Abstract

Solid state lighting is becoming a popular light source for color vision experiments. One of the advantages of light emitting diodes (LEDs) is the possibility to shape the target light spectrum according to the experimenter’s needs. In this paper, we present a method for creating metameric lights with an LED-based spectrally tunable illuminator. The equipment we use consists of six Gamma Scientific RS-5B lamps, each containing nine different LEDs and a 1 m integrating sphere. We provide a method for describing the (almost) entire set of illuminant metamers. It will be shown that the main difficulty in describing this set arises as the result of the intensity dependent peak-wavelength shift, which is manifested by the majority of the LEDs used by the illuminators of this type. We define the normalized metamer set describing all illuminator spectra that colorimetrically match a given chromaticity. Finally, we describe a method for choosing the smoothest or least smooth metamer from the entire set.

© 2014 Optical Society of America

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References

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  1. M. Mackiewicz, S. Crichton, S. Newsome, R. Gazerro, G. Finlayson, and A. Hurlbert, “Spectrally tunable led illuminator for vision research,” in Proceedings of the 6th Colour in Graphics, Imaging and Vision (CGIV) (2012), Vol. 6, pp. 372–377.
  2. G. Wyszecki and W. Styles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 1982).
  3. G. Finlayson and P. Morovic, “Metamer sets,” J. Opt. Soc. Am. A 22, 810–819, 2005.
    [CrossRef]
  4. P. Morovic, “Metamer sets,” Ph.D. dissertation (School of Computing Sciences, University of East Anglia, 2002).
  5. C. Lawson and R. Hanson, Solving Least-Squares Problems (Prentice-Hall, 1974).
  6. M. S. Peercy, “Linear color representations for full speed spectral rendering,” in Proceedings of the 20th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH ’93) (ACM, 1993), pp. 191–198.
  7. H. J. Trussell, Foundations of Digital Imaging (Cambridge University, 2008).
  8. G. Golub and C. van Loan, Matrix Computations, 4th ed. (Johns Hopkins University, 2012).
  9. J. B. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychon. Sci. 1, 369–370 (1967).
  10. 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]
  11. P. S. Parkkinen, J. Hallikainen, and T. Jaaskelainen, “Characteristic spectra of munsell colors,” J. Opt. Soc. Am. A 6, 318–322 (1989).
    [CrossRef]
  12. S. Westland, J. Shaw, and H. Owens, “Colour statistics of natural and man-made surfaces,” Sensor Rev. 20, 50–55 (2000).
  13. G. Wyszecki, “Evaluation of metameric colors,” J. Opt. Soc. Am. 48, 451–454 (1958).
    [CrossRef]
  14. J. B. Cohen and W. E. Kappauf, “Metameric color stimuli, fundamental metamers, and Wyszecki’s metameric blacks,” Am. J. Psychol. 95, 537–564 (1982).
    [CrossRef]
  15. J. B. Cohen and W. E. Kappauf, “Color mixture and fundamental metamers: theory, algebra, geometry, application,” Am. J. Psychol. 98, 171–259 (1985).
    [CrossRef]
  16. J. B. Fraleigh and R. A. Beauregard, Linear Algebra (Addison-Wesley, 1990).
  17. Finlayson and Morovic also describe an additional naturalness constraint, which is not relevant in this context. The interested reader can refer to [3].
  18. B. Grünbaum, Convex Polytopes (Springer, 1967).
  19. C. Barber, D. P. Dobkin, and H. T. Huhdanpaa, “The quickhull algorithm for convex hulls,” ACM Trans. Mathematical Softw. 22, 469–483 (1996).
  20. F. P. Preparata and M. I. Shamos, Computational Geometry An Introduction, 2nd ed. (Springer-Verlag, 1988).
  21. Gamma Scientific, San Diego, USA. Available: http://www.gamma-sci.com .
  22. S. Muthu, F. J. Schuurmans, and M. D. Pashley, “Red, green, and blue LED based white light generation: issues and control,” in Proceedings of the Institute of Electrical and Electronics Engineers Industry Applications Society Annual Meeting (IEEE IAS) (IEEE, 2002), Vol. 1, pp. 327–333.
  23. C. Li and M. R. Luo, “The estimation of spectral reflectances using smoothness constraint condition,” in Proceedings of the 9th Color and Imaging Conference (CIC) (2001), pp. 62–67.
  24. P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, 1981).
  25. R. Hunt and M. R. Pointer, Measuring Colour, 4th ed. (Wiley, 2011).
  26. N. Ohta and G. Wyszecki, “Theoretical chromaticity mismatch limits of metamers viewed under different illuminants,” J. Opt. Soc. Am. 65, 327–333 (1975).
    [CrossRef]
  27. A. Logvinenko, C. Godau, and B. Funt, “Metamer mismatch volumes,” in Proceedings of the 6th Colour in Graphics, Imaging and Vision (CGIV), Amsterdam, 2012.
  28. W. D. van Driel and X. J. Fan, Solid State Lighting Reliability (Springer, 2013).

2005 (1)

2000 (1)

S. Westland, J. Shaw, and H. Owens, “Colour statistics of natural and man-made surfaces,” Sensor Rev. 20, 50–55 (2000).

1996 (1)

C. Barber, D. P. Dobkin, and H. T. Huhdanpaa, “The quickhull algorithm for convex hulls,” ACM Trans. Mathematical Softw. 22, 469–483 (1996).

1989 (1)

1986 (1)

1985 (1)

J. B. Cohen and W. E. Kappauf, “Color mixture and fundamental metamers: theory, algebra, geometry, application,” Am. J. Psychol. 98, 171–259 (1985).
[CrossRef]

1982 (1)

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

1975 (1)

1967 (1)

J. B. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychon. Sci. 1, 369–370 (1967).

1958 (1)

Barber, C.

C. Barber, D. P. Dobkin, and H. T. Huhdanpaa, “The quickhull algorithm for convex hulls,” ACM Trans. Mathematical Softw. 22, 469–483 (1996).

Beauregard, R. A.

J. B. Fraleigh and R. A. Beauregard, Linear Algebra (Addison-Wesley, 1990).

Cohen, J. B.

J. B. Cohen and W. E. Kappauf, “Color mixture and fundamental metamers: theory, algebra, geometry, application,” Am. J. Psychol. 98, 171–259 (1985).
[CrossRef]

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

J. B. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychon. Sci. 1, 369–370 (1967).

Crichton, S.

M. Mackiewicz, S. Crichton, S. Newsome, R. Gazerro, G. Finlayson, and A. Hurlbert, “Spectrally tunable led illuminator for vision research,” in Proceedings of the 6th Colour in Graphics, Imaging and Vision (CGIV) (2012), Vol. 6, pp. 372–377.

Dobkin, D. P.

C. Barber, D. P. Dobkin, and H. T. Huhdanpaa, “The quickhull algorithm for convex hulls,” ACM Trans. Mathematical Softw. 22, 469–483 (1996).

Fan, X. J.

W. D. van Driel and X. J. Fan, Solid State Lighting Reliability (Springer, 2013).

Finlayson, G.

G. Finlayson and P. Morovic, “Metamer sets,” J. Opt. Soc. Am. A 22, 810–819, 2005.
[CrossRef]

M. Mackiewicz, S. Crichton, S. Newsome, R. Gazerro, G. Finlayson, and A. Hurlbert, “Spectrally tunable led illuminator for vision research,” in Proceedings of the 6th Colour in Graphics, Imaging and Vision (CGIV) (2012), Vol. 6, pp. 372–377.

Fraleigh, J. B.

J. B. Fraleigh and R. A. Beauregard, Linear Algebra (Addison-Wesley, 1990).

Funt, B.

A. Logvinenko, C. Godau, and B. Funt, “Metamer mismatch volumes,” in Proceedings of the 6th Colour in Graphics, Imaging and Vision (CGIV), Amsterdam, 2012.

Gazerro, R.

M. Mackiewicz, S. Crichton, S. Newsome, R. Gazerro, G. Finlayson, and A. Hurlbert, “Spectrally tunable led illuminator for vision research,” in Proceedings of the 6th Colour in Graphics, Imaging and Vision (CGIV) (2012), Vol. 6, pp. 372–377.

Gill, P. E.

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, 1981).

Godau, C.

A. Logvinenko, C. Godau, and B. Funt, “Metamer mismatch volumes,” in Proceedings of the 6th Colour in Graphics, Imaging and Vision (CGIV), Amsterdam, 2012.

Golub, G.

G. Golub and C. van Loan, Matrix Computations, 4th ed. (Johns Hopkins University, 2012).

Grünbaum, B.

B. Grünbaum, Convex Polytopes (Springer, 1967).

Hallikainen, J.

Hanson, R.

C. Lawson and R. Hanson, Solving Least-Squares Problems (Prentice-Hall, 1974).

Huhdanpaa, H. T.

C. Barber, D. P. Dobkin, and H. T. Huhdanpaa, “The quickhull algorithm for convex hulls,” ACM Trans. Mathematical Softw. 22, 469–483 (1996).

Hunt, R.

R. Hunt and M. R. Pointer, Measuring Colour, 4th ed. (Wiley, 2011).

Hurlbert, A.

M. Mackiewicz, S. Crichton, S. Newsome, R. Gazerro, G. Finlayson, and A. Hurlbert, “Spectrally tunable led illuminator for vision research,” in Proceedings of the 6th Colour in Graphics, Imaging and Vision (CGIV) (2012), Vol. 6, pp. 372–377.

Jaaskelainen, T.

Kappauf, W. E.

J. B. Cohen and W. E. Kappauf, “Color mixture and fundamental metamers: theory, algebra, geometry, application,” Am. J. Psychol. 98, 171–259 (1985).
[CrossRef]

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

Lawson, C.

C. Lawson and R. Hanson, Solving Least-Squares Problems (Prentice-Hall, 1974).

Li, C.

C. Li and M. R. Luo, “The estimation of spectral reflectances using smoothness constraint condition,” in Proceedings of the 9th Color and Imaging Conference (CIC) (2001), pp. 62–67.

Logvinenko, A.

A. Logvinenko, C. Godau, and B. Funt, “Metamer mismatch volumes,” in Proceedings of the 6th Colour in Graphics, Imaging and Vision (CGIV), Amsterdam, 2012.

Luo, M. R.

C. Li and M. R. Luo, “The estimation of spectral reflectances using smoothness constraint condition,” in Proceedings of the 9th Color and Imaging Conference (CIC) (2001), pp. 62–67.

Mackiewicz, M.

M. Mackiewicz, S. Crichton, S. Newsome, R. Gazerro, G. Finlayson, and A. Hurlbert, “Spectrally tunable led illuminator for vision research,” in Proceedings of the 6th Colour in Graphics, Imaging and Vision (CGIV) (2012), Vol. 6, pp. 372–377.

Maloney, L. T.

Morovic, P.

G. Finlayson and P. Morovic, “Metamer sets,” J. Opt. Soc. Am. A 22, 810–819, 2005.
[CrossRef]

P. Morovic, “Metamer sets,” Ph.D. dissertation (School of Computing Sciences, University of East Anglia, 2002).

Murray, W.

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, 1981).

Muthu, S.

S. Muthu, F. J. Schuurmans, and M. D. Pashley, “Red, green, and blue LED based white light generation: issues and control,” in Proceedings of the Institute of Electrical and Electronics Engineers Industry Applications Society Annual Meeting (IEEE IAS) (IEEE, 2002), Vol. 1, pp. 327–333.

Newsome, S.

M. Mackiewicz, S. Crichton, S. Newsome, R. Gazerro, G. Finlayson, and A. Hurlbert, “Spectrally tunable led illuminator for vision research,” in Proceedings of the 6th Colour in Graphics, Imaging and Vision (CGIV) (2012), Vol. 6, pp. 372–377.

Ohta, N.

Owens, H.

S. Westland, J. Shaw, and H. Owens, “Colour statistics of natural and man-made surfaces,” Sensor Rev. 20, 50–55 (2000).

Parkkinen, P. S.

Pashley, M. D.

S. Muthu, F. J. Schuurmans, and M. D. Pashley, “Red, green, and blue LED based white light generation: issues and control,” in Proceedings of the Institute of Electrical and Electronics Engineers Industry Applications Society Annual Meeting (IEEE IAS) (IEEE, 2002), Vol. 1, pp. 327–333.

Peercy, M. S.

M. S. Peercy, “Linear color representations for full speed spectral rendering,” in Proceedings of the 20th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH ’93) (ACM, 1993), pp. 191–198.

Pointer, M. R.

R. Hunt and M. R. Pointer, Measuring Colour, 4th ed. (Wiley, 2011).

Preparata, F. P.

F. P. Preparata and M. I. Shamos, Computational Geometry An Introduction, 2nd ed. (Springer-Verlag, 1988).

Schuurmans, F. J.

S. Muthu, F. J. Schuurmans, and M. D. Pashley, “Red, green, and blue LED based white light generation: issues and control,” in Proceedings of the Institute of Electrical and Electronics Engineers Industry Applications Society Annual Meeting (IEEE IAS) (IEEE, 2002), Vol. 1, pp. 327–333.

Shamos, M. I.

F. P. Preparata and M. I. Shamos, Computational Geometry An Introduction, 2nd ed. (Springer-Verlag, 1988).

Shaw, J.

S. Westland, J. Shaw, and H. Owens, “Colour statistics of natural and man-made surfaces,” Sensor Rev. 20, 50–55 (2000).

Styles, W.

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

Trussell, H. J.

H. J. Trussell, Foundations of Digital Imaging (Cambridge University, 2008).

van Driel, W. D.

W. D. van Driel and X. J. Fan, Solid State Lighting Reliability (Springer, 2013).

van Loan, C.

G. Golub and C. van Loan, Matrix Computations, 4th ed. (Johns Hopkins University, 2012).

Westland, S.

S. Westland, J. Shaw, and H. Owens, “Colour statistics of natural and man-made surfaces,” Sensor Rev. 20, 50–55 (2000).

Wright, M. H.

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, 1981).

Wyszecki, G.

ACM Trans. Mathematical Softw. (1)

C. Barber, D. P. Dobkin, and H. T. Huhdanpaa, “The quickhull algorithm for convex hulls,” ACM Trans. Mathematical Softw. 22, 469–483 (1996).

Am. J. Psychol. (2)

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

J. B. Cohen and W. E. Kappauf, “Color mixture and fundamental metamers: theory, algebra, geometry, application,” Am. J. Psychol. 98, 171–259 (1985).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

Psychon. Sci. (1)

J. B. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychon. Sci. 1, 369–370 (1967).

Sensor Rev. (1)

S. Westland, J. Shaw, and H. Owens, “Colour statistics of natural and man-made surfaces,” Sensor Rev. 20, 50–55 (2000).

Other (18)

A. Logvinenko, C. Godau, and B. Funt, “Metamer mismatch volumes,” in Proceedings of the 6th Colour in Graphics, Imaging and Vision (CGIV), Amsterdam, 2012.

W. D. van Driel and X. J. Fan, Solid State Lighting Reliability (Springer, 2013).

J. B. Fraleigh and R. A. Beauregard, Linear Algebra (Addison-Wesley, 1990).

Finlayson and Morovic also describe an additional naturalness constraint, which is not relevant in this context. The interested reader can refer to [3].

B. Grünbaum, Convex Polytopes (Springer, 1967).

P. Morovic, “Metamer sets,” Ph.D. dissertation (School of Computing Sciences, University of East Anglia, 2002).

C. Lawson and R. Hanson, Solving Least-Squares Problems (Prentice-Hall, 1974).

M. S. Peercy, “Linear color representations for full speed spectral rendering,” in Proceedings of the 20th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH ’93) (ACM, 1993), pp. 191–198.

H. J. Trussell, Foundations of Digital Imaging (Cambridge University, 2008).

G. Golub and C. van Loan, Matrix Computations, 4th ed. (Johns Hopkins University, 2012).

F. P. Preparata and M. I. Shamos, Computational Geometry An Introduction, 2nd ed. (Springer-Verlag, 1988).

Gamma Scientific, San Diego, USA. Available: http://www.gamma-sci.com .

S. Muthu, F. J. Schuurmans, and M. D. Pashley, “Red, green, and blue LED based white light generation: issues and control,” in Proceedings of the Institute of Electrical and Electronics Engineers Industry Applications Society Annual Meeting (IEEE IAS) (IEEE, 2002), Vol. 1, pp. 327–333.

C. Li and M. R. Luo, “The estimation of spectral reflectances using smoothness constraint condition,” in Proceedings of the 9th Color and Imaging Conference (CIC) (2001), pp. 62–67.

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, 1981).

R. Hunt and M. R. Pointer, Measuring Colour, 4th ed. (Wiley, 2011).

M. Mackiewicz, S. Crichton, S. Newsome, R. Gazerro, G. Finlayson, and A. Hurlbert, “Spectrally tunable led illuminator for vision research,” in Proceedings of the 6th Colour in Graphics, Imaging and Vision (CGIV) (2012), Vol. 6, pp. 372–377.

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

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Figures (18)

Fig. 1.
Fig. 1.

Spectra of one of the LEDs for 10 intensities, ranging from the 0.1 of the maximum intensity to the maximum.

Fig. 2.
Fig. 2.

RS-5B optical heads mounted onto the 1 m integrating sphere.

Fig. 3.
Fig. 3.

RS-5B illuminating the viewing cabinet through the 34×68cm aperture.

Fig. 4.
Fig. 4.

Spectra of nine LEDs in RS-5B illuminator at their maximum intensity.

Fig. 5.
Fig. 5.

Spectra from Fig. 1 after normalization.

Fig. 6.
Fig. 6.

First versus second principal component of the 10 LED spectra from Fig. 1 for channel 5. The 10 plotted points correspond to the 10 LED spectra of increasing intensities (from left to right).

Fig. 7.
Fig. 7.

Outline of the u’v’ chromaticity diagram: (blue) location of LED lights for different intensities; (red x) RS-5B gamut.

Fig. 8.
Fig. 8.

Illuminant A and D65 metamer particular solutions (at 246cd/m2).

Fig. 9.
Fig. 9.

Metameric black (orthogonal to Λ) basis spectra.

Fig. 10.
Fig. 10.

Illuminant A metamers (at 246cd/m2).

Fig. 11.
Fig. 11.

Illuminant D65 metamers (at 246cd/m2).

Fig. 12.
Fig. 12.

Illuminant D65 and A CIE 1931 chromaticities, x; solid and dashed lines represent the convex hulls of metamers at vertices of the metamer sets for 246 and 123cd/m2, respectively.

Fig. 13.
Fig. 13.

Illuminant A metamers (at 246cd/m2) calculated for one of the interpolation bands. The thick red and blue lines denote the lower and upper envelopes of this particular interpolation band.

Fig. 14.
Fig. 14.

Illuminant D65 metamers (at 246cd/m2) calculated for one of the interpolation bands.

Fig. 15.
Fig. 15.

Illuminant A and D65 (both at 246cd/m2) 2D metamer mismatch volumes for the fourth (dark green) patch from the Macbeth CC. Thick black lines denote the two volumes calculated with the basic model, and the color lines denote the subvolumes calculated with the complex model.

Fig. 16.
Fig. 16.

Illuminant A normalized metamers calculated for one of the interpolation bands.

Fig. 17.
Fig. 17.

Illuminant D65 normalized metamers calculated for one of the interpolation bands.

Fig. 18.
Fig. 18.

Smoothest and least smooth illuminant A and D65 metamers at 246cd/m2 [according to the model (solid lines) and measured (dashed lines)].

Tables (1)

Tables Icon

Table 1. Mean ΔE Errors Calculated for All Vertices of the Metamer Sets (v) and for Random Convex Combinations of These Vertices (cc) for Basic and Complex Algorithms at Three Luminance Levels

Equations (54)

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

χ1=ωR(λ)E(λ)X1(λ)dλ,
χ2=ωR(λ)E(λ)X2(λ)dλ,
χ3=ωR(λ)E(λ)X3(λ)dλ;
X1(λ)=X(λ),X2(λ)=Y(λ),X3(λ)=Z(λ).
χ¯=ωR(λ)E(λ)X¯(λ)dλ.
χi=jxi,jejrjΔλ,
χ¯=XTD(e¯)r¯.
r¯=Bσ¯,
χ¯=ΛTσ¯.
σ¯=σ¯p+σ¯0,
χ¯=ΛTσ¯p,
0¯=ΛTσ¯0.
σ¯p=Λ(ΛTΛ)1χ¯.
M(χ¯)={σ¯|σ¯=σ¯p+σ¯0:ΛTσ¯=χ¯}.
σ¯0=Ψα¯,
P={σ¯|Bσ¯0¯Bσ¯1¯},
M(χ¯)P.
B(σ¯p+Ψα¯)0¯,
BΨα¯Bσ¯p.
BΨα¯1¯Bσ¯p.
χ¯=XTD(r¯)e¯.
e¯=Cσ¯,
χ¯=ΛTσ¯.
PI={σ¯|σ¯0¯σ¯1¯},
M(χ¯)PI.
σ¯p+Ψα¯0¯,
Ψα¯σ¯p.
Ψα¯1¯σ¯p.
γ¯l(w)={γ¯l,k(1a)+γ¯l,k1a,ifk{2,,Kl}γ¯l,ka,ifk=1,
Γ(k¯)=[γ¯1,k¯(1)1Tγ¯1,k¯(1)Tγ¯2,k¯(2)1Tγ¯2,k¯(2)Tγ¯L,k¯(L)1Tγ¯L,k¯(L)T]
Δ(k¯)=[γ¯1,k¯(1)Tγ¯2,k¯(2)Tγ¯L,k¯(L)T].
e¯=Γ(k¯)a¯+Δ(k¯).
χ¯XTD(r¯)Δ(k¯)=XTD(r¯)Γ(k¯)a¯.
χ¯=ΛTa¯.
M(χ¯,k¯)={a¯|a¯=a¯p+a¯0:ΛTa¯=χ¯}.
PI={a¯|a¯0¯a¯1¯}.
C(χ¯,k¯)=M(χ¯,k¯)PI,
Ψα¯a¯p,
Ψα¯1¯a¯p,
wl={wl,k(1al)+wl,k1al,ifk{2,,Kl}wl,kal,ifk=1.
F(χ¯)=i=1NC(χ¯,k¯i).
βχ¯=XTD(r¯)e¯.
βχ¯XTD(r¯)Δ(k¯)=XTD(r¯)Γ(k¯)a¯.
XTD(r¯)Γ(k¯)a¯βχ¯=XTD(r¯)Δ(k¯).
[XTD(r¯)Γ(k¯),χ¯][a¯β]=XTD(r¯)Δ(k¯).
ΛTa¯=χ¯.
MN(χ¯,k¯)={a¯|a¯=a¯p+a¯0:ΛTa¯=χ¯}.
Ψα¯a¯p,
Ψα¯1¯a¯p.
FN(χ¯)=i=1NCN(χ¯,k¯i).
ω(dEdλ)2dλGe¯2/Δλ,
G=[110001100011],
mine¯Ge¯2.
minimizea¯GMa¯2subject to0¯a¯1¯i=1pai=1.

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