Abstract

For digital cameras, device-dependent pixel values describe the camera’s response to the incoming spectrum of light. We convert device-dependent RGB values to device- and illuminant-independent reflectance spectra. Simple regularization methods with widely used polynomial modeling provide an efficient approach for this conversion. We also introduce a more general framework for spectral estimation: regularized least-squares regression in reproducing kernel Hilbert spaces (RKHS). Obtained results show that the regularization framework provides an efficient approach for enhancing the generalization properties of the models.

© 2007 Optical Society of America

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  1. M. D. Fairchild, Color Appearance Models (Addison-Wesley, 1998).
  2. M. J. Vrhel and H. J. Trussel, "Color device calibration: a mathematical formulation," IEEE Trans. Image Process. 8, 1796-1806 (1999).
    [CrossRef]
  3. H. Haneishi, T. Hasegawa, A. Hosoi, Y. Yokoyama, N. Tsumura, and Y. Miyake, "System design for accurately estimating the reflectance spectra of art paintings," Appl. Opt. 39, 6621-6632 (2000).
    [CrossRef]
  4. 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]
  5. J. Parkkinen, J. Hallikainen, and T. Jaaskelainen, "Characteristic spectra of Munsell colors," J. Opt. Soc. Am. A 6, 318-322 (1989).
    [CrossRef]
  6. D. Connah and J. Y. Hardeberg, "Spectral recovery using polynomial models," Proc. SPIE 5667, pp. 65-75 (2005).
    [CrossRef]
  7. P. Stigell, K. Miyata, and M. Hauta-Kasari, "Wiener estimation method in estimation of spectral reflectance from rgb images," Pattern Recogn. Image Anal. 15, 327-329 (2005).
  8. M. Bertero, T. Poggio, and V. Torre, "Ill-posed problems in early vision," Proc. IEEE 76, 869-889 (1988).
    [CrossRef]
  9. Å. Björck, Numerical Methods for Least Squares Problems (SIAM, 1996).
    [CrossRef]
  10. J. Y. Hardeberg, Acquisition and Reproduction of Color Images--Colorimetric and Multispectral Approaches (Dissertation.com, 2001).
  11. G. Hong, M. R. Luo, and P. A. Rhodes, "A study of digital camera colorimetric characterization based on polynomial modeling," Color Res. Appl. 26, 76-84 (2001).
    [CrossRef]
  12. A. Neumaier, "Solving ill-conditioned and singular linear systems, a tutorial on regularization," SIAM Rev. 99, 636-666 (1998).
    [CrossRef]
  13. T. Jetsu, V. Heikkinen, J. Parkkinen, M. Hauta-Kasari, B. Martinkauppi, S. D. Lee, H. W. Ok, and C. Y. Kim, "Color calibration of digital camera using polynomial transformation," in CGIV, Third European Conference on Color in Graphics, Imaging and Vision, (IS&T, 2006), pp. 163-166.
  14. G. Wahba, Spline Models for Observational Data, Vol. 59 of SIAM CBMS-NSF Regional Conference Series in Applied Mathematics (SIAM, 1990).
    [CrossRef]
  15. T. Poggio and F. Girosi, "Networks for approximation and learning," Proc. IEEE 78, 1481-1497 (1990).
    [CrossRef]
  16. T. Evgeniou, M. Pontil, and T. Poggio, "Regularization networks and support vector machines," Adv. Comput. Math. 13, 1-50 (2000).
    [CrossRef]
  17. B. Schölkopf and A. J. Smola, Learning with Kernels (MIT, 2002).
  18. F. Girosi, M. Jones, and T. Poggio, "Regularization theory and neural network architectures," Neural Comput. 7, 219-269 (1995).
    [CrossRef]
  19. J. Parkkinen, "Subspace methods in two machine vision problems," Ph.D. thesis (University of Kuopio, 1989).
  20. N. Aronszajn, "Theory of reproducing kernels," Trans. Am. Math. Soc. 68, 337-404 (1950).
    [CrossRef]
  21. S. Saitoh, Theory of Reproducing Kernels and Its Applications (Longman, 1988).
  22. J. Shawe-Taylor and N. Christianini, Kernel Methods for Pattern Analysis (Cambridge U. Press, 2004).
    [CrossRef]
  23. V. I. Lebedev, An Introduction to Functional Analysis and Computational Mathematics (Birkhäuser, 1997).
  24. F. Cucker and S. Smale, "On the mathematical foundations of learning," Bull. Am. Math. Soc. 39, 1-49 (2002).
    [CrossRef]
  25. T. Poggio, S. Mukherjee, R. Rifkin, A. Rakhlin, and A. Verri, "B," in Uncertainty in Geometric Computations, J.Winkler and M.Niranjan, eds., (Kluwer, 2002), pp. 131-141.
    [CrossRef]
  26. F. Girosi, "An equivalence between sparse approximation and support vector machines," Neural Comput. 10, 1455-1480 (1998).
    [CrossRef] [PubMed]
  27. V. Vapnik, Statistical Learning Theory (Wiley, 1998).
  28. B. Hamers, "Kernel models for large scale applications," Ph.D. thesis (Katholieke Universiteit Leuven, 2004).
  29. T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference and Prediction (Springer-Verlag, 2001).
  30. M. R. Luo, G. Cui, and B. Rigg, "The development of the CIE 2000 colour-difference formula," Color Res. Appl. 26, 340-350 (2000).
    [CrossRef]

2005 (2)

D. Connah and J. Y. Hardeberg, "Spectral recovery using polynomial models," Proc. SPIE 5667, pp. 65-75 (2005).
[CrossRef]

P. Stigell, K. Miyata, and M. Hauta-Kasari, "Wiener estimation method in estimation of spectral reflectance from rgb images," Pattern Recogn. Image Anal. 15, 327-329 (2005).

2002 (1)

F. Cucker and S. Smale, "On the mathematical foundations of learning," Bull. Am. Math. Soc. 39, 1-49 (2002).
[CrossRef]

2001 (1)

G. Hong, M. R. Luo, and P. A. Rhodes, "A study of digital camera colorimetric characterization based on polynomial modeling," Color Res. Appl. 26, 76-84 (2001).
[CrossRef]

2000 (3)

T. Evgeniou, M. Pontil, and T. Poggio, "Regularization networks and support vector machines," Adv. Comput. Math. 13, 1-50 (2000).
[CrossRef]

M. R. Luo, G. Cui, and B. Rigg, "The development of the CIE 2000 colour-difference formula," Color Res. Appl. 26, 340-350 (2000).
[CrossRef]

H. Haneishi, T. Hasegawa, A. Hosoi, Y. Yokoyama, N. Tsumura, and Y. Miyake, "System design for accurately estimating the reflectance spectra of art paintings," Appl. Opt. 39, 6621-6632 (2000).
[CrossRef]

1999 (1)

M. J. Vrhel and H. J. Trussel, "Color device calibration: a mathematical formulation," IEEE Trans. Image Process. 8, 1796-1806 (1999).
[CrossRef]

1998 (2)

F. Girosi, "An equivalence between sparse approximation and support vector machines," Neural Comput. 10, 1455-1480 (1998).
[CrossRef] [PubMed]

A. Neumaier, "Solving ill-conditioned and singular linear systems, a tutorial on regularization," SIAM Rev. 99, 636-666 (1998).
[CrossRef]

1995 (1)

F. Girosi, M. Jones, and T. Poggio, "Regularization theory and neural network architectures," Neural Comput. 7, 219-269 (1995).
[CrossRef]

1990 (1)

T. Poggio and F. Girosi, "Networks for approximation and learning," Proc. IEEE 78, 1481-1497 (1990).
[CrossRef]

1989 (1)

1988 (1)

M. Bertero, T. Poggio, and V. Torre, "Ill-posed problems in early vision," Proc. IEEE 76, 869-889 (1988).
[CrossRef]

1986 (1)

1950 (1)

N. Aronszajn, "Theory of reproducing kernels," Trans. Am. Math. Soc. 68, 337-404 (1950).
[CrossRef]

Aronszajn, N.

N. Aronszajn, "Theory of reproducing kernels," Trans. Am. Math. Soc. 68, 337-404 (1950).
[CrossRef]

Bertero, M.

M. Bertero, T. Poggio, and V. Torre, "Ill-posed problems in early vision," Proc. IEEE 76, 869-889 (1988).
[CrossRef]

Björck, Å.

Å. Björck, Numerical Methods for Least Squares Problems (SIAM, 1996).
[CrossRef]

Christianini, N.

J. Shawe-Taylor and N. Christianini, Kernel Methods for Pattern Analysis (Cambridge U. Press, 2004).
[CrossRef]

Connah, D.

D. Connah and J. Y. Hardeberg, "Spectral recovery using polynomial models," Proc. SPIE 5667, pp. 65-75 (2005).
[CrossRef]

Cucker, F.

F. Cucker and S. Smale, "On the mathematical foundations of learning," Bull. Am. Math. Soc. 39, 1-49 (2002).
[CrossRef]

Cui, G.

M. R. Luo, G. Cui, and B. Rigg, "The development of the CIE 2000 colour-difference formula," Color Res. Appl. 26, 340-350 (2000).
[CrossRef]

Evgeniou, T.

T. Evgeniou, M. Pontil, and T. Poggio, "Regularization networks and support vector machines," Adv. Comput. Math. 13, 1-50 (2000).
[CrossRef]

Fairchild, M. D.

M. D. Fairchild, Color Appearance Models (Addison-Wesley, 1998).

Friedman, J.

T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference and Prediction (Springer-Verlag, 2001).

Girosi, F.

F. Girosi, "An equivalence between sparse approximation and support vector machines," Neural Comput. 10, 1455-1480 (1998).
[CrossRef] [PubMed]

F. Girosi, M. Jones, and T. Poggio, "Regularization theory and neural network architectures," Neural Comput. 7, 219-269 (1995).
[CrossRef]

T. Poggio and F. Girosi, "Networks for approximation and learning," Proc. IEEE 78, 1481-1497 (1990).
[CrossRef]

Hallikainen, J.

Hamers, B.

B. Hamers, "Kernel models for large scale applications," Ph.D. thesis (Katholieke Universiteit Leuven, 2004).

Haneishi, H.

Hardeberg, J. Y.

D. Connah and J. Y. Hardeberg, "Spectral recovery using polynomial models," Proc. SPIE 5667, pp. 65-75 (2005).
[CrossRef]

J. Y. Hardeberg, Acquisition and Reproduction of Color Images--Colorimetric and Multispectral Approaches (Dissertation.com, 2001).

Hasegawa, T.

Hastie, T.

T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference and Prediction (Springer-Verlag, 2001).

Hauta-Kasari, M.

P. Stigell, K. Miyata, and M. Hauta-Kasari, "Wiener estimation method in estimation of spectral reflectance from rgb images," Pattern Recogn. Image Anal. 15, 327-329 (2005).

T. Jetsu, V. Heikkinen, J. Parkkinen, M. Hauta-Kasari, B. Martinkauppi, S. D. Lee, H. W. Ok, and C. Y. Kim, "Color calibration of digital camera using polynomial transformation," in CGIV, Third European Conference on Color in Graphics, Imaging and Vision, (IS&T, 2006), pp. 163-166.

Heikkinen, V.

T. Jetsu, V. Heikkinen, J. Parkkinen, M. Hauta-Kasari, B. Martinkauppi, S. D. Lee, H. W. Ok, and C. Y. Kim, "Color calibration of digital camera using polynomial transformation," in CGIV, Third European Conference on Color in Graphics, Imaging and Vision, (IS&T, 2006), pp. 163-166.

Hong, G.

G. Hong, M. R. Luo, and P. A. Rhodes, "A study of digital camera colorimetric characterization based on polynomial modeling," Color Res. Appl. 26, 76-84 (2001).
[CrossRef]

Hosoi, A.

Jaaskelainen, T.

Jetsu, T.

T. Jetsu, V. Heikkinen, J. Parkkinen, M. Hauta-Kasari, B. Martinkauppi, S. D. Lee, H. W. Ok, and C. Y. Kim, "Color calibration of digital camera using polynomial transformation," in CGIV, Third European Conference on Color in Graphics, Imaging and Vision, (IS&T, 2006), pp. 163-166.

Jones, M.

F. Girosi, M. Jones, and T. Poggio, "Regularization theory and neural network architectures," Neural Comput. 7, 219-269 (1995).
[CrossRef]

Kim, C. Y.

T. Jetsu, V. Heikkinen, J. Parkkinen, M. Hauta-Kasari, B. Martinkauppi, S. D. Lee, H. W. Ok, and C. Y. Kim, "Color calibration of digital camera using polynomial transformation," in CGIV, Third European Conference on Color in Graphics, Imaging and Vision, (IS&T, 2006), pp. 163-166.

Lebedev, V. I.

V. I. Lebedev, An Introduction to Functional Analysis and Computational Mathematics (Birkhäuser, 1997).

Lee, S. D.

T. Jetsu, V. Heikkinen, J. Parkkinen, M. Hauta-Kasari, B. Martinkauppi, S. D. Lee, H. W. Ok, and C. Y. Kim, "Color calibration of digital camera using polynomial transformation," in CGIV, Third European Conference on Color in Graphics, Imaging and Vision, (IS&T, 2006), pp. 163-166.

Luo, M. R.

G. Hong, M. R. Luo, and P. A. Rhodes, "A study of digital camera colorimetric characterization based on polynomial modeling," Color Res. Appl. 26, 76-84 (2001).
[CrossRef]

M. R. Luo, G. Cui, and B. Rigg, "The development of the CIE 2000 colour-difference formula," Color Res. Appl. 26, 340-350 (2000).
[CrossRef]

Maloney, L. T.

Martinkauppi, B.

T. Jetsu, V. Heikkinen, J. Parkkinen, M. Hauta-Kasari, B. Martinkauppi, S. D. Lee, H. W. Ok, and C. Y. Kim, "Color calibration of digital camera using polynomial transformation," in CGIV, Third European Conference on Color in Graphics, Imaging and Vision, (IS&T, 2006), pp. 163-166.

Miyake, Y.

Miyata, K.

P. Stigell, K. Miyata, and M. Hauta-Kasari, "Wiener estimation method in estimation of spectral reflectance from rgb images," Pattern Recogn. Image Anal. 15, 327-329 (2005).

Mukherjee, S.

T. Poggio, S. Mukherjee, R. Rifkin, A. Rakhlin, and A. Verri, "B," in Uncertainty in Geometric Computations, J.Winkler and M.Niranjan, eds., (Kluwer, 2002), pp. 131-141.
[CrossRef]

Neumaier, A.

A. Neumaier, "Solving ill-conditioned and singular linear systems, a tutorial on regularization," SIAM Rev. 99, 636-666 (1998).
[CrossRef]

Ok, H. W.

T. Jetsu, V. Heikkinen, J. Parkkinen, M. Hauta-Kasari, B. Martinkauppi, S. D. Lee, H. W. Ok, and C. Y. Kim, "Color calibration of digital camera using polynomial transformation," in CGIV, Third European Conference on Color in Graphics, Imaging and Vision, (IS&T, 2006), pp. 163-166.

Parkkinen, J.

J. Parkkinen, J. Hallikainen, and T. Jaaskelainen, "Characteristic spectra of Munsell colors," J. Opt. Soc. Am. A 6, 318-322 (1989).
[CrossRef]

J. Parkkinen, "Subspace methods in two machine vision problems," Ph.D. thesis (University of Kuopio, 1989).

T. Jetsu, V. Heikkinen, J. Parkkinen, M. Hauta-Kasari, B. Martinkauppi, S. D. Lee, H. W. Ok, and C. Y. Kim, "Color calibration of digital camera using polynomial transformation," in CGIV, Third European Conference on Color in Graphics, Imaging and Vision, (IS&T, 2006), pp. 163-166.

Poggio, T.

T. Evgeniou, M. Pontil, and T. Poggio, "Regularization networks and support vector machines," Adv. Comput. Math. 13, 1-50 (2000).
[CrossRef]

F. Girosi, M. Jones, and T. Poggio, "Regularization theory and neural network architectures," Neural Comput. 7, 219-269 (1995).
[CrossRef]

T. Poggio and F. Girosi, "Networks for approximation and learning," Proc. IEEE 78, 1481-1497 (1990).
[CrossRef]

M. Bertero, T. Poggio, and V. Torre, "Ill-posed problems in early vision," Proc. IEEE 76, 869-889 (1988).
[CrossRef]

T. Poggio, S. Mukherjee, R. Rifkin, A. Rakhlin, and A. Verri, "B," in Uncertainty in Geometric Computations, J.Winkler and M.Niranjan, eds., (Kluwer, 2002), pp. 131-141.
[CrossRef]

Pontil, M.

T. Evgeniou, M. Pontil, and T. Poggio, "Regularization networks and support vector machines," Adv. Comput. Math. 13, 1-50 (2000).
[CrossRef]

Rakhlin, A.

T. Poggio, S. Mukherjee, R. Rifkin, A. Rakhlin, and A. Verri, "B," in Uncertainty in Geometric Computations, J.Winkler and M.Niranjan, eds., (Kluwer, 2002), pp. 131-141.
[CrossRef]

Rhodes, P. A.

G. Hong, M. R. Luo, and P. A. Rhodes, "A study of digital camera colorimetric characterization based on polynomial modeling," Color Res. Appl. 26, 76-84 (2001).
[CrossRef]

Rifkin, R.

T. Poggio, S. Mukherjee, R. Rifkin, A. Rakhlin, and A. Verri, "B," in Uncertainty in Geometric Computations, J.Winkler and M.Niranjan, eds., (Kluwer, 2002), pp. 131-141.
[CrossRef]

Rigg, B.

M. R. Luo, G. Cui, and B. Rigg, "The development of the CIE 2000 colour-difference formula," Color Res. Appl. 26, 340-350 (2000).
[CrossRef]

Saitoh, S.

S. Saitoh, Theory of Reproducing Kernels and Its Applications (Longman, 1988).

Schölkopf, B.

B. Schölkopf and A. J. Smola, Learning with Kernels (MIT, 2002).

Shawe-Taylor, J.

J. Shawe-Taylor and N. Christianini, Kernel Methods for Pattern Analysis (Cambridge U. Press, 2004).
[CrossRef]

Smale, S.

F. Cucker and S. Smale, "On the mathematical foundations of learning," Bull. Am. Math. Soc. 39, 1-49 (2002).
[CrossRef]

Smola, A. J.

B. Schölkopf and A. J. Smola, Learning with Kernels (MIT, 2002).

Stigell, P.

P. Stigell, K. Miyata, and M. Hauta-Kasari, "Wiener estimation method in estimation of spectral reflectance from rgb images," Pattern Recogn. Image Anal. 15, 327-329 (2005).

Tibshirani, R.

T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference and Prediction (Springer-Verlag, 2001).

Torre, V.

M. Bertero, T. Poggio, and V. Torre, "Ill-posed problems in early vision," Proc. IEEE 76, 869-889 (1988).
[CrossRef]

Trussel, H. J.

M. J. Vrhel and H. J. Trussel, "Color device calibration: a mathematical formulation," IEEE Trans. Image Process. 8, 1796-1806 (1999).
[CrossRef]

Tsumura, N.

Vapnik, V.

V. Vapnik, Statistical Learning Theory (Wiley, 1998).

Verri, A.

T. Poggio, S. Mukherjee, R. Rifkin, A. Rakhlin, and A. Verri, "B," in Uncertainty in Geometric Computations, J.Winkler and M.Niranjan, eds., (Kluwer, 2002), pp. 131-141.
[CrossRef]

Vrhel, M. J.

M. J. Vrhel and H. J. Trussel, "Color device calibration: a mathematical formulation," IEEE Trans. Image Process. 8, 1796-1806 (1999).
[CrossRef]

Wahba, G.

G. Wahba, Spline Models for Observational Data, Vol. 59 of SIAM CBMS-NSF Regional Conference Series in Applied Mathematics (SIAM, 1990).
[CrossRef]

Yokoyama, Y.

Adv. Comput. Math. (1)

T. Evgeniou, M. Pontil, and T. Poggio, "Regularization networks and support vector machines," Adv. Comput. Math. 13, 1-50 (2000).
[CrossRef]

Appl. Opt. (1)

Bull. Am. Math. Soc. (1)

F. Cucker and S. Smale, "On the mathematical foundations of learning," Bull. Am. Math. Soc. 39, 1-49 (2002).
[CrossRef]

Color Res. Appl. (2)

M. R. Luo, G. Cui, and B. Rigg, "The development of the CIE 2000 colour-difference formula," Color Res. Appl. 26, 340-350 (2000).
[CrossRef]

G. Hong, M. R. Luo, and P. A. Rhodes, "A study of digital camera colorimetric characterization based on polynomial modeling," Color Res. Appl. 26, 76-84 (2001).
[CrossRef]

IEEE Trans. Image Process. (1)

M. J. Vrhel and H. J. Trussel, "Color device calibration: a mathematical formulation," IEEE Trans. Image Process. 8, 1796-1806 (1999).
[CrossRef]

J. Opt. Soc. Am. A (2)

Neural Comput. (2)

F. Girosi, M. Jones, and T. Poggio, "Regularization theory and neural network architectures," Neural Comput. 7, 219-269 (1995).
[CrossRef]

F. Girosi, "An equivalence between sparse approximation and support vector machines," Neural Comput. 10, 1455-1480 (1998).
[CrossRef] [PubMed]

Pattern Recogn. Image Anal. (1)

P. Stigell, K. Miyata, and M. Hauta-Kasari, "Wiener estimation method in estimation of spectral reflectance from rgb images," Pattern Recogn. Image Anal. 15, 327-329 (2005).

Proc. IEEE (2)

M. Bertero, T. Poggio, and V. Torre, "Ill-posed problems in early vision," Proc. IEEE 76, 869-889 (1988).
[CrossRef]

T. Poggio and F. Girosi, "Networks for approximation and learning," Proc. IEEE 78, 1481-1497 (1990).
[CrossRef]

Proc. SPIE (1)

D. Connah and J. Y. Hardeberg, "Spectral recovery using polynomial models," Proc. SPIE 5667, pp. 65-75 (2005).
[CrossRef]

SIAM Rev. (1)

A. Neumaier, "Solving ill-conditioned and singular linear systems, a tutorial on regularization," SIAM Rev. 99, 636-666 (1998).
[CrossRef]

Trans. Am. Math. Soc. (1)

N. Aronszajn, "Theory of reproducing kernels," Trans. Am. Math. Soc. 68, 337-404 (1950).
[CrossRef]

Other (14)

S. Saitoh, Theory of Reproducing Kernels and Its Applications (Longman, 1988).

J. Shawe-Taylor and N. Christianini, Kernel Methods for Pattern Analysis (Cambridge U. Press, 2004).
[CrossRef]

V. I. Lebedev, An Introduction to Functional Analysis and Computational Mathematics (Birkhäuser, 1997).

B. Schölkopf and A. J. Smola, Learning with Kernels (MIT, 2002).

T. Jetsu, V. Heikkinen, J. Parkkinen, M. Hauta-Kasari, B. Martinkauppi, S. D. Lee, H. W. Ok, and C. Y. Kim, "Color calibration of digital camera using polynomial transformation," in CGIV, Third European Conference on Color in Graphics, Imaging and Vision, (IS&T, 2006), pp. 163-166.

G. Wahba, Spline Models for Observational Data, Vol. 59 of SIAM CBMS-NSF Regional Conference Series in Applied Mathematics (SIAM, 1990).
[CrossRef]

Å. Björck, Numerical Methods for Least Squares Problems (SIAM, 1996).
[CrossRef]

J. Y. Hardeberg, Acquisition and Reproduction of Color Images--Colorimetric and Multispectral Approaches (Dissertation.com, 2001).

V. Vapnik, Statistical Learning Theory (Wiley, 1998).

B. Hamers, "Kernel models for large scale applications," Ph.D. thesis (Katholieke Universiteit Leuven, 2004).

T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference and Prediction (Springer-Verlag, 2001).

J. Parkkinen, "Subspace methods in two machine vision problems," Ph.D. thesis (University of Kuopio, 1989).

T. Poggio, S. Mukherjee, R. Rifkin, A. Rakhlin, and A. Verri, "B," in Uncertainty in Geometric Computations, J.Winkler and M.Niranjan, eds., (Kluwer, 2002), pp. 131-141.
[CrossRef]

M. D. Fairchild, Color Appearance Models (Addison-Wesley, 1998).

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Figures (4)

Fig. 1
Fig. 1

Reconstruction of Munsell spectra using the polynomial model (35 terms) with a training set of 100 samples. (a) Reconstruction result for spectrum whose Δ E difference without regularization is the largest in the test set. Regularization with the Tikhonov method lowers the Δ E value of this spectrum from 53.17 to 6.02. (b) Reconstruction result for the spectrum whose root-mean-squared-error (RMSE) difference without regularization is the largest in the test set. Regularization with the Tikhonov method lowers the RMSE value of this spectrum from 0.9015 to 0.0615.

Fig. 2
Fig. 2

Distributions of training, validation, and test set values.

Fig. 3
Fig. 3

Examples of RMSE values and Δ E values for different spectra. Solid curves present the original spectra.

Fig. 4
Fig. 4

Reconstruction of Munsell spectra using the kernel model. The Duchon spline ( d = 3 ) with training set 600 is used. This figure presents the reconstruction result for the spectrum of the test set with maximal Δ E difference when the training set is interpolated ( λ = 0 ) and when regularization is used ( λ 0 ) .

Tables (2)

Tables Icon

Table 1 Errors in Spectral Estimation with a Training Set of 600 Samples a

Tables Icon

Table 2 Errors in Spectral Estimation with a Training Set of 100 Samples a

Equations (39)

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

X W Y ,
min w X w y 2 ,
L ( f , S ) = i = 1 l ( f ( x i ) y i ) 2 = i = 1 l ( w , x i y i ) 2 ,
L ( f , S ) = i = 1 l ( f ( x i ) y i ) 2 = i = 1 l ( w , Φ p k ( x i ) y i ) 2 ,
X = U S V T , S = [ S k 0 0 0 ] ,
w ̃ = i = 1 k u i T y σ i v i .
X = U S V T = i = 1 p σ i u i v i T + i = p + 1 k σ i u i v i T = U 1 S 1 V 1 T + U 2 S 2 V 2 T .
min w X p w y 2 , X p = U 1 S 1 V 1 T ,
w ̃ p = i = 1 p u i T y σ i v i .
L ( f , S , λ , P ) = i ( f ( x i ) y i ) 2 + λ P f 2 2 ,
w ̃ ( λ ) = i = 1 k u i T y f i σ i v i , f i = σ i 2 σ i 2 + λ .
Φ ( x ) , Φ ( z ) = k ( x , z ) .
f ( x ) h ( x ) M x f h H .
i , j l α i α j k ( x i , x i ) 0 .
f ( ) , k ( x , ) H = f ( x ) ,
k ( x , ) , k ( z , ) H = k ( x , z ) .
min f H 1 2 i = 1 l ( y i f ( x i ) ) 2 + λ f H 2 ,
k ( x , z ) = i = 1 λ i ϕ i ( x ) ϕ i ( z ) ,
f , g H = i = 1 l j = 1 k α i β j k ( x i , z j ) ,
f H 2 = i = 1 l j = 1 l α i α j k ( x i , x j ) = α T K α .
min α 1 2 ( y K α ) T ( y K α ) + λ α T K α ,
( K + λ I l ) α = y ,
Φ ( x ) , w = K test ( K + λ I l ) 1 y = y ̃ ,
K test ( K + λ I l ) 1 Y = Y ̃ .
k ( x , z ) = exp ( x z 2 2 σ 2 ) ,
k ( x , z ) = n = 1 λ n ϕ n ( x ) ϕ n ( z ) = n = 1 λ n exp ( i 2 π n , x ) exp ( i 2 π n , z ) .
k ( x , z ) = ( x , z + 1 ) d ,
{ ϕ i ( x ) } i = 1 10 = ( 1 , x 1 , x 2 , x 3 , x 1 2 , x 2 2 , x 3 2 , x 1 x 2 , x 1 x 3 , x 2 x 3 ) ,
{ λ i } i = 1 10 = ( 1 , ( 2 ) , ( 2 ) , ( 2 ) , 1 , 1 , 1 , ( ) 2 ) , ( 2 ) , ( 2 ) ) .
k d ( x , z ) = r = 0 d [ d r ] 2 d r + 1 min ( x , z ) 2 d r + 1 x z r + r = 0 d x r z r ,
k d ( x , z ) = i = 1 n k ( x i , z i ) .
k ( x , z ) = x z 2 2 m n .
f H m 2 = α 1 + + α n = m m ! α 1 ! α n ! ( m f x 1 α 1 x n α n ) 2 i = 1 n d x i .
f H 2 2 = Ω ( f x 1 x 1 2 + f x 2 x 2 2 + f x 3 x 3 2 + 2 [ f x 1 x 2 2 + f x 1 x 3 2 + f x 2 x 3 2 ] ) d x 1 d x 2 d x 3 .
r = [ n + m 1 n ] .
min { α , a } 1 2 ( y K α C a ) T ( y K α C a ) + λ α T K α ,
[ K + λ I C C T 0 ] w = [ y 0 ] ,
Δ E a b * = ( Δ L * ) 2 + ( Δ a * ) 2 + ( Δ b * ) 2 .
RMSE = r r ̃ 2 m ,

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