Abstract

A new algorithm is suggested for color-recipe formulation. Whereas previous algorithms solved for recipes using, in turn, all possible combinations of three colorants from a given list, the new algorithm finds directly both the colorants to be used and their proportions. This is done by fitting the problem to a linear program and using existing solution algorithms.

© 1974 Optical Society of America

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References

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  1. C. Preston and D. Tough, Color Engineering 3, 19 (1965).
  2. E. Allen, American Dyestuff Reporter 54, 57 (1965).
  3. E. Allen, J. Opt. Soc. Am. 56, 1256 (1966).
    [Crossref]

1966 (1)

1965 (2)

C. Preston and D. Tough, Color Engineering 3, 19 (1965).

E. Allen, American Dyestuff Reporter 54, 57 (1965).

Allen, E.

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

E. Allen, American Dyestuff Reporter 54, 57 (1965).

Preston, C.

C. Preston and D. Tough, Color Engineering 3, 19 (1965).

Tough, D.

C. Preston and D. Tough, Color Engineering 3, 19 (1965).

American Dyestuff Reporter (1)

E. Allen, American Dyestuff Reporter 54, 57 (1965).

Color Engineering (1)

C. Preston and D. Tough, Color Engineering 3, 19 (1965).

J. Opt. Soc. Am. (1)

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Figures (1)

Fig. 1
Fig. 1

Illustrating the metamerism-constraint ellipse and the rectangular approximation in two dimensions.

Equations (31)

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X C = λ E C ( λ ) x ¯ ( λ ) R ( λ ) d λ ,
Y C = λ E C ( λ ) y ¯ ( λ ) R ( λ ) d λ ,
Z C = λ E C ( λ ) z ¯ ( λ ) R ( λ ) d λ .
R T = [ R ( λ 1 ) R ( λ 2 ) R ( λ N ) ] , θ T = [ X C Y C Z C ] , [ E C ] = [ E C ( λ 1 ) x ¯ ( λ 1 ) E C ( λ N ) x ¯ ( λ N ) E C ( λ 1 ) y ¯ ( λ 1 ) E C ( λ N ) y ¯ ( λ N ) E C ( λ 1 ) z ¯ ( λ 1 ) E C ( λ N ) z ¯ ( λ N ) ] ,
θ C = [ E C ] R .
θ A = [ E A ] R .
Δ E 2 = Δ θ T [ Q ] Δ θ ,
f { R ( λ i ) } = α 0 ( λ i ) + c 1 α 1 ( λ i ) + + c n α n ( λ i ) , i = 1 , 2 , N .
f { R ( λ i ) } = { 1 - R ( λ i ) } 2 2 R ( λ i ) ;
R ( λ i ) = - { 1 + f ( λ i ) } + { f 2 ( λ i ) + 2 f ( λ i ) } 1 2 .
f T = [ f { R ( λ 1 ) } , f { R ( λ 2 ) } , f { R ( λ N ) } ] , [ A ] = [ α 1 ( λ 1 ) α 2 ( λ 1 ) α n ( λ 1 ) α 1 ( λ 2 ) α 2 ( λ 2 ) α n ( λ 2 ) α 1 ( λ N ) α 2 ( λ N ) α n ( λ N ) ] , α 0 T = [ α 0 ( λ 1 ) , α 0 ( λ 2 ) , α 0 ( λ N ) ] ,
c T = [ c 1 , c 2 , c n ] .
f = α 0 + [ A ] c
R = R ( c ) .
β T = [ β 1 , β 2 , β n ] ,
Δ θ A = θ A * - [ E A ] R ( c ) .
J = β T c ,
c i 0 ,             i = 1 , 2 , n
[ E C ] R ( c ) = θ C *
Δ θ A T [ Q ] Δ θ A M max 2
[ E C ] R * + [ E C ] ( R f ) { α 0 - f * } + [ E C ] ( R f ) [ A ] c = θ C *
[ E C ] ( R f ) { α 0 - f * } + [ E C ] ( R f ) [ A ] c = 0.
Δ θ A = θ A - θ A * = θ A - [ E A ] R * ,
Δ θ A = [ E A ] ( R f ) { α 0 - f * } + [ E A ] ( R f ) [ A ] c .
Δ θ A T [ Q ] Δ θ A = M max 2 .
- M max ρ i v i T Δ θ A M max ρ i ,             i = 1 , 2 , 3.
J = β T c ,
c j 0 ,             j = 1 , 2 , n ,
[ E C ] ( R f ) [ A ] c + [ E C ] ( R f ) { α 0 - f * } = 0 ,
- v i T [ E A ] ( R f ) { α 0 - f * } - v i T [ E A ] ( R f ) [ A ] c M max ρ i ,             i = 1 , 2 , 3
v i T [ E A ] ( R f ) { α 0 - f * } + v i T [ E A ] ( R f ) [ A ] c M max ρ i ,             i = 1 , 2 , 3 ,