Abstract

Accurate scanning of a color image, which is absolutely essential for good color reproduction, can ensure that all relevant information about the color stimulus of a signal is obtained. The set of scanning filters is hence an important component of a color reproduction system. In this paper we introduce a measure of the goodness of a set of color-scanning filters. This measure relates the space spanned by the scanning filters to the human visual subspace. The q factor of a single color-scanning filter is shown to be a particular case of the measure. Experimental results are presented to justify the appropriateness of the measure.

© 1993 Optical Society of America

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References

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  1. H. J. Trussell, “Applications of set theoretic methods to color systems,” Color Res. Appl. 16, 31–41 (1991).
    [CrossRef]
  2. J. B. Cohen, W. E. Kappauf, “Metameric color stimuli, fundamental metamers and Wyszecki’s metameric blacks,” Am. J. Psychol. 95, 537–564 (1982).
    [CrossRef] [PubMed]
  3. B. K. P. Horn, “Exact reproduction of color images,” Comput. Vision Graphics Image Process. 26, 135–167 (1984).
    [CrossRef]
  4. B. A. Wandell, “The synthesis and analysis of color images,”IEEE Trans. Anal. Mach. Intell. 9, 2–13 (1987).
    [CrossRef]
  5. M. J. Vrhel, H. J. Trussell, “Color correction using principal components,” Color Res. Appl. 17, 328–338 (1992).
    [CrossRef]
  6. W. A. Shapiro, “Generalization of tristimulus coordinates,”J. Opt. Soc. Am. 56, 795–802 (1966).
    [CrossRef]
  7. H. E. J. Neugebauer, “Quality factor for filters whose spectral transmittances are different from color mixture curves, and Its application to color photography,”J. Opt. Soc. Am. 46, 821–824 (1956).
    [CrossRef]
  8. G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1989).
  9. T. Kato, A Short Introduction to Perturbation Theory for Linear Operators (Springer-Verlag, New York, 1982), pp. 20–21.
  10. S. R. Searle, Matrix Algebra Useful for Statistics (Wiley, New York, 1982), p. 63.
  11. J. R. Magnus, H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometrics (Wiley, New York, 1988) p. 211.

1992 (1)

M. J. Vrhel, H. J. Trussell, “Color correction using principal components,” Color Res. Appl. 17, 328–338 (1992).
[CrossRef]

1991 (1)

H. J. Trussell, “Applications of set theoretic methods to color systems,” Color Res. Appl. 16, 31–41 (1991).
[CrossRef]

1987 (1)

B. A. Wandell, “The synthesis and analysis of color images,”IEEE Trans. Anal. Mach. Intell. 9, 2–13 (1987).
[CrossRef]

1984 (1)

B. K. P. Horn, “Exact reproduction of color images,” Comput. Vision Graphics Image Process. 26, 135–167 (1984).
[CrossRef]

1982 (1)

J. B. Cohen, W. E. Kappauf, “Metameric color stimuli, fundamental metamers and Wyszecki’s metameric blacks,” Am. J. Psychol. 95, 537–564 (1982).
[CrossRef] [PubMed]

1966 (1)

1956 (1)

Cohen, J. B.

J. B. Cohen, W. E. Kappauf, “Metameric color stimuli, fundamental metamers and Wyszecki’s metameric blacks,” Am. J. Psychol. 95, 537–564 (1982).
[CrossRef] [PubMed]

Golub, G. H.

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1989).

Horn, B. K. P.

B. K. P. Horn, “Exact reproduction of color images,” Comput. Vision Graphics Image Process. 26, 135–167 (1984).
[CrossRef]

Kappauf, W. E.

J. B. Cohen, W. E. Kappauf, “Metameric color stimuli, fundamental metamers and Wyszecki’s metameric blacks,” Am. J. Psychol. 95, 537–564 (1982).
[CrossRef] [PubMed]

Kato, T.

T. Kato, A Short Introduction to Perturbation Theory for Linear Operators (Springer-Verlag, New York, 1982), pp. 20–21.

Magnus, J. R.

J. R. Magnus, H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometrics (Wiley, New York, 1988) p. 211.

Neudecker, H.

J. R. Magnus, H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometrics (Wiley, New York, 1988) p. 211.

Neugebauer, H. E. J.

Searle, S. R.

S. R. Searle, Matrix Algebra Useful for Statistics (Wiley, New York, 1982), p. 63.

Shapiro, W. A.

Trussell, H. J.

M. J. Vrhel, H. J. Trussell, “Color correction using principal components,” Color Res. Appl. 17, 328–338 (1992).
[CrossRef]

H. J. Trussell, “Applications of set theoretic methods to color systems,” Color Res. Appl. 16, 31–41 (1991).
[CrossRef]

Van Loan, C. F.

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1989).

Vrhel, M. J.

M. J. Vrhel, H. J. Trussell, “Color correction using principal components,” Color Res. Appl. 17, 328–338 (1992).
[CrossRef]

Wandell, B. A.

B. A. Wandell, “The synthesis and analysis of color images,”IEEE Trans. Anal. Mach. Intell. 9, 2–13 (1987).
[CrossRef]

Am. J. Psychol. (1)

J. B. Cohen, W. E. Kappauf, “Metameric color stimuli, fundamental metamers and Wyszecki’s metameric blacks,” Am. J. Psychol. 95, 537–564 (1982).
[CrossRef] [PubMed]

Color Res. Appl. (2)

M. J. Vrhel, H. J. Trussell, “Color correction using principal components,” Color Res. Appl. 17, 328–338 (1992).
[CrossRef]

H. J. Trussell, “Applications of set theoretic methods to color systems,” Color Res. Appl. 16, 31–41 (1991).
[CrossRef]

Comput. Vision Graphics Image Process. (1)

B. K. P. Horn, “Exact reproduction of color images,” Comput. Vision Graphics Image Process. 26, 135–167 (1984).
[CrossRef]

IEEE Trans. Anal. Mach. Intell. (1)

B. A. Wandell, “The synthesis and analysis of color images,”IEEE Trans. Anal. Mach. Intell. 9, 2–13 (1987).
[CrossRef]

J. Opt. Soc. Am. (2)

Other (4)

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1989).

T. Kato, A Short Introduction to Perturbation Theory for Linear Operators (Springer-Verlag, New York, 1982), pp. 20–21.

S. R. Searle, Matrix Algebra Useful for Statistics (Wiley, New York, 1982), p. 63.

J. R. Magnus, H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometrics (Wiley, New York, 1988) p. 211.

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Figures (7)

Fig. 1
Fig. 1

R(M2) is closer to the HVSS than is R(M1).

Fig. 2
Fig. 2

Measure is the sum of q factors of orthogonal scanning filters.

Fig. 3
Fig. 3

Measure is not the sum of q factors of nonorthogonal scanning filters.

Fig. 4
Fig. 4

Filter 1.

Fig. 5
Fig. 5

Filter 2.

Fig. 6
Fig. 6

Filter 3.

Fig. 7
Fig. 7

Filter 4.

Tables (5)

Tables Icon

Table 1 Munsell-Chip Set

Tables Icon

Table 2 Color-Copier Data Set

Tables Icon

Table 3 Lithographic-Printer Data Set

Tables Icon

Table 4 Thermal-Printer Data Set

Tables Icon

Table 5 Inkjet-Printer Data Set

Equations (58)

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P V f = A L ( A L T A L ) - 1 A L T f
P V f = P v g A L T f = A L T g .
q ( m ) = P V ( m ) 2 m 2 ,
R ( M ) R ( A L ) ,
v 1 = [ 1 , 0 , 0 , 0 , , 0 ] T , v 2 = [ 0 , 1 , 0 , 0 , , 0 ] T , v 3 = [ 0 , 0 , 1 , 0 , , 0 ] T
m 1 = [ 1 , 0 , 0 , 0 , , 0 ] T , m 2 = [ 0 , 0 , 1 , 0 , , 0 ] T , m 3 = [ 0 , 0 , 0 , 1 , , 0 ] T
m 1 = [ 1 , 0 , 0 , 0 , , 0 ] T , m 2 = [ 0 , 0.95 , 0 , 0.05 , , 0 ] T m 3 = [ 0 , 0 , 1.0 , 0 , , 0 ] T
N T N = I .
O T O = I .
P V ( f ) = N N T f .
P M ( f ) = O O T f .
P V P M ( f ) = N N T O O T f .
e = N N T ( I - O O T ) f .
e = 0 R ( M ) R ( V ) ,
E [ e 2 ] = Trace [ N T ( I - O O T ) R ( I - O O T ) N ] .
0 E [ e 2 ] Trace R .
Trace R = γ .
{ f f i = u             if i = 2 ,             f i = 0 else } ,
E [ e 2 ] = σ 2 { i = 1 α [ 1 - λ i 2 ( O T N ) ] } ,
λ i 2 ( O T N ) = cos 2 ( θ i )             i = 1 , α ,
1 - λ i 2 ( O T N ) 0             i = 1 , α ,
0 λ i 2 ( O T N ) 1             i = 1 , α ,
0 E [ e 2 ] = σ 2 [ α - i = 1 α λ i 2 ( O T N ) ] σ 2 α .
α = i = 1 α λ i 2 ( O T N )
λ i ( O T N ) = 1             i = 1 , α .
ν ( V , M ) = [ i = 1 α λ i 2 ( O T N ) ] / α .
ν ( V , M ) = 1.
ν ( V , M ) = [ i = 1 β q ( o i ) ] / α .
ϕ ( V , M ) = i = 1 2 q ( m i ) .
cos ( θ k ) = max u R ( V ) max v R ( M ) u T v = u k T v k ,
u = v = 1 , u T u i = 0             i = 1 , , k - 1 , v T v i = 0             i = 1 , , k - 1.
cos ( θ k ) = λ k ( O T N ) .
ν ( V , M ) = [ i = 1 α cos 2 ( θ i ) ] / α .
h = ( A T RM ( M T RM ) - 1 ) ( M T f ) ,
P ^ V f = [ A ( A T A ) - 1 ] h ,
P ^ V f = [ A ( A T A ) - 1 A T RM ( M T RM ) - 1 ] ( M T f ) .
P ^ V f = P V P M f = [ A ( A T A ) - 1 A T M ( M T M ) - 1 ] ( M T f ) ,
A T P M f = [ A T M ( M T M ) - 1 ] ( M T f ) ,
R ( M ) R ( V )
P V P M f = P V f , e = P V f - P V P M f = 0 .
R ( O ) R ( N ) ,
Null ( N ) Null ( O ) .
O T f = 0 .
f T N N T f = ( N T f ) T N T f = 0 ,
N T f = 0
e 2 = [ N N T ( I - O O T ) f ] T N N T ( I - O O T ) f .
e 2 = f T ( I - O O T ) N N T ( I - O O T ) f ,
e 2 = Trace [ N T ( I - O O T ) f f T ( I - O O T ) T ( N T ) T ] .
E [ e 2 ] = σ 2 Trace [ N T ( I - O O T ) ( I - O O T ) N ] .
E [ e 2 ] = σ 2 Trace ( I α - N T O O T N ) .
E [ e 2 ] = σ 2 { i = 1 α [ 1 - λ i 2 ( O T N ) ] } ,
0 Trace X T QX Trace Q
Trace X T QX = 0.
0 E [ e 2 ] Trace ( I - O O T ) R ( I - O O T ) ,
i = 1 β q ( o i ) = i = 1 β P V ( o i ) 2 = Trace { [ P V ( O ) ] T P V ( O ) } .
Trace [ P V ( O ) ] T P V ( O ) = Trace ( O T N N T N N T O ) = Trace ( O T N N T O ) .
i = 1 β q ( o i ) = Trace ( O T N ) ( O T N ) T = Trace ( N T O ) ( N T O ) T ,
i = 1 β q ( o i ) = i = 1 α λ i 2 ( O T N ) .

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