Abstract

A computer algorithm has been developed by means of a minimax approximation to approximate an absorption band by linear combination of several dyes. The minimax approximation always gives feasible solutions and smaller errors than a least-squares method. Suitable computer programs have been developed based on the linear programming theory, and same practical approximations have been demonstrated for spectrally matching colors of human skin, blue sky, and nonselective gray of 50% luminous reflectance with twelve component dyes.

© 1972 Optical Society of America

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References

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  1. R. W. G. Hunt, J. Photogr. Sci. 18, 205 (1970).
  2. R. H. Park, E. I. Stearns, J. Opt. Soc. Am. 34, 112 (1944).
    [CrossRef]
  3. F. W. Billmeyer, J. K. Beasley, J. A. Sheldon, J. Opt. Soc. Am. 50, 70 (1960).
    [CrossRef]
  4. J. V. Alderson, E. Atherton, A. N. Derbyshire, J. Soc. Dyers Colourist 77, 657 (1961).
    [CrossRef]
  5. E. Allen, J. Opt. Soc. Am. 56, 1256 (1966).
    [CrossRef]
  6. T. Bridgeman, Die Farbe 17, 114 (1968).
  7. E. I. Stearns, The Practice of Absorption Spectrophotometry (Wiley-Interscience, New York, 1969).
  8. N. Ohta, Appl. Opt. 10, 2183 (1971).
    [CrossRef] [PubMed]
  9. N. Ohta, J. Opt. Soc. Am. 62, 129 (1972).
    [CrossRef]
  10. N. Ohta, Photogr. Sci. Eng. 16, 136 (1972).
  11. N. Ohta, Die Farbe21, in press (1972).
  12. N. Ohta, Photogr. Sci. Eng.16 (4), in press (1972).
  13. R. H. Hamming, Numerical Methods for Scientists and Engineers (McGraw-Hill, New York, 1962).
  14. G. S. G. Beveridge, R. S. Schechter, Optimization: Theory and Practice (McGraw-Hill, New York, 1970).
  15. A. Ohga of these Research Laboratories, private communication.

1972 (2)

N. Ohta, J. Opt. Soc. Am. 62, 129 (1972).
[CrossRef]

N. Ohta, Photogr. Sci. Eng. 16, 136 (1972).

1971 (1)

1970 (1)

R. W. G. Hunt, J. Photogr. Sci. 18, 205 (1970).

1968 (1)

T. Bridgeman, Die Farbe 17, 114 (1968).

1966 (1)

1961 (1)

J. V. Alderson, E. Atherton, A. N. Derbyshire, J. Soc. Dyers Colourist 77, 657 (1961).
[CrossRef]

1960 (1)

1944 (1)

Alderson, J. V.

J. V. Alderson, E. Atherton, A. N. Derbyshire, J. Soc. Dyers Colourist 77, 657 (1961).
[CrossRef]

Allen, E.

Atherton, E.

J. V. Alderson, E. Atherton, A. N. Derbyshire, J. Soc. Dyers Colourist 77, 657 (1961).
[CrossRef]

Beasley, J. K.

Beveridge, G. S. G.

G. S. G. Beveridge, R. S. Schechter, Optimization: Theory and Practice (McGraw-Hill, New York, 1970).

Billmeyer, F. W.

Bridgeman, T.

T. Bridgeman, Die Farbe 17, 114 (1968).

Derbyshire, A. N.

J. V. Alderson, E. Atherton, A. N. Derbyshire, J. Soc. Dyers Colourist 77, 657 (1961).
[CrossRef]

Hamming, R. H.

R. H. Hamming, Numerical Methods for Scientists and Engineers (McGraw-Hill, New York, 1962).

Hunt, R. W. G.

R. W. G. Hunt, J. Photogr. Sci. 18, 205 (1970).

Ohga, A.

A. Ohga of these Research Laboratories, private communication.

Ohta, N.

N. Ohta, Photogr. Sci. Eng. 16, 136 (1972).

N. Ohta, J. Opt. Soc. Am. 62, 129 (1972).
[CrossRef]

N. Ohta, Appl. Opt. 10, 2183 (1971).
[CrossRef] [PubMed]

N. Ohta, Die Farbe21, in press (1972).

N. Ohta, Photogr. Sci. Eng.16 (4), in press (1972).

Park, R. H.

Schechter, R. S.

G. S. G. Beveridge, R. S. Schechter, Optimization: Theory and Practice (McGraw-Hill, New York, 1970).

Sheldon, J. A.

Stearns, E. I.

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

E. I. Stearns, The Practice of Absorption Spectrophotometry (Wiley-Interscience, New York, 1969).

Appl. Opt. (1)

Die Farbe (1)

T. Bridgeman, Die Farbe 17, 114 (1968).

J. Opt. Soc. Am. (4)

J. Photogr. Sci. (1)

R. W. G. Hunt, J. Photogr. Sci. 18, 205 (1970).

J. Soc. Dyers Colourist (1)

J. V. Alderson, E. Atherton, A. N. Derbyshire, J. Soc. Dyers Colourist 77, 657 (1961).
[CrossRef]

Photogr. Sci. Eng. (1)

N. Ohta, Photogr. Sci. Eng. 16, 136 (1972).

Other (6)

N. Ohta, Die Farbe21, in press (1972).

N. Ohta, Photogr. Sci. Eng.16 (4), in press (1972).

R. H. Hamming, Numerical Methods for Scientists and Engineers (McGraw-Hill, New York, 1962).

G. S. G. Beveridge, R. S. Schechter, Optimization: Theory and Practice (McGraw-Hill, New York, 1970).

A. Ohga of these Research Laboratories, private communication.

E. I. Stearns, The Practice of Absorption Spectrophotometry (Wiley-Interscience, New York, 1969).

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Figures (4)

Fig. 1
Fig. 1

Spectral density distributions of twelve component dyes.

Fig. 2
Fig. 2

Spectral color matching of human skin (solid line is observed and dotted line is calculated spectral reflectance).

Fig. 3
Fig. 3

Spectral color matching of blue sky (solid line is observed and dotted line is calculated spectral reflectance).

Fig. 4
Fig. 4

Spectral color matching of nonselective gray of 50% luminous reflectance (solid line is aimed and dotted line is calculated spectral reflectance).

Tables (1)

Tables Icon

Table I Calculated Dye Amounts in Peak Density by Minimax Approximation (MA) and Least-Squares Method (LS)

Equations (9)

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D ( λ i ) = j = 1 m c j · d j ( λ i )             ( i = 1 , n ) ,
- E max F ( λ i ) - j = 1 m c j · d j ( λ i ) E max             ( i = 1 , n ) .
j = 1 m c j · d j ( λ i ) + E max F ( λ i ) ,
- j = 1 m c j · d j ( λ i ) + E max - F ( λ i )             ( i = 1 , n ) .
Z = j = 1 m 0 · c j + E max = E max .
i = 1 n x j · d i ( λ j ) - j = n + 1 2 n x j · d i ( λ j - n ) 0             ( i = 1 , m ) ,
i = 1 2 n x i 1 .
Z = i = 1 n x i · F ( λ i ) - i = n + 1 2 n x i · F ( λ i - n ) .
N = i = 1 12 C 12 i = 2 12 - 1 = 4095 .

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