Abstract

A whole-slide imaging (WSI) device is a color medical imaging system whose application in digital pathology is to digitalize stained tissue samples into electronic images for pathologists to diagnose without using a conventional light microscope. Testing the color performance of a WSI device usually implies a color target with known truth that is compared with the device output to estimate color differences. Using stained tissue samples as color targets is challenging because the cellular features cannot be measured with ordinary spectroradiometers unless a hyperspectral imaging microscopy system (HIMS) is used. The goal of this study is to determine the colorimetrical uncertainty of such a reference HIMS that is designed to assess the color performance of WSI devices. A set of optical filters are used for that purpose. The color truth, in terms of spectral transmittance in the visible band, of the optical filters is measured by a reference spectroradiometer. The spectral transmittance is combined with a standard illuminant to generate colorimetrical measures using the CIEXYZ and CIELAB formulas. The differences between the reference HIMS and the reference spectroradiometer are evaluated using the CIE 1976 color difference formulas.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. “Medical Devices; Hematology and Pathology Devices; Classification of the Whole Slide Imaging System. Final order,” Federal register 83, 20 (2018).
  2. E. L. Clarke and D. Treanor, “Colour in digital pathology: a review,” Histopathology 70(2), 153–163 (2017).
    [Crossref]
  3. E. L. Clarke, C. Revie, D. Brettle, M. Shires, P. Jackson, R. Cochrane, R. Wilson, C. Mello-Thoms, and D. Treanor, “Development of a novel tissue-mimicking color calibration slide for digital microscopy,” Color Res. Appl. 43(2), 184–197 (2018).
    [Crossref]
  4. A. Badano, C. Revie, A. Casertano, W.-C. Cheng, P. Green, T. Kimpe, E. Krupinski, C. Sisson, S. Skrøvseth, and D. Treanor, “Consistency and standardization of color in medical imaging: a consensus report,” J. Digit Imaging 28(1), 41–52 (2015).
    [Crossref]
  5. W. C. Cheng, F. Saleheen, and A. Badano, “Assessing color performance of whole-slide imaging scanners for digital pathology,” Color Res. Appl. 44(3), 322–334 (2019).
    [Crossref]
  6. A. R. Robertson, “The CIE 1977 color-difference formulae,” Color Res. Appl. 2(1), 7–11 (1977).
    [Crossref]
  7. P. Shrestha and B. Hulsken, “Color accuracy and reproducibility in whole slide imaging scanners,” J. Med. Imag. 1(2), 027501 (2014).
    [Crossref]
  8. M. E. Nadal, E. A. Early, and R. R. Bousquet, “0: 45 Surface Color,” NIST Special Publication SP250-71 (2008).
  9. “CIE S014-1/E: 2006: Colorimetry - Part I: CIE Standard Colorimetric Observer” (2007).
  10. “CIE S014-1/E: 2006: Colorimetry - Part II: CIE Standard Illuminant” (2007).
  11. B. N. Taylor and C. E. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” NIST Technical Report 1297 (1994).
  12. “Guide to the Expression of Uncertainty in Measurement (GUM)–Supplement 1: Numerical Methods for the Propagation of Distributions,” International Organization for Standardization (2004).
  13. J. Gardner and R. Frenkel, “Correlation coefficients for tristimulus response value uncertainties,” Metrologica 36(5), 477–480 (1999).
    [Crossref]
  14. G. Wübbeler, J. Campos Acosta, and C. Elster, “Evaluation of uncertainties for CIELAB color coordinates,” Color Res. Appl. 42(5), 564–570 (2017).
    [Crossref]
  15. M. Anderson, R. Motta, S. Chandrasekar, and M. Stokes, “Proposal for a standard default color space for the internet—srgb,” in Color and imaging conference (Society for Imaging Science and Technology, 1996), pp. 238–245.

2019 (1)

W. C. Cheng, F. Saleheen, and A. Badano, “Assessing color performance of whole-slide imaging scanners for digital pathology,” Color Res. Appl. 44(3), 322–334 (2019).
[Crossref]

2018 (1)

E. L. Clarke, C. Revie, D. Brettle, M. Shires, P. Jackson, R. Cochrane, R. Wilson, C. Mello-Thoms, and D. Treanor, “Development of a novel tissue-mimicking color calibration slide for digital microscopy,” Color Res. Appl. 43(2), 184–197 (2018).
[Crossref]

2017 (2)

E. L. Clarke and D. Treanor, “Colour in digital pathology: a review,” Histopathology 70(2), 153–163 (2017).
[Crossref]

G. Wübbeler, J. Campos Acosta, and C. Elster, “Evaluation of uncertainties for CIELAB color coordinates,” Color Res. Appl. 42(5), 564–570 (2017).
[Crossref]

2015 (1)

A. Badano, C. Revie, A. Casertano, W.-C. Cheng, P. Green, T. Kimpe, E. Krupinski, C. Sisson, S. Skrøvseth, and D. Treanor, “Consistency and standardization of color in medical imaging: a consensus report,” J. Digit Imaging 28(1), 41–52 (2015).
[Crossref]

2014 (1)

P. Shrestha and B. Hulsken, “Color accuracy and reproducibility in whole slide imaging scanners,” J. Med. Imag. 1(2), 027501 (2014).
[Crossref]

1999 (1)

J. Gardner and R. Frenkel, “Correlation coefficients for tristimulus response value uncertainties,” Metrologica 36(5), 477–480 (1999).
[Crossref]

1977 (1)

A. R. Robertson, “The CIE 1977 color-difference formulae,” Color Res. Appl. 2(1), 7–11 (1977).
[Crossref]

Anderson, M.

M. Anderson, R. Motta, S. Chandrasekar, and M. Stokes, “Proposal for a standard default color space for the internet—srgb,” in Color and imaging conference (Society for Imaging Science and Technology, 1996), pp. 238–245.

Badano, A.

W. C. Cheng, F. Saleheen, and A. Badano, “Assessing color performance of whole-slide imaging scanners for digital pathology,” Color Res. Appl. 44(3), 322–334 (2019).
[Crossref]

A. Badano, C. Revie, A. Casertano, W.-C. Cheng, P. Green, T. Kimpe, E. Krupinski, C. Sisson, S. Skrøvseth, and D. Treanor, “Consistency and standardization of color in medical imaging: a consensus report,” J. Digit Imaging 28(1), 41–52 (2015).
[Crossref]

Bousquet, R. R.

M. E. Nadal, E. A. Early, and R. R. Bousquet, “0: 45 Surface Color,” NIST Special Publication SP250-71 (2008).

Brettle, D.

E. L. Clarke, C. Revie, D. Brettle, M. Shires, P. Jackson, R. Cochrane, R. Wilson, C. Mello-Thoms, and D. Treanor, “Development of a novel tissue-mimicking color calibration slide for digital microscopy,” Color Res. Appl. 43(2), 184–197 (2018).
[Crossref]

Campos Acosta, J.

G. Wübbeler, J. Campos Acosta, and C. Elster, “Evaluation of uncertainties for CIELAB color coordinates,” Color Res. Appl. 42(5), 564–570 (2017).
[Crossref]

Casertano, A.

A. Badano, C. Revie, A. Casertano, W.-C. Cheng, P. Green, T. Kimpe, E. Krupinski, C. Sisson, S. Skrøvseth, and D. Treanor, “Consistency and standardization of color in medical imaging: a consensus report,” J. Digit Imaging 28(1), 41–52 (2015).
[Crossref]

Chandrasekar, S.

M. Anderson, R. Motta, S. Chandrasekar, and M. Stokes, “Proposal for a standard default color space for the internet—srgb,” in Color and imaging conference (Society for Imaging Science and Technology, 1996), pp. 238–245.

Cheng, W. C.

W. C. Cheng, F. Saleheen, and A. Badano, “Assessing color performance of whole-slide imaging scanners for digital pathology,” Color Res. Appl. 44(3), 322–334 (2019).
[Crossref]

Cheng, W.-C.

A. Badano, C. Revie, A. Casertano, W.-C. Cheng, P. Green, T. Kimpe, E. Krupinski, C. Sisson, S. Skrøvseth, and D. Treanor, “Consistency and standardization of color in medical imaging: a consensus report,” J. Digit Imaging 28(1), 41–52 (2015).
[Crossref]

Clarke, E. L.

E. L. Clarke, C. Revie, D. Brettle, M. Shires, P. Jackson, R. Cochrane, R. Wilson, C. Mello-Thoms, and D. Treanor, “Development of a novel tissue-mimicking color calibration slide for digital microscopy,” Color Res. Appl. 43(2), 184–197 (2018).
[Crossref]

E. L. Clarke and D. Treanor, “Colour in digital pathology: a review,” Histopathology 70(2), 153–163 (2017).
[Crossref]

Cochrane, R.

E. L. Clarke, C. Revie, D. Brettle, M. Shires, P. Jackson, R. Cochrane, R. Wilson, C. Mello-Thoms, and D. Treanor, “Development of a novel tissue-mimicking color calibration slide for digital microscopy,” Color Res. Appl. 43(2), 184–197 (2018).
[Crossref]

Early, E. A.

M. E. Nadal, E. A. Early, and R. R. Bousquet, “0: 45 Surface Color,” NIST Special Publication SP250-71 (2008).

Elster, C.

G. Wübbeler, J. Campos Acosta, and C. Elster, “Evaluation of uncertainties for CIELAB color coordinates,” Color Res. Appl. 42(5), 564–570 (2017).
[Crossref]

Frenkel, R.

J. Gardner and R. Frenkel, “Correlation coefficients for tristimulus response value uncertainties,” Metrologica 36(5), 477–480 (1999).
[Crossref]

Gardner, J.

J. Gardner and R. Frenkel, “Correlation coefficients for tristimulus response value uncertainties,” Metrologica 36(5), 477–480 (1999).
[Crossref]

Green, P.

A. Badano, C. Revie, A. Casertano, W.-C. Cheng, P. Green, T. Kimpe, E. Krupinski, C. Sisson, S. Skrøvseth, and D. Treanor, “Consistency and standardization of color in medical imaging: a consensus report,” J. Digit Imaging 28(1), 41–52 (2015).
[Crossref]

Hulsken, B.

P. Shrestha and B. Hulsken, “Color accuracy and reproducibility in whole slide imaging scanners,” J. Med. Imag. 1(2), 027501 (2014).
[Crossref]

Jackson, P.

E. L. Clarke, C. Revie, D. Brettle, M. Shires, P. Jackson, R. Cochrane, R. Wilson, C. Mello-Thoms, and D. Treanor, “Development of a novel tissue-mimicking color calibration slide for digital microscopy,” Color Res. Appl. 43(2), 184–197 (2018).
[Crossref]

Kimpe, T.

A. Badano, C. Revie, A. Casertano, W.-C. Cheng, P. Green, T. Kimpe, E. Krupinski, C. Sisson, S. Skrøvseth, and D. Treanor, “Consistency and standardization of color in medical imaging: a consensus report,” J. Digit Imaging 28(1), 41–52 (2015).
[Crossref]

Krupinski, E.

A. Badano, C. Revie, A. Casertano, W.-C. Cheng, P. Green, T. Kimpe, E. Krupinski, C. Sisson, S. Skrøvseth, and D. Treanor, “Consistency and standardization of color in medical imaging: a consensus report,” J. Digit Imaging 28(1), 41–52 (2015).
[Crossref]

Kuyatt, C. E.

B. N. Taylor and C. E. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” NIST Technical Report 1297 (1994).

Mello-Thoms, C.

E. L. Clarke, C. Revie, D. Brettle, M. Shires, P. Jackson, R. Cochrane, R. Wilson, C. Mello-Thoms, and D. Treanor, “Development of a novel tissue-mimicking color calibration slide for digital microscopy,” Color Res. Appl. 43(2), 184–197 (2018).
[Crossref]

Motta, R.

M. Anderson, R. Motta, S. Chandrasekar, and M. Stokes, “Proposal for a standard default color space for the internet—srgb,” in Color and imaging conference (Society for Imaging Science and Technology, 1996), pp. 238–245.

Nadal, M. E.

M. E. Nadal, E. A. Early, and R. R. Bousquet, “0: 45 Surface Color,” NIST Special Publication SP250-71 (2008).

Revie, C.

E. L. Clarke, C. Revie, D. Brettle, M. Shires, P. Jackson, R. Cochrane, R. Wilson, C. Mello-Thoms, and D. Treanor, “Development of a novel tissue-mimicking color calibration slide for digital microscopy,” Color Res. Appl. 43(2), 184–197 (2018).
[Crossref]

A. Badano, C. Revie, A. Casertano, W.-C. Cheng, P. Green, T. Kimpe, E. Krupinski, C. Sisson, S. Skrøvseth, and D. Treanor, “Consistency and standardization of color in medical imaging: a consensus report,” J. Digit Imaging 28(1), 41–52 (2015).
[Crossref]

Robertson, A. R.

A. R. Robertson, “The CIE 1977 color-difference formulae,” Color Res. Appl. 2(1), 7–11 (1977).
[Crossref]

Saleheen, F.

W. C. Cheng, F. Saleheen, and A. Badano, “Assessing color performance of whole-slide imaging scanners for digital pathology,” Color Res. Appl. 44(3), 322–334 (2019).
[Crossref]

Shires, M.

E. L. Clarke, C. Revie, D. Brettle, M. Shires, P. Jackson, R. Cochrane, R. Wilson, C. Mello-Thoms, and D. Treanor, “Development of a novel tissue-mimicking color calibration slide for digital microscopy,” Color Res. Appl. 43(2), 184–197 (2018).
[Crossref]

Shrestha, P.

P. Shrestha and B. Hulsken, “Color accuracy and reproducibility in whole slide imaging scanners,” J. Med. Imag. 1(2), 027501 (2014).
[Crossref]

Sisson, C.

A. Badano, C. Revie, A. Casertano, W.-C. Cheng, P. Green, T. Kimpe, E. Krupinski, C. Sisson, S. Skrøvseth, and D. Treanor, “Consistency and standardization of color in medical imaging: a consensus report,” J. Digit Imaging 28(1), 41–52 (2015).
[Crossref]

Skrøvseth, S.

A. Badano, C. Revie, A. Casertano, W.-C. Cheng, P. Green, T. Kimpe, E. Krupinski, C. Sisson, S. Skrøvseth, and D. Treanor, “Consistency and standardization of color in medical imaging: a consensus report,” J. Digit Imaging 28(1), 41–52 (2015).
[Crossref]

Stokes, M.

M. Anderson, R. Motta, S. Chandrasekar, and M. Stokes, “Proposal for a standard default color space for the internet—srgb,” in Color and imaging conference (Society for Imaging Science and Technology, 1996), pp. 238–245.

Taylor, B. N.

B. N. Taylor and C. E. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” NIST Technical Report 1297 (1994).

Treanor, D.

E. L. Clarke, C. Revie, D. Brettle, M. Shires, P. Jackson, R. Cochrane, R. Wilson, C. Mello-Thoms, and D. Treanor, “Development of a novel tissue-mimicking color calibration slide for digital microscopy,” Color Res. Appl. 43(2), 184–197 (2018).
[Crossref]

E. L. Clarke and D. Treanor, “Colour in digital pathology: a review,” Histopathology 70(2), 153–163 (2017).
[Crossref]

A. Badano, C. Revie, A. Casertano, W.-C. Cheng, P. Green, T. Kimpe, E. Krupinski, C. Sisson, S. Skrøvseth, and D. Treanor, “Consistency and standardization of color in medical imaging: a consensus report,” J. Digit Imaging 28(1), 41–52 (2015).
[Crossref]

Wilson, R.

E. L. Clarke, C. Revie, D. Brettle, M. Shires, P. Jackson, R. Cochrane, R. Wilson, C. Mello-Thoms, and D. Treanor, “Development of a novel tissue-mimicking color calibration slide for digital microscopy,” Color Res. Appl. 43(2), 184–197 (2018).
[Crossref]

Wübbeler, G.

G. Wübbeler, J. Campos Acosta, and C. Elster, “Evaluation of uncertainties for CIELAB color coordinates,” Color Res. Appl. 42(5), 564–570 (2017).
[Crossref]

Color Res. Appl. (4)

W. C. Cheng, F. Saleheen, and A. Badano, “Assessing color performance of whole-slide imaging scanners for digital pathology,” Color Res. Appl. 44(3), 322–334 (2019).
[Crossref]

A. R. Robertson, “The CIE 1977 color-difference formulae,” Color Res. Appl. 2(1), 7–11 (1977).
[Crossref]

E. L. Clarke, C. Revie, D. Brettle, M. Shires, P. Jackson, R. Cochrane, R. Wilson, C. Mello-Thoms, and D. Treanor, “Development of a novel tissue-mimicking color calibration slide for digital microscopy,” Color Res. Appl. 43(2), 184–197 (2018).
[Crossref]

G. Wübbeler, J. Campos Acosta, and C. Elster, “Evaluation of uncertainties for CIELAB color coordinates,” Color Res. Appl. 42(5), 564–570 (2017).
[Crossref]

Histopathology (1)

E. L. Clarke and D. Treanor, “Colour in digital pathology: a review,” Histopathology 70(2), 153–163 (2017).
[Crossref]

J. Digit Imaging (1)

A. Badano, C. Revie, A. Casertano, W.-C. Cheng, P. Green, T. Kimpe, E. Krupinski, C. Sisson, S. Skrøvseth, and D. Treanor, “Consistency and standardization of color in medical imaging: a consensus report,” J. Digit Imaging 28(1), 41–52 (2015).
[Crossref]

J. Med. Imag. (1)

P. Shrestha and B. Hulsken, “Color accuracy and reproducibility in whole slide imaging scanners,” J. Med. Imag. 1(2), 027501 (2014).
[Crossref]

Metrologica (1)

J. Gardner and R. Frenkel, “Correlation coefficients for tristimulus response value uncertainties,” Metrologica 36(5), 477–480 (1999).
[Crossref]

Other (7)

M. Anderson, R. Motta, S. Chandrasekar, and M. Stokes, “Proposal for a standard default color space for the internet—srgb,” in Color and imaging conference (Society for Imaging Science and Technology, 1996), pp. 238–245.

“Medical Devices; Hematology and Pathology Devices; Classification of the Whole Slide Imaging System. Final order,” Federal register 83, 20 (2018).

M. E. Nadal, E. A. Early, and R. R. Bousquet, “0: 45 Surface Color,” NIST Special Publication SP250-71 (2008).

“CIE S014-1/E: 2006: Colorimetry - Part I: CIE Standard Colorimetric Observer” (2007).

“CIE S014-1/E: 2006: Colorimetry - Part II: CIE Standard Illuminant” (2007).

B. N. Taylor and C. E. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” NIST Technical Report 1297 (1994).

“Guide to the Expression of Uncertainty in Measurement (GUM)–Supplement 1: Numerical Methods for the Propagation of Distributions,” International Organization for Standardization (2004).

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Figures (10)

Fig. 1.
Fig. 1. Schematic layout of the hyperspectral microscope equipped with a PR730 spectroradiometer; SRM: spectroradiometer; FP: fiber probe; MSC: motorized stage control; LG: light guide; TLS: tunable light source
Fig. 2.
Fig. 2. OL490 light engine output of the 41 spectral bands (scaled to the same light intensity parameter) used to obtain the hyperspectral images.
Fig. 3.
Fig. 3. Color phantom with 23 Roscoloux color filter dots glued on a cardboard slab set with an adequate series of punched holes. The empty position is the 100% transmittance slot.
Fig. 4.
Fig. 4. (a) Comparison of ${T_S}$, the transmittance spectra measured with a spectroradiometer, and ${T_{SA}}$, the transmittance measured with the camera (spatial average over the image) for a set of KW gelatin neutral density filters with $OD = [{0.1,\; 0.2,\; 0.3,\; 0.6,\; 1.0,\; 2.0} ]$. ${T_{SA}}$ and ${T_S}$ overlap within the error bars (coverage factor $k = 2$) for $\lambda = 390\; \textrm{nm}\; to\; 770\; \textrm{nm}$ for most ND filters; (b) ${T_{SA}}$ versus ${T_S}$ fitted with a linear model for ${\lambda } = \; 550\; \textrm{nm}$.
Fig. 5.
Fig. 5. The CIE 1931 color matching functions $\bar{x}(\lambda )$, $\bar{y}(\lambda )$ and $\bar{z}(\lambda )$ the relative cumulative weight.
Fig. 6.
Fig. 6. (a) Comparison of ${T_S}$, the transmittance spectra measured with a spectroradiometer, and ${T_{SA}}$, the transmittance measured with the camera (spatial average over the image) for KW gelatin neutral density filter $OD = 0.3$ (error bars with a coverage factor $k = 2$); (b) (c) and (d) CIELAB coordinates and 95% confidence regions from ${T_S}$ (blue) and ${T_{SA}}$ (red) in the (${a^\ast },{L^\ast }$), (${b^\ast },{L^\ast }$) and (${b^\ast },{a^\ast }$) projection planes, respectively.
Fig. 7.
Fig. 7. Transmittance spectra of five KW color gelatin filters (#12: yellow; #25: red: #32: magenta; #47: deep blue: #58: green) measured by the spectroradiometer (${T_S}$: color, plain) and the camera (${T_{SA}}$: grey, dash). ${T_{SA}}$ and ${T_S}$ overlap within the error bars (coverage factor $k = 2$) for ${\lambda } = \; 390\; \textrm{to}\; 770\; \textrm{nm}$.
Fig. 8.
Fig. 8. (a) Comparison of ${T_S}$, the transmittance spectra measured with a spectroradiometer, and ${T_{SA}}$, the transmittance measured with the camera (spatial average over the image) for KW color gelatin filter #32 (magenta) (error bars with a coverage factor $k = 2$); (b) (c) and (d) CIELAB coordinates and 95% confidence regions from ${T_S}$ (blue) and ${T_{SA}}$ (red) in the (${a^\ast },{L^\ast }$), (${b^\ast },{L^\ast }$) and (${b^\ast },{a^\ast }$) projection planes, respectively.
Fig. 9.
Fig. 9. (a) CIE LAB representation of the 23 patches composing the color filter phantom using measurements by the reference spectroradiometer; (b) Corresponding ${\Delta }E_{ab}^\ast $ issued from ${T_S}$ (spectroradiometer) and ${T_{SA}}$ (camera) measurements.
Fig. 10.
Fig. 10. Image of a H&E stained tissue (H&E stained, cancer adjacent cervix tissue, CNS801 B10) obtained by the HIMS.

Tables (2)

Tables Icon

Table 1. CIELAB coordinates for the KW gelatin neutral density filters as measured by the spectroradiometer, T S , and the spatially averaged images acquired by the camera, T S A . The statistical distribution of the Euclidian distance in the CIELAB color space between both types of measurements is computed by Monte Carlo simulations of the color point positions using the covariance matrices of the CIELAB coordinates. The median value me d Δ E a b of these distributions are reported. The uncertainties are presented with a coverage factor k = 2 .

Tables Icon

Table 2. CIELAB coordinates of five KW color gelatin filters (#12: yellow; #25: red: #32: magenta; #47: deep blue: #58: green) derived from the spectra measured by the spectroradiometer, T S , and the spatially averaged images acquired by the camera, T S A . The statistical distribution of the Euclidian distance in the CIELAB color space between both types of measurements was computed by Monte Carlo simulations of the color point positions using the covariance matrices of the CIELAB coordinates. The median value, me d Δ E a b , of these distributions are reported. The uncertainties are presented with a coverage factor k = 2 .

Equations (13)

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T ( λ ) = I S ( λ ) I S B g ( λ ) I W ( λ ) I W B g ( λ ) ,
X = K T ( λ ) S ( λ ) x ¯ ( λ ) d λ Y = K T ( λ ) S ( λ ) y ¯ ( λ ) d λ Z = K T ( λ ) S ( λ ) z ¯ ( λ ) d λ ,
X = K i = 1 N T ( λ i ) S ( λ i ) x ¯ ( λ i ) Δ λ = f 1 ( T ) Y = K i = 1 N T ( λ i ) S ( λ i ) y ¯ ( λ i ) Δ λ = f 2 ( T ) Z = K i = 1 N T ( λ i ) S ( λ i ) z ¯ ( λ i ) Δ λ = f 3 ( T ) ,
K = 100 i = 1 N S ( λ i ) y ¯ ( λ i ) Δ λ .
L = 116 f ( Y Y n ) 16 a = 500 ( f ( X X n ) f ( Y Y n ) ) , b = 200 ( f ( Y Y n ) f ( Z Z n ) )
f ( t ) = { t 3 if t > δ 3 t 3 δ 2 + 4 29 otherwise with δ = 6 29 ,
Δ E a b = Δ L 2 + Δ a 2 + Δ b 2
C y J C x J t
J 1 = [ T I S T I W T I S B g T I W B g ] .
s T ( λ ) 2 = i = 1 4 ( T ( λ ) x i ) 2 s x i 2 .
J 2 = [ f 1 ( T ) T ( λ 1 ) f 1 ( T ) T ( λ N ) f 2 ( T ) T ( λ 1 ) f 2 ( T ) T ( λ N ) f 3 ( T ) T ( λ 1 ) f 3 ( T ) T ( λ N ) ] = K [ S ( λ 1 ) x ¯ ( λ 1 ) S ( λ N ) x ¯ ( λ N ) S ( λ 1 ) y ¯ ( λ 1 ) S ( λ N ) y ¯ ( λ N ) S ( λ 1 ) z ¯ ( λ 1 ) S ( λ N ) z ¯ ( λ N ) ] ,
J 3 = [ 0 116 Y n f ( Y Y n ) 0 500 X n f ( X X n ) 500 Y n f ( Y Y n ) 0 0 200 Y n f ( Y Y n ) 200 Z n f ( Z Z n ) ] ,
f ( t ) = { 1 3 t 2 / 3 if t > δ 3 1 3 δ 2 otherwise w i t h δ = 6 29

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