Abstract

We describe a class of unsupervised systems that extract features from databases of reflectance spectra that sample color space in a way that reflects the properties of human color perception. The systems find the internal weight coefficients by optimizing an energy function. We describe several energy functions based on second- and fourth-order statistical moments of the computed output values. We also investigate the effects of imposing boundary conditions on the filter coefficients and the performance of the resulting systems for the databases with the reflectance spectra. The experiments show that the weight matrix for one of the systems is very similar to the eigenvector system, whereas the second type of system tries to rotate the eigenvector system in such a way that the resulting filters partition the spectrum into different bands. We also show how the system can be forced to use weight vectors with positive coefficients. Systems consisting of positive weight vectors are then approximated with Gaussian quadrature methods. In the experimental part of the paper we investigate the properties of three databases consisting of reflectance spectra. We compare the statistical structure of the different databases and investigate how these systems can be used to explore the structure of the space of reflectance spectra.

© 1996 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. F. W. Nuboer, “A comparative view on colour vision,” Neth. J. Zoo. 36, 344–380 (1986).
    [CrossRef]
  2. G. Wyszecki, W. S. Stiles, Color Science, 2nd ed. (Wiley, London, 1982).
  3. J. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychon. Sci. 1, 369–370 (1964).
  4. J. P. S. Parkkinen, J. Hallikainen, T. Jaaskelainen, “Characteristic spectra of Munsell colors,” J. Opt. Soc. Am. A 6, 318–322 (1989).
    [CrossRef]
  5. L. T. Maloney, B. A. Wandell, “Color constancy: a method for recovering surface spectral reflectance,” J. Opt. Soc. Am. A 3, 29–33 (1986).
    [CrossRef] [PubMed]
  6. M. D’Zmura, G. Iverson, “Color constancy. III. General linear recovery of spectral descriptions for lights and surfaces,” J. Opt. Soc. Am. A 11, 2389–2400 (1994).
    [CrossRef]
  7. T. Jaaskelainen, J. Parkkinen, S. Toyooka, “Vector subspace model for color representation,” J. Opt. Soc. Am. A 7, 725–730 (1990).
    [CrossRef]
  8. S. Usui, S. Nakauchi, M. Nakano, “Reconstruction of Munsell color space by a five-layer neural network,” J. Opt. Soc. Am. A 9, 516–520 (1992).
    [CrossRef]
  9. S. Usui, S. Nakauchi, “Computational color vision models by neural networks,” in Computational Intelligence, Imitating Life, T. M. Zurada, R. J. Marks, C. J. Robinson, eds. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1994), pp. 252–263.
  10. F. Attneave, “Some informational aspects of visual perception,” Psychol. Rev. 61, 183–193 (1954).
    [CrossRef] [PubMed]
  11. H. B. Barlow, “The coding of sensory messages,” in Current Problems in Animal Behavior, W. H. Thorpe, O. L. Zangwill, eds. (Cambridge. U. Press, Cambridge, 1961), pp. 331–360.
  12. J. J. Atick, “Could information theory provide an ecological theory of sensory processing?” Network 3, 213–251 (1992).
    [CrossRef]
  13. D. J. Field, “What is the goal of sensory coding,” Neural Comput. 6, 559–601 (1994).
    [CrossRef]
  14. J. G. Taylor, M. D. Plumbley, “Information theory and neural networks,” in Mathematical Applications to Neural Networks, J. G. Taylor, ed. (Elsevier, New York, 1993), pp. 307–340.
  15. K. Mantere, J. Parkkinen, M. Mäntyjärvi, T. Jaaskelainen, “An eigenvector interpretation of the Farnsworth Munsell 100-hue test,” J. Opt. Soc. Am. A 12, 2237–2243 (1995).
    [CrossRef]
  16. T. Jaaskelainen, S. Tyooka, S. Izawa, H. Kadono, “Color classification by vector subspace method and its implementation using liquid crystal spatial light modulator,” Opt. Commun. 89, 23–29 (1992).
    [CrossRef]
  17. N. Hayasaka, S. Toyooka, T. Jaaskelainen, “Iterative feedback method to make a spatial filter on a liquid crystal spatial light modulator for 2D spectroscopic pattern recognition,” Opt. Commun. 119, 643–651 (1995).
    [CrossRef]
  18. Munsell Book of Color, Matte Finish Collection (Munsell Color, Baltimore, 1976).
  19. B. Flury, Common Principal Components and Related Multivariate Models (Wiley, New York, 1988).
  20. R. Lenz, M. Österberg, “Computing the Karhunen–Loéve expansion with a parallel, unsupervised filter system,” Neural Comput. 4, 382–392 (1992).
    [CrossRef]
  21. R. Lenz, M. Österberg, “A new method for unsupervised linear feature extraction using fourth order moments,” Pattern Recognition Lett. 13, 827–836 (1992).
    [CrossRef]
  22. M. Österberg, “Quality functions for parallel selective principal component analysis,” Ph.D. dissertation (Linköping University, Linkoping, Sweden, 1994).
  23. M. Österberg, R. Lenz, “Unsupervised parallel feature extraction from first principles,” in Advances in Neural Information Processing Systems 6, J. D. Cowan, G. Tesauro, J. Alspector, eds. (Morgan, Kaufmann, San Francisco, Calif., 1994), pp. 136–144.
  24. J. A. Cadzow, X. Li, “Blind deconvolution,” Digital Signal Process. 5, 3–20 (1995).
    [CrossRef]
  25. C. F. Borges, “Numerical determination of tristimulus values,” J. Opt. Soc. Am. A 11, 3152–3161 (1994).
    [CrossRef]
  26. J. Stoer, R. Bulirsch, Introduction of Numerical Analysis (Springer-Verlag, New York, 1980).
  27. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, 1992).
  28. J. Nocedal, “Theory of algorithms for unconstrained optimization,” Acta Numer. 1, 199–242 (1992).
    [CrossRef]
  29. J. E. Dennis, R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, N.J., 1983).
  30. J. Moré, D. J. Thuente, “Line search algorithms with guaranteed sufficient decrease,”ACM Trans. Math. Software 20, 286–307 (1994).
    [CrossRef]
  31. D. F. Shanno, K. H. Phua, “Remark on algorithm 500,”ACM Trans. Math. Software 6, 618–622 (1980).
    [CrossRef]

1995 (3)

K. Mantere, J. Parkkinen, M. Mäntyjärvi, T. Jaaskelainen, “An eigenvector interpretation of the Farnsworth Munsell 100-hue test,” J. Opt. Soc. Am. A 12, 2237–2243 (1995).
[CrossRef]

N. Hayasaka, S. Toyooka, T. Jaaskelainen, “Iterative feedback method to make a spatial filter on a liquid crystal spatial light modulator for 2D spectroscopic pattern recognition,” Opt. Commun. 119, 643–651 (1995).
[CrossRef]

J. A. Cadzow, X. Li, “Blind deconvolution,” Digital Signal Process. 5, 3–20 (1995).
[CrossRef]

1994 (4)

C. F. Borges, “Numerical determination of tristimulus values,” J. Opt. Soc. Am. A 11, 3152–3161 (1994).
[CrossRef]

D. J. Field, “What is the goal of sensory coding,” Neural Comput. 6, 559–601 (1994).
[CrossRef]

M. D’Zmura, G. Iverson, “Color constancy. III. General linear recovery of spectral descriptions for lights and surfaces,” J. Opt. Soc. Am. A 11, 2389–2400 (1994).
[CrossRef]

J. Moré, D. J. Thuente, “Line search algorithms with guaranteed sufficient decrease,”ACM Trans. Math. Software 20, 286–307 (1994).
[CrossRef]

1992 (6)

S. Usui, S. Nakauchi, M. Nakano, “Reconstruction of Munsell color space by a five-layer neural network,” J. Opt. Soc. Am. A 9, 516–520 (1992).
[CrossRef]

J. J. Atick, “Could information theory provide an ecological theory of sensory processing?” Network 3, 213–251 (1992).
[CrossRef]

T. Jaaskelainen, S. Tyooka, S. Izawa, H. Kadono, “Color classification by vector subspace method and its implementation using liquid crystal spatial light modulator,” Opt. Commun. 89, 23–29 (1992).
[CrossRef]

J. Nocedal, “Theory of algorithms for unconstrained optimization,” Acta Numer. 1, 199–242 (1992).
[CrossRef]

R. Lenz, M. Österberg, “Computing the Karhunen–Loéve expansion with a parallel, unsupervised filter system,” Neural Comput. 4, 382–392 (1992).
[CrossRef]

R. Lenz, M. Österberg, “A new method for unsupervised linear feature extraction using fourth order moments,” Pattern Recognition Lett. 13, 827–836 (1992).
[CrossRef]

1990 (1)

1989 (1)

1986 (2)

1980 (1)

D. F. Shanno, K. H. Phua, “Remark on algorithm 500,”ACM Trans. Math. Software 6, 618–622 (1980).
[CrossRef]

1964 (1)

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

1954 (1)

F. Attneave, “Some informational aspects of visual perception,” Psychol. Rev. 61, 183–193 (1954).
[CrossRef] [PubMed]

Atick, J. J.

J. J. Atick, “Could information theory provide an ecological theory of sensory processing?” Network 3, 213–251 (1992).
[CrossRef]

Attneave, F.

F. Attneave, “Some informational aspects of visual perception,” Psychol. Rev. 61, 183–193 (1954).
[CrossRef] [PubMed]

Barlow, H. B.

H. B. Barlow, “The coding of sensory messages,” in Current Problems in Animal Behavior, W. H. Thorpe, O. L. Zangwill, eds. (Cambridge. U. Press, Cambridge, 1961), pp. 331–360.

Borges, C. F.

Bulirsch, R.

J. Stoer, R. Bulirsch, Introduction of Numerical Analysis (Springer-Verlag, New York, 1980).

Cadzow, J. A.

J. A. Cadzow, X. Li, “Blind deconvolution,” Digital Signal Process. 5, 3–20 (1995).
[CrossRef]

Cohen, J.

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

D’Zmura, M.

Dennis, J. E.

J. E. Dennis, R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, N.J., 1983).

Field, D. J.

D. J. Field, “What is the goal of sensory coding,” Neural Comput. 6, 559–601 (1994).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, 1992).

Flury, B.

B. Flury, Common Principal Components and Related Multivariate Models (Wiley, New York, 1988).

Hallikainen, J.

Hayasaka, N.

N. Hayasaka, S. Toyooka, T. Jaaskelainen, “Iterative feedback method to make a spatial filter on a liquid crystal spatial light modulator for 2D spectroscopic pattern recognition,” Opt. Commun. 119, 643–651 (1995).
[CrossRef]

Iverson, G.

Izawa, S.

T. Jaaskelainen, S. Tyooka, S. Izawa, H. Kadono, “Color classification by vector subspace method and its implementation using liquid crystal spatial light modulator,” Opt. Commun. 89, 23–29 (1992).
[CrossRef]

Jaaskelainen, T.

K. Mantere, J. Parkkinen, M. Mäntyjärvi, T. Jaaskelainen, “An eigenvector interpretation of the Farnsworth Munsell 100-hue test,” J. Opt. Soc. Am. A 12, 2237–2243 (1995).
[CrossRef]

N. Hayasaka, S. Toyooka, T. Jaaskelainen, “Iterative feedback method to make a spatial filter on a liquid crystal spatial light modulator for 2D spectroscopic pattern recognition,” Opt. Commun. 119, 643–651 (1995).
[CrossRef]

T. Jaaskelainen, S. Tyooka, S. Izawa, H. Kadono, “Color classification by vector subspace method and its implementation using liquid crystal spatial light modulator,” Opt. Commun. 89, 23–29 (1992).
[CrossRef]

T. Jaaskelainen, J. Parkkinen, S. Toyooka, “Vector subspace model for color representation,” J. Opt. Soc. Am. A 7, 725–730 (1990).
[CrossRef]

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

Kadono, H.

T. Jaaskelainen, S. Tyooka, S. Izawa, H. Kadono, “Color classification by vector subspace method and its implementation using liquid crystal spatial light modulator,” Opt. Commun. 89, 23–29 (1992).
[CrossRef]

Lenz, R.

R. Lenz, M. Österberg, “Computing the Karhunen–Loéve expansion with a parallel, unsupervised filter system,” Neural Comput. 4, 382–392 (1992).
[CrossRef]

R. Lenz, M. Österberg, “A new method for unsupervised linear feature extraction using fourth order moments,” Pattern Recognition Lett. 13, 827–836 (1992).
[CrossRef]

M. Österberg, R. Lenz, “Unsupervised parallel feature extraction from first principles,” in Advances in Neural Information Processing Systems 6, J. D. Cowan, G. Tesauro, J. Alspector, eds. (Morgan, Kaufmann, San Francisco, Calif., 1994), pp. 136–144.

Li, X.

J. A. Cadzow, X. Li, “Blind deconvolution,” Digital Signal Process. 5, 3–20 (1995).
[CrossRef]

Maloney, L. T.

Mantere, K.

Mäntyjärvi, M.

Moré, J.

J. Moré, D. J. Thuente, “Line search algorithms with guaranteed sufficient decrease,”ACM Trans. Math. Software 20, 286–307 (1994).
[CrossRef]

Nakano, M.

Nakauchi, S.

S. Usui, S. Nakauchi, M. Nakano, “Reconstruction of Munsell color space by a five-layer neural network,” J. Opt. Soc. Am. A 9, 516–520 (1992).
[CrossRef]

S. Usui, S. Nakauchi, “Computational color vision models by neural networks,” in Computational Intelligence, Imitating Life, T. M. Zurada, R. J. Marks, C. J. Robinson, eds. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1994), pp. 252–263.

Nocedal, J.

J. Nocedal, “Theory of algorithms for unconstrained optimization,” Acta Numer. 1, 199–242 (1992).
[CrossRef]

Nuboer, J. F. W.

J. F. W. Nuboer, “A comparative view on colour vision,” Neth. J. Zoo. 36, 344–380 (1986).
[CrossRef]

Österberg, M.

R. Lenz, M. Österberg, “Computing the Karhunen–Loéve expansion with a parallel, unsupervised filter system,” Neural Comput. 4, 382–392 (1992).
[CrossRef]

R. Lenz, M. Österberg, “A new method for unsupervised linear feature extraction using fourth order moments,” Pattern Recognition Lett. 13, 827–836 (1992).
[CrossRef]

M. Österberg, “Quality functions for parallel selective principal component analysis,” Ph.D. dissertation (Linköping University, Linkoping, Sweden, 1994).

M. Österberg, R. Lenz, “Unsupervised parallel feature extraction from first principles,” in Advances in Neural Information Processing Systems 6, J. D. Cowan, G. Tesauro, J. Alspector, eds. (Morgan, Kaufmann, San Francisco, Calif., 1994), pp. 136–144.

Parkkinen, J.

Parkkinen, J. P. S.

Phua, K. H.

D. F. Shanno, K. H. Phua, “Remark on algorithm 500,”ACM Trans. Math. Software 6, 618–622 (1980).
[CrossRef]

Plumbley, M. D.

J. G. Taylor, M. D. Plumbley, “Information theory and neural networks,” in Mathematical Applications to Neural Networks, J. G. Taylor, ed. (Elsevier, New York, 1993), pp. 307–340.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, 1992).

Schnabel, R. B.

J. E. Dennis, R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, N.J., 1983).

Shanno, D. F.

D. F. Shanno, K. H. Phua, “Remark on algorithm 500,”ACM Trans. Math. Software 6, 618–622 (1980).
[CrossRef]

Stiles, W. S.

G. Wyszecki, W. S. Stiles, Color Science, 2nd ed. (Wiley, London, 1982).

Stoer, J.

J. Stoer, R. Bulirsch, Introduction of Numerical Analysis (Springer-Verlag, New York, 1980).

Taylor, J. G.

J. G. Taylor, M. D. Plumbley, “Information theory and neural networks,” in Mathematical Applications to Neural Networks, J. G. Taylor, ed. (Elsevier, New York, 1993), pp. 307–340.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, 1992).

Thuente, D. J.

J. Moré, D. J. Thuente, “Line search algorithms with guaranteed sufficient decrease,”ACM Trans. Math. Software 20, 286–307 (1994).
[CrossRef]

Toyooka, S.

N. Hayasaka, S. Toyooka, T. Jaaskelainen, “Iterative feedback method to make a spatial filter on a liquid crystal spatial light modulator for 2D spectroscopic pattern recognition,” Opt. Commun. 119, 643–651 (1995).
[CrossRef]

T. Jaaskelainen, J. Parkkinen, S. Toyooka, “Vector subspace model for color representation,” J. Opt. Soc. Am. A 7, 725–730 (1990).
[CrossRef]

Tyooka, S.

T. Jaaskelainen, S. Tyooka, S. Izawa, H. Kadono, “Color classification by vector subspace method and its implementation using liquid crystal spatial light modulator,” Opt. Commun. 89, 23–29 (1992).
[CrossRef]

Usui, S.

S. Usui, S. Nakauchi, M. Nakano, “Reconstruction of Munsell color space by a five-layer neural network,” J. Opt. Soc. Am. A 9, 516–520 (1992).
[CrossRef]

S. Usui, S. Nakauchi, “Computational color vision models by neural networks,” in Computational Intelligence, Imitating Life, T. M. Zurada, R. J. Marks, C. J. Robinson, eds. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1994), pp. 252–263.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, 1992).

Wandell, B. A.

Wyszecki, G.

G. Wyszecki, W. S. Stiles, Color Science, 2nd ed. (Wiley, London, 1982).

ACM Trans. Math. Software (2)

J. Moré, D. J. Thuente, “Line search algorithms with guaranteed sufficient decrease,”ACM Trans. Math. Software 20, 286–307 (1994).
[CrossRef]

D. F. Shanno, K. H. Phua, “Remark on algorithm 500,”ACM Trans. Math. Software 6, 618–622 (1980).
[CrossRef]

Acta Numer. (1)

J. Nocedal, “Theory of algorithms for unconstrained optimization,” Acta Numer. 1, 199–242 (1992).
[CrossRef]

Digital Signal Process. (1)

J. A. Cadzow, X. Li, “Blind deconvolution,” Digital Signal Process. 5, 3–20 (1995).
[CrossRef]

J. Opt. Soc. Am. A (7)

Neth. J. Zoo. (1)

J. F. W. Nuboer, “A comparative view on colour vision,” Neth. J. Zoo. 36, 344–380 (1986).
[CrossRef]

Network (1)

J. J. Atick, “Could information theory provide an ecological theory of sensory processing?” Network 3, 213–251 (1992).
[CrossRef]

Neural Comput. (2)

D. J. Field, “What is the goal of sensory coding,” Neural Comput. 6, 559–601 (1994).
[CrossRef]

R. Lenz, M. Österberg, “Computing the Karhunen–Loéve expansion with a parallel, unsupervised filter system,” Neural Comput. 4, 382–392 (1992).
[CrossRef]

Opt. Commun. (2)

T. Jaaskelainen, S. Tyooka, S. Izawa, H. Kadono, “Color classification by vector subspace method and its implementation using liquid crystal spatial light modulator,” Opt. Commun. 89, 23–29 (1992).
[CrossRef]

N. Hayasaka, S. Toyooka, T. Jaaskelainen, “Iterative feedback method to make a spatial filter on a liquid crystal spatial light modulator for 2D spectroscopic pattern recognition,” Opt. Commun. 119, 643–651 (1995).
[CrossRef]

Pattern Recognition Lett. (1)

R. Lenz, M. Österberg, “A new method for unsupervised linear feature extraction using fourth order moments,” Pattern Recognition Lett. 13, 827–836 (1992).
[CrossRef]

Psychol. Rev. (1)

F. Attneave, “Some informational aspects of visual perception,” Psychol. Rev. 61, 183–193 (1954).
[CrossRef] [PubMed]

Psychon. Sci. (1)

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

Other (11)

J. G. Taylor, M. D. Plumbley, “Information theory and neural networks,” in Mathematical Applications to Neural Networks, J. G. Taylor, ed. (Elsevier, New York, 1993), pp. 307–340.

J. E. Dennis, R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, N.J., 1983).

J. Stoer, R. Bulirsch, Introduction of Numerical Analysis (Springer-Verlag, New York, 1980).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, 1992).

H. B. Barlow, “The coding of sensory messages,” in Current Problems in Animal Behavior, W. H. Thorpe, O. L. Zangwill, eds. (Cambridge. U. Press, Cambridge, 1961), pp. 331–360.

G. Wyszecki, W. S. Stiles, Color Science, 2nd ed. (Wiley, London, 1982).

S. Usui, S. Nakauchi, “Computational color vision models by neural networks,” in Computational Intelligence, Imitating Life, T. M. Zurada, R. J. Marks, C. J. Robinson, eds. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1994), pp. 252–263.

M. Österberg, “Quality functions for parallel selective principal component analysis,” Ph.D. dissertation (Linköping University, Linkoping, Sweden, 1994).

M. Österberg, R. Lenz, “Unsupervised parallel feature extraction from first principles,” in Advances in Neural Information Processing Systems 6, J. D. Cowan, G. Tesauro, J. Alspector, eds. (Morgan, Kaufmann, San Francisco, Calif., 1994), pp. 136–144.

Munsell Book of Color, Matte Finish Collection (Munsell Color, Baltimore, 1976).

B. Flury, Common Principal Components and Related Multivariate Models (Wiley, New York, 1988).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (13)

Fig. 1
Fig. 1

CIE tristimulus functions.

Fig. 2
Fig. 2

First four eigenvectors for the three databases: Munsell-I (dashed), Munsell (dotted), NCS (solid).

Fig. 3
Fig. 3

First three eigenvectors: first (solid), second (dashed), third (dotted).

Fig. 4
Fig. 4

Learned filter functions: conmin with use of Q4.

Fig. 5
Fig. 5

Learned filter functions: conmin with use of QV.

Fig. 6
Fig. 6

Learned filter functions: conmin with use of QV and positive weights.

Fig. 7
Fig. 7

Learned filter functions: conmin with use of QV and positive weights: NCS (solid), Munsell-I (dashed–dotted), Munsell (dashed).

Fig. 8
Fig. 8

System with two filter functions.

Fig. 9
Fig. 9

System with six filter functions.

Fig. 10
Fig. 10

Approximation errors: eigenvector system (), unrestricted filter systems learned from the Munsell database (×), positive filter functions computed from the same data set (+).

Fig. 11
Fig. 11

Spectra with largest approximation errors: (a) eigenvector system; (b) learned (general) filters, (c) learned (positive) filters, (d) typical approximation error for learned (positive) filters. See the text for an explanation of the different curves.

Fig. 12
Fig. 12

Approximation errors for Gaussian quadrature: weight vectors (*), 2 (+), and 3 ().

Fig. 13
Fig. 13

Gaussian quadrature approximations: original spectrum (solid); reconstructions based on full vectors and eight-node approximation (dashed), two-node approximation (dotted), four-node approximation (dashed–dotted).

Tables (2)

Tables Icon

Table 1 Correlations of Eigenvectors from the Different Databases

Tables Icon

Table 2 Eigenvalues from the Full Munsell Database

Equations (8)

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

X = k S ( λ ) x ¯ ( λ ) d λ , Y = k S ( λ ) y ¯ ( λ ) d λ , Z = k S ( λ ) z ¯ ( λ ) d λ ,
Q D = - det ( C o ) .
Q A = Q V = n = 1 N var ( o n 2 ) ,
1 Q A = Q 4 = n = 1 N E [ o n 2 ( 1 - o n 2 ) ] .
f , g = - f ( λ ) g ( λ ) w ( λ ) d λ ,
p k + 1 ( λ ) = ( λ - a k ) p k ( λ ) - b k p k - 1 ( λ ) ,
f ( λ ) w ( λ ) d λ k = 1 n f ( x k ) ω k ,
π l ( λ ) d λ = 1 2 l + 1 [ π l + 1 ( λ ) - π l - 1 ( λ ) ] .

Metrics