Abstract

The Bayesian inference approach to the inverse problem of spectral signal recovery has been extended to mixtures of Gaussian probability distributions of a training dataset in order to increase the efficiency of estimating the spectral signal from the response of a transformation system. Bayesian (BIC) and Akaike (AIC) information criteria were assessed in order to provide the Gaussian mixture model (GMM) with the optimum number of clusters within the spectral space. The spectra of 2600 solar illuminations measured in Granada (Spain) were recovered over the range of 360–830 nm from their corresponding tristimulus values using a linear model of basis functions, the Wiener inverse (WI) method, and the Bayesian inverse approach extended to the GMM (BGMM). A model of Gaussian mixtures for solar irradiance was deemed to be more appropriate than a single Gaussian distribution for representing the probability distribution of the solar spectral data. The results showed that the estimation performance of the BGMM method was better than either the linear model or the WI method for the spectral approximation of daylight from the three-dimensional tristimulus values.

© 2012 Optical Society of America

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  1. J. Parkkinen, J. Hallikainen, and T. Jääskeläinen, “Characteristic spectra of Munsell colors,” J. Opt. Soc. Am. A 6, 318–322 (1989).
    [CrossRef]
  2. A. Garcia-Beltrán, J. L. Nieves, J. Hernández-Andrés, and J. Romero, “Linear bases for spectral reflectance functions of acrylic paints,” Color Res. Appl. 23, 39–45 (1998).
    [CrossRef]
  3. G. Healey and L. Benites, “Linear models for spectral reflectance functions over the mid-wave and long-wave infrared,” J. Opt. Soc. Am. A 15, 2216–2227 (1998).
    [CrossRef]
  4. C. Chiao, D. Osorio, M. Vorobyev, and T. W. Cronin, “Characterization of natural illuminants in forests and the use of digital video data to reconstruct illuminant spectra,” J. Opt. Soc. Am. A 17, 1713–1721 (2000).
    [CrossRef]
  5. A. K. Romney and T. Indow, “A model for the simultaneous analysis of reflectance spectra and basis factors of Munsell color samples under D65 illumination in three-dimensional euclidean space,” Proc. Natl. Acad. Sci. USA 99, 11543–11546 (2002).
    [CrossRef]
  6. O. Kohonen, J. Parkkinen, and T. Jääskeläinen, “Databases for spectral color science,” Color Res. Appl. 31, 381–390 (2006).
    [CrossRef]
  7. T. Indow and A. K. Romney, “Reflectance spectra of Munsell standard chips and their appearance,” Color Res. Appl. 33, 229–237 (2008).
    [CrossRef]
  8. G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formula (Wiley, 1982).
  9. S. R. Das and V. D. P. Sastri, “Spectral distribution and color of tropical daylight,” J. Opt. Soc. Am. 55, 319–322 (1965).
    [CrossRef]
  10. S. R. Das and V. D. P. Sastri, “Spectral distribution and color of north sky at Delhi,” J. Opt. Soc. Am. 56, 829–830 (1966).
    [CrossRef]
  11. S. R. Das and V. D. P. Sastri, “Typical spectral distributions and color for tropical daylight,” J. Opt. Soc. Am. 58, 391–398 (1968).
    [CrossRef]
  12. E. Dixon, “Spectral distribution of Australian daylight,” J. Opt. Soc. Am. 68, 437–450 (1978).
    [CrossRef]
  13. J. Hernández-Andrés, J. Romero, J. L. Nieves, and R. L. Lee, “Color and spectral analysis of daylight in southern Europe,” J. Opt. Soc. Am. A 18, 1325–1335 (2001).
    [CrossRef]
  14. J. B. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychonomic Sci. 1, 369–370(1994).
  15. D. B. Judd, D. L. MacAdam, G. Wyszecki, H. W. Budde, H. R. Condit, S. T. Henderson, and J. L. Simonds, “Spectral distribution of typical daylight as a function of correlated color temperature,” J. Opt. Soc. Am. 54, 1031–1040 (1964).
    [CrossRef]
  16. L. T. Maloney, and B. A. Wandell, “Color constancy: a method for recovering surface spectral reflectance,” J. Opt. Soc. Am. A 3, 29–33 (1986).
    [CrossRef]
  17. 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]
  18. D. H. Marimont, and B. A. Wandell, “Linear models of surface and illuminant spectra,” J. Opt. Soc. Am. A 9, 1905–1913(1992).
    [CrossRef]
  19. J. Romero, A. Garcia-Beltrán, and J. Hernández-Andrés, “Linear bases for representation of natural and artificial illuminants,” J. Opt. Soc. Am. A 14, 1007–1014 (1997).
    [CrossRef]
  20. J. Hernández-Andrés, J. Romero, A. Garcia-Beltrán, and J. L. Nieves, “Testing linear models on spectral daylight measurements,” Appl. Opt. 37, 971–977 (1998).
    [CrossRef]
  21. J. Hernández-Andrés, J. L. Nieves, E. M. Valero, and J. Romero, “Spectral-daylight recovery by use of only a few sensors,” J. Opt. Soc. Am. A 21, 13–23 (2004).
    [CrossRef]
  22. D. Slater, and G. Healey, “Analyzing the spectral dimensionality of outdoor visible and near-infrared illumination functions,” J. Opt. Soc. Am. A 15, 2913–2920 (1998).
    [CrossRef]
  23. Z. Pan, G. Healey, and D. Slater, “Global spectral irradiance variability and material discrimination at Boulder, Colorado,” J. Opt. Soc. Am. A 20, 513–521 (2003).
    [CrossRef]
  24. D. H. Brainard, “Bayesian method for reconstructing color images from trichromatic samples,” in Proceedings of the IS&T 47th Annual Conference (IS&T, 1995), pp. 375–380.
  25. H. S. Fairman, and M. H. Brill, “The principal components of reflectances,” Color Res. Appl. 29, 104–110 (2004).
    [CrossRef]
  26. F. M. Abed, S. H. Amirshahi, S. Peyvandi, and M. R. M. Abed, “Reconstruction of the reflectance curves by using interpolation method,” in Proceedings of AIC 2007—Color Science for Industry, midterm meeting of the International Color Association (2007).
  27. F. Agahian, S. A. Amirshahi, and S. H. Amirshahi, “Reconstruction of reflectance spectra using weighted principal component analysis,” Color Res. Appl. 33, 360–371 (2008).
    [CrossRef]
  28. T. Harfi, S. H. Amirshahi, and F. Agahian, “Recovery of reflectance spectra from colorimetric data using principal component analysis embedded regression technique,” Opt. Rev. 15, 302–308 (2008).
    [CrossRef]
  29. T. Eslahi, S. H. Amirshahi, and F. Agahian, “Recovery of spectral data using weighted canonical correlation regression,” Opt. Rev. 16, 296–303 (2009).
    [CrossRef]
  30. S. A. Amirshahi, and S. H. Amirshahi, “Adaptive non-negative bases for reconstruction of spectral data from colorimetric information,” Opt. Rev. 17, 562–569 (2010).
    [CrossRef]
  31. S. Peyvandi, and S. H. Amirshahi, “Generalized spectral decomposition: a theory and practice to spectral reconstruction,” J. Opt. Soc. Am. A 28, 1545–1553 (2011).
    [CrossRef]
  32. Y. Murakami, K. Fukura, M. Yamaguchi, and N. Ohyama, “Color reproduction from low-snr multispectral images using spatio-spectral wiener estimation,” Opt. Express 16, 4106–4120(2008).
    [CrossRef]
  33. N. Shimano, “Recovery of spectral reflectances of objects being imaged without prior knowledge,” IEEE Trans. Image Process. 15, 1848–1856 (2006).
    [CrossRef]
  34. N. Shimano, K. Terai, and M. Hironaga, “Recovery of spectral reflectances of objects being imaged by multispectral cameras,” J. Opt. Soc. Am. A 24, 3211–3219 (2007).
    [CrossRef]
  35. N. Shimano, “Optimization of spectral sensitivities with gaussian distribution functions for a color image acquisition device in the presence of noise,” Opt. Eng. 45, 013201 (2006).
    [CrossRef]
  36. N. Shimano, and M. Hironaga, “Recovery of spectral reflectances of imaged objects by the use of features of spectral reflectances,” J. Opt. Soc. Am. A 27, 251–258 (2010).
    [CrossRef]
  37. A. Mohammad-Djafari, “Bayesian inference for inverse problems in signal and image processing and applications,” Int. J. Imaging Syst. Tech. 16, 209–214 (2006).
    [CrossRef]
  38. M. Bertero, and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).
  39. D. Tzeng, and R. S. Berns, “A review of principal component analysis and its applications to color technology,” Color Res. Appl. 30, 84–98 (2005).
    [CrossRef]
  40. D. H. Brainard and W. T. Freeman, “Bayesian color constancy,” J. Opt. Soc. Am. A 14, 1393–1411 (1997).
    [CrossRef]
  41. V. Heikkinen, R. Lenz, T. Jetsu, J. Parkkinen, M. Hauta-Kasari, and T. Jääskeläinen, “Evaluation and unification of some methods for estimating reflectance spectra from RGB images,” J. Opt. Soc. Am. A 25, 2444–2458 (2008).
    [CrossRef]
  42. Y. Murakami, K. Ietom, M. Yamaguchi, and N. Ohyama, “Maximum a posteriori estimation of spectral reflectance from color image and multipoint spectral measurements,” Appl. Opt. 46, 7068–7082 (2007).
    [CrossRef]
  43. D. Attewell and R. J. Baddeley, “The distribution of reflectances within the visual environment,” Vis. Res. 47, 548–554 (2007).
    [CrossRef]
  44. Y. Murakami, T. Obi, M. Yamaguchi, and N. Ohyama, “Nonlinear estimation of spectral reflectance based on gaussian mixture distribution for color image reproduction,” Appl. Opt. 41, 4840–4847 (2002).
    [CrossRef]
  45. J. MacQueen, “Some methods for classification and analysis of multivariate observations,” in Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability(University of California, 1967), pp. 281–297.
  46. A. Dempster, N. Laird, and D. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B. Methodol. 39B, 1–38 (1977).
  47. M. A. T. Figueiredo and A. K. Jain, “Unsupervised learning of finite mixture models,” IEEE Trans. Pattern Anal. Machine Intell. 24, 381–396 (2002).
    [CrossRef]
  48. G. McLachlan and D. Peel, Finite Mixture Models (Wiley, 2000).
  49. A. R. Liddle, “Information criteria for astrophysical model selection,” Mon. Not. R. Astron. Soc. 377, L74–L78 (2007).
    [CrossRef]
  50. H. Akaike, “A new look at the statistical model identification,” IEEE Trans. Autom. Control 19, 716–723 (1974).
    [CrossRef]
  51. G. Schwarz, “Estimating the dimension of a model,” Ann. Stat. 6, 461–464 (1978).
    [CrossRef]
  52. D. A. Harville, Matrix Algebra From A Statistician’s Prespective (Springer, 2008), pp. 419–420.
  53. A. J. Laub, Matrix Analysis for Scientists & Engineers (SIAM, 2005), p. 48.
  54. K. B. Petersen and M. S. Pedersen, “The matrix cookbook,” http://matrixcookbook.com (14Nov.2008).
  55. M. F. Huber, T. Bailey, H. Durrant-Whyte, and U. D. Hanebeck, “On entropy approximation for gaussian mixture random vectors,” in IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (IEEE, 2008), pp. 181–188.
  56. P. S. Dwyer, “Some applications of matrix derivatives in multivariate analysis,” J. Am. Stat. Assoc. 62, 607–625 (1967).
    [CrossRef]

2011

2010

S. A. Amirshahi, and S. H. Amirshahi, “Adaptive non-negative bases for reconstruction of spectral data from colorimetric information,” Opt. Rev. 17, 562–569 (2010).
[CrossRef]

N. Shimano, and M. Hironaga, “Recovery of spectral reflectances of imaged objects by the use of features of spectral reflectances,” J. Opt. Soc. Am. A 27, 251–258 (2010).
[CrossRef]

2009

T. Eslahi, S. H. Amirshahi, and F. Agahian, “Recovery of spectral data using weighted canonical correlation regression,” Opt. Rev. 16, 296–303 (2009).
[CrossRef]

2008

Y. Murakami, K. Fukura, M. Yamaguchi, and N. Ohyama, “Color reproduction from low-snr multispectral images using spatio-spectral wiener estimation,” Opt. Express 16, 4106–4120(2008).
[CrossRef]

F. Agahian, S. A. Amirshahi, and S. H. Amirshahi, “Reconstruction of reflectance spectra using weighted principal component analysis,” Color Res. Appl. 33, 360–371 (2008).
[CrossRef]

T. Harfi, S. H. Amirshahi, and F. Agahian, “Recovery of reflectance spectra from colorimetric data using principal component analysis embedded regression technique,” Opt. Rev. 15, 302–308 (2008).
[CrossRef]

T. Indow and A. K. Romney, “Reflectance spectra of Munsell standard chips and their appearance,” Color Res. Appl. 33, 229–237 (2008).
[CrossRef]

V. Heikkinen, R. Lenz, T. Jetsu, J. Parkkinen, M. Hauta-Kasari, and T. Jääskeläinen, “Evaluation and unification of some methods for estimating reflectance spectra from RGB images,” J. Opt. Soc. Am. A 25, 2444–2458 (2008).
[CrossRef]

2007

2006

N. Shimano, “Optimization of spectral sensitivities with gaussian distribution functions for a color image acquisition device in the presence of noise,” Opt. Eng. 45, 013201 (2006).
[CrossRef]

O. Kohonen, J. Parkkinen, and T. Jääskeläinen, “Databases for spectral color science,” Color Res. Appl. 31, 381–390 (2006).
[CrossRef]

N. Shimano, “Recovery of spectral reflectances of objects being imaged without prior knowledge,” IEEE Trans. Image Process. 15, 1848–1856 (2006).
[CrossRef]

A. Mohammad-Djafari, “Bayesian inference for inverse problems in signal and image processing and applications,” Int. J. Imaging Syst. Tech. 16, 209–214 (2006).
[CrossRef]

2005

D. Tzeng, and R. S. Berns, “A review of principal component analysis and its applications to color technology,” Color Res. Appl. 30, 84–98 (2005).
[CrossRef]

2004

2003

2002

A. K. Romney and T. Indow, “A model for the simultaneous analysis of reflectance spectra and basis factors of Munsell color samples under D65 illumination in three-dimensional euclidean space,” Proc. Natl. Acad. Sci. USA 99, 11543–11546 (2002).
[CrossRef]

Y. Murakami, T. Obi, M. Yamaguchi, and N. Ohyama, “Nonlinear estimation of spectral reflectance based on gaussian mixture distribution for color image reproduction,” Appl. Opt. 41, 4840–4847 (2002).
[CrossRef]

M. A. T. Figueiredo and A. K. Jain, “Unsupervised learning of finite mixture models,” IEEE Trans. Pattern Anal. Machine Intell. 24, 381–396 (2002).
[CrossRef]

2001

2000

1998

1997

1994

J. B. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychonomic Sci. 1, 369–370(1994).

1992

1989

1986

1978

E. Dixon, “Spectral distribution of Australian daylight,” J. Opt. Soc. Am. 68, 437–450 (1978).
[CrossRef]

G. Schwarz, “Estimating the dimension of a model,” Ann. Stat. 6, 461–464 (1978).
[CrossRef]

1977

A. Dempster, N. Laird, and D. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B. Methodol. 39B, 1–38 (1977).

1974

H. Akaike, “A new look at the statistical model identification,” IEEE Trans. Autom. Control 19, 716–723 (1974).
[CrossRef]

1968

1967

P. S. Dwyer, “Some applications of matrix derivatives in multivariate analysis,” J. Am. Stat. Assoc. 62, 607–625 (1967).
[CrossRef]

1966

1965

1964

Abed, F. M.

F. M. Abed, S. H. Amirshahi, S. Peyvandi, and M. R. M. Abed, “Reconstruction of the reflectance curves by using interpolation method,” in Proceedings of AIC 2007—Color Science for Industry, midterm meeting of the International Color Association (2007).

Abed, M. R. M.

F. M. Abed, S. H. Amirshahi, S. Peyvandi, and M. R. M. Abed, “Reconstruction of the reflectance curves by using interpolation method,” in Proceedings of AIC 2007—Color Science for Industry, midterm meeting of the International Color Association (2007).

Agahian, F.

T. Eslahi, S. H. Amirshahi, and F. Agahian, “Recovery of spectral data using weighted canonical correlation regression,” Opt. Rev. 16, 296–303 (2009).
[CrossRef]

T. Harfi, S. H. Amirshahi, and F. Agahian, “Recovery of reflectance spectra from colorimetric data using principal component analysis embedded regression technique,” Opt. Rev. 15, 302–308 (2008).
[CrossRef]

F. Agahian, S. A. Amirshahi, and S. H. Amirshahi, “Reconstruction of reflectance spectra using weighted principal component analysis,” Color Res. Appl. 33, 360–371 (2008).
[CrossRef]

Akaike, H.

H. Akaike, “A new look at the statistical model identification,” IEEE Trans. Autom. Control 19, 716–723 (1974).
[CrossRef]

Amirshahi, S. A.

S. A. Amirshahi, and S. H. Amirshahi, “Adaptive non-negative bases for reconstruction of spectral data from colorimetric information,” Opt. Rev. 17, 562–569 (2010).
[CrossRef]

F. Agahian, S. A. Amirshahi, and S. H. Amirshahi, “Reconstruction of reflectance spectra using weighted principal component analysis,” Color Res. Appl. 33, 360–371 (2008).
[CrossRef]

Amirshahi, S. H.

S. Peyvandi, and S. H. Amirshahi, “Generalized spectral decomposition: a theory and practice to spectral reconstruction,” J. Opt. Soc. Am. A 28, 1545–1553 (2011).
[CrossRef]

S. A. Amirshahi, and S. H. Amirshahi, “Adaptive non-negative bases for reconstruction of spectral data from colorimetric information,” Opt. Rev. 17, 562–569 (2010).
[CrossRef]

T. Eslahi, S. H. Amirshahi, and F. Agahian, “Recovery of spectral data using weighted canonical correlation regression,” Opt. Rev. 16, 296–303 (2009).
[CrossRef]

T. Harfi, S. H. Amirshahi, and F. Agahian, “Recovery of reflectance spectra from colorimetric data using principal component analysis embedded regression technique,” Opt. Rev. 15, 302–308 (2008).
[CrossRef]

F. Agahian, S. A. Amirshahi, and S. H. Amirshahi, “Reconstruction of reflectance spectra using weighted principal component analysis,” Color Res. Appl. 33, 360–371 (2008).
[CrossRef]

F. M. Abed, S. H. Amirshahi, S. Peyvandi, and M. R. M. Abed, “Reconstruction of the reflectance curves by using interpolation method,” in Proceedings of AIC 2007—Color Science for Industry, midterm meeting of the International Color Association (2007).

Attewell, D.

D. Attewell and R. J. Baddeley, “The distribution of reflectances within the visual environment,” Vis. Res. 47, 548–554 (2007).
[CrossRef]

Baddeley, R. J.

D. Attewell and R. J. Baddeley, “The distribution of reflectances within the visual environment,” Vis. Res. 47, 548–554 (2007).
[CrossRef]

Bailey, T.

M. F. Huber, T. Bailey, H. Durrant-Whyte, and U. D. Hanebeck, “On entropy approximation for gaussian mixture random vectors,” in IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (IEEE, 2008), pp. 181–188.

Benites, L.

Berns, R. S.

D. Tzeng, and R. S. Berns, “A review of principal component analysis and its applications to color technology,” Color Res. Appl. 30, 84–98 (2005).
[CrossRef]

Bertero, M.

M. Bertero, and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).

Boccacci, P.

M. Bertero, and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).

Brainard, D. H.

D. H. Brainard and W. T. Freeman, “Bayesian color constancy,” J. Opt. Soc. Am. A 14, 1393–1411 (1997).
[CrossRef]

D. H. Brainard, “Bayesian method for reconstructing color images from trichromatic samples,” in Proceedings of the IS&T 47th Annual Conference (IS&T, 1995), pp. 375–380.

Brill, M. H.

H. S. Fairman, and M. H. Brill, “The principal components of reflectances,” Color Res. Appl. 29, 104–110 (2004).
[CrossRef]

Budde, H. W.

Chiao, C.

Cohen, J. B.

J. B. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychonomic Sci. 1, 369–370(1994).

Condit, H. R.

Cronin, T. W.

Das, S. R.

Dempster, A.

A. Dempster, N. Laird, and D. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B. Methodol. 39B, 1–38 (1977).

Dixon, E.

Durrant-Whyte, H.

M. F. Huber, T. Bailey, H. Durrant-Whyte, and U. D. Hanebeck, “On entropy approximation for gaussian mixture random vectors,” in IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (IEEE, 2008), pp. 181–188.

Dwyer, P. S.

P. S. Dwyer, “Some applications of matrix derivatives in multivariate analysis,” J. Am. Stat. Assoc. 62, 607–625 (1967).
[CrossRef]

Eslahi, T.

T. Eslahi, S. H. Amirshahi, and F. Agahian, “Recovery of spectral data using weighted canonical correlation regression,” Opt. Rev. 16, 296–303 (2009).
[CrossRef]

Fairman, H. S.

H. S. Fairman, and M. H. Brill, “The principal components of reflectances,” Color Res. Appl. 29, 104–110 (2004).
[CrossRef]

Figueiredo, M. A. T.

M. A. T. Figueiredo and A. K. Jain, “Unsupervised learning of finite mixture models,” IEEE Trans. Pattern Anal. Machine Intell. 24, 381–396 (2002).
[CrossRef]

Freeman, W. T.

Fukura, K.

Garcia-Beltrán, A.

Hallikainen, J.

Hanebeck, U. D.

M. F. Huber, T. Bailey, H. Durrant-Whyte, and U. D. Hanebeck, “On entropy approximation for gaussian mixture random vectors,” in IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (IEEE, 2008), pp. 181–188.

Harfi, T.

T. Harfi, S. H. Amirshahi, and F. Agahian, “Recovery of reflectance spectra from colorimetric data using principal component analysis embedded regression technique,” Opt. Rev. 15, 302–308 (2008).
[CrossRef]

Harville, D. A.

D. A. Harville, Matrix Algebra From A Statistician’s Prespective (Springer, 2008), pp. 419–420.

Hauta-Kasari, M.

Healey, G.

Heikkinen, V.

Henderson, S. T.

Hernández-Andrés, J.

Hironaga, M.

Huber, M. F.

M. F. Huber, T. Bailey, H. Durrant-Whyte, and U. D. Hanebeck, “On entropy approximation for gaussian mixture random vectors,” in IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (IEEE, 2008), pp. 181–188.

Ietom, K.

Indow, T.

T. Indow and A. K. Romney, “Reflectance spectra of Munsell standard chips and their appearance,” Color Res. Appl. 33, 229–237 (2008).
[CrossRef]

A. K. Romney and T. Indow, “A model for the simultaneous analysis of reflectance spectra and basis factors of Munsell color samples under D65 illumination in three-dimensional euclidean space,” Proc. Natl. Acad. Sci. USA 99, 11543–11546 (2002).
[CrossRef]

Jääskeläinen, T.

Jain, A. K.

M. A. T. Figueiredo and A. K. Jain, “Unsupervised learning of finite mixture models,” IEEE Trans. Pattern Anal. Machine Intell. 24, 381–396 (2002).
[CrossRef]

Jetsu, T.

Judd, D. B.

Kohonen, O.

O. Kohonen, J. Parkkinen, and T. Jääskeläinen, “Databases for spectral color science,” Color Res. Appl. 31, 381–390 (2006).
[CrossRef]

Laird, N.

A. Dempster, N. Laird, and D. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B. Methodol. 39B, 1–38 (1977).

Laub, A. J.

A. J. Laub, Matrix Analysis for Scientists & Engineers (SIAM, 2005), p. 48.

Lee, R. L.

Lenz, R.

Liddle, A. R.

A. R. Liddle, “Information criteria for astrophysical model selection,” Mon. Not. R. Astron. Soc. 377, L74–L78 (2007).
[CrossRef]

MacAdam, D. L.

MacQueen, J.

J. MacQueen, “Some methods for classification and analysis of multivariate observations,” in Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability(University of California, 1967), pp. 281–297.

Maloney, L. T.

Marimont, D. H.

McLachlan, G.

G. McLachlan and D. Peel, Finite Mixture Models (Wiley, 2000).

Mohammad-Djafari, A.

A. Mohammad-Djafari, “Bayesian inference for inverse problems in signal and image processing and applications,” Int. J. Imaging Syst. Tech. 16, 209–214 (2006).
[CrossRef]

Murakami, Y.

Nieves, J. L.

Obi, T.

Ohyama, N.

Osorio, D.

Pan, Z.

Parkkinen, J.

Peel, D.

G. McLachlan and D. Peel, Finite Mixture Models (Wiley, 2000).

Peyvandi, S.

S. Peyvandi, and S. H. Amirshahi, “Generalized spectral decomposition: a theory and practice to spectral reconstruction,” J. Opt. Soc. Am. A 28, 1545–1553 (2011).
[CrossRef]

F. M. Abed, S. H. Amirshahi, S. Peyvandi, and M. R. M. Abed, “Reconstruction of the reflectance curves by using interpolation method,” in Proceedings of AIC 2007—Color Science for Industry, midterm meeting of the International Color Association (2007).

Romero, J.

Romney, A. K.

T. Indow and A. K. Romney, “Reflectance spectra of Munsell standard chips and their appearance,” Color Res. Appl. 33, 229–237 (2008).
[CrossRef]

A. K. Romney and T. Indow, “A model for the simultaneous analysis of reflectance spectra and basis factors of Munsell color samples under D65 illumination in three-dimensional euclidean space,” Proc. Natl. Acad. Sci. USA 99, 11543–11546 (2002).
[CrossRef]

Rubin, D.

A. Dempster, N. Laird, and D. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B. Methodol. 39B, 1–38 (1977).

Sastri, V. D. P.

Schwarz, G.

G. Schwarz, “Estimating the dimension of a model,” Ann. Stat. 6, 461–464 (1978).
[CrossRef]

Shimano, N.

N. Shimano, and M. Hironaga, “Recovery of spectral reflectances of imaged objects by the use of features of spectral reflectances,” J. Opt. Soc. Am. A 27, 251–258 (2010).
[CrossRef]

N. Shimano, K. Terai, and M. Hironaga, “Recovery of spectral reflectances of objects being imaged by multispectral cameras,” J. Opt. Soc. Am. A 24, 3211–3219 (2007).
[CrossRef]

N. Shimano, “Recovery of spectral reflectances of objects being imaged without prior knowledge,” IEEE Trans. Image Process. 15, 1848–1856 (2006).
[CrossRef]

N. Shimano, “Optimization of spectral sensitivities with gaussian distribution functions for a color image acquisition device in the presence of noise,” Opt. Eng. 45, 013201 (2006).
[CrossRef]

Simonds, J. L.

Slater, D.

Stiles, W. S.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formula (Wiley, 1982).

Terai, K.

Tzeng, D.

D. Tzeng, and R. S. Berns, “A review of principal component analysis and its applications to color technology,” Color Res. Appl. 30, 84–98 (2005).
[CrossRef]

Valero, E. M.

Vorobyev, M.

Wandell, B. A.

Wyszecki, G.

Yamaguchi, M.

Ann. Stat.

G. Schwarz, “Estimating the dimension of a model,” Ann. Stat. 6, 461–464 (1978).
[CrossRef]

Appl. Opt.

Color Res. Appl.

O. Kohonen, J. Parkkinen, and T. Jääskeläinen, “Databases for spectral color science,” Color Res. Appl. 31, 381–390 (2006).
[CrossRef]

T. Indow and A. K. Romney, “Reflectance spectra of Munsell standard chips and their appearance,” Color Res. Appl. 33, 229–237 (2008).
[CrossRef]

A. Garcia-Beltrán, J. L. Nieves, J. Hernández-Andrés, and J. Romero, “Linear bases for spectral reflectance functions of acrylic paints,” Color Res. Appl. 23, 39–45 (1998).
[CrossRef]

F. Agahian, S. A. Amirshahi, and S. H. Amirshahi, “Reconstruction of reflectance spectra using weighted principal component analysis,” Color Res. Appl. 33, 360–371 (2008).
[CrossRef]

H. S. Fairman, and M. H. Brill, “The principal components of reflectances,” Color Res. Appl. 29, 104–110 (2004).
[CrossRef]

D. Tzeng, and R. S. Berns, “A review of principal component analysis and its applications to color technology,” Color Res. Appl. 30, 84–98 (2005).
[CrossRef]

IEEE Trans. Autom. Control

H. Akaike, “A new look at the statistical model identification,” IEEE Trans. Autom. Control 19, 716–723 (1974).
[CrossRef]

IEEE Trans. Image Process.

N. Shimano, “Recovery of spectral reflectances of objects being imaged without prior knowledge,” IEEE Trans. Image Process. 15, 1848–1856 (2006).
[CrossRef]

IEEE Trans. Pattern Anal. Machine Intell.

M. A. T. Figueiredo and A. K. Jain, “Unsupervised learning of finite mixture models,” IEEE Trans. Pattern Anal. Machine Intell. 24, 381–396 (2002).
[CrossRef]

Int. J. Imaging Syst. Tech.

A. Mohammad-Djafari, “Bayesian inference for inverse problems in signal and image processing and applications,” Int. J. Imaging Syst. Tech. 16, 209–214 (2006).
[CrossRef]

J. Am. Stat. Assoc.

P. S. Dwyer, “Some applications of matrix derivatives in multivariate analysis,” J. Am. Stat. Assoc. 62, 607–625 (1967).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

C. Chiao, D. Osorio, M. Vorobyev, and T. W. Cronin, “Characterization of natural illuminants in forests and the use of digital video data to reconstruct illuminant spectra,” J. Opt. Soc. Am. A 17, 1713–1721 (2000).
[CrossRef]

D. H. Marimont, and B. A. Wandell, “Linear models of surface and illuminant spectra,” J. Opt. Soc. Am. A 9, 1905–1913(1992).
[CrossRef]

J. Hernández-Andrés, J. Romero, J. L. Nieves, and R. L. Lee, “Color and spectral analysis of daylight in southern Europe,” J. Opt. Soc. Am. A 18, 1325–1335 (2001).
[CrossRef]

V. Heikkinen, R. Lenz, T. Jetsu, J. Parkkinen, M. Hauta-Kasari, and T. Jääskeläinen, “Evaluation and unification of some methods for estimating reflectance spectra from RGB images,” J. Opt. Soc. Am. A 25, 2444–2458 (2008).
[CrossRef]

N. Shimano, and M. Hironaga, “Recovery of spectral reflectances of imaged objects by the use of features of spectral reflectances,” J. Opt. Soc. Am. A 27, 251–258 (2010).
[CrossRef]

S. Peyvandi, and S. H. Amirshahi, “Generalized spectral decomposition: a theory and practice to spectral reconstruction,” J. Opt. Soc. Am. A 28, 1545–1553 (2011).
[CrossRef]

Z. Pan, G. Healey, and D. Slater, “Global spectral irradiance variability and material discrimination at Boulder, Colorado,” J. Opt. Soc. Am. A 20, 513–521 (2003).
[CrossRef]

J. Hernández-Andrés, J. L. Nieves, E. M. Valero, and J. Romero, “Spectral-daylight recovery by use of only a few sensors,” J. Opt. Soc. Am. A 21, 13–23 (2004).
[CrossRef]

N. Shimano, K. Terai, and M. Hironaga, “Recovery of spectral reflectances of objects being imaged by multispectral cameras,” J. Opt. Soc. Am. A 24, 3211–3219 (2007).
[CrossRef]

G. Healey and L. Benites, “Linear models for spectral reflectance functions over the mid-wave and long-wave infrared,” J. Opt. Soc. Am. A 15, 2216–2227 (1998).
[CrossRef]

D. Slater, and G. Healey, “Analyzing the spectral dimensionality of outdoor visible and near-infrared illumination functions,” J. Opt. Soc. Am. A 15, 2913–2920 (1998).
[CrossRef]

J. Romero, A. Garcia-Beltrán, and J. Hernández-Andrés, “Linear bases for representation of natural and artificial illuminants,” J. Opt. Soc. Am. A 14, 1007–1014 (1997).
[CrossRef]

D. H. Brainard and W. T. Freeman, “Bayesian color constancy,” J. Opt. Soc. Am. A 14, 1393–1411 (1997).
[CrossRef]

L. T. Maloney, and B. A. Wandell, “Color constancy: a method for recovering surface spectral reflectance,” J. Opt. Soc. Am. A 3, 29–33 (1986).
[CrossRef]

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]

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

J. R. Stat. Soc. Ser. B. Methodol.

A. Dempster, N. Laird, and D. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B. Methodol. 39B, 1–38 (1977).

Mon. Not. R. Astron. Soc.

A. R. Liddle, “Information criteria for astrophysical model selection,” Mon. Not. R. Astron. Soc. 377, L74–L78 (2007).
[CrossRef]

Opt. Eng.

N. Shimano, “Optimization of spectral sensitivities with gaussian distribution functions for a color image acquisition device in the presence of noise,” Opt. Eng. 45, 013201 (2006).
[CrossRef]

Opt. Express

Opt. Rev.

T. Harfi, S. H. Amirshahi, and F. Agahian, “Recovery of reflectance spectra from colorimetric data using principal component analysis embedded regression technique,” Opt. Rev. 15, 302–308 (2008).
[CrossRef]

T. Eslahi, S. H. Amirshahi, and F. Agahian, “Recovery of spectral data using weighted canonical correlation regression,” Opt. Rev. 16, 296–303 (2009).
[CrossRef]

S. A. Amirshahi, and S. H. Amirshahi, “Adaptive non-negative bases for reconstruction of spectral data from colorimetric information,” Opt. Rev. 17, 562–569 (2010).
[CrossRef]

Proc. Natl. Acad. Sci. USA

A. K. Romney and T. Indow, “A model for the simultaneous analysis of reflectance spectra and basis factors of Munsell color samples under D65 illumination in three-dimensional euclidean space,” Proc. Natl. Acad. Sci. USA 99, 11543–11546 (2002).
[CrossRef]

Psychonomic Sci.

J. B. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychonomic Sci. 1, 369–370(1994).

Vis. Res.

D. Attewell and R. J. Baddeley, “The distribution of reflectances within the visual environment,” Vis. Res. 47, 548–554 (2007).
[CrossRef]

Other

D. H. Brainard, “Bayesian method for reconstructing color images from trichromatic samples,” in Proceedings of the IS&T 47th Annual Conference (IS&T, 1995), pp. 375–380.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formula (Wiley, 1982).

J. MacQueen, “Some methods for classification and analysis of multivariate observations,” in Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability(University of California, 1967), pp. 281–297.

M. Bertero, and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).

F. M. Abed, S. H. Amirshahi, S. Peyvandi, and M. R. M. Abed, “Reconstruction of the reflectance curves by using interpolation method,” in Proceedings of AIC 2007—Color Science for Industry, midterm meeting of the International Color Association (2007).

G. McLachlan and D. Peel, Finite Mixture Models (Wiley, 2000).

D. A. Harville, Matrix Algebra From A Statistician’s Prespective (Springer, 2008), pp. 419–420.

A. J. Laub, Matrix Analysis for Scientists & Engineers (SIAM, 2005), p. 48.

K. B. Petersen and M. S. Pedersen, “The matrix cookbook,” http://matrixcookbook.com (14Nov.2008).

M. F. Huber, T. Bailey, H. Durrant-Whyte, and U. D. Hanebeck, “On entropy approximation for gaussian mixture random vectors,” in IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (IEEE, 2008), pp. 181–188.

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Figures (5)

Fig. 1.
Fig. 1.

The CIE 1931 chromaticity coordinates of 2600 natural-daylight spectra (open blue circles) overlaid with the CIE daylight (red solid curve) and Planckian (blue solid curve with black circles) loci.

Fig. 2.
Fig. 2.

BIC and AIC of GMM with different numbers of clusters L=1,,8 created to represent the probability distribution of the primary spectral daylight illumination.

Fig. 3.
Fig. 3.

The log-pdf functions for the CIE 1931 chromaticity coordinates of 2600 natural-daylight spectra, the probability distributions of which were illustrated separately by the five Gaussian mixtures and a single Gaussian distribution. The contour lines for the two models are also shown in the figure.

Fig. 4.
Fig. 4.

The spectra of the centers μj*, j=1,,5 of the five Gaussian clusters (a), and the corresponding chromaticity coordinates (b).

Fig. 5.
Fig. 5.

Spectral recovery of the six natural outdoor illuminations (solid curve) using the linear model (dashed curve), the WI method (crossed curve), and the BGMM (dotted curve), presented for the spectral illuminations with CCT equal to (a) 3680, (b) 4117, (c) 7657, (d) 15812, (e) 23303, and (f) 33354 K.

Tables (1)

Tables Icon

Table 1. Spectral and Colorimetric Performance of Spectral Reconstruction Using the Linear Model, the WI Approach, and the Bayesian Inverse Model Extended to the GMM (BGMM)a

Equations (41)

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c=ATr+ϵ,
r^=Sc+(ISAT)μr,
p(r|c)=p(c|r)p(r)p(c),
μr|c=Ξ(cμϵATμr)+μr,
Σr|c=ΣrΞATΣr,
p(r)=j=1LωjN(r;μj,Σj),
p(r|c)=j=1Lωj*N(r;μj*,Σj*),
r^=j=1Lωj*μj*.
r^MAP=argmaxr{p(r|c)}=argmaxr{j=1Lωj*N(r;μj*,Σj*)},
r^MAP{j=1Lωj*2pj*(μ˜)}1j=1Lωj*{pj*(μ˜)μ˜2pj*(μ˜)}.
μ˜=argmaxmM{p(m|c)},
AIC=2lnL+2K,
BIC=2lnL+KlnN,
GFC=r,r^r,r1/2r^,r^1/2,
p(r)=j=1Lωj((2π)n|Σj|)1/2exp(Γj),
Γj=12(rμj)TΣj1(rμj)
p(c|r)=((2π)p|Σϵ|)1/2exp(ϒ),
ϒ=12[c(ATr+μϵ)]TΣϵ1[c(ATr+μϵ)].
p(c)=j=1Lωj((2π)p|Kj|)1/2exp(Φj),
Φj=12(cκj)TKj1(cκj),
p(r|c)=((2π)p|Σϵ|)1/2exp(ϒ)j=1Lωj((2π)n|Σj|)1/2exp(Γj)j=1Lωj((2π)p|Kj|)1/2exp(Φj)=((2π)p|Σϵ|)1/2j=1Lωj((2π)n|Σj|)1/2exp(Φj)exp(ϒ+ΓjΦj)j=1Lωj((2π)p|Kj|)1/2exp(Φj).
Γj*=12[c(ATr+μϵ)]TΣϵ1[c(ATr+μϵ)]12(rμj)TΣj1(rμj)+12(cκj)TKj1(cκj),
Γj*=12(rμj*)TΣ*j1(rμj*),
Σj*=(Σj1+AΣϵ1AT)1,
μj*=Σj*(Σj1μj+AΣϵ1cAΣϵ1μϵ).
((2π)p|Σϵ|)1/2((2π)n|Σj|)1/2((2π)p|Kj|)1/2=((2π)n|(Σj1+AΣϵ1AT)1|)1/2=((2π)n|Σj*|)1/2.
p(r|c)=j=1Lωj((2π)p|Kj|)1/2((2π)n|Σj*|)1/2exp(Φj)exp(Γj*)j=1Lωj((2π)p|Kj|)1/2exp(Φj),
p(r|c)=j=1Lωj*((2π)n|Σj*|)1/2exp(Γj*),
ωj*=ωj((2π)p|Kj|)1/2exp(Φj)j=1Lωj((2π)p|Kj|)1/2exp(Φj)=ωjN(c;κj,Kj)j=1LωjN(c;κj,Kj)
Σj*=ΣjSjATΣj,
μj*=Sj(cμϵATμj)+μj,
Sj=ΣjA(ATΣjA+Σϵ)1=ΣjAKj1.
p(x)=k=0R1k!((xμ˜))kp(x)|x=μ˜+OR,
pj*(r)=pj*(μ˜)+(rμ˜)Tpj*(μ˜)+12!(rμ˜)T2pj*(μ˜)(rμ˜)+13!{[(rμ˜)(rμ˜)T](rμ˜)}3pj*(μ˜)+,
pj*(x)=Σ*j1(xμj*)pj*(x),
2pj*(x)=Hj(x)=Σ*j1[(xμj*)(pj*(x))T+pj*(x)],
3pj*(x)=Hj(x)(Σ*j1(xμj*))pj*(x)Σ*j1vec(Σ*j1)(pj*(x))T,
p(r|c)j=1Lωj*{pj*(μ˜)+(rμ˜)Tpj*(μ˜)+12(rμ˜)T2pj*(μ˜)(rμ˜)}.
p(r|c)r=0.
p(r|c)rj=1Lωj*{pj*(μ˜)+Hj(μ˜)(rμ˜)}=0.
r^MAP{j=1Lωj*Hj(μ˜)}1j=1Lωj*{pj*(μ˜)μ˜Hj(μ˜)},

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