Abstract

We describe procedures for creating efficient spectral representations for color. The representations generalize conventional tristimulus representations, which are based on the peripheral encoding by the human eye. We use low-dimensional linear models to approximate the spectral properties of surfaces and illuminants with respect to a collection of sensing devices. We choose the linear-model basis functions by minimizing the error in approximating sensor responses for collections of surfaces and illuminants. These linear models offer some conceptual simplifications for applications such as printer calibration; they also perform substantially better than principal-components approximations for computer-graphics applications.

© 1992 Optical Society of America

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References

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  1. P. S. Parkkinen, J. Hallikainen, T. Jaaskelainen, “Characteristic spectra of Munsell colors,” J. Opt. Soc. Am. A 6, 318–322 (1989).
    [CrossRef]
  2. J. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychonom. Sci. 1, 369–370 (1964).
  3. L. T. Maloney, “Evaluation of linear models of surface spectral reflectance with small numbers of parameters,” J. Opt. Soc. Am. A 3, 1673–1683 (1986).
    [CrossRef] [PubMed]
  4. D. B. Judd, D. L. MacAdam, G. W. Wyszecki, “Spectral distribution of typical daylight as a function of correlated color temperature,”J. Opt. Soc. Am. 54, 1031 (1964).
    [CrossRef]
  5. L. R. Tucker, “Some mathematical notes on three-mode factor analysis,” Psychometrika 31, 279–311 (1966).
    [CrossRef] [PubMed]
  6. J. R. Magnus, H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometrics (Wiley, New York, 1988).
  7. A. Kapteyn, H. Neudecker, T. Wansbeek, “An approach to n-mode components analysis,” Psychometrika 51, 269–275 (1986).
    [CrossRef]
  8. P. M. Kroonenberg, J. de Leeuw, “Principal component analyses of three-mode data by means of alternating least squares algorithms,” Psychometrika 45, 69–97 (1980).
    [CrossRef]
  9. G. H. Golub, C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983).
  10. M. Tsukada, Y. Ohta, “An approach to color constancy using multiple images,” presented at the Third International Conference on Computer Vision, December 4–7, 1990, Osaka, Japan.
  11. M. D’Zmura, “Color constancy: surface color from changing illumination,” J. Opt. Soc. Am. A 9, 490–493 (1992).
    [CrossRef]
  12. K. Takahama, Y. Nayatani, “New method for generating metameric stimuli of object colors,”J. Opt. Soc. Am. 62, 1516–1520 (1972).
    [CrossRef]
  13. J. B. Cohen, W. E. Kappauf, “Metameric color stimuli, fundamental metamers, and Wyszecki’s metameric blacks: theory, algebra, geometry, application,” Am. J. Psychol. 95, 537–564 (1982).
    [CrossRef] [PubMed]
  14. J. B. Cohen, W. E. Kappauf, “Color mixture and fundamental metamers: theory, algebra, geometry, application,” Am. J. Psychol. 98, 171–259 (1985).
    [CrossRef]
  15. J. B. Cohen, “Color and color mixture: scalar and vector fundamentals,” Color Res. Appl. 13, 4–39 (1988).
    [CrossRef]
  16. H. J. Trussell, “Applications of set theoretic methods to color systems,” Color Res. Appl. 16, 31–41 (1991).
    [CrossRef]
  17. S. A. Burns, J. B. Cohen, E. N. Kuznetsov, “Multiple metamers: preserving color matches under diverse illuminants,” Color Res. Appl. 14, 16–22 (1989).
    [CrossRef]
  18. E. H. Adelson, E. Simoncelli, R. Hingorani, “Orthogonal pyramid transforms for image coding,” in Visual Communications and Image Processing II, T. R. Hsing, ed., Proc. Soc. Photo-Opt. Instrum. Eng.845, 50–57 (1987).
    [CrossRef]
  19. J. Dongarra, J. R. Bunch, C. B. Moler, G. W. Stewart, LINPACK Users Guide (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1978).
  20. C. S. McCamy, H. Marcus, J. G. Davidson, “A color-rendition chart,”J. Appl. Photog. 48, 777–784 (1976).
  21. L. T. Maloney, “Photoreceptor spectral sensitivities and color correction,” in Perceiving, Measuring, and Using Color, M. H. Brill, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1250, 103–110 (1990).
    [CrossRef]
  22. M. S. Drew, B. V. Funt, “Natural metamers,” Computer Vision Graphics Image Process. (to be published).
  23. M. J. Vrhel, H. J. Trussell, “Color correction using principal components,” Color Res. Appl. (to be published).
  24. D. H. Brainard, B. A. Wandell, W. B. Cowan, “Black light: how sensors filter spectral variation of the illuminant,”IEEE Trans. Biomed. Eng. 36, 140–149 (1989).
    [CrossRef] [PubMed]
  25. D. Nickerson, Spectrophotometric Data for a Collection of Munsell Samples (U.S. Department of Agriculture, Washington, D.C., 1957).
  26. C. F. Borges, “Trichromatic approximation method for surface illumination,” J. Opt. Soc. Am. A 8, 1319–1323 (1991).
    [CrossRef]
  27. R. Wallis, “Fast computation of tristimulus values by use of Gaussian quadrature,”J. Opt. Soc. Am. 65, 91–94 (1975).
    [CrossRef]
  28. G. W. Meyer, “Wavelength selection for synthetic image generation,” Comput. Vision Graphics Image Process. 41, 57–79 (1988).
    [CrossRef]

1992 (1)

1991 (2)

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

C. F. Borges, “Trichromatic approximation method for surface illumination,” J. Opt. Soc. Am. A 8, 1319–1323 (1991).
[CrossRef]

1989 (3)

S. A. Burns, J. B. Cohen, E. N. Kuznetsov, “Multiple metamers: preserving color matches under diverse illuminants,” Color Res. Appl. 14, 16–22 (1989).
[CrossRef]

P. S. Parkkinen, J. Hallikainen, T. Jaaskelainen, “Characteristic spectra of Munsell colors,” J. Opt. Soc. Am. A 6, 318–322 (1989).
[CrossRef]

D. H. Brainard, B. A. Wandell, W. B. Cowan, “Black light: how sensors filter spectral variation of the illuminant,”IEEE Trans. Biomed. Eng. 36, 140–149 (1989).
[CrossRef] [PubMed]

1988 (2)

G. W. Meyer, “Wavelength selection for synthetic image generation,” Comput. Vision Graphics Image Process. 41, 57–79 (1988).
[CrossRef]

J. B. Cohen, “Color and color mixture: scalar and vector fundamentals,” Color Res. Appl. 13, 4–39 (1988).
[CrossRef]

1986 (2)

L. T. Maloney, “Evaluation of linear models of surface spectral reflectance with small numbers of parameters,” J. Opt. Soc. Am. A 3, 1673–1683 (1986).
[CrossRef] [PubMed]

A. Kapteyn, H. Neudecker, T. Wansbeek, “An approach to n-mode components analysis,” Psychometrika 51, 269–275 (1986).
[CrossRef]

1985 (1)

J. B. Cohen, W. E. Kappauf, “Color mixture and fundamental metamers: theory, algebra, geometry, application,” Am. J. Psychol. 98, 171–259 (1985).
[CrossRef]

1982 (1)

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

1980 (1)

P. M. Kroonenberg, J. de Leeuw, “Principal component analyses of three-mode data by means of alternating least squares algorithms,” Psychometrika 45, 69–97 (1980).
[CrossRef]

1976 (1)

C. S. McCamy, H. Marcus, J. G. Davidson, “A color-rendition chart,”J. Appl. Photog. 48, 777–784 (1976).

1975 (1)

1972 (1)

1966 (1)

L. R. Tucker, “Some mathematical notes on three-mode factor analysis,” Psychometrika 31, 279–311 (1966).
[CrossRef] [PubMed]

1964 (2)

D. B. Judd, D. L. MacAdam, G. W. Wyszecki, “Spectral distribution of typical daylight as a function of correlated color temperature,”J. Opt. Soc. Am. 54, 1031 (1964).
[CrossRef]

J. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychonom. Sci. 1, 369–370 (1964).

Adelson, E. H.

E. H. Adelson, E. Simoncelli, R. Hingorani, “Orthogonal pyramid transforms for image coding,” in Visual Communications and Image Processing II, T. R. Hsing, ed., Proc. Soc. Photo-Opt. Instrum. Eng.845, 50–57 (1987).
[CrossRef]

Borges, C. F.

Brainard, D. H.

D. H. Brainard, B. A. Wandell, W. B. Cowan, “Black light: how sensors filter spectral variation of the illuminant,”IEEE Trans. Biomed. Eng. 36, 140–149 (1989).
[CrossRef] [PubMed]

Bunch, J. R.

J. Dongarra, J. R. Bunch, C. B. Moler, G. W. Stewart, LINPACK Users Guide (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1978).

Burns, S. A.

S. A. Burns, J. B. Cohen, E. N. Kuznetsov, “Multiple metamers: preserving color matches under diverse illuminants,” Color Res. Appl. 14, 16–22 (1989).
[CrossRef]

Cohen, J.

J. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychonom. Sci. 1, 369–370 (1964).

Cohen, J. B.

S. A. Burns, J. B. Cohen, E. N. Kuznetsov, “Multiple metamers: preserving color matches under diverse illuminants,” Color Res. Appl. 14, 16–22 (1989).
[CrossRef]

J. B. Cohen, “Color and color mixture: scalar and vector fundamentals,” Color Res. Appl. 13, 4–39 (1988).
[CrossRef]

J. B. Cohen, W. E. Kappauf, “Color mixture and fundamental metamers: theory, algebra, geometry, application,” Am. J. Psychol. 98, 171–259 (1985).
[CrossRef]

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

Cowan, W. B.

D. H. Brainard, B. A. Wandell, W. B. Cowan, “Black light: how sensors filter spectral variation of the illuminant,”IEEE Trans. Biomed. Eng. 36, 140–149 (1989).
[CrossRef] [PubMed]

D’Zmura, M.

Davidson, J. G.

C. S. McCamy, H. Marcus, J. G. Davidson, “A color-rendition chart,”J. Appl. Photog. 48, 777–784 (1976).

de Leeuw, J.

P. M. Kroonenberg, J. de Leeuw, “Principal component analyses of three-mode data by means of alternating least squares algorithms,” Psychometrika 45, 69–97 (1980).
[CrossRef]

Dongarra, J.

J. Dongarra, J. R. Bunch, C. B. Moler, G. W. Stewart, LINPACK Users Guide (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1978).

Drew, M. S.

M. S. Drew, B. V. Funt, “Natural metamers,” Computer Vision Graphics Image Process. (to be published).

Funt, B. V.

M. S. Drew, B. V. Funt, “Natural metamers,” Computer Vision Graphics Image Process. (to be published).

Golub, G. H.

G. H. Golub, C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983).

Hallikainen, J.

Hingorani, R.

E. H. Adelson, E. Simoncelli, R. Hingorani, “Orthogonal pyramid transforms for image coding,” in Visual Communications and Image Processing II, T. R. Hsing, ed., Proc. Soc. Photo-Opt. Instrum. Eng.845, 50–57 (1987).
[CrossRef]

Jaaskelainen, T.

Judd, D. B.

Kappauf, W. E.

J. B. Cohen, W. E. Kappauf, “Color mixture and fundamental metamers: theory, algebra, geometry, application,” Am. J. Psychol. 98, 171–259 (1985).
[CrossRef]

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

Kapteyn, A.

A. Kapteyn, H. Neudecker, T. Wansbeek, “An approach to n-mode components analysis,” Psychometrika 51, 269–275 (1986).
[CrossRef]

Kroonenberg, P. M.

P. M. Kroonenberg, J. de Leeuw, “Principal component analyses of three-mode data by means of alternating least squares algorithms,” Psychometrika 45, 69–97 (1980).
[CrossRef]

Kuznetsov, E. N.

S. A. Burns, J. B. Cohen, E. N. Kuznetsov, “Multiple metamers: preserving color matches under diverse illuminants,” Color Res. Appl. 14, 16–22 (1989).
[CrossRef]

MacAdam, D. L.

Magnus, J. R.

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

Maloney, L. T.

L. T. Maloney, “Evaluation of linear models of surface spectral reflectance with small numbers of parameters,” J. Opt. Soc. Am. A 3, 1673–1683 (1986).
[CrossRef] [PubMed]

L. T. Maloney, “Photoreceptor spectral sensitivities and color correction,” in Perceiving, Measuring, and Using Color, M. H. Brill, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1250, 103–110 (1990).
[CrossRef]

Marcus, H.

C. S. McCamy, H. Marcus, J. G. Davidson, “A color-rendition chart,”J. Appl. Photog. 48, 777–784 (1976).

McCamy, C. S.

C. S. McCamy, H. Marcus, J. G. Davidson, “A color-rendition chart,”J. Appl. Photog. 48, 777–784 (1976).

Meyer, G. W.

G. W. Meyer, “Wavelength selection for synthetic image generation,” Comput. Vision Graphics Image Process. 41, 57–79 (1988).
[CrossRef]

Moler, C. B.

J. Dongarra, J. R. Bunch, C. B. Moler, G. W. Stewart, LINPACK Users Guide (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1978).

Nayatani, Y.

Neudecker, H.

A. Kapteyn, H. Neudecker, T. Wansbeek, “An approach to n-mode components analysis,” Psychometrika 51, 269–275 (1986).
[CrossRef]

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

Nickerson, D.

D. Nickerson, Spectrophotometric Data for a Collection of Munsell Samples (U.S. Department of Agriculture, Washington, D.C., 1957).

Ohta, Y.

M. Tsukada, Y. Ohta, “An approach to color constancy using multiple images,” presented at the Third International Conference on Computer Vision, December 4–7, 1990, Osaka, Japan.

Parkkinen, P. S.

Simoncelli, E.

E. H. Adelson, E. Simoncelli, R. Hingorani, “Orthogonal pyramid transforms for image coding,” in Visual Communications and Image Processing II, T. R. Hsing, ed., Proc. Soc. Photo-Opt. Instrum. Eng.845, 50–57 (1987).
[CrossRef]

Stewart, G. W.

J. Dongarra, J. R. Bunch, C. B. Moler, G. W. Stewart, LINPACK Users Guide (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1978).

Takahama, K.

Trussell, H. J.

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

M. J. Vrhel, H. J. Trussell, “Color correction using principal components,” Color Res. Appl. (to be published).

Tsukada, M.

M. Tsukada, Y. Ohta, “An approach to color constancy using multiple images,” presented at the Third International Conference on Computer Vision, December 4–7, 1990, Osaka, Japan.

Tucker, L. R.

L. R. Tucker, “Some mathematical notes on three-mode factor analysis,” Psychometrika 31, 279–311 (1966).
[CrossRef] [PubMed]

van Loan, C. F.

G. H. Golub, C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983).

Vrhel, M. J.

M. J. Vrhel, H. J. Trussell, “Color correction using principal components,” Color Res. Appl. (to be published).

Wallis, R.

Wandell, B. A.

D. H. Brainard, B. A. Wandell, W. B. Cowan, “Black light: how sensors filter spectral variation of the illuminant,”IEEE Trans. Biomed. Eng. 36, 140–149 (1989).
[CrossRef] [PubMed]

Wansbeek, T.

A. Kapteyn, H. Neudecker, T. Wansbeek, “An approach to n-mode components analysis,” Psychometrika 51, 269–275 (1986).
[CrossRef]

Wyszecki, G. W.

Am. J. Psychol. (2)

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

J. B. Cohen, W. E. Kappauf, “Color mixture and fundamental metamers: theory, algebra, geometry, application,” Am. J. Psychol. 98, 171–259 (1985).
[CrossRef]

Color Res. Appl. (3)

J. B. Cohen, “Color and color mixture: scalar and vector fundamentals,” Color Res. Appl. 13, 4–39 (1988).
[CrossRef]

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

S. A. Burns, J. B. Cohen, E. N. Kuznetsov, “Multiple metamers: preserving color matches under diverse illuminants,” Color Res. Appl. 14, 16–22 (1989).
[CrossRef]

Comput. Vision Graphics Image Process. (1)

G. W. Meyer, “Wavelength selection for synthetic image generation,” Comput. Vision Graphics Image Process. 41, 57–79 (1988).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

D. H. Brainard, B. A. Wandell, W. B. Cowan, “Black light: how sensors filter spectral variation of the illuminant,”IEEE Trans. Biomed. Eng. 36, 140–149 (1989).
[CrossRef] [PubMed]

J. Appl. Photog. (1)

C. S. McCamy, H. Marcus, J. G. Davidson, “A color-rendition chart,”J. Appl. Photog. 48, 777–784 (1976).

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (4)

Psychometrika (3)

L. R. Tucker, “Some mathematical notes on three-mode factor analysis,” Psychometrika 31, 279–311 (1966).
[CrossRef] [PubMed]

A. Kapteyn, H. Neudecker, T. Wansbeek, “An approach to n-mode components analysis,” Psychometrika 51, 269–275 (1986).
[CrossRef]

P. M. Kroonenberg, J. de Leeuw, “Principal component analyses of three-mode data by means of alternating least squares algorithms,” Psychometrika 45, 69–97 (1980).
[CrossRef]

Psychonom. Sci. (1)

J. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychonom. Sci. 1, 369–370 (1964).

Other (9)

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

G. H. Golub, C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983).

M. Tsukada, Y. Ohta, “An approach to color constancy using multiple images,” presented at the Third International Conference on Computer Vision, December 4–7, 1990, Osaka, Japan.

L. T. Maloney, “Photoreceptor spectral sensitivities and color correction,” in Perceiving, Measuring, and Using Color, M. H. Brill, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1250, 103–110 (1990).
[CrossRef]

M. S. Drew, B. V. Funt, “Natural metamers,” Computer Vision Graphics Image Process. (to be published).

M. J. Vrhel, H. J. Trussell, “Color correction using principal components,” Color Res. Appl. (to be published).

E. H. Adelson, E. Simoncelli, R. Hingorani, “Orthogonal pyramid transforms for image coding,” in Visual Communications and Image Processing II, T. R. Hsing, ed., Proc. Soc. Photo-Opt. Instrum. Eng.845, 50–57 (1987).
[CrossRef]

J. Dongarra, J. R. Bunch, C. B. Moler, G. W. Stewart, LINPACK Users Guide (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1978).

D. Nickerson, Spectrophotometric Data for a Collection of Munsell Samples (U.S. Department of Agriculture, Washington, D.C., 1957).

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Figures (10)

Fig. 1
Fig. 1

Device sensor responses. We group the surface-reflectance vectors s in the columns of a matrix S. Similarly, we group the sensor-response vectors r in the columns of a matrix R. (a) The sensor responses are determined by the product of the surface-reflectance function (columns of the rightmost matrix) and a diagonal matrix containing the illuminant spectral power distribution and a matrix whose rows contain the sensor responsivities. (b) We group the sensor matrix and the illuminant matrix to define a surface-transfer matrix.

Fig. 2
Fig. 2

Linear models as projection operators. These models define a mapping from the original surface reflectance to an approximation that falls in a subspace. We can conceive of the projection as a linear sampling, Ls, followed by a basis reconstruction, Lb. The product, PD = LbLs, is a projection, i.e., PD = PD2.

Fig. 3
Fig. 3

Pooled sensor responses grouped across devices and surfaces. We group the sensor responsivities into a single matrix, T. We group the surface-reflectance functions in the collection into the columns of a matrix, S. The sensor responses are equal to TS. We analyze the sensor responses to derive the one-mode representation of the surface-reflectance functions.

Fig. 4
Fig. 4

Comparison of the sampling functions for (a) the principal-components linear model and (b) the one-mode linear model that were used to describe the Macbeth color-checker surfaces. The principal-components representation is independent of the sensors. The one-mode representation is chosen with respect to the sensors described in the text.

Fig. 5
Fig. 5

Root mean-squared error when the sensor responses are predicted by using the principal-components model (dark bars) and the one-mode model (light bars) for different linear model dimensions: (a) errors for the scanner data, (b) errors for the XYZ values.

Fig. 6
Fig. 6

Geometric interpretation of the one-mode linear model and the principal components. The surfaces in the collection are indicated by their end points; the sensor vector is indicated by a line. The sensor response to a surface, s, is found by dropping a perpendicular from the surface vector end point to the sensor line. The first principal component passes through the data points, minimizing the distance between the vector and the data. The principal-component approximation introduces error in the prediction of the sensor response. The one-mode component is the same as the sensor line. In this example, the one-mode approximation is ŝ, a vector on the sensor line. The vector s, which is invisible to the sensor, is perpendicular to the sensor line and is shown added to the vector ŝ.

Fig. 7
Fig. 7

This matrix tableau illustrates how to convert the data from the format used for the one-mode analysis of surfaces (left) to the one-mode analysis of illuminants (right). The operation is essentially a transposition, but it is applied to the rgb vectors of data rather than to the individual elements.

Fig. 8
Fig. 8

This figure contains the spectral-power distributions of the illuminants used in our calculation. There are five black-body radiators (3K, 4K, 5K, 6K, 9K) and three CIE standard illuminants (a, b, and c). The vectors representing the illuminants were normalized to unit length.

Fig. 9
Fig. 9

(a) Sampling functions for three-dimensional surface and illuminant models, respectively, with the use of principal-components methods. (b) Sampling functions for the surface and illuminant models, respectively, with the use of two-mode methods. The linear models were built for a collection of 462 Munsell chips. The collection of illuminants is described in the caption for Fig. 8.

Fig. 10
Fig. 10

Comparison of ΔEab values for different dimensions of the two-mode and the principal-components linear models. The sensor data are the XYZ values of Munsell chips rendered under the illuminants plotted in Fig. 8. The filled bars are the principal-component errrors, and the open bars are the two-mode errors. The height of the bars defines the mean error from the 462 × 8 = 3696 illuminant–surface pairs. The horizontal lines define the twenty-fifth-, fiftieth- (mode), and seventy-fifth-percentile errors.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

S ( λ ) i = 1 i = d σ i S i ( λ ) .
S [ S ( λ ) - i = 1 i = d σ i S i ( λ ) ] 2 d λ .
r = T E s .
0 = T E s .
s ^ = T E t w .
r = T E s , s ^ = T E t ( T E T E t ) - 1 r .
T E s = r = T E ( s ^ + s ) = T E s ^ .
r = T E s ^ = ( T E T E t ) w .
s L b w .
w = L b + s ,
E p c = S - P D S ,
E o m = TS - T P D S .
TS T S ^ = ( T L b ) ( L s S ) ,
R ( U d 0 ) [ D d 0 0 0 ] [ V d t 0 ] = ( U d D d ) ( V d t ) .
L S S = W .
R s = T E S .

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