [ X / C 1 X / C 2 X / C 3 Y / C 1 Y / C 2 Y / C 3 Z / C 1 Z / C 2 Z / C 3 ] .
P = [ X / R 400 ( m ) X / R 420 ( m ) X / R 700 ( m ) Y / R 400 ( m ) Y / R 420 ( m ) Y / R 700 ( m ) Z / R 400 ( m ) Z / R 420 ( m ) Z / R 700 ( m ) ] ,
Q = [ R 400 ( m ) / C 1 R 400 ( m ) / C 2 R 400 ( m ) / C 3 R 420 ( m ) / C 1 R 420 ( m ) / C 2 R 420 ( m ) / C 3 R 700 ( m ) / C 1 R 700 ( m ) / C 2 R 700 ( m ) / C 3 ] ,
B = PQ .
X = x ¯ 400 E 400 R 400 ( m ) + + x ¯ 700 E 700 R 700 ( m ) ,
P = [ x ¯ 400 E 400 x ¯ 420 E 420 x ¯ 700 E 700 y ¯ 400 E 400 y ¯ 420 E 420 y ¯ 700 E 700 z ¯ 400 E 400 z ¯ 420 E 420 z ¯ 700 E 700 ] ,
P = TE .
R 400 ( m ) C 1 = ( d R d f ( R ) ) 400 f ( R ) 400 ( m ) C 1 = d 400 f ( R ) 400 ( m ) C 1 .
f ( R ) 400 ( m ) = f ( R ) 400 ( t ) + C 1 ϕ 400 , 1 + C 2 ϕ 400 , 2 + C 3 ϕ 400 , 3 ,
f ( R ) 400 ( m ) / C 1 = ϕ 400 , 1 ,
R 400 ( m ) / C 1 = d 400 ϕ 400 , 1 ,
Q = [ d 400 ϕ 400 , 1 d 400 ϕ 400 , 2 d 400 ϕ 400 , 3 d 420 ϕ 420 , 1 d 420 ϕ 420 , 2 d 420 ϕ 420 , 3 d 700 ϕ 700 , 1 d 700 ϕ 700 , 2 d 700 ϕ 700 , 3 ] ,
Q = D Φ .
B = TED Φ ,
Δ T = TED Φ Δ c ,
Δ c = ( TED Φ ) 1 Δ T .
c = ( TE ϕ ) 1 TE ( f ( s ) f ( t ) ) .
E = E D 1 D = E o D ,
c = ( T E o D ϕ ) 1 T E o D [ f ( s ) f ( t ) ] .