Abstract

The measured light spectrum is the result of an illuminant interacting with a surface. The illuminant spectral power distribution multiplies the surface spectral reflectance function to form a color signal—the light spectrum that gives rise to our perception. Disambiguation of the two factors, illuminant and surface, is difficult without prior knowledge. Previously [IEEE Trans. Pattern Anal. Mach. Intell. 12, 966 (1990) ; J. Opt. Soc. Am. A 21, 1825 (2004) ], one approach to this problem applied a finite-dimensional basis function model to recover the separate illuminant and surface reflectance components that make up the color signal, using principal component bases for lights and for reflectances. We introduce the idea of making use of finite-dimensional models of logarithms of spectra for this problem. Recognizing that multiplications turn into additions in such a formulation, we can replace the original iterative method with a direct, analytic algorithm with no iteration, resulting in a speedup of several orders of magnitude. Moreover, in the new, logarithm-based approach, it is straightforward to further design new basis functions, for both illuminant and reflectance simultaneously, such that the initial basis function coefficients derived from the input color signal are optimally mapped onto separate coefficients that produce spectra that more closely approximate the illuminant and the surface reflectance for any given dimensionality. This is accomplished by using an extra bias correction step that maps the analytically determined basis function coefficients onto the optimal coefficient set, separately for lights and surfaces, for the training set. The analytic equation plus the bias correction is then used for unknown input color signals.

© 2007 Optical Society of America

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References

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  1. S. Nevas, F. Manoocheri, and E. Ikonen, "Gonioreflectometer for measuring spectral diffuse reflectance," Appl. Opt. 9, 6391-6399 (2004).
    [CrossRef]
  2. These data are available at http://www.multispectral.org/.
  3. S. Bergner, T. Möller, M. K. Tory, and M. Drew, "A practical approach to spectral volume rendering," IEEE Trans. Vis. Comput. Graph. 11, 207-216 (2005).
    [CrossRef] [PubMed]
  4. Q. Wu, L. Zeng, H. Ke, W. Xie, H. Zheng, and Y. Zhang, "Analysis of blood and bone marrow smears using multispectral imaging analysis techniques," in Medical Imaging 2005: Image Processing, J.M. Fitzpatrick and J.M. Reinhardt, eds., Proc. SPIE 5747, 1872-1882 (2005).
  5. M. Roula, J. Diamond, A. Bouridane, P. Miller, and A. Amira, "A multispectral computer vision system for automatic grading of prostatic neoplasia," in Proceedings of IEEE 2002 International Symposium on Biomedical Imaging (IEEE, 2002), pp. 193-196.
    [CrossRef]
  6. N. Tsumura, "Appearance reproduction and multi-spectral imaging," in AIC Colour 05: Proceedings of the 10th Congress of the International Colour Association, J.L. Nieves and J. Hernández-Andrés, eds. (AIC, Granada, 2005), pp. 119-123.
  7. R. Berns, "Rejuvenating Seurat's palette using color and imaging science: a simulation," in Seurat and the Making of La Grande Jatte, R.L. Herbert, ed. (University of California Press, 2004), pp. 214-227.
  8. H. Barrow and J. Tenenbaum, "Recovering intrinsic scene characteristics from images," in Computer Vision Systems, A. Hanson and E. Riseman, eds. (Academic, 1978), pp. 3-26.
  9. J. Ho, B. Funt, and M. Drew, "Separating a color signal into illumination and surface reflectance components: theory and applications," IEEE Trans. Pattern Anal. Mach. Intell. 12, 966-977 (1990). Reprinted in Physics-Based Vision. Principles and Practice, G.E. Healey, S.A. Shafer, and L.B., eds. (Wolff, Jones, and Bartlett, 1992), Vol. 2, p. 272.
    [CrossRef]
  10. B. Wandell, "The synthesis and analysis of color images," IEEE Trans. Pattern Anal. Mach. Intell. PAMI-9, 2-13 (1987).
    [CrossRef]
  11. D. H. Marimont and B. A. Wandell, "Linear models of surface and illuminant spectra," J. Opt. Soc. Am. A 9, 1905-1913 (1992).
    [CrossRef] [PubMed]
  12. J. Marchant and C. Onyango, "Spectral invariance under daylight illumination changes," J. Opt. Soc. Am. A 19, 840-848 (2002).
    [CrossRef]
  13. M. Drew and B. V. Funt, "Natural metamers," CVGIP: Image Understand. 56, 139-151 (1992).
    [CrossRef]
  14. B. Funt, "Modeling reflectance by logarithmic basis functions," in Color Imaging Conference: Color, Science, Systems and Applications [Society for Imaging Science & Technology (IS&T)/Society for Information Display (SID), 1993], pp. 68-71.
  15. E. Angelopoulou, "Objective colour from multispectral imaging," in ECCV 2000: European Conference on Computer Vision, D.Vernon, ed. (Springer, 2000), pp. 359-374.
    [CrossRef]
  16. R. Lenz, P. Meer, and M. Hauta-Kasari, "Spectral-based illumination estimation and color correction," Color Res. Appl. 24, 98-111 (1999).
    [CrossRef]
  17. G. Healey and K. Chandra, "Estimating visible through near-infrared spectral reflectance from a sensor radiance spectrum," J. Opt. Soc. Am. A 21, 1825-1833 (2004).
    [CrossRef]
  18. D. Judd, D. MacAdam, and G. Wyszecki, "Spectral distribution of typical daylight as a function of correlated color temperature," J. Opt. Soc. Am. 54, 1031-1040 (1964).
    [CrossRef]
  19. S. Nascimento, D. Foster, and K. Amano, "Psychophysical estimates of the number of spectral-reflectance basis functions needed to reproduce natural scenes," J. Opt. Soc. Am. A 22, 1017-1022 (2005).
    [CrossRef]
  20. J. Faires and R. Burden, Numerical Analysis, 3rd ed. (Brooks/Cole, 2003).
  21. Z.-N. Li and M. Drew, Fundamentals of Multimedia (Prentice Hall, 2004).
  22. A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley, 2001).
    [CrossRef]
  23. A. Tikhonov and V. Arsenin, Solutions of Ill-Posed Problems (Wiley, 1977).
  24. D. Brainard, B. Wandell, and W. Cowan, "Black light: how sensors filter spectral variation of illuminant," IEEE Trans. Biomed. Eng. 36, 140-149 (1989).
    [CrossRef] [PubMed]
  25. K. Barnard, L. Martin, B. Funt, and A. Coath, "A data set for colour research," Color Res. Appl. 27, 147-151 (2002).
    [CrossRef]
  26. M. Vrhel, R. Gershon, and L. Iwan, "Measurement and analysis of object reflectance spectra," Color Res. Appl. 19, 4-9 (1994).
  27. G. Wyszecki and W. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulas, 2nd ed. (Wiley, 1982).
  28. C. McCamy, H. Marcus, and J. Davidson, "A color-rendition chart," J. Appl. Photogr. Eng. 2, 95-99 (1976).

2005

S. Bergner, T. Möller, M. K. Tory, and M. Drew, "A practical approach to spectral volume rendering," IEEE Trans. Vis. Comput. Graph. 11, 207-216 (2005).
[CrossRef] [PubMed]

S. Nascimento, D. Foster, and K. Amano, "Psychophysical estimates of the number of spectral-reflectance basis functions needed to reproduce natural scenes," J. Opt. Soc. Am. A 22, 1017-1022 (2005).
[CrossRef]

2004

G. Healey and K. Chandra, "Estimating visible through near-infrared spectral reflectance from a sensor radiance spectrum," J. Opt. Soc. Am. A 21, 1825-1833 (2004).
[CrossRef]

S. Nevas, F. Manoocheri, and E. Ikonen, "Gonioreflectometer for measuring spectral diffuse reflectance," Appl. Opt. 9, 6391-6399 (2004).
[CrossRef]

2002

K. Barnard, L. Martin, B. Funt, and A. Coath, "A data set for colour research," Color Res. Appl. 27, 147-151 (2002).
[CrossRef]

J. Marchant and C. Onyango, "Spectral invariance under daylight illumination changes," J. Opt. Soc. Am. A 19, 840-848 (2002).
[CrossRef]

1999

R. Lenz, P. Meer, and M. Hauta-Kasari, "Spectral-based illumination estimation and color correction," Color Res. Appl. 24, 98-111 (1999).
[CrossRef]

1994

M. Vrhel, R. Gershon, and L. Iwan, "Measurement and analysis of object reflectance spectra," Color Res. Appl. 19, 4-9 (1994).

1992

1990

J. Ho, B. Funt, and M. Drew, "Separating a color signal into illumination and surface reflectance components: theory and applications," IEEE Trans. Pattern Anal. Mach. Intell. 12, 966-977 (1990). Reprinted in Physics-Based Vision. Principles and Practice, G.E. Healey, S.A. Shafer, and L.B., eds. (Wolff, Jones, and Bartlett, 1992), Vol. 2, p. 272.
[CrossRef]

1989

D. Brainard, B. Wandell, and W. Cowan, "Black light: how sensors filter spectral variation of illuminant," IEEE Trans. Biomed. Eng. 36, 140-149 (1989).
[CrossRef] [PubMed]

1987

B. Wandell, "The synthesis and analysis of color images," IEEE Trans. Pattern Anal. Mach. Intell. PAMI-9, 2-13 (1987).
[CrossRef]

1976

C. McCamy, H. Marcus, and J. Davidson, "A color-rendition chart," J. Appl. Photogr. Eng. 2, 95-99 (1976).

1964

Amano, K.

Amira, A.

M. Roula, J. Diamond, A. Bouridane, P. Miller, and A. Amira, "A multispectral computer vision system for automatic grading of prostatic neoplasia," in Proceedings of IEEE 2002 International Symposium on Biomedical Imaging (IEEE, 2002), pp. 193-196.
[CrossRef]

Angelopoulou, E.

E. Angelopoulou, "Objective colour from multispectral imaging," in ECCV 2000: European Conference on Computer Vision, D.Vernon, ed. (Springer, 2000), pp. 359-374.
[CrossRef]

Arsenin, V.

A. Tikhonov and V. Arsenin, Solutions of Ill-Posed Problems (Wiley, 1977).

Barnard, K.

K. Barnard, L. Martin, B. Funt, and A. Coath, "A data set for colour research," Color Res. Appl. 27, 147-151 (2002).
[CrossRef]

Barrow, H.

H. Barrow and J. Tenenbaum, "Recovering intrinsic scene characteristics from images," in Computer Vision Systems, A. Hanson and E. Riseman, eds. (Academic, 1978), pp. 3-26.

Bergner, S.

S. Bergner, T. Möller, M. K. Tory, and M. Drew, "A practical approach to spectral volume rendering," IEEE Trans. Vis. Comput. Graph. 11, 207-216 (2005).
[CrossRef] [PubMed]

Berns, R.

R. Berns, "Rejuvenating Seurat's palette using color and imaging science: a simulation," in Seurat and the Making of La Grande Jatte, R.L. Herbert, ed. (University of California Press, 2004), pp. 214-227.

Bouridane, A.

M. Roula, J. Diamond, A. Bouridane, P. Miller, and A. Amira, "A multispectral computer vision system for automatic grading of prostatic neoplasia," in Proceedings of IEEE 2002 International Symposium on Biomedical Imaging (IEEE, 2002), pp. 193-196.
[CrossRef]

Brainard, D.

D. Brainard, B. Wandell, and W. Cowan, "Black light: how sensors filter spectral variation of illuminant," IEEE Trans. Biomed. Eng. 36, 140-149 (1989).
[CrossRef] [PubMed]

Burden, R.

J. Faires and R. Burden, Numerical Analysis, 3rd ed. (Brooks/Cole, 2003).

Chandra, K.

Coath, A.

K. Barnard, L. Martin, B. Funt, and A. Coath, "A data set for colour research," Color Res. Appl. 27, 147-151 (2002).
[CrossRef]

Cowan, W.

D. Brainard, B. Wandell, and W. Cowan, "Black light: how sensors filter spectral variation of illuminant," IEEE Trans. Biomed. Eng. 36, 140-149 (1989).
[CrossRef] [PubMed]

Davidson, J.

C. McCamy, H. Marcus, and J. Davidson, "A color-rendition chart," J. Appl. Photogr. Eng. 2, 95-99 (1976).

Diamond, J.

M. Roula, J. Diamond, A. Bouridane, P. Miller, and A. Amira, "A multispectral computer vision system for automatic grading of prostatic neoplasia," in Proceedings of IEEE 2002 International Symposium on Biomedical Imaging (IEEE, 2002), pp. 193-196.
[CrossRef]

Drew, M.

S. Bergner, T. Möller, M. K. Tory, and M. Drew, "A practical approach to spectral volume rendering," IEEE Trans. Vis. Comput. Graph. 11, 207-216 (2005).
[CrossRef] [PubMed]

M. Drew and B. V. Funt, "Natural metamers," CVGIP: Image Understand. 56, 139-151 (1992).
[CrossRef]

J. Ho, B. Funt, and M. Drew, "Separating a color signal into illumination and surface reflectance components: theory and applications," IEEE Trans. Pattern Anal. Mach. Intell. 12, 966-977 (1990). Reprinted in Physics-Based Vision. Principles and Practice, G.E. Healey, S.A. Shafer, and L.B., eds. (Wolff, Jones, and Bartlett, 1992), Vol. 2, p. 272.
[CrossRef]

Z.-N. Li and M. Drew, Fundamentals of Multimedia (Prentice Hall, 2004).

Faires, J.

J. Faires and R. Burden, Numerical Analysis, 3rd ed. (Brooks/Cole, 2003).

Foster, D.

Funt, B.

K. Barnard, L. Martin, B. Funt, and A. Coath, "A data set for colour research," Color Res. Appl. 27, 147-151 (2002).
[CrossRef]

J. Ho, B. Funt, and M. Drew, "Separating a color signal into illumination and surface reflectance components: theory and applications," IEEE Trans. Pattern Anal. Mach. Intell. 12, 966-977 (1990). Reprinted in Physics-Based Vision. Principles and Practice, G.E. Healey, S.A. Shafer, and L.B., eds. (Wolff, Jones, and Bartlett, 1992), Vol. 2, p. 272.
[CrossRef]

B. Funt, "Modeling reflectance by logarithmic basis functions," in Color Imaging Conference: Color, Science, Systems and Applications [Society for Imaging Science & Technology (IS&T)/Society for Information Display (SID), 1993], pp. 68-71.

Funt, B. V.

M. Drew and B. V. Funt, "Natural metamers," CVGIP: Image Understand. 56, 139-151 (1992).
[CrossRef]

Gershon, R.

M. Vrhel, R. Gershon, and L. Iwan, "Measurement and analysis of object reflectance spectra," Color Res. Appl. 19, 4-9 (1994).

Hauta-Kasari, M.

R. Lenz, P. Meer, and M. Hauta-Kasari, "Spectral-based illumination estimation and color correction," Color Res. Appl. 24, 98-111 (1999).
[CrossRef]

Healey, G.

Ho, J.

J. Ho, B. Funt, and M. Drew, "Separating a color signal into illumination and surface reflectance components: theory and applications," IEEE Trans. Pattern Anal. Mach. Intell. 12, 966-977 (1990). Reprinted in Physics-Based Vision. Principles and Practice, G.E. Healey, S.A. Shafer, and L.B., eds. (Wolff, Jones, and Bartlett, 1992), Vol. 2, p. 272.
[CrossRef]

Hyvärinen, A.

A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley, 2001).
[CrossRef]

Ikonen, E.

S. Nevas, F. Manoocheri, and E. Ikonen, "Gonioreflectometer for measuring spectral diffuse reflectance," Appl. Opt. 9, 6391-6399 (2004).
[CrossRef]

Iwan, L.

M. Vrhel, R. Gershon, and L. Iwan, "Measurement and analysis of object reflectance spectra," Color Res. Appl. 19, 4-9 (1994).

Judd, D.

Karhunen, J.

A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley, 2001).
[CrossRef]

Ke, H.

Q. Wu, L. Zeng, H. Ke, W. Xie, H. Zheng, and Y. Zhang, "Analysis of blood and bone marrow smears using multispectral imaging analysis techniques," in Medical Imaging 2005: Image Processing, J.M. Fitzpatrick and J.M. Reinhardt, eds., Proc. SPIE 5747, 1872-1882 (2005).

Lenz, R.

R. Lenz, P. Meer, and M. Hauta-Kasari, "Spectral-based illumination estimation and color correction," Color Res. Appl. 24, 98-111 (1999).
[CrossRef]

Li, Z.-N.

Z.-N. Li and M. Drew, Fundamentals of Multimedia (Prentice Hall, 2004).

MacAdam, D.

Manoocheri, F.

S. Nevas, F. Manoocheri, and E. Ikonen, "Gonioreflectometer for measuring spectral diffuse reflectance," Appl. Opt. 9, 6391-6399 (2004).
[CrossRef]

Marchant, J.

Marcus, H.

C. McCamy, H. Marcus, and J. Davidson, "A color-rendition chart," J. Appl. Photogr. Eng. 2, 95-99 (1976).

Marimont, D. H.

Martin, L.

K. Barnard, L. Martin, B. Funt, and A. Coath, "A data set for colour research," Color Res. Appl. 27, 147-151 (2002).
[CrossRef]

McCamy, C.

C. McCamy, H. Marcus, and J. Davidson, "A color-rendition chart," J. Appl. Photogr. Eng. 2, 95-99 (1976).

Meer, P.

R. Lenz, P. Meer, and M. Hauta-Kasari, "Spectral-based illumination estimation and color correction," Color Res. Appl. 24, 98-111 (1999).
[CrossRef]

Miller, P.

M. Roula, J. Diamond, A. Bouridane, P. Miller, and A. Amira, "A multispectral computer vision system for automatic grading of prostatic neoplasia," in Proceedings of IEEE 2002 International Symposium on Biomedical Imaging (IEEE, 2002), pp. 193-196.
[CrossRef]

Möller, T.

S. Bergner, T. Möller, M. K. Tory, and M. Drew, "A practical approach to spectral volume rendering," IEEE Trans. Vis. Comput. Graph. 11, 207-216 (2005).
[CrossRef] [PubMed]

Nascimento, S.

Nevas, S.

S. Nevas, F. Manoocheri, and E. Ikonen, "Gonioreflectometer for measuring spectral diffuse reflectance," Appl. Opt. 9, 6391-6399 (2004).
[CrossRef]

Oja, E.

A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley, 2001).
[CrossRef]

Onyango, C.

Roula, M.

M. Roula, J. Diamond, A. Bouridane, P. Miller, and A. Amira, "A multispectral computer vision system for automatic grading of prostatic neoplasia," in Proceedings of IEEE 2002 International Symposium on Biomedical Imaging (IEEE, 2002), pp. 193-196.
[CrossRef]

Stiles, W.

G. Wyszecki and W. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulas, 2nd ed. (Wiley, 1982).

Tenenbaum, J.

H. Barrow and J. Tenenbaum, "Recovering intrinsic scene characteristics from images," in Computer Vision Systems, A. Hanson and E. Riseman, eds. (Academic, 1978), pp. 3-26.

Tikhonov, A.

A. Tikhonov and V. Arsenin, Solutions of Ill-Posed Problems (Wiley, 1977).

Tory, M. K.

S. Bergner, T. Möller, M. K. Tory, and M. Drew, "A practical approach to spectral volume rendering," IEEE Trans. Vis. Comput. Graph. 11, 207-216 (2005).
[CrossRef] [PubMed]

Tsumura, N.

N. Tsumura, "Appearance reproduction and multi-spectral imaging," in AIC Colour 05: Proceedings of the 10th Congress of the International Colour Association, J.L. Nieves and J. Hernández-Andrés, eds. (AIC, Granada, 2005), pp. 119-123.

Vrhel, M.

M. Vrhel, R. Gershon, and L. Iwan, "Measurement and analysis of object reflectance spectra," Color Res. Appl. 19, 4-9 (1994).

Wandell, B.

D. Brainard, B. Wandell, and W. Cowan, "Black light: how sensors filter spectral variation of illuminant," IEEE Trans. Biomed. Eng. 36, 140-149 (1989).
[CrossRef] [PubMed]

B. Wandell, "The synthesis and analysis of color images," IEEE Trans. Pattern Anal. Mach. Intell. PAMI-9, 2-13 (1987).
[CrossRef]

Wandell, B. A.

Wu, Q.

Q. Wu, L. Zeng, H. Ke, W. Xie, H. Zheng, and Y. Zhang, "Analysis of blood and bone marrow smears using multispectral imaging analysis techniques," in Medical Imaging 2005: Image Processing, J.M. Fitzpatrick and J.M. Reinhardt, eds., Proc. SPIE 5747, 1872-1882 (2005).

Wyszecki, G.

D. Judd, D. MacAdam, and G. Wyszecki, "Spectral distribution of typical daylight as a function of correlated color temperature," J. Opt. Soc. Am. 54, 1031-1040 (1964).
[CrossRef]

G. Wyszecki and W. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulas, 2nd ed. (Wiley, 1982).

Xie, W.

Q. Wu, L. Zeng, H. Ke, W. Xie, H. Zheng, and Y. Zhang, "Analysis of blood and bone marrow smears using multispectral imaging analysis techniques," in Medical Imaging 2005: Image Processing, J.M. Fitzpatrick and J.M. Reinhardt, eds., Proc. SPIE 5747, 1872-1882 (2005).

Zeng, L.

Q. Wu, L. Zeng, H. Ke, W. Xie, H. Zheng, and Y. Zhang, "Analysis of blood and bone marrow smears using multispectral imaging analysis techniques," in Medical Imaging 2005: Image Processing, J.M. Fitzpatrick and J.M. Reinhardt, eds., Proc. SPIE 5747, 1872-1882 (2005).

Zhang, Y.

Q. Wu, L. Zeng, H. Ke, W. Xie, H. Zheng, and Y. Zhang, "Analysis of blood and bone marrow smears using multispectral imaging analysis techniques," in Medical Imaging 2005: Image Processing, J.M. Fitzpatrick and J.M. Reinhardt, eds., Proc. SPIE 5747, 1872-1882 (2005).

Zheng, H.

Q. Wu, L. Zeng, H. Ke, W. Xie, H. Zheng, and Y. Zhang, "Analysis of blood and bone marrow smears using multispectral imaging analysis techniques," in Medical Imaging 2005: Image Processing, J.M. Fitzpatrick and J.M. Reinhardt, eds., Proc. SPIE 5747, 1872-1882 (2005).

Appl. Opt.

S. Nevas, F. Manoocheri, and E. Ikonen, "Gonioreflectometer for measuring spectral diffuse reflectance," Appl. Opt. 9, 6391-6399 (2004).
[CrossRef]

Color Res. Appl.

K. Barnard, L. Martin, B. Funt, and A. Coath, "A data set for colour research," Color Res. Appl. 27, 147-151 (2002).
[CrossRef]

M. Vrhel, R. Gershon, and L. Iwan, "Measurement and analysis of object reflectance spectra," Color Res. Appl. 19, 4-9 (1994).

R. Lenz, P. Meer, and M. Hauta-Kasari, "Spectral-based illumination estimation and color correction," Color Res. Appl. 24, 98-111 (1999).
[CrossRef]

CVGIP: Image Understand.

M. Drew and B. V. Funt, "Natural metamers," CVGIP: Image Understand. 56, 139-151 (1992).
[CrossRef]

IEEE Trans. Biomed. Eng.

D. Brainard, B. Wandell, and W. Cowan, "Black light: how sensors filter spectral variation of illuminant," IEEE Trans. Biomed. Eng. 36, 140-149 (1989).
[CrossRef] [PubMed]

IEEE Trans. Pattern Anal. Mach. Intell.

J. Ho, B. Funt, and M. Drew, "Separating a color signal into illumination and surface reflectance components: theory and applications," IEEE Trans. Pattern Anal. Mach. Intell. 12, 966-977 (1990). Reprinted in Physics-Based Vision. Principles and Practice, G.E. Healey, S.A. Shafer, and L.B., eds. (Wolff, Jones, and Bartlett, 1992), Vol. 2, p. 272.
[CrossRef]

B. Wandell, "The synthesis and analysis of color images," IEEE Trans. Pattern Anal. Mach. Intell. PAMI-9, 2-13 (1987).
[CrossRef]

IEEE Trans. Vis. Comput. Graph.

S. Bergner, T. Möller, M. K. Tory, and M. Drew, "A practical approach to spectral volume rendering," IEEE Trans. Vis. Comput. Graph. 11, 207-216 (2005).
[CrossRef] [PubMed]

J. Appl. Photogr. Eng.

C. McCamy, H. Marcus, and J. Davidson, "A color-rendition chart," J. Appl. Photogr. Eng. 2, 95-99 (1976).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Other

B. Funt, "Modeling reflectance by logarithmic basis functions," in Color Imaging Conference: Color, Science, Systems and Applications [Society for Imaging Science & Technology (IS&T)/Society for Information Display (SID), 1993], pp. 68-71.

E. Angelopoulou, "Objective colour from multispectral imaging," in ECCV 2000: European Conference on Computer Vision, D.Vernon, ed. (Springer, 2000), pp. 359-374.
[CrossRef]

G. Wyszecki and W. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulas, 2nd ed. (Wiley, 1982).

These data are available at http://www.multispectral.org/.

Q. Wu, L. Zeng, H. Ke, W. Xie, H. Zheng, and Y. Zhang, "Analysis of blood and bone marrow smears using multispectral imaging analysis techniques," in Medical Imaging 2005: Image Processing, J.M. Fitzpatrick and J.M. Reinhardt, eds., Proc. SPIE 5747, 1872-1882 (2005).

M. Roula, J. Diamond, A. Bouridane, P. Miller, and A. Amira, "A multispectral computer vision system for automatic grading of prostatic neoplasia," in Proceedings of IEEE 2002 International Symposium on Biomedical Imaging (IEEE, 2002), pp. 193-196.
[CrossRef]

N. Tsumura, "Appearance reproduction and multi-spectral imaging," in AIC Colour 05: Proceedings of the 10th Congress of the International Colour Association, J.L. Nieves and J. Hernández-Andrés, eds. (AIC, Granada, 2005), pp. 119-123.

R. Berns, "Rejuvenating Seurat's palette using color and imaging science: a simulation," in Seurat and the Making of La Grande Jatte, R.L. Herbert, ed. (University of California Press, 2004), pp. 214-227.

H. Barrow and J. Tenenbaum, "Recovering intrinsic scene characteristics from images," in Computer Vision Systems, A. Hanson and E. Riseman, eds. (Academic, 1978), pp. 3-26.

J. Faires and R. Burden, Numerical Analysis, 3rd ed. (Brooks/Cole, 2003).

Z.-N. Li and M. Drew, Fundamentals of Multimedia (Prentice Hall, 2004).

A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley, 2001).
[CrossRef]

A. Tikhonov and V. Arsenin, Solutions of Ill-Posed Problems (Wiley, 1977).

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Figures (4)

Fig. 1
Fig. 1

Training illuminants and reflectances: (a) set of 102 standard illuminant spectra, (b) singular values for illuminant set, (c) singular values for reflectance set.

Fig. 2
Fig. 2

Plot of an illuminant recovered with about 15% error (actual, solid curve; approximation, dashed curve) and of reflectance recovered with about 13% error (actual, dotted–dashed curve; approximation, dashed curve).

Fig. 3
Fig. 3

Median errors for Method REGLOGSEP: (a) median errors for recovered illuminants, (b) median surface reflectance errors.

Fig. 4
Fig. 4

Median errors for Method REGUNILOGSEP: (a) median errors for recovered illuminants, (b) median surface reflectance errors.

Tables (4)

Tables Icon

Table 1 Average Relative Timings for Algorithms Considered a

Tables Icon

Table 2 Minima over Methods Examined of Median RMS Errors a

Tables Icon

Table 3 Dimensionalities for Minima for Reflectance Errors and Errors for Illumination and Reflectance at Those Dimensions a

Tables Icon

Table 4 RMS Errors for Uniform Basis Regression Methods of Section 5 for Testing Set a

Equations (30)

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C ̂ ( λ ) = E ̂ ( λ ) S ̂ ( λ ) .
E ̂ ( λ ) i = 1 m ϵ ̂ i E ̂ i ( λ ) , S ̂ ( λ ) j = 1 n σ ̂ j S ̂ j ( λ ) ,
C ̂ ( λ ) i = 1 m j = 1 n E ̂ i ( λ ) S ̂ i ( λ ) ϵ ̂ i σ ̂ j ,
min I ̂ = [ i = 1 m j = 1 n ϵ ̂ i σ ̂ j E ̂ i ( λ ) S ̂ i ( λ ) C ̂ ( λ ) ] 2 d λ ,
M ̂ σ ̂ = F ̂ ϵ ̂ , N ̂ ϵ ̂ = F ̂ σ ̂ ,
M ̂ q j p i ϵ ̂ i ϵ ̂ p H ̂ i p j q d λ , N ̂ p i q j σ ̂ j σ ̂ q H ̂ i p j q d λ ,
H ̂ i p j q E ̂ p E ̂ i S ̂ q S ̂ j d λ , F ̂ i j E ̂ i S ̂ j C ̂ d λ .
E ( λ ) = log ( E ̂ ( λ ) ) ,
S ( λ ) = log ( S ̂ ( λ ) ) .
E ( λ ) i ϵ i E i ( λ ) ,
S ( λ ) j σ j S j ( λ ) ,
C ( λ ) i ϵ i E i ( λ ) + j S j ( λ ) σ j .
min I = [ i = 1 m ϵ i E i ( λ ) + j = 1 n σ j S j ( λ ) C ( λ ) ] 2 d λ .
min I = [ E ϵ + S σ c ] 2 d λ ,
{ M ϵ + N σ = f , O ϵ + P σ = g , }
M = E T E , N = E T S ,
O = N T , P = S T S ,
f = E T c , g = S T c ,
A α = h ,
A = [ M N N T P ] , α = ( ϵ σ ) , h = ( f g ) .
min I 1 = ϵ ̃ Q α ̃ 2 ,
min I 2 = σ ̃ R α ̃ 2 .
Q = ϵ ̃ α ̃ T ( α ̃ α ̃ T ) 1 ,
R = σ ̃ α ̃ T ( α ̃ α ̃ T ) 1 .
α ̃ α ̃ new = T α ̃ , T = [ Q 0 0 R ] .
W = 1 ( m + n ) diag ( α ̃ α ̃ T ) I ( m + n ) ,
Q = ϵ ̃ α ̃ T ( α ̃ α ̃ T + λ W ) 1 ,
R = σ ̃ α ̃ T ( α ̃ α ̃ T + λ W ) 1 ,
m i n I 3 = α ̃ U C ̃ 2
error = E ̃ ( λ ) E ̂ ( λ ) E ̂ ( λ )

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