Abstract

An improved spectral reflectance estimation method is developed to transform raw camera RGB responses to spectral reflectance. The novelty of our method is to apply a local weighted linear regression model for spectral reflectance estimation and construct the weighting matrix using a Gaussian function in CIELAB uniform color space. The proposed method was tested using both a standard color chart and a set of textile samples, with a digital RGB camera and by ten times ten-fold cross-validation. The results demonstrate that our method gives the best accuracy in estimating both the spectral reflectance and the colorimetric values in comparison with existing methods.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref]
  10. K. Xiao, Y. Zhu, C. Li, D. Connah, J. M. Yates, and S. Wuerger, “Improved method for skin reflectance reconstruction from camera images,” Opt. Express 24(13), 14934–14950 (2016).
    [Crossref] [PubMed]
  11. X. Zhang, Q. Wang, J. Li, X. Zhou, Y. Yang, and H. Xu, “Estimating spectral reflectance from camera responses based on CIE XYZ tristimulus values under multi-illuminants,” Color Res. Appl. 42(1), 68–77 (2017).
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  20. V. Babaei, S. H. Amirshahi, and F. Agahian, “Using weighted pseudo-inverse method for reconstruction of reflectance spectra and analyzing the dataset in terms of normality,” Color Res. Appl. 36(4), 295–305 (2011).
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  21. G. Paschos, “Perceptually uniform color spaces for color texture analysis: an empirical evaluation,” IEEE Trans. Image Process. 10(6), 932–937 (2001).
    [Crossref]
  22. R. Kohavi, “A study of cross-validation and bootstrap for accuracy estimation and model selection,” inProceedings of the Fourteenth International Joint Conference on Artificial Intelligence, C. S. Mellish, ed. (Academic,1995), pp. 1137–1143.
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    [Crossref]
  27. J. E. Garcia, A. G. Dyer, A. D. Greentree, G. Spring, and P. A. Wilksch, “Linearisation of RGB camera responses for quantitative image analysis of visible and UV photography: a comparison of two techniques,” PLoS One 8(11), e79534 (2013).
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  32. CIE, CIE 15:24 Colorimetry (Vienna, 2004).
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2017 (3)

X. Zhang, Q. Wang, J. Li, X. Zhou, Y. Yang, and H. Xu, “Estimating spectral reflectance from camera responses based on CIE XYZ tristimulus values under multi-illuminants,” Color Res. Appl. 42(1), 68–77 (2017).
[Crossref]

J. Liang and X. Wan, “Optimized method for spectral reflectance reconstruction from camera responses,” Opt. Express 25(23), 28273–28287 (2017).
[Crossref]

B. Cao, N. Liao, and H. Cheng, “Spectral reflectance reconstruction from RGB images based on weighting smaller color difference group,” Color Res. Appl. 42(3), 327–332 (2017).
[Crossref]

2016 (2)

J. Liang, X. Wan, Q. Liu, C. Li, and J. Li, “Research on filter selection method for broadband spectral imaging system based on ancient murals,” Color Res. Appl. 41(6), 585–595 (2016).
[Crossref]

K. Xiao, Y. Zhu, C. Li, D. Connah, J. M. Yates, and S. Wuerger, “Improved method for skin reflectance reconstruction from camera images,” Opt. Express 24(13), 14934–14950 (2016).
[Crossref] [PubMed]

2014 (3)

2013 (1)

J. E. Garcia, A. G. Dyer, A. D. Greentree, G. Spring, and P. A. Wilksch, “Linearisation of RGB camera responses for quantitative image analysis of visible and UV photography: a comparison of two techniques,” PLoS One 8(11), e79534 (2013).
[Crossref] [PubMed]

2011 (1)

V. Babaei, S. H. Amirshahi, and F. Agahian, “Using weighted pseudo-inverse method for reconstruction of reflectance spectra and analyzing the dataset in terms of normality,” Color Res. Appl. 36(4), 295–305 (2011).
[Crossref]

2010 (1)

H. Shen, H. Wan, and Z. Zhang, “Estimating reflectance from multispectral camera responses based on partial least-squares regression,” J. Electron. Imaging 19(2), 020501 (2010).
[Crossref]

2008 (3)

V. Heikkinen, R. Lenz, T. Jetsu, J. Parkkinen, M. Hauta-Kasari, and T. Jääskeläinen, “Evaluation and unification of some methods for estimating reflectance spectra from RGB images,” J. Opt. Soc. Am. A 25(10), 2444–2458 (2008).
[Crossref] [PubMed]

J. Brauers, N. Schulte, and T. Aach, “Multispectral filter-wheel cameras: geometric distortion model and compensation algorithms,” IEEE Trans. Image Process. 17(12), 2368–2380 (2008).
[Crossref] [PubMed]

A. Ribes and F. Schmitt, “Linear inverse problems in imaging,” IEEE Signal Process. Mag. 25(4), 84–99 (2008).
[Crossref]

2007 (1)

2006 (2)

2005 (2)

R. Ramanath, W. E. Snyder, Y. Yoo, and M. S. Drew, “Color image processing pipeline,” IEEE Signal Process. Mag. 22(1), 34–43 (2005).
[Crossref]

D. Connah and J. Y. Hardeberg, “Spectral recovery using polynomial models,” Proc. SPIE 5667, 65–75 (2005).
[Crossref]

2004 (1)

H. Malvar, L. He, and R. Cutler, “High-quality linear interpolation for demosaicing of Bayer-patterned color images,” Proc. IEEE 3, 485–488 (2004).

2002 (1)

2001 (1)

G. Paschos, “Perceptually uniform color spaces for color texture analysis: an empirical evaluation,” IEEE Trans. Image Process. 10(6), 932–937 (2001).
[Crossref]

Aach, T.

J. Brauers, N. Schulte, and T. Aach, “Multispectral filter-wheel cameras: geometric distortion model and compensation algorithms,” IEEE Trans. Image Process. 17(12), 2368–2380 (2008).
[Crossref] [PubMed]

Agahian, F.

V. Babaei, S. H. Amirshahi, and F. Agahian, “Using weighted pseudo-inverse method for reconstruction of reflectance spectra and analyzing the dataset in terms of normality,” Color Res. Appl. 36(4), 295–305 (2011).
[Crossref]

Amiri, M. M.

M. M. Amiri and S. H. Amirshahi, “A hybrid of weighted regression and linear models for extraction of reflectance spectra from CIEXYZ tristimulus values,” Opt. Rev. 21(6), 816–825 (2014).
[Crossref]

Amirshahi, S. H.

M. M. Amiri and S. H. Amirshahi, “A hybrid of weighted regression and linear models for extraction of reflectance spectra from CIEXYZ tristimulus values,” Opt. Rev. 21(6), 816–825 (2014).
[Crossref]

V. Babaei, S. H. Amirshahi, and F. Agahian, “Using weighted pseudo-inverse method for reconstruction of reflectance spectra and analyzing the dataset in terms of normality,” Color Res. Appl. 36(4), 295–305 (2011).
[Crossref]

Babaei, V.

V. Babaei, S. H. Amirshahi, and F. Agahian, “Using weighted pseudo-inverse method for reconstruction of reflectance spectra and analyzing the dataset in terms of normality,” Color Res. Appl. 36(4), 295–305 (2011).
[Crossref]

Brauers, J.

J. Brauers, N. Schulte, and T. Aach, “Multispectral filter-wheel cameras: geometric distortion model and compensation algorithms,” IEEE Trans. Image Process. 17(12), 2368–2380 (2008).
[Crossref] [PubMed]

Cao, B.

B. Cao, N. Liao, and H. Cheng, “Spectral reflectance reconstruction from RGB images based on weighting smaller color difference group,” Color Res. Appl. 42(3), 327–332 (2017).
[Crossref]

Cheng, H.

B. Cao, N. Liao, and H. Cheng, “Spectral reflectance reconstruction from RGB images based on weighting smaller color difference group,” Color Res. Appl. 42(3), 327–332 (2017).
[Crossref]

Cheung, V.

V. Cheung and S. Westland, “Methods for optimal color selection,” J. Imaging Sci. Technol. 50(5), 481–488 (2006).
[Crossref]

Connah, D.

Cutler, R.

H. Malvar, L. He, and R. Cutler, “High-quality linear interpolation for demosaicing of Bayer-patterned color images,” Proc. IEEE 3, 485–488 (2004).

Drew, M. S.

R. Ramanath, W. E. Snyder, Y. Yoo, and M. S. Drew, “Color image processing pipeline,” IEEE Signal Process. Mag. 22(1), 34–43 (2005).
[Crossref]

Dyer, A. G.

J. E. Garcia, A. G. Dyer, A. D. Greentree, G. Spring, and P. A. Wilksch, “Linearisation of RGB camera responses for quantitative image analysis of visible and UV photography: a comparison of two techniques,” PLoS One 8(11), e79534 (2013).
[Crossref] [PubMed]

Eckhard, T.

Garcia, J. E.

J. E. Garcia, A. G. Dyer, A. D. Greentree, G. Spring, and P. A. Wilksch, “Linearisation of RGB camera responses for quantitative image analysis of visible and UV photography: a comparison of two techniques,” PLoS One 8(11), e79534 (2013).
[Crossref] [PubMed]

Greentree, A. D.

J. E. Garcia, A. G. Dyer, A. D. Greentree, G. Spring, and P. A. Wilksch, “Linearisation of RGB camera responses for quantitative image analysis of visible and UV photography: a comparison of two techniques,” PLoS One 8(11), e79534 (2013).
[Crossref] [PubMed]

Hardeberg, J. Y.

Hauta-Kasari, M.

He, L.

H. Malvar, L. He, and R. Cutler, “High-quality linear interpolation for demosaicing of Bayer-patterned color images,” Proc. IEEE 3, 485–488 (2004).

Heikkinen, V.

Hernández-Andrés, J.

Jaaskelainen, T.

Jääskeläinen, T.

Jetsu, T.

Lee, S. D.

Lenz, R.

Li, C.

J. Liang, X. Wan, Q. Liu, C. Li, and J. Li, “Research on filter selection method for broadband spectral imaging system based on ancient murals,” Color Res. Appl. 41(6), 585–595 (2016).
[Crossref]

K. Xiao, Y. Zhu, C. Li, D. Connah, J. M. Yates, and S. Wuerger, “Improved method for skin reflectance reconstruction from camera images,” Opt. Express 24(13), 14934–14950 (2016).
[Crossref] [PubMed]

Li, H.

H. Li, Z. Wu, L. Zhang, and J. Parkkinen, “SR-LLA: A novel spectral reconstruction method based on locally linear approximation,” inProceedings of IEEE Conference on Image Processing (IEEE, 2013), pp. 2029–2033.
[Crossref]

Li, J.

X. Zhang, Q. Wang, J. Li, X. Zhou, Y. Yang, and H. Xu, “Estimating spectral reflectance from camera responses based on CIE XYZ tristimulus values under multi-illuminants,” Color Res. Appl. 42(1), 68–77 (2017).
[Crossref]

J. Liang, X. Wan, Q. Liu, C. Li, and J. Li, “Research on filter selection method for broadband spectral imaging system based on ancient murals,” Color Res. Appl. 41(6), 585–595 (2016).
[Crossref]

Liang, J.

J. Liang and X. Wan, “Optimized method for spectral reflectance reconstruction from camera responses,” Opt. Express 25(23), 28273–28287 (2017).
[Crossref]

J. Liang, X. Wan, Q. Liu, C. Li, and J. Li, “Research on filter selection method for broadband spectral imaging system based on ancient murals,” Color Res. Appl. 41(6), 585–595 (2016).
[Crossref]

Liao, N.

B. Cao, N. Liao, and H. Cheng, “Spectral reflectance reconstruction from RGB images based on weighting smaller color difference group,” Color Res. Appl. 42(3), 327–332 (2017).
[Crossref]

Liu, Q.

J. Liang, X. Wan, Q. Liu, C. Li, and J. Li, “Research on filter selection method for broadband spectral imaging system based on ancient murals,” Color Res. Appl. 41(6), 585–595 (2016).
[Crossref]

Malvar, H.

H. Malvar, L. He, and R. Cutler, “High-quality linear interpolation for demosaicing of Bayer-patterned color images,” Proc. IEEE 3, 485–488 (2004).

Murakami, Y.

Obi, T.

Ohyama, N.

Parkkinen, J.

Paschos, G.

G. Paschos, “Perceptually uniform color spaces for color texture analysis: an empirical evaluation,” IEEE Trans. Image Process. 10(6), 932–937 (2001).
[Crossref]

Ramanath, R.

R. Ramanath, W. E. Snyder, Y. Yoo, and M. S. Drew, “Color image processing pipeline,” IEEE Signal Process. Mag. 22(1), 34–43 (2005).
[Crossref]

Ribes, A.

A. Ribes and F. Schmitt, “Linear inverse problems in imaging,” IEEE Signal Process. Mag. 25(4), 84–99 (2008).
[Crossref]

Schmitt, F.

A. Ribes and F. Schmitt, “Linear inverse problems in imaging,” IEEE Signal Process. Mag. 25(4), 84–99 (2008).
[Crossref]

Schnitzlein, M.

Schulte, N.

J. Brauers, N. Schulte, and T. Aach, “Multispectral filter-wheel cameras: geometric distortion model and compensation algorithms,” IEEE Trans. Image Process. 17(12), 2368–2380 (2008).
[Crossref] [PubMed]

Shen, H.

H. Shen, H. Wan, and Z. Zhang, “Estimating reflectance from multispectral camera responses based on partial least-squares regression,” J. Electron. Imaging 19(2), 020501 (2010).
[Crossref]

Shen, H. L.

Shrestha, R.

Snyder, W. E.

R. Ramanath, W. E. Snyder, Y. Yoo, and M. S. Drew, “Color image processing pipeline,” IEEE Signal Process. Mag. 22(1), 34–43 (2005).
[Crossref]

Spring, G.

J. E. Garcia, A. G. Dyer, A. D. Greentree, G. Spring, and P. A. Wilksch, “Linearisation of RGB camera responses for quantitative image analysis of visible and UV photography: a comparison of two techniques,” PLoS One 8(11), e79534 (2013).
[Crossref] [PubMed]

Valero, E. M.

Wan, H.

H. Shen, H. Wan, and Z. Zhang, “Estimating reflectance from multispectral camera responses based on partial least-squares regression,” J. Electron. Imaging 19(2), 020501 (2010).
[Crossref]

Wan, X.

J. Liang and X. Wan, “Optimized method for spectral reflectance reconstruction from camera responses,” Opt. Express 25(23), 28273–28287 (2017).
[Crossref]

J. Liang, X. Wan, Q. Liu, C. Li, and J. Li, “Research on filter selection method for broadband spectral imaging system based on ancient murals,” Color Res. Appl. 41(6), 585–595 (2016).
[Crossref]

Wang, Q.

X. Zhang, Q. Wang, J. Li, X. Zhou, Y. Yang, and H. Xu, “Estimating spectral reflectance from camera responses based on CIE XYZ tristimulus values under multi-illuminants,” Color Res. Appl. 42(1), 68–77 (2017).
[Crossref]

Westland, S.

V. Cheung and S. Westland, “Methods for optimal color selection,” J. Imaging Sci. Technol. 50(5), 481–488 (2006).
[Crossref]

Wilksch, P. A.

J. E. Garcia, A. G. Dyer, A. D. Greentree, G. Spring, and P. A. Wilksch, “Linearisation of RGB camera responses for quantitative image analysis of visible and UV photography: a comparison of two techniques,” PLoS One 8(11), e79534 (2013).
[Crossref] [PubMed]

Wu, Z.

H. Li, Z. Wu, L. Zhang, and J. Parkkinen, “SR-LLA: A novel spectral reconstruction method based on locally linear approximation,” inProceedings of IEEE Conference on Image Processing (IEEE, 2013), pp. 2029–2033.
[Crossref]

Wuerger, S.

Xiao, K.

Xin, J. H.

Xu, H.

X. Zhang, Q. Wang, J. Li, X. Zhou, Y. Yang, and H. Xu, “Estimating spectral reflectance from camera responses based on CIE XYZ tristimulus values under multi-illuminants,” Color Res. Appl. 42(1), 68–77 (2017).
[Crossref]

Yamaguchi, M.

Yang, Y.

X. Zhang, Q. Wang, J. Li, X. Zhou, Y. Yang, and H. Xu, “Estimating spectral reflectance from camera responses based on CIE XYZ tristimulus values under multi-illuminants,” Color Res. Appl. 42(1), 68–77 (2017).
[Crossref]

Yates, J. M.

Yoo, Y.

R. Ramanath, W. E. Snyder, Y. Yoo, and M. S. Drew, “Color image processing pipeline,” IEEE Signal Process. Mag. 22(1), 34–43 (2005).
[Crossref]

Zhang, L.

H. Li, Z. Wu, L. Zhang, and J. Parkkinen, “SR-LLA: A novel spectral reconstruction method based on locally linear approximation,” inProceedings of IEEE Conference on Image Processing (IEEE, 2013), pp. 2029–2033.
[Crossref]

Zhang, X.

X. Zhang, Q. Wang, J. Li, X. Zhou, Y. Yang, and H. Xu, “Estimating spectral reflectance from camera responses based on CIE XYZ tristimulus values under multi-illuminants,” Color Res. Appl. 42(1), 68–77 (2017).
[Crossref]

Zhang, Z.

H. Shen, H. Wan, and Z. Zhang, “Estimating reflectance from multispectral camera responses based on partial least-squares regression,” J. Electron. Imaging 19(2), 020501 (2010).
[Crossref]

Zhou, X.

X. Zhang, Q. Wang, J. Li, X. Zhou, Y. Yang, and H. Xu, “Estimating spectral reflectance from camera responses based on CIE XYZ tristimulus values under multi-illuminants,” Color Res. Appl. 42(1), 68–77 (2017).
[Crossref]

Zhu, Y.

Appl. Opt. (2)

Color Res. Appl. (4)

B. Cao, N. Liao, and H. Cheng, “Spectral reflectance reconstruction from RGB images based on weighting smaller color difference group,” Color Res. Appl. 42(3), 327–332 (2017).
[Crossref]

V. Babaei, S. H. Amirshahi, and F. Agahian, “Using weighted pseudo-inverse method for reconstruction of reflectance spectra and analyzing the dataset in terms of normality,” Color Res. Appl. 36(4), 295–305 (2011).
[Crossref]

X. Zhang, Q. Wang, J. Li, X. Zhou, Y. Yang, and H. Xu, “Estimating spectral reflectance from camera responses based on CIE XYZ tristimulus values under multi-illuminants,” Color Res. Appl. 42(1), 68–77 (2017).
[Crossref]

J. Liang, X. Wan, Q. Liu, C. Li, and J. Li, “Research on filter selection method for broadband spectral imaging system based on ancient murals,” Color Res. Appl. 41(6), 585–595 (2016).
[Crossref]

IEEE Signal Process. Mag. (2)

A. Ribes and F. Schmitt, “Linear inverse problems in imaging,” IEEE Signal Process. Mag. 25(4), 84–99 (2008).
[Crossref]

R. Ramanath, W. E. Snyder, Y. Yoo, and M. S. Drew, “Color image processing pipeline,” IEEE Signal Process. Mag. 22(1), 34–43 (2005).
[Crossref]

IEEE Trans. Image Process. (2)

J. Brauers, N. Schulte, and T. Aach, “Multispectral filter-wheel cameras: geometric distortion model and compensation algorithms,” IEEE Trans. Image Process. 17(12), 2368–2380 (2008).
[Crossref] [PubMed]

G. Paschos, “Perceptually uniform color spaces for color texture analysis: an empirical evaluation,” IEEE Trans. Image Process. 10(6), 932–937 (2001).
[Crossref]

J. Electron. Imaging (1)

H. Shen, H. Wan, and Z. Zhang, “Estimating reflectance from multispectral camera responses based on partial least-squares regression,” J. Electron. Imaging 19(2), 020501 (2010).
[Crossref]

J. Imaging Sci. Technol. (1)

V. Cheung and S. Westland, “Methods for optimal color selection,” J. Imaging Sci. Technol. 50(5), 481–488 (2006).
[Crossref]

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

Opt. Express (3)

Opt. Rev. (1)

M. M. Amiri and S. H. Amirshahi, “A hybrid of weighted regression and linear models for extraction of reflectance spectra from CIEXYZ tristimulus values,” Opt. Rev. 21(6), 816–825 (2014).
[Crossref]

PLoS One (1)

J. E. Garcia, A. G. Dyer, A. D. Greentree, G. Spring, and P. A. Wilksch, “Linearisation of RGB camera responses for quantitative image analysis of visible and UV photography: a comparison of two techniques,” PLoS One 8(11), e79534 (2013).
[Crossref] [PubMed]

Proc. IEEE (1)

H. Malvar, L. He, and R. Cutler, “High-quality linear interpolation for demosaicing of Bayer-patterned color images,” Proc. IEEE 3, 485–488 (2004).

Proc. SPIE (1)

D. Connah and J. Y. Hardeberg, “Spectral recovery using polynomial models,” Proc. SPIE 5667, 65–75 (2005).
[Crossref]

Other (11)

F. H. Imai and R. S. Berns, “Spectral estimation using trichromatic digital cameras, ”in Proceedings of the International Symposium on Multispectral Imaging and Color Reproduction for Digital Archives (Society of Multispectral Imaging of Japan, 1999), pp. 42–49.

H. Li, Z. Wu, L. Zhang, and J. Parkkinen, “SR-LLA: A novel spectral reconstruction method based on locally linear approximation,” inProceedings of IEEE Conference on Image Processing (IEEE, 2013), pp. 2029–2033.
[Crossref]

M. M. Amiri and M. D. Fairchild, “A strategy toward spectral and colorimetric color reproduction using ordinary digital cameras,” Color Res. Appl. (first published 15 April 2018, in press).

R. Kohavi, “A study of cross-validation and bootstrap for accuracy estimation and model selection,” inProceedings of the Fourteenth International Joint Conference on Artificial Intelligence, C. S. Mellish, ed. (Academic,1995), pp. 1137–1143.

L. Swirski, “CFA Interpolation Detection,” https://pdfs.semanticscholar.org/2ff3/8c5a1aab7e8ed1ff9e1c11c4a26690b56e21.pdf

CIE, CIE 15:24 Colorimetry (Vienna, 2004).

P. Harrington, Machine Learning in Action (Manning Publications, 2012), Chap. 8.

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Figures (8)

Fig. 1
Fig. 1 Linearity comparison between raw and post-processed camera responses as a function luminance factor: (a) R-channel, (b) G-channel, and (c) B-channel.

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Fig. 2
Fig. 2 Framework of the proposed spectral reflectance estimation method from raw camera responses based on Gaussian weighted linear regression model.

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Fig. 3
Fig. 3 Color distribution of the samples in an a*-b* plane of CIELAB color space: (a) color distribution of the CCSG chart, and (b) color distribution of the textile samples.

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Fig. 4
Fig. 4 (a) the relationship between spectral estimation accuracy and k for the CCSG chart, and (b) the relationship between spectral estimation accuracy and k for the textile samples.

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Fig. 5
Fig. 5 Boxplot distributions of spectral error and colorimetric error of the proposed method and the existing methods: (a) boxplot distribution of the RMSE of CCSG chart, (b) boxplot distribution of the CIELAB color difference of CCSG chart, (c) boxplot distribution of the RMSE of textile samples, and (d) boxplot distribution of the CIELAB color difference of textile samples.

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Fig. 6
Fig. 6 (a) Mean RMSE of the proposed method solved by first-, second-, third-, and fourth-order polynomial regression model, and (b) Mean CIELAB color difference of the proposed method solved by first-, second-, third-, and fourth-order polynomial regression model.

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Fig. 7
Fig. 7 (a) the relationship between spectral estimation accuracy and k for the CCSG chart in device-dependent camera RGB color space, and (b) the relationship between spectral estimation accuracy and k for the textile samples in device-dependent camera RGB color space.

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Fig. 8
Fig. 8 (a) the stability of the proposed method for CCSG chart with ten times ten-fold cross-validation, and (b) the stability of the proposed method for textile samples with ten times ten-fold cross-validation.

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Tables (4)

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Table 1 Estimation accuracy of the methods for CCSG and textile samples in terms of RMSE, GFC and CIELAB color difference using ten times ten-fold cross validation

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Table 2 The comparison of computational times (in seconds) between the different methods for CCSG chart and textile samples using ten times ten-fold cross-validation

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Table 3 Spectral estimation results of the proposed method solved by first-, second-, third-, and fourth-order polynomial regression model

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Table 4 Spectral estimation results of the proposed method with weighting matrix calculated in CIELAB color space and device-dependent camera RGB color space

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Equations (11)

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d i = ∫ λ min λ max l ( λ ) s i ( λ ) r ( λ ) d λ + n i .
d = M r .
r ˜ = Q d .
s j = ( L test * − L train , j * ) 2 + ( a test * − a train , j * ) 2 + ( b test * − b train , j * ) 2 ( j = 1 , 2 , ⋯ , N ) .
w j = exp ( s j − 2 k 2 ) ( j = 1 , 2 , ⋯ , N ) .
W = [ w 1 0 ⋯ 0 0 w 2 0 ⋮ ⋮ 0 ⋱ 0 0 ⋯ 0 w N ] N × N .
d exp = [ 1 r g b ] ' .
Q = R train W ( D train,exp W ) + .
r ˜ test = Q d test,exp .
RMSE = 1 n ( r ˜ − r ) T ( r ˜ − r ) ,
GFC = r ˜ T r ‖ r ˜ T r ˜ ‖ ⋅ ‖ r T r ‖ .

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