Abstract

For colorimetric imaging the tristimulus technique is still the best practical method to keep the measurement time within reasonable limits. However, the achievable color measurement uncertainties for special sources can be large. It is described how the systematic errors can be significantly reduced by using matrix-based color corrections and how the matrix elements can be optimized to obtain the smallest spectral mismatch errors for different light-source distributions. An approach for decreasing the systematic errors is to increase the number of the colorimeter channels (or filters) used for a measurement. Using five channels in a colorimeter is an optimum choice. Determination of the optimum matrices for the five channels is discussed. The correction matrices are designed such that the spectral mismatch errors of the realized functions are minimized relative to the CIE standard color matching functions for several selected test-source distributions. The optimum matrix depends on the (test) light source to be measured. Adaptive matrix values are determined by using the channel outputs and the spectral power distribution of color LEDs approximated with a simple approximation function. The systematic errors are evaluated for a number of colored and white LEDs. The noise propagation with the applied matrix corrections is investigated.

© 2010 Optical Society of America

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References

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  1. G. P. Eppeldauer, G. Lux, and J. Schanda, “A hibák elektronikus javítása a fényforrások tristimulusos szinmérésében (Electronic correction of errors in tristimulus color measurement of light sources),” Mérés Autom.  22/10, 383–386 (1974).
  2. G. P. Eppeldauer and J. Schanda, “Colorimeter with matrix transformation,” in Proceedings of the AIC Color Dynamics Conference (OMKDK, 1976), pp. 403–413.
  3. G. P. Eppeldauer, “Spectral response based calibration method of tristimulus colorimeters,” J. Res. Natl. Inst. Stand. Technol.  103(6), 615–619 (1998).
  4. G.P.Eppeldauer, ed., “Optical radiation measurements based on detector standards,” NIST Technical Note 1621 (National Institute of Standards and Technology, March 2009).
  5. J. Schanda and G. Lux, “On the electronic correction of errors in a tristimulus colorimeter,” in AIC Colour 73—the 2nd AIC Congress (Hilger, 1973), pp. 466–469.
  6. Y. Ohno and J. E. Hardis, “Four-color matrix method for correction of tristimulus colorimeter,” in Proceedings of the Fifth Color Imaging Conference: Color Science, Systems and Applications (Society for Imaging Science and Technology, 1997), pp. 301–305.
  7. Y. Ohno and S. W. Brown, “Four-color matrix method for correction of tristimulus colorimeters—part 2,” in Proceedings of the Sixth Imaging Conference: Color Science, Systems, and Applications (Society for Imaging Science and Technology, 1998) , pp. 65–68.
  8. I. Fryc, S. W. Brown, and Y. Ohno, “A spectrally tunable led sphere source enables accurate calibration of tristimulus colorimeters,” Proc. SPIE  6158, 67580E (2006).
  9. K. Muray, Y. Ohno, B. Kranicz, and J. Schanda, “Comparison measurements of LEDs: spectral power distribution,” in Proceedings of the CIE 2nd LED Measurement Symposium (CIE, 2001), pp. 52–55.
  10. Zs. Kosztyán, S. Sturm, D. Müller, and J. Schanda, “Decreasing colour measuring systematic error in image taking tristimulus colorimeters,” in CIE Expert Symposium on Advances in Photometry and Colorimetry, Conference Proceedings (CIE, 2008), pp. 21–25.
  11. J. Schanda, C. Sik-Lányi, Zs. Kosztyán, P. Csuti, and Gy. Schanda, “Colour measurement of LEDs, problems and corrections,” presented at Midterm Meeting of the International Color Association, Hangzhou, China, 12–14 July 2007.
  12. B. E. Bayer, “Color imaging array,” U.S. patent 3,971,065 (20 July 1976).
  13. G. P. Eppeldauer and M. Rácz, “Design and characterization of a photometer-colorimeter standard,” Appl. Opt.  43, 2621–2631 (2004).
    [CrossRef] [PubMed]
  14. G. P. Eppeldauer, Zs. Kosztyan, J. D. Schanda, Gy. Schanda, C. C. Miller, T. C. Larason, and Y. Ohno, “Extension of the NIST tristimulus colorimeter for solid-state light source measurements,” in CIE Light and Lighting Conference, Proceedings (CIE, 2009), p. 56.
  15. D. S. Hochbaum and J. G. Shanthikumar, “Convex separable optimization is not much harder than linear optimization,” J. ACM  37, 843–862 (1990).
    [CrossRef]
  16. R. J. Leveque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics (Cambridge Univ. Press, 2002), pp. 158–187.
  17. E. F. Schubert, Light-Emitting Diodes (Cambridge Univ. Press, 2003), pp. 88–90, 103, 116.
  18. Y. Ohno, “Simulation analysis of white LED spectra and color rendering,” in CIE LED Light Sources Symposium (CIE, 2005), pp. 28–32.
  19. Zs. Kosztyan and J. Schanda, “Using adaptive matrix transformation for decreasing colour measuring systematic error in image taking tristimulus colorimeters,” in CIE Light and Lighting Conference (CIE, 2009), pp. 131–133.
  20. “Methods of characterizing illuminance meters and luminance meters,” CIE Publication No. 69 (CIE, 1987).

2006 (1)

I. Fryc, S. W. Brown, and Y. Ohno, “A spectrally tunable led sphere source enables accurate calibration of tristimulus colorimeters,” Proc. SPIE  6158, 67580E (2006).

2004 (1)

1998 (1)

G. P. Eppeldauer, “Spectral response based calibration method of tristimulus colorimeters,” J. Res. Natl. Inst. Stand. Technol.  103(6), 615–619 (1998).

1990 (1)

D. S. Hochbaum and J. G. Shanthikumar, “Convex separable optimization is not much harder than linear optimization,” J. ACM  37, 843–862 (1990).
[CrossRef]

1974 (1)

G. P. Eppeldauer, G. Lux, and J. Schanda, “A hibák elektronikus javítása a fényforrások tristimulusos szinmérésében (Electronic correction of errors in tristimulus color measurement of light sources),” Mérés Autom.  22/10, 383–386 (1974).

Bayer, B. E.

B. E. Bayer, “Color imaging array,” U.S. patent 3,971,065 (20 July 1976).

Brown, S. W.

I. Fryc, S. W. Brown, and Y. Ohno, “A spectrally tunable led sphere source enables accurate calibration of tristimulus colorimeters,” Proc. SPIE  6158, 67580E (2006).

Y. Ohno and S. W. Brown, “Four-color matrix method for correction of tristimulus colorimeters—part 2,” in Proceedings of the Sixth Imaging Conference: Color Science, Systems, and Applications (Society for Imaging Science and Technology, 1998) , pp. 65–68.

Csuti, P.

J. Schanda, C. Sik-Lányi, Zs. Kosztyán, P. Csuti, and Gy. Schanda, “Colour measurement of LEDs, problems and corrections,” presented at Midterm Meeting of the International Color Association, Hangzhou, China, 12–14 July 2007.

Eppeldauer, G. P.

G. P. Eppeldauer and M. Rácz, “Design and characterization of a photometer-colorimeter standard,” Appl. Opt.  43, 2621–2631 (2004).
[CrossRef] [PubMed]

G. P. Eppeldauer, “Spectral response based calibration method of tristimulus colorimeters,” J. Res. Natl. Inst. Stand. Technol.  103(6), 615–619 (1998).

G. P. Eppeldauer, G. Lux, and J. Schanda, “A hibák elektronikus javítása a fényforrások tristimulusos szinmérésében (Electronic correction of errors in tristimulus color measurement of light sources),” Mérés Autom.  22/10, 383–386 (1974).

G. P. Eppeldauer and J. Schanda, “Colorimeter with matrix transformation,” in Proceedings of the AIC Color Dynamics Conference (OMKDK, 1976), pp. 403–413.

G. P. Eppeldauer, Zs. Kosztyan, J. D. Schanda, Gy. Schanda, C. C. Miller, T. C. Larason, and Y. Ohno, “Extension of the NIST tristimulus colorimeter for solid-state light source measurements,” in CIE Light and Lighting Conference, Proceedings (CIE, 2009), p. 56.

Fryc, I.

I. Fryc, S. W. Brown, and Y. Ohno, “A spectrally tunable led sphere source enables accurate calibration of tristimulus colorimeters,” Proc. SPIE  6158, 67580E (2006).

Hardis, J. E.

Y. Ohno and J. E. Hardis, “Four-color matrix method for correction of tristimulus colorimeter,” in Proceedings of the Fifth Color Imaging Conference: Color Science, Systems and Applications (Society for Imaging Science and Technology, 1997), pp. 301–305.

Hochbaum, D. S.

D. S. Hochbaum and J. G. Shanthikumar, “Convex separable optimization is not much harder than linear optimization,” J. ACM  37, 843–862 (1990).
[CrossRef]

Kosztyan, Zs.

Zs. Kosztyan and J. Schanda, “Using adaptive matrix transformation for decreasing colour measuring systematic error in image taking tristimulus colorimeters,” in CIE Light and Lighting Conference (CIE, 2009), pp. 131–133.

G. P. Eppeldauer, Zs. Kosztyan, J. D. Schanda, Gy. Schanda, C. C. Miller, T. C. Larason, and Y. Ohno, “Extension of the NIST tristimulus colorimeter for solid-state light source measurements,” in CIE Light and Lighting Conference, Proceedings (CIE, 2009), p. 56.

Kosztyán, Zs.

J. Schanda, C. Sik-Lányi, Zs. Kosztyán, P. Csuti, and Gy. Schanda, “Colour measurement of LEDs, problems and corrections,” presented at Midterm Meeting of the International Color Association, Hangzhou, China, 12–14 July 2007.

Zs. Kosztyán, S. Sturm, D. Müller, and J. Schanda, “Decreasing colour measuring systematic error in image taking tristimulus colorimeters,” in CIE Expert Symposium on Advances in Photometry and Colorimetry, Conference Proceedings (CIE, 2008), pp. 21–25.

Kranicz, B.

K. Muray, Y. Ohno, B. Kranicz, and J. Schanda, “Comparison measurements of LEDs: spectral power distribution,” in Proceedings of the CIE 2nd LED Measurement Symposium (CIE, 2001), pp. 52–55.

Larason, T. C.

G. P. Eppeldauer, Zs. Kosztyan, J. D. Schanda, Gy. Schanda, C. C. Miller, T. C. Larason, and Y. Ohno, “Extension of the NIST tristimulus colorimeter for solid-state light source measurements,” in CIE Light and Lighting Conference, Proceedings (CIE, 2009), p. 56.

Leveque, R. J.

R. J. Leveque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics (Cambridge Univ. Press, 2002), pp. 158–187.

Lux, G.

G. P. Eppeldauer, G. Lux, and J. Schanda, “A hibák elektronikus javítása a fényforrások tristimulusos szinmérésében (Electronic correction of errors in tristimulus color measurement of light sources),” Mérés Autom.  22/10, 383–386 (1974).

J. Schanda and G. Lux, “On the electronic correction of errors in a tristimulus colorimeter,” in AIC Colour 73—the 2nd AIC Congress (Hilger, 1973), pp. 466–469.

Miller, C. C.

G. P. Eppeldauer, Zs. Kosztyan, J. D. Schanda, Gy. Schanda, C. C. Miller, T. C. Larason, and Y. Ohno, “Extension of the NIST tristimulus colorimeter for solid-state light source measurements,” in CIE Light and Lighting Conference, Proceedings (CIE, 2009), p. 56.

Müller, D.

Zs. Kosztyán, S. Sturm, D. Müller, and J. Schanda, “Decreasing colour measuring systematic error in image taking tristimulus colorimeters,” in CIE Expert Symposium on Advances in Photometry and Colorimetry, Conference Proceedings (CIE, 2008), pp. 21–25.

Muray, K.

K. Muray, Y. Ohno, B. Kranicz, and J. Schanda, “Comparison measurements of LEDs: spectral power distribution,” in Proceedings of the CIE 2nd LED Measurement Symposium (CIE, 2001), pp. 52–55.

Ohno, Y.

I. Fryc, S. W. Brown, and Y. Ohno, “A spectrally tunable led sphere source enables accurate calibration of tristimulus colorimeters,” Proc. SPIE  6158, 67580E (2006).

K. Muray, Y. Ohno, B. Kranicz, and J. Schanda, “Comparison measurements of LEDs: spectral power distribution,” in Proceedings of the CIE 2nd LED Measurement Symposium (CIE, 2001), pp. 52–55.

Y. Ohno and S. W. Brown, “Four-color matrix method for correction of tristimulus colorimeters—part 2,” in Proceedings of the Sixth Imaging Conference: Color Science, Systems, and Applications (Society for Imaging Science and Technology, 1998) , pp. 65–68.

Y. Ohno and J. E. Hardis, “Four-color matrix method for correction of tristimulus colorimeter,” in Proceedings of the Fifth Color Imaging Conference: Color Science, Systems and Applications (Society for Imaging Science and Technology, 1997), pp. 301–305.

Y. Ohno, “Simulation analysis of white LED spectra and color rendering,” in CIE LED Light Sources Symposium (CIE, 2005), pp. 28–32.

G. P. Eppeldauer, Zs. Kosztyan, J. D. Schanda, Gy. Schanda, C. C. Miller, T. C. Larason, and Y. Ohno, “Extension of the NIST tristimulus colorimeter for solid-state light source measurements,” in CIE Light and Lighting Conference, Proceedings (CIE, 2009), p. 56.

Rácz, M.

Schanda, Gy.

J. Schanda, C. Sik-Lányi, Zs. Kosztyán, P. Csuti, and Gy. Schanda, “Colour measurement of LEDs, problems and corrections,” presented at Midterm Meeting of the International Color Association, Hangzhou, China, 12–14 July 2007.

G. P. Eppeldauer, Zs. Kosztyan, J. D. Schanda, Gy. Schanda, C. C. Miller, T. C. Larason, and Y. Ohno, “Extension of the NIST tristimulus colorimeter for solid-state light source measurements,” in CIE Light and Lighting Conference, Proceedings (CIE, 2009), p. 56.

Schanda, J.

G. P. Eppeldauer, G. Lux, and J. Schanda, “A hibák elektronikus javítása a fényforrások tristimulusos szinmérésében (Electronic correction of errors in tristimulus color measurement of light sources),” Mérés Autom.  22/10, 383–386 (1974).

G. P. Eppeldauer and J. Schanda, “Colorimeter with matrix transformation,” in Proceedings of the AIC Color Dynamics Conference (OMKDK, 1976), pp. 403–413.

J. Schanda and G. Lux, “On the electronic correction of errors in a tristimulus colorimeter,” in AIC Colour 73—the 2nd AIC Congress (Hilger, 1973), pp. 466–469.

Zs. Kosztyán, S. Sturm, D. Müller, and J. Schanda, “Decreasing colour measuring systematic error in image taking tristimulus colorimeters,” in CIE Expert Symposium on Advances in Photometry and Colorimetry, Conference Proceedings (CIE, 2008), pp. 21–25.

J. Schanda, C. Sik-Lányi, Zs. Kosztyán, P. Csuti, and Gy. Schanda, “Colour measurement of LEDs, problems and corrections,” presented at Midterm Meeting of the International Color Association, Hangzhou, China, 12–14 July 2007.

K. Muray, Y. Ohno, B. Kranicz, and J. Schanda, “Comparison measurements of LEDs: spectral power distribution,” in Proceedings of the CIE 2nd LED Measurement Symposium (CIE, 2001), pp. 52–55.

Zs. Kosztyan and J. Schanda, “Using adaptive matrix transformation for decreasing colour measuring systematic error in image taking tristimulus colorimeters,” in CIE Light and Lighting Conference (CIE, 2009), pp. 131–133.

Schanda, J. D.

G. P. Eppeldauer, Zs. Kosztyan, J. D. Schanda, Gy. Schanda, C. C. Miller, T. C. Larason, and Y. Ohno, “Extension of the NIST tristimulus colorimeter for solid-state light source measurements,” in CIE Light and Lighting Conference, Proceedings (CIE, 2009), p. 56.

Schubert, E. F.

E. F. Schubert, Light-Emitting Diodes (Cambridge Univ. Press, 2003), pp. 88–90, 103, 116.

Shanthikumar, J. G.

D. S. Hochbaum and J. G. Shanthikumar, “Convex separable optimization is not much harder than linear optimization,” J. ACM  37, 843–862 (1990).
[CrossRef]

Sik-Lányi, C.

J. Schanda, C. Sik-Lányi, Zs. Kosztyán, P. Csuti, and Gy. Schanda, “Colour measurement of LEDs, problems and corrections,” presented at Midterm Meeting of the International Color Association, Hangzhou, China, 12–14 July 2007.

Sturm, S.

Zs. Kosztyán, S. Sturm, D. Müller, and J. Schanda, “Decreasing colour measuring systematic error in image taking tristimulus colorimeters,” in CIE Expert Symposium on Advances in Photometry and Colorimetry, Conference Proceedings (CIE, 2008), pp. 21–25.

Appl. Opt. (1)

J. ACM (1)

D. S. Hochbaum and J. G. Shanthikumar, “Convex separable optimization is not much harder than linear optimization,” J. ACM  37, 843–862 (1990).
[CrossRef]

J. Res. Natl. Inst. Stand. Technol. (1)

G. P. Eppeldauer, “Spectral response based calibration method of tristimulus colorimeters,” J. Res. Natl. Inst. Stand. Technol.  103(6), 615–619 (1998).

Mérés Autom. (1)

G. P. Eppeldauer, G. Lux, and J. Schanda, “A hibák elektronikus javítása a fényforrások tristimulusos szinmérésében (Electronic correction of errors in tristimulus color measurement of light sources),” Mérés Autom.  22/10, 383–386 (1974).

Proc. SPIE (1)

I. Fryc, S. W. Brown, and Y. Ohno, “A spectrally tunable led sphere source enables accurate calibration of tristimulus colorimeters,” Proc. SPIE  6158, 67580E (2006).

Other (15)

K. Muray, Y. Ohno, B. Kranicz, and J. Schanda, “Comparison measurements of LEDs: spectral power distribution,” in Proceedings of the CIE 2nd LED Measurement Symposium (CIE, 2001), pp. 52–55.

Zs. Kosztyán, S. Sturm, D. Müller, and J. Schanda, “Decreasing colour measuring systematic error in image taking tristimulus colorimeters,” in CIE Expert Symposium on Advances in Photometry and Colorimetry, Conference Proceedings (CIE, 2008), pp. 21–25.

J. Schanda, C. Sik-Lányi, Zs. Kosztyán, P. Csuti, and Gy. Schanda, “Colour measurement of LEDs, problems and corrections,” presented at Midterm Meeting of the International Color Association, Hangzhou, China, 12–14 July 2007.

B. E. Bayer, “Color imaging array,” U.S. patent 3,971,065 (20 July 1976).

G.P.Eppeldauer, ed., “Optical radiation measurements based on detector standards,” NIST Technical Note 1621 (National Institute of Standards and Technology, March 2009).

J. Schanda and G. Lux, “On the electronic correction of errors in a tristimulus colorimeter,” in AIC Colour 73—the 2nd AIC Congress (Hilger, 1973), pp. 466–469.

Y. Ohno and J. E. Hardis, “Four-color matrix method for correction of tristimulus colorimeter,” in Proceedings of the Fifth Color Imaging Conference: Color Science, Systems and Applications (Society for Imaging Science and Technology, 1997), pp. 301–305.

Y. Ohno and S. W. Brown, “Four-color matrix method for correction of tristimulus colorimeters—part 2,” in Proceedings of the Sixth Imaging Conference: Color Science, Systems, and Applications (Society for Imaging Science and Technology, 1998) , pp. 65–68.

G. P. Eppeldauer and J. Schanda, “Colorimeter with matrix transformation,” in Proceedings of the AIC Color Dynamics Conference (OMKDK, 1976), pp. 403–413.

G. P. Eppeldauer, Zs. Kosztyan, J. D. Schanda, Gy. Schanda, C. C. Miller, T. C. Larason, and Y. Ohno, “Extension of the NIST tristimulus colorimeter for solid-state light source measurements,” in CIE Light and Lighting Conference, Proceedings (CIE, 2009), p. 56.

R. J. Leveque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics (Cambridge Univ. Press, 2002), pp. 158–187.

E. F. Schubert, Light-Emitting Diodes (Cambridge Univ. Press, 2003), pp. 88–90, 103, 116.

Y. Ohno, “Simulation analysis of white LED spectra and color rendering,” in CIE LED Light Sources Symposium (CIE, 2005), pp. 28–32.

Zs. Kosztyan and J. Schanda, “Using adaptive matrix transformation for decreasing colour measuring systematic error in image taking tristimulus colorimeters,” in CIE Light and Lighting Conference (CIE, 2009), pp. 131–133.

“Methods of characterizing illuminance meters and luminance meters,” CIE Publication No. 69 (CIE, 1987).

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Figures (7)

Fig. 1
Fig. 1

CIE 2 ° CMFs [ x ¯ ( λ ) , y ¯ ( λ ) , z ¯ ( λ ) ] and the channel spectral responsivities of a tristimulus colorimeter [ x M short ( λ ) , x M long ( λ ) , y M ( λ ) , z M ( λ ) ].

Fig. 2
Fig. 2

Picture of the single-element Si photodiode-based colorimeter.

Fig. 3
Fig. 3

Measured spectral responsivity of the colorimeter channels.

Fig. 4
Fig. 4

Relative SPD of LEDs from the test database.

Fig. 5
Fig. 5

CIE 2 ° CMFs, the Gaussian curve, and the realized spectral responsivities of a tristimulus camera.

Fig. 6
Fig. 6

SPD of a blue LED and important spectral parameters. λ p is the peak wavelength, Δ λ 0.5 is the spectral bandwidth at half intensity, λ c is the centroid wavelength of the SPD.

Fig. 7
Fig. 7

Relative SPD of an LED (DW, 605.4 nm ) and its approximations using (a) triangle, (b) Gaussian, (c) generalized Lorentzian, and (d) beta distributions.

Tables (6)

Tables Icon

Table 1 Average Colorimetric Errors for Planckian Sources and a Number of LEDs a

Tables Icon

Table 2 Global Matrices for Different Kinds of Tristimulus Colorimetric Systems

Tables Icon

Table 3 Results of Unique Matrix Transformations for Cyan LEDs a

Tables Icon

Table 4 Optimal Unique Matrices Optimized to Cyan LED Sources

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Table 5 Comparison of Results for Detector-Based and Matrix-Corrected Calibration Methods of Tristimulus Colorimeters, Measuring Color LEDs (Four Filters)

Tables Icon

Table 6 Comparison of Results of Detector-Based and Matrix-Corrected Calibration Methods of Tristimulus Colorimeters, Measuring Color LEDs (Five Filters)

Equations (20)

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

k X 1 = X 1 X M 1 = K m Λ E ( λ ) x ¯ 1 ( λ ) d λ Λ E ( λ ) s X 1 ( λ ) d λ , k X 2 = X 2 X M 2 = K m Λ E ( λ ) x ¯ 2 ( λ ) d λ Λ E ( λ ) s X 2 ( λ ) d λ , k Y = Y Y M = K m Λ E ( λ ) y ¯ ( λ ) d λ Λ E ( λ ) s Y ( λ ) d λ , k Z = Z Z M = K m Λ E ( λ ) z ¯ ( λ ) d λ Λ E ( λ ) s Z ( λ ) d λ ,
[ X T s Y T s Z T s ] = [ a 11 a 12 a 13 a 14 a 15 a 1 m a 21 a 22 a 23 a 24 a 25 a 2 m a 31 a 32 a 33 a 34 a 35 a 3 m ] · [ X M , short s X M , long s Y M s Z M s K M 1 s K M m 4 s ] , where     a i j A R 3 x m .
X s = Λ x ¯ ( λ ) S s ( λ ) d λ , s 1 , 2 , , n ,
Y s = Λ y ¯ ( λ ) S s ( λ ) d λ , s 1 , 2 , , n ,
Z s = Λ z ¯ ( λ ) S s ( λ ) d λ , s 1 , 2 , , n .
X M , short s = Λ x M , short ( λ ) S s ( λ ) d λ , s 1 , 2 , , n ,
X M , long s = Λ x M , long ( λ ) S s ( λ ) d λ , s 1 , 2 , , n ,
Y M s = Λ y M ( λ ) S s ( λ ) d λ , s 1 , 2 , , n ,
Z M s = Λ z M ( λ ) S s ( λ ) d λ , s 1 , 2 , , n .
K M k s = Λ k M k ( λ ) S s ( λ ) d λ , s 1 , 2 , , n ; k 1 , 2 , , m 4..
z * ( u ) i 1 n Δ E * ab ( C s , C T s ) n min or
z ( u ) i 1 n Δ ( u , v ) ( C s , C T s ) n min ,
K M s = Λ k M ( λ ) S s ( λ ) d λ , s 1 , 2 , , n .
k ( λ ) = α · σ 2 π · f N ( λ m , σ ) ( λ ) = α exp [ 1 2 ( λ λ m σ ) 2 ] , where     λ m , λ Λ = [ 380 nm , 780 nm ] , f N ( λ m , σ ) ( λ ) = 1 σ 2 π exp [ 1 2 ( λ λ m σ ) 2 ] .
k k ( λ ) = α k · σ k 2 π · f N ( λ m k , σ k ) ( λ ) = α k exp [ 1 2 ( λ λ m k σ k ) 2 ] , where    λ m k , λ Λ = [ 380 nm , 780 nm ] , k 1 , 2 , m 4 , f N ( λ m k , σ k ) ( λ ) = 1 σ 2 π exp [ 1 2 ( λ λ m k σ k ) 2 ] .
g λ m , σ ( λ ) = α exp [ 1 2 ( λ λ m σ ) 2 ] ,
L λ m , γ , p ( λ ) = α 1 + γ | λ λ m | p ,
t λ a , λ m , λ b ( λ ) = { 0 λ [ λ a , λ b ] λ λ a λ m λ a λ a < λ λ m λ b λ λ b λ m λ m < λ < λ b .
f ( x ) = { 0 x ( 0 , 1 ) 1 B ( α , β ) x α 1 ( 1 x ) β 1 x ( 0 , 1 ) ,
B ( α , β ) = 0 1 y α 1 ( 1 y ) β 1 d y .

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