Abstract

One of the most important problems in photoelectric colorimetry is the duplication, by physical means, of a given set of color-mixture functions. Several methods are outlined in this paper for the numerical derivation of close approximations to given color-mixture functions using the relative spectral-sensitivity function of a photoelectric receiver and the spectral transmittances of glass filters. The filters are placed either in series or side by side between the source and the receiver. Numerical examples illustrate the methods.

© 1962 Optical Society of America

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References

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  1. A. Dresler, Licht 3, 41 (1933).
  2. G. T. Winch and E. H. Palmer, Trans. Illum. Eng. Soc. (London) 2, 137 (1937).
  3. B. T. Barnes, J. Opt. Soc. Am. 29, 448 (1939).
    [Crossref]
  4. R. S. Hunter, J. Opt. Soc. Am. 32, 509 (1942).
    [Crossref]
  5. H. König, Helv. Phys. Acta 17, 571 (1944).
  6. H. G. W. Harding, J. Sci. Instr. 27, 132 (1950).
    [Crossref]
  7. K. H. Bachmann, Z. Meteorol. 4, 176 (1950).
  8. L. G. Glasser and D. J. Troy, J. Opt. Soc. Am. 42, 652 (1952).
    [Crossref]
  9. I. Nimeroff and S. W. Wilson, J. Research Natl. Bur. Standards 52, 195 (1954); RP 2488.
    [Crossref]
  10. H. G. Frühling and F. Krempel, Farbe 3, 137 (1955).
  11. K. Corduan, Tech. Wiss. Abhandl. Osram-Ges. 7, 314 (1957).
  12. T. Azuma, L. Mori, and I. Niikura, J. Illum. Eng. Inst. Japan 41, 26 (1957).
  13. L. Mori, J. Illum. Eng. Japan,  42, 18 (1958).
  14. G. Geutler, Farbe 7, 153, (1958).
  15. KCS Limited, 20 Spadina Road, Toronto 4, Ontario.
  16. G. Wyszecki, J. Opt. Soc. Am. 50, 992 (1960).
    [Crossref]

1960 (1)

1958 (2)

L. Mori, J. Illum. Eng. Japan,  42, 18 (1958).

G. Geutler, Farbe 7, 153, (1958).

1957 (2)

K. Corduan, Tech. Wiss. Abhandl. Osram-Ges. 7, 314 (1957).

T. Azuma, L. Mori, and I. Niikura, J. Illum. Eng. Inst. Japan 41, 26 (1957).

1955 (1)

H. G. Frühling and F. Krempel, Farbe 3, 137 (1955).

1954 (1)

I. Nimeroff and S. W. Wilson, J. Research Natl. Bur. Standards 52, 195 (1954); RP 2488.
[Crossref]

1952 (1)

1950 (2)

H. G. W. Harding, J. Sci. Instr. 27, 132 (1950).
[Crossref]

K. H. Bachmann, Z. Meteorol. 4, 176 (1950).

1944 (1)

H. König, Helv. Phys. Acta 17, 571 (1944).

1942 (1)

1939 (1)

1937 (1)

G. T. Winch and E. H. Palmer, Trans. Illum. Eng. Soc. (London) 2, 137 (1937).

1933 (1)

A. Dresler, Licht 3, 41 (1933).

Azuma, T.

T. Azuma, L. Mori, and I. Niikura, J. Illum. Eng. Inst. Japan 41, 26 (1957).

Bachmann, K. H.

K. H. Bachmann, Z. Meteorol. 4, 176 (1950).

Barnes, B. T.

Corduan, K.

K. Corduan, Tech. Wiss. Abhandl. Osram-Ges. 7, 314 (1957).

Dresler, A.

A. Dresler, Licht 3, 41 (1933).

Frühling, H. G.

H. G. Frühling and F. Krempel, Farbe 3, 137 (1955).

Geutler, G.

G. Geutler, Farbe 7, 153, (1958).

Glasser, L. G.

Harding, H. G. W.

H. G. W. Harding, J. Sci. Instr. 27, 132 (1950).
[Crossref]

Hunter, R. S.

König, H.

H. König, Helv. Phys. Acta 17, 571 (1944).

Krempel, F.

H. G. Frühling and F. Krempel, Farbe 3, 137 (1955).

Mori, L.

L. Mori, J. Illum. Eng. Japan,  42, 18 (1958).

T. Azuma, L. Mori, and I. Niikura, J. Illum. Eng. Inst. Japan 41, 26 (1957).

Niikura, I.

T. Azuma, L. Mori, and I. Niikura, J. Illum. Eng. Inst. Japan 41, 26 (1957).

Nimeroff, I.

I. Nimeroff and S. W. Wilson, J. Research Natl. Bur. Standards 52, 195 (1954); RP 2488.
[Crossref]

Palmer, E. H.

G. T. Winch and E. H. Palmer, Trans. Illum. Eng. Soc. (London) 2, 137 (1937).

Troy, D. J.

Wilson, S. W.

I. Nimeroff and S. W. Wilson, J. Research Natl. Bur. Standards 52, 195 (1954); RP 2488.
[Crossref]

Winch, G. T.

G. T. Winch and E. H. Palmer, Trans. Illum. Eng. Soc. (London) 2, 137 (1937).

Wyszecki, G.

Farbe (2)

H. G. Frühling and F. Krempel, Farbe 3, 137 (1955).

G. Geutler, Farbe 7, 153, (1958).

Helv. Phys. Acta (1)

H. König, Helv. Phys. Acta 17, 571 (1944).

J. Illum. Eng. Inst. Japan (1)

T. Azuma, L. Mori, and I. Niikura, J. Illum. Eng. Inst. Japan 41, 26 (1957).

J. Illum. Eng. Japan (1)

L. Mori, J. Illum. Eng. Japan,  42, 18 (1958).

J. Opt. Soc. Am. (4)

J. Research Natl. Bur. Standards (1)

I. Nimeroff and S. W. Wilson, J. Research Natl. Bur. Standards 52, 195 (1954); RP 2488.
[Crossref]

J. Sci. Instr. (1)

H. G. W. Harding, J. Sci. Instr. 27, 132 (1950).
[Crossref]

Licht (1)

A. Dresler, Licht 3, 41 (1933).

Tech. Wiss. Abhandl. Osram-Ges. (1)

K. Corduan, Tech. Wiss. Abhandl. Osram-Ges. 7, 314 (1957).

Trans. Illum. Eng. Soc. (London) (1)

G. T. Winch and E. H. Palmer, Trans. Illum. Eng. Soc. (London) 2, 137 (1937).

Z. Meteorol. (1)

K. H. Bachmann, Z. Meteorol. 4, 176 (1950).

Other (1)

KCS Limited, 20 Spadina Road, Toronto 4, Ontario.

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Figures (11)

Fig. 1
Fig. 1

Filter arrangements I and II.

Fig. 2
Fig. 2

1931 CIE color-mixture functions x ¯ λ , y ¯ λ , z ¯ λ.

Fig. 3
Fig. 3

Relative spectral-sensitivity function Sλ of a Gillod-Boutry photocell.

Fig. 4
Fig. 4

Spectral transmittance functions tλ of 8 glass filters used in calculations based on filter arrangement I.

Fig. 5
Fig. 5

Relative spectral transmittances of ideal correction filters Tλ(X1), Tλ(X2) Tλ(Y), Tλ(Z) for a Gillod-Boutry photocell to give x ¯ λ , y ¯ λ , z ¯ λ responses, respectively.

Fig. 6
Fig. 6

Effect of different weighting functions on least squares approximation of Tλ(Y) function. The difference functions numbered 1 to 4 of ΔTλ(Y) refer to weighting functions T λ ( Y ) 1 2, Tλ(Y), Tλ(Y)2, Tλ(Y)4, respectively.

Fig. 7
Fig. 7

Functions ΔTλ(X1,X2), ΔTλ(Y), ΔTλ(Z) which give differences of spectral transmittances between approximating filter combinations and corresponding ideal correction filters (see Table II). The approximations are based on filter arrangement I

Fig. 8
Fig. 8

Spectral transmittance functions tλ of 28 cutoff filters used in calculations based on filter arrangement II.

Fig. 9
Fig. 9

Functions Δ x ¯ λ , Δ y ¯ λ , Δ z ¯ λ, which give differences between normalized CIE color-mixture functions and physical approximations by cutoff filters combined with a photocell. The approximations are based on filter arrangement II.

Fig. 10
Fig. 10

Spectral transmittance functions tλ of 14 glass filter combinations used in calculations based on filter arrangement II.

Fig. 11
Fig. 11

Functions Δ y ¯ λ which give differences of the spectral response between a filter-photocell combination (arrangement II) and y ¯ λ of the standard observer. Solid dots give Δ y ¯ λ with respect to all 14 narrow-band filters shown in Fig. 10. Open circles give Δ y ¯ λ with respect to a subset of the 14-narrow band filters.

Tables (4)

Tables Icon

Table I Manufacturer’s specifications and unit thicknesses of glass filters used in filter arrangement I.

Tables Icon

Table II Results of approximating ideal correction filters Tλ on the basis of filter arrangement I.

Tables Icon

Table III Manufacturer’s specifications and thicknesses of cutoff filters used in filter arrangement II.

Tables Icon

Table IV Selected numerical examples of approximating color-mixture functions on the basis of filter arrangement II.

Equations (19)

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α T λ = S λ * / S λ ,
T λ = i = 1 n t i λ δ i .
λ ( T λ α T λ ) 2 d λ = M = minimum .
δ i D i λ = δ i log 10 ( 1 / t i λ ) , P λ = log 10 ( 1 / T λ ) ,
P λ + A = log 10 ( 1 / T λ ) + A ,
A = log 10 ( 1 / α ) ,
P λ = i = 1 n δ i D i λ .
λ [ P λ ( P λ + A ) ] 2 d λ = M = minimum .
λ T λ [ P λ ( P λ + A ) ] 2 d λ = M ¯ = minimum .
i δ i D i k + A = P k , i = 1 , 2 , , n , k = 1 , 2 , 3 , , n + 1.
λ ( T λ α T λ ) 2 d λ = M 1 = minimum .
λ ( T λ S λ S λ * ) 2 d λ = M 2 = minimum .
λ ( T λ H λ S λ H λ S λ * ) 2 d λ = M 3 = minimum .
T λ = a i t i λ , i = 1 , 2 , , n .
a 1 R 11 + a 2 R 12 + + a n R 1 n = R 1 , a 1 R 21 + a 2 R 22 + + a n R 2 n = R 2 , a 1 R n 1 a 2 R n 2 + + a n R n n = R n ,
R i j = R j i = λ t i λ t j λ H λ 2 S λ 2 d λ
R i = λ t i λ H λ S λ * H λ S λ d λ .
σ = [ ( u ¯ λ u ¯ λ ) 2 / 77 ] 1 2 ,
| Δ r | max = | u ¯ λ u ¯ λ | max