Abstract

Based on the theory of orthogonal color mixture curves first published by D. L. MacAdam, a quality factor q of color filters can be defined which is unity for color mixture curves and a positive proper fraction for all the other filters. The difference 1−q measures the average amount of errors when an arbitrary filter is used instead of one whose spectral energy response is a color mixture curve. The theory leads to a rather precise definition of photographic filters that match the usual primaries of color photography and introduce less errors than other filters due to the fact that they fail to be color mixture curves.

© 1956 Optical Society of America

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References

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  1. H. E. Ives, J. Franklin Inst. 180, 673 (1915).
    [Crossref]
  2. R. Luther, Z. tech. Phys. 8, 540 (1927).
  3. A. C. Hardy and F. Wurzburg, J. Opt. Soc. Am. 27, 227 (1937).
    [Crossref]
  4. D. L. MacAdam, J. Opt. Soc. Am. 43, 533 (1953).
    [Crossref] [PubMed]
  5. H. E. J. Neugebauer, Physik. Bl. 4, 258 (1948).
  6. H. E. J. Neugebauer, J. Opt. Soc. Am. 42, 740 (1952).
    [Crossref]
  7. D. L. MacAdam, J. Opt. Soc. Am. 25, 361 (1935).

1953 (1)

1952 (1)

1948 (1)

H. E. J. Neugebauer, Physik. Bl. 4, 258 (1948).

1937 (1)

1935 (1)

1927 (1)

R. Luther, Z. tech. Phys. 8, 540 (1927).

1915 (1)

H. E. Ives, J. Franklin Inst. 180, 673 (1915).
[Crossref]

Hardy, A. C.

Ives, H. E.

H. E. Ives, J. Franklin Inst. 180, 673 (1915).
[Crossref]

Luther, R.

R. Luther, Z. tech. Phys. 8, 540 (1927).

MacAdam, D. L.

Neugebauer, H. E. J.

H. E. J. Neugebauer, J. Opt. Soc. Am. 42, 740 (1952).
[Crossref]

H. E. J. Neugebauer, Physik. Bl. 4, 258 (1948).

Wurzburg, F.

J. Franklin Inst. (1)

H. E. Ives, J. Franklin Inst. 180, 673 (1915).
[Crossref]

J. Opt. Soc. Am. (4)

Physik. Bl. (1)

H. E. J. Neugebauer, Physik. Bl. 4, 258 (1948).

Z. tech. Phys. (1)

R. Luther, Z. tech. Phys. 8, 540 (1927).

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Figures (4)

Fig. 1
Fig. 1

Set of normalized orthogonal color mixture curves, U1, U2, U3.

Fig. 2
Fig. 2

Quality factor q of narrow transmission band filters as function of wavelength.

Fig. 3
Fig. 3

Set of normalized orthogonal color mixture curves, U3, ( U 1 - U 2 ) / 2 , ( U 1 + U 2 ) / 2, approximating the curve of Fig. 2 near its peaks.

Fig. 4
Fig. 4

Proposed filters for color photography.

Equations (26)

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f ( λ ) = f x x ¯ ( λ ) + f y y ¯ ( λ ) + f z z ¯ ( λ ) .
U 1 ( λ ) , U 2 ( λ ) , U n ( λ )             ( n = 1 , 2 , )
f = i = 1 , 2 , f i U i ( λ ) , where f i = ( f U i ) = = σ f ( λ ) U i ( λ ) d λ .
a i = ( a U i ) = σ a ( λ ) U i ( λ ) d λ ,
( f a ) = σ f ( λ ) a ( λ ) d λ = i = 1 , 2 , f i a i ,
( f 2 ) = σ f 2 d λ = i = 1 , 2 , f i 2 .
V 1 = y ¯ ,             V 2 = x ¯ ,             V 3 = z ¯ ,             V 4 = F 1 ,             V 5 = F 2 ,            
U i ( λ ) = [ V i ( λ ) - k = 1 , 2 , i - 1 ( V i U k ) U k ( λ ) ] / [ ( V i 2 - k = 1 , 2 , i - 1 ( V i U k ) 2 ] 1 2
Δ = σ ( f - φ ) 2 d λ = ( f 1 - φ 1 ) 2 + ( f 2 - φ 2 ) 2 + ( f 3 - φ 3 ) 2 + f 4 2 + f 5 2 + .
Δ = f 4 2 + f 5 2 + .
q = 1 - Δ / ( f 2 ) .
q = [ f 1 2 + f 2 2 + f 3 2 ] / ( f 2 ) ,
( a f ) = i = 1 , 2 , a i f i .
( a φ ) = a 1 f 1 + a 2 f 2 + a 3 f 3 .
E = a 4 f 4 + a 5 f 5 + a 6 f 6 + .
E 2 = f 4 2 + f 5 2 + = Δ .
E 2 = ( 1 - q ) ( f 2 ) .
E = a 4 f 4 + a 5 f 5 + a 6 f 6 + ,
U 1 = t i 1 W 1 + t i 2 W 2 + t i 3 W 3             ( i = 1 , 2 , 3 )
t i 1 t k 1 + t i 2 t k 2 + t i 3 t k 3 = δ i k .
q = [ ( f U 1 ) 2 + ( f U 2 ) 2 + ( f U 3 ) 2 ] / ( f ) 2 = [ ( f W 1 ) 2 + ( f W 2 ) 2 + ( f W 3 ) 2 ] / ( f ) 2 .
{ U 1 = 0.359 y ¯ , U 2 = 0.575 x ¯ - 0.421 y ¯ , U 3 = - 0.1791 x ¯ + 0.1018 y ¯ + 0.281 z ¯ .
{ U = 0.0016 U 1 + 0.796 U 2 - 0.604 U 3 , V = U 1 , W = - 0.0047 U 1 + 0.602 U 2 + 0.797 U 3 .
q ( λ ) = [ U 1 ( λ ) 2 + U 2 ( λ ) 2 + U 3 ( λ ) 2 ] δ λ .
U 3 , ( U 1 - U 2 ) / 2 , ( U 1 + U 2 ) / 2
0.407 x ¯ - 0.081 y ¯ - 0.041 z ¯ .