Abstract

We examine the use of linear spectral reflectance models for calibrating a color scanner to generate device-independent CIE XYZ values from scanner vectors. Polynomial regression approaches to color scanner calibration use parameterized functions to approximate the calibration mapping over a set of training colors. These approaches can perform poorly if the parameterized functions do not accurately model the structure of the desired calibration mapping. Several studies have shown that linear reflectance models accurately characterize a wide range of materials. By viewing color scanner calibration as reflectance estimation, we can incorporate linear reflectance models into the calibration process. We show that in most cases linear models do not constrain the calibration problem sufficiently to allow exact recovery of X, Y, Z from a scanner vector obtained with three filters. By examining a series of methods that exploit information about reflectance functions, however, we show that reflectance information can be used to improve the accuracy of calibration over that of standard methods applied to the same set of inputs.

© 2002 Optical Society of America

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References

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  1. G. Wyszecki, W. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, New York, 1982).
  2. M. J. Vrhel, H. J. Trussell, “Color device calibration: a mathematical formulation,” IEEE Trans. Image Process. 8, 1796–1806 (1999).
    [CrossRef]
  3. P. C. Hung, “Calorimetric calibration in electronic imaging devices using a look-up table model and interpolation,” J. Electron. Imag. 2, 53–61 (1993).
    [CrossRef]
  4. H. R. Kang, “Color scanner calibration,” J. Imaging Sci. Technol. 36, 162–170 (1992).
  5. H. R. Kang, P. G. Anderson, “Neural Network applications to the color scanner and printer calibrations,” J. Electron. Imag. 1, 125–134 (1992).
    [CrossRef]
  6. G. Sharma, H. J. Trussell, “Figures of merit for color scanners,” IEEE Trans. Image Process. 6, 990–1001 (1997).
    [CrossRef] [PubMed]
  7. G. Sharma, H. J. Trussell, M. J. Vrhel, “Optimal nonnegative color scanning filters,” IEEE Trans. Image Process. 7, 129–133 (1998).
    [CrossRef]
  8. P. Vora, H. J. Trussell, “Measure of goodness of a set of color scanning filters,” J. Opt. Soc. Am. A 10, 1499–1508 (1993).
    [CrossRef]
  9. P. L. Vora, H. J. Trussell, “Mathematical methods for the analysis of color scanning filters,” IEEE Trans. Image Process. 6, 312–320 (1997).
    [CrossRef]
  10. M. J. Vrhel, H. J. Trussell, “Optimal color filters in the presence of noise,” IEEE Trans. Image Process. 4, 814–823 (1995).
    [CrossRef] [PubMed]
  11. M. J. Vrhel, H. J. Trussell, “Filter considerations in color correction,” IEEE Trans. Image Process. 3, 147–161 (1994).
    [CrossRef] [PubMed]
  12. G. Sharma, H. J. Trussell, “Characterization of scanner sensitivity,” in Proceedings of the IS&T/SID Color Imaging Conference: Transforms and Transportability of Color (Society for Imaging Science and Technology, Springfield, Va., 1993), pp. 103–107.
  13. J. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psycho. Sci. 1, 369–370 (1964).
    [CrossRef]
  14. L. 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]
  15. J. P. S. Parkkinen, J. Hallikainen, T. Jaaskelainen, “Characteristic spectra of Munsell colors,” J. Opt. Soc. Am. A 6, 318–322 (1989).
    [CrossRef]
  16. M. J. Vrhel, R. Gershon, L. Iwan, “Measurement and analysis of object reflectance spectra,” Color Res. Appl. 19, 4–9 (1994).
  17. D. Marimont, B. Wandell, “Linear models of surface and illuminant spectra,” J. Opt. Soc. Am. A 9, 1905–1913 (1992).
    [CrossRef] [PubMed]
  18. B. Wandell, “The synthesis and analysis of color images,” IEEE Trans. Pattern Anal. Mach. Intell. 9, 2–13 (1987).
    [CrossRef] [PubMed]
  19. B. K. P. Horn, “Exact reproduction of colored images,” Comput. Vision Graph. Image Process. 26, 135–167 (1984).
    [CrossRef]
  20. G. Hong, M. R. Luo, P. A. Rhodes, “A study of digital camera colorimetric characterization based on polynomial modeling,” Color Res. Appl. 26, 76–84 (2001).
    [CrossRef]
  21. J. Cohen, W. Kappauf, “Metameric color stimuli, fundamental metamers and Wyszecki’s metameric blacks,” Am. J. Psychol. 95, 537–564 (1982).
    [CrossRef] [PubMed]
  22. G. H. Golub, C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore Md., 1983).
  23. M. S. Drew, B. V. Funt, “Natural metamers,” Comput. Vision Graph. Image Process. 56, 139–151 (1992).
  24. C. McCamy, H. Marcus, J. Davidson, “A color-rendition chart,” J. Appl. Photogr. Eng. 2, 95–99 (1976).
  25. G. Healey, R. Kondepudy, “Radiometric CCD camera calibration and noise estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 267–276 (1994).
    [CrossRef]
  26. H. R. Kang, Color Technology for Electronic Imaging Devices (SPIE Press, Bellingham, Wash., 1997).
  27. M. Mahy, L. Van Eyckden, A. Oosterlinck, “Evaluation of uniform color spaces developed after the adoption of CIELab and CIELuv,” Color Res. Appl. 19, 105–121 (1994).

2001 (1)

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

1999 (1)

M. J. Vrhel, H. J. Trussell, “Color device calibration: a mathematical formulation,” IEEE Trans. Image Process. 8, 1796–1806 (1999).
[CrossRef]

1998 (1)

G. Sharma, H. J. Trussell, M. J. Vrhel, “Optimal nonnegative color scanning filters,” IEEE Trans. Image Process. 7, 129–133 (1998).
[CrossRef]

1997 (2)

G. Sharma, H. J. Trussell, “Figures of merit for color scanners,” IEEE Trans. Image Process. 6, 990–1001 (1997).
[CrossRef] [PubMed]

P. L. Vora, H. J. Trussell, “Mathematical methods for the analysis of color scanning filters,” IEEE Trans. Image Process. 6, 312–320 (1997).
[CrossRef]

1995 (1)

M. J. Vrhel, H. J. Trussell, “Optimal color filters in the presence of noise,” IEEE Trans. Image Process. 4, 814–823 (1995).
[CrossRef] [PubMed]

1994 (4)

M. J. Vrhel, H. J. Trussell, “Filter considerations in color correction,” IEEE Trans. Image Process. 3, 147–161 (1994).
[CrossRef] [PubMed]

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

G. Healey, R. Kondepudy, “Radiometric CCD camera calibration and noise estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 267–276 (1994).
[CrossRef]

M. Mahy, L. Van Eyckden, A. Oosterlinck, “Evaluation of uniform color spaces developed after the adoption of CIELab and CIELuv,” Color Res. Appl. 19, 105–121 (1994).

1993 (2)

P. Vora, H. J. Trussell, “Measure of goodness of a set of color scanning filters,” J. Opt. Soc. Am. A 10, 1499–1508 (1993).
[CrossRef]

P. C. Hung, “Calorimetric calibration in electronic imaging devices using a look-up table model and interpolation,” J. Electron. Imag. 2, 53–61 (1993).
[CrossRef]

1992 (4)

H. R. Kang, “Color scanner calibration,” J. Imaging Sci. Technol. 36, 162–170 (1992).

H. R. Kang, P. G. Anderson, “Neural Network applications to the color scanner and printer calibrations,” J. Electron. Imag. 1, 125–134 (1992).
[CrossRef]

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

M. S. Drew, B. V. Funt, “Natural metamers,” Comput. Vision Graph. Image Process. 56, 139–151 (1992).

1989 (1)

1987 (1)

B. Wandell, “The synthesis and analysis of color images,” IEEE Trans. Pattern Anal. Mach. Intell. 9, 2–13 (1987).
[CrossRef] [PubMed]

1986 (1)

1984 (1)

B. K. P. Horn, “Exact reproduction of colored images,” Comput. Vision Graph. Image Process. 26, 135–167 (1984).
[CrossRef]

1982 (1)

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

1976 (1)

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

1964 (1)

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

Anderson, P. G.

H. R. Kang, P. G. Anderson, “Neural Network applications to the color scanner and printer calibrations,” J. Electron. Imag. 1, 125–134 (1992).
[CrossRef]

Cohen, J.

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

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

Davidson, J.

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

Drew, M. S.

M. S. Drew, B. V. Funt, “Natural metamers,” Comput. Vision Graph. Image Process. 56, 139–151 (1992).

Funt, B. V.

M. S. Drew, B. V. Funt, “Natural metamers,” Comput. Vision Graph. Image Process. 56, 139–151 (1992).

Gershon, R.

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

Golub, G. H.

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

Hallikainen, J.

Healey, G.

G. Healey, R. Kondepudy, “Radiometric CCD camera calibration and noise estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 267–276 (1994).
[CrossRef]

Hong, G.

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

Horn, B. K. P.

B. K. P. Horn, “Exact reproduction of colored images,” Comput. Vision Graph. Image Process. 26, 135–167 (1984).
[CrossRef]

Hung, P. C.

P. C. Hung, “Calorimetric calibration in electronic imaging devices using a look-up table model and interpolation,” J. Electron. Imag. 2, 53–61 (1993).
[CrossRef]

Iwan, L.

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

Jaaskelainen, T.

Kang, H. R.

H. R. Kang, “Color scanner calibration,” J. Imaging Sci. Technol. 36, 162–170 (1992).

H. R. Kang, P. G. Anderson, “Neural Network applications to the color scanner and printer calibrations,” J. Electron. Imag. 1, 125–134 (1992).
[CrossRef]

H. R. Kang, Color Technology for Electronic Imaging Devices (SPIE Press, Bellingham, Wash., 1997).

Kappauf, W.

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

Kondepudy, R.

G. Healey, R. Kondepudy, “Radiometric CCD camera calibration and noise estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 267–276 (1994).
[CrossRef]

Luo, M. R.

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

Mahy, M.

M. Mahy, L. Van Eyckden, A. Oosterlinck, “Evaluation of uniform color spaces developed after the adoption of CIELab and CIELuv,” Color Res. Appl. 19, 105–121 (1994).

Maloney, L.

Marcus, H.

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

Marimont, D.

McCamy, C.

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

Oosterlinck, A.

M. Mahy, L. Van Eyckden, A. Oosterlinck, “Evaluation of uniform color spaces developed after the adoption of CIELab and CIELuv,” Color Res. Appl. 19, 105–121 (1994).

Parkkinen, J. P. S.

Rhodes, P. A.

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

Sharma, G.

G. Sharma, H. J. Trussell, M. J. Vrhel, “Optimal nonnegative color scanning filters,” IEEE Trans. Image Process. 7, 129–133 (1998).
[CrossRef]

G. Sharma, H. J. Trussell, “Figures of merit for color scanners,” IEEE Trans. Image Process. 6, 990–1001 (1997).
[CrossRef] [PubMed]

G. Sharma, H. J. Trussell, “Characterization of scanner sensitivity,” in Proceedings of the IS&T/SID Color Imaging Conference: Transforms and Transportability of Color (Society for Imaging Science and Technology, Springfield, Va., 1993), pp. 103–107.

Stiles, W.

G. Wyszecki, W. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, New York, 1982).

Trussell, H. J.

M. J. Vrhel, H. J. Trussell, “Color device calibration: a mathematical formulation,” IEEE Trans. Image Process. 8, 1796–1806 (1999).
[CrossRef]

G. Sharma, H. J. Trussell, M. J. Vrhel, “Optimal nonnegative color scanning filters,” IEEE Trans. Image Process. 7, 129–133 (1998).
[CrossRef]

P. L. Vora, H. J. Trussell, “Mathematical methods for the analysis of color scanning filters,” IEEE Trans. Image Process. 6, 312–320 (1997).
[CrossRef]

G. Sharma, H. J. Trussell, “Figures of merit for color scanners,” IEEE Trans. Image Process. 6, 990–1001 (1997).
[CrossRef] [PubMed]

M. J. Vrhel, H. J. Trussell, “Optimal color filters in the presence of noise,” IEEE Trans. Image Process. 4, 814–823 (1995).
[CrossRef] [PubMed]

M. J. Vrhel, H. J. Trussell, “Filter considerations in color correction,” IEEE Trans. Image Process. 3, 147–161 (1994).
[CrossRef] [PubMed]

P. Vora, H. J. Trussell, “Measure of goodness of a set of color scanning filters,” J. Opt. Soc. Am. A 10, 1499–1508 (1993).
[CrossRef]

G. Sharma, H. J. Trussell, “Characterization of scanner sensitivity,” in Proceedings of the IS&T/SID Color Imaging Conference: Transforms and Transportability of Color (Society for Imaging Science and Technology, Springfield, Va., 1993), pp. 103–107.

Van Eyckden, L.

M. Mahy, L. Van Eyckden, A. Oosterlinck, “Evaluation of uniform color spaces developed after the adoption of CIELab and CIELuv,” Color Res. Appl. 19, 105–121 (1994).

van Loan, C. F.

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

Vora, P.

Vora, P. L.

P. L. Vora, H. J. Trussell, “Mathematical methods for the analysis of color scanning filters,” IEEE Trans. Image Process. 6, 312–320 (1997).
[CrossRef]

Vrhel, M. J.

M. J. Vrhel, H. J. Trussell, “Color device calibration: a mathematical formulation,” IEEE Trans. Image Process. 8, 1796–1806 (1999).
[CrossRef]

G. Sharma, H. J. Trussell, M. J. Vrhel, “Optimal nonnegative color scanning filters,” IEEE Trans. Image Process. 7, 129–133 (1998).
[CrossRef]

M. J. Vrhel, H. J. Trussell, “Optimal color filters in the presence of noise,” IEEE Trans. Image Process. 4, 814–823 (1995).
[CrossRef] [PubMed]

M. J. Vrhel, H. J. Trussell, “Filter considerations in color correction,” IEEE Trans. Image Process. 3, 147–161 (1994).
[CrossRef] [PubMed]

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

Wandell, B.

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

B. Wandell, “The synthesis and analysis of color images,” IEEE Trans. Pattern Anal. Mach. Intell. 9, 2–13 (1987).
[CrossRef] [PubMed]

Wyszecki, G.

G. Wyszecki, W. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, New York, 1982).

Am. J. Psychol. (1)

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

Color Res. Appl. (3)

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

M. Mahy, L. Van Eyckden, A. Oosterlinck, “Evaluation of uniform color spaces developed after the adoption of CIELab and CIELuv,” Color Res. Appl. 19, 105–121 (1994).

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

Comput. Vision Graph. Image Process. (2)

B. K. P. Horn, “Exact reproduction of colored images,” Comput. Vision Graph. Image Process. 26, 135–167 (1984).
[CrossRef]

M. S. Drew, B. V. Funt, “Natural metamers,” Comput. Vision Graph. Image Process. 56, 139–151 (1992).

IEEE Trans. Image Process. (6)

G. Sharma, H. J. Trussell, “Figures of merit for color scanners,” IEEE Trans. Image Process. 6, 990–1001 (1997).
[CrossRef] [PubMed]

G. Sharma, H. J. Trussell, M. J. Vrhel, “Optimal nonnegative color scanning filters,” IEEE Trans. Image Process. 7, 129–133 (1998).
[CrossRef]

M. J. Vrhel, H. J. Trussell, “Color device calibration: a mathematical formulation,” IEEE Trans. Image Process. 8, 1796–1806 (1999).
[CrossRef]

P. L. Vora, H. J. Trussell, “Mathematical methods for the analysis of color scanning filters,” IEEE Trans. Image Process. 6, 312–320 (1997).
[CrossRef]

M. J. Vrhel, H. J. Trussell, “Optimal color filters in the presence of noise,” IEEE Trans. Image Process. 4, 814–823 (1995).
[CrossRef] [PubMed]

M. J. Vrhel, H. J. Trussell, “Filter considerations in color correction,” IEEE Trans. Image Process. 3, 147–161 (1994).
[CrossRef] [PubMed]

IEEE Trans. Pattern Anal. Mach. Intell. (2)

B. Wandell, “The synthesis and analysis of color images,” IEEE Trans. Pattern Anal. Mach. Intell. 9, 2–13 (1987).
[CrossRef] [PubMed]

G. Healey, R. Kondepudy, “Radiometric CCD camera calibration and noise estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 267–276 (1994).
[CrossRef]

J. Appl. Photogr. Eng. (1)

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

J. Electron. Imag. (2)

P. C. Hung, “Calorimetric calibration in electronic imaging devices using a look-up table model and interpolation,” J. Electron. Imag. 2, 53–61 (1993).
[CrossRef]

H. R. Kang, P. G. Anderson, “Neural Network applications to the color scanner and printer calibrations,” J. Electron. Imag. 1, 125–134 (1992).
[CrossRef]

J. Imaging Sci. Technol. (1)

H. R. Kang, “Color scanner calibration,” J. Imaging Sci. Technol. 36, 162–170 (1992).

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

Psycho. Sci. (1)

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

Other (4)

G. Wyszecki, W. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, New York, 1982).

G. Sharma, H. J. Trussell, “Characterization of scanner sensitivity,” in Proceedings of the IS&T/SID Color Imaging Conference: Transforms and Transportability of Color (Society for Imaging Science and Technology, Springfield, Va., 1993), pp. 103–107.

H. R. Kang, Color Technology for Electronic Imaging Devices (SPIE Press, Bellingham, Wash., 1997).

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

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Figures (6)

Fig. 1
Fig. 1

Reflectance functions giving the same scanner R, G, B and different X, Y, Z (example 1).

Fig. 2
Fig. 2

Reflectance functions giving the same scanner R, G, B and different X, Y, Z (example 2).

Fig. 3
Fig. 3

Reflectance functions giving the same scanner R, G, B and different X, Y, Z (example 3).

Fig. 4
Fig. 4

Kodak Q60 color input target.

Fig. 5
Fig. 5

Sensor response functions.

Fig. 6
Fig. 6

Basis functions in a four-dimensional linear model.

Tables (15)

Tables Icon

Table 1 Results with Three-Term Polynomial Regression Used for IT8

Tables Icon

Table 2 Results with Five-Term Polynomial Regression Used for IT8

Tables Icon

Table 3 Results with Six-Term Polynomial Regression Used for IT8

Tables Icon

Table 4 Results with Eight-Term Polynomial Regression Used for IT8

Tables Icon

Table 5 Results with Nine-Term Polynomial Regression Used for IT8

Tables Icon

Table 6 Results with Eleven-Term Polynomial Regression Used for IT8

Tables Icon

Table 7 Results with Scanner Subspace Projection Used for IT8

Tables Icon

Table 8 Results with Four-Dimensional Linear Model Used for IT8

Tables Icon

Table 9 Results with Five-Dimensional Linear Model Used for IT8

Tables Icon

Table 10 Results with Six-Dimensional Linear Model Used for IT8

Tables Icon

Table 11 Results with Seven-Dimensional Linear Model Used for IT8

Tables Icon

Table 12 Results with Regression-Based Calibration Used for MCC

Tables Icon

Table 13 Results with Higher-Dimensional Linear Model Calibration Used for MCC

Tables Icon

Table 14 Results with Regression-Based Calibration Used for MUN

Tables Icon

Table 15 Results with Higher-Dimensional Linear Model Calibration Used for MUN

Equations (41)

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

X=k  X(λ)l(λ)f(λ)dλ=v1(λ)f(λ)dλ,
Y=k  Y(λ)l(λ)f(λ)dλ=v2(λ)f(λ)dλ,
Z=k  Z(λ)l(λ)f(λ)dλ=v3(λ)f(λ)dλ,
k=100/ Y(λ)l(λ)dλ,
si= ti(λ)q(λ)p(λ)e(λ)f(λ)dλ,1i3,
si= mi(λ)f(λ)dλ,1i3.
XYZ=Vf,
S=Mf,
Pv=1i3[f·ni]ni.
X=Pv·v1,
Y=Pv·v2,
Z=Pv·v3.
Pm=1i3[f·oi]oi.
XˆYˆZˆ=VPm.
f(λ)1iBaibi(λ),
f=La=[b1,b2,, bB][a1,a2,, aB]T,
1jKfj-i=1B aijbi2,
S=MLa.
a=(ML)-1S
XYZ=VL(ML)-1S.
E=i=1KVfi-VTMfi2.
S=M[b1,b2,, bB-3][a1,a2,, aB-3]T+M[bB-2,bB-1,bB][aB-2,aB-1,aB]T.
L1=[b1,b2,, bB-3],L2=[bB-2,bB-1,bB],
A1=[a1,a2,, aB-3],A2=[aB-2,aB-1,aB],
S=ML1A1T+ML2A2T,
A2T=(M L2)-1(S-M L1A1T).
f=L1A1T+L2(ML2)-1(S-ML1A1T),
fi*=L1(A1*)T+L2(ML2)-1[S-ML1(A1*)T],
(A1*)T=[L1-L2(ML2)-1ML1]+×[fi-L2(ML2)-1S],
Di*=fi*-fi.
X=c11s1+c12s2+c13s3+c14s1s2+c15s2s3+c16s1s3,
Y=c21s1+c22s2+c23s3+c24s1s2+c25s2s3+c26s1s3,
Z=c31s1+c32s2+c33s3+c34s1s2+c35s2s3+c36s1s3.
Q=X1X2XKY1Y2YKZ1Z2ZK,
U=s11s21sK1s12s22sK2s13s23sK3s11s12s21s22sK1sK2s12s13s22s23sK2sK3s11s13s21s23sK1sK3,
C=QUT(UUT)-1.
XYZ=Cs1s2s3s1s2s2s3s1s3.
(s1s2s3),(s1s2s3s1s2s31),(s1s2s3s1s2s2s3s1s3),(s1s2s3s1s2s2s3s1s3s1s2s31),(s1s2s3s1s2s2s3s1s3s12s22s32),(s1s2s3s1s2s2s3s1s3s12s22s32s1s2s31).
VF-VTMF
VT=VF(MF)+,
(MF)+=(MF)T[MF(MF)T]-1.

Metrics