Abstract

Regression methods are widely used to estimate the spectral reflectance of object surfaces from camera responses. These methods are under the same problem setting as that to build an estimation function for each sampled wavelength separately, which means that the accuracy of the spectral estimation will be reduced when the training set is small. To improve the spectral estimation accuracy, we propose a novel estimating approach based on the support vector regression method. The proposed approach utilizes a composite modeling scheme, which formulates the RGB values and the sampled wavelength together as the input term to make the most use of the information from the training samples. Experimental results show that the proposed method can improve the recovery accuracy when the training set is small.

© 2008 Optical Society of America

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References

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2007 (6)

2006 (2)

X. Wang, A. Li, Z. H. Jiang, and H. Q. Feng, “Missing value estimation for DNA microarray gene expression data by support vector regression imputation and orthogonal coding scheme,” BMC Bioinf. 7, 32 (2006).
[CrossRef]

N. Shimano, “Recovery of spectral reflectances of objects being imaged without prior knowledge,” IEEE Trans. Image Process. 15, 1848-1856 (2006).
[CrossRef] [PubMed]

2005 (5)

P. Stigell, K. Miyata, and M. Hauta-Kasari, “Wiener estimation method in estimation of spectral reflectance from RGB images,” Pattern Recogn. Image Anal. 15, 327-329 (2005).

A. Rakotomamonjy and S. Canu, “Frames, reproducing kernels, regularization and learning,” J. Mach. Learn. Res. 6, 1485-1515 (2005).

R. E. Fan, P. H. Chen, and C. J. Lin, “Working set selection using second order information for training SVM,” J. Mach. Learn. Res. 6, 1889-1918 (2005).

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

V. Cheung, S. Westland, C. Li, J. Hardeberg, and D. Connah, “Characterization of trichromatic color cameras by using a new multispectral imaging technique,” J. Opt. Soc. Am. A 22, 1231-1240 (2005).
[CrossRef]

2004 (3)

J. Y. Hardeberg, “Filter selection for multispectral color image acquistion,” J. Imaging Sci. Technol. 48, 105-110 (2004).

A. Smola and B. Schölkopf, “A tutorial on support vector regression,” Stat. Comput. 14, 199-222 (2004).
[CrossRef]

T. L. V. Cheung, S. Westland, D. R. Connah, and C. Ripamonti, “A comparative study of the characterization of color cameras by means of neural networks and polynomial transforms,” J. Coloration Technol. 120, 19-25 (2004).
[CrossRef]

2003 (1)

L. J. Cao and F. E. H. Tay, “Support vector machine with adaptive parameters in financial time series forecasting,” IEEE Trans. Neural Netw. 14, 1506-1518 (2003).
[CrossRef]

2001 (1)

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

2000 (2)

1988 (1)

M. Bertero, T. Poggio, and V. Torre, “Ill-posed problems in early vision,” Proc. IEEE 76, 869-889 (1988).
[CrossRef]

1986 (2)

1964 (1)

Arsenin, V. Y.

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (W. H. Winston, 1977).

Bertero, M.

M. Bertero, T. Poggio, and V. Torre, “Ill-posed problems in early vision,” Proc. IEEE 76, 869-889 (1988).
[CrossRef]

Cai, P. Q.

Canu, S.

A. Rakotomamonjy and S. Canu, “Frames, reproducing kernels, regularization and learning,” J. Mach. Learn. Res. 6, 1485-1515 (2005).

Cao, L. J.

L. J. Cao and F. E. H. Tay, “Support vector machine with adaptive parameters in financial time series forecasting,” IEEE Trans. Neural Netw. 14, 1506-1518 (2003).
[CrossRef]

Chang, C. C.

C. C. Chang and C. J. Lin, “LIBSVM: a library for support vector machines,” 2001. Software available at: http://www.csie.ntu.edu.tw/~cjlin/libsvm.

Chen, P. H.

R. E. Fan, P. H. Chen, and C. J. Lin, “Working set selection using second order information for training SVM,” J. Mach. Learn. Res. 6, 1889-1918 (2005).

Cheung, T. L. V.

T. L. V. Cheung, S. Westland, D. R. Connah, and C. Ripamonti, “A comparative study of the characterization of color cameras by means of neural networks and polynomial transforms,” J. Coloration Technol. 120, 19-25 (2004).
[CrossRef]

Cheung, V.

Cohen, J.

Connah, D.

Connah, D. R.

T. L. V. Cheung, S. Westland, D. R. Connah, and C. Ripamonti, “A comparative study of the characterization of color cameras by means of neural networks and polynomial transforms,” J. Coloration Technol. 120, 19-25 (2004).
[CrossRef]

Cristianini, N.

J. Shawe-Taylor and N. Cristianini, Kernel Methods for Pattern Analysis (Cambridge U. Press, 2004).
[CrossRef]

Dai, D. Q.

W. F. Zhang, D. Q. Dai, and H. Yan, “On a new class of framelet kernels for support vector regression and regularization networks,” in Advances in Knowledge Discovery and Data Mining (Springer, 2007), Vol. 4426, pp. 355-366.
[CrossRef]

Evgeniou, T.

T. Evgeniou, M. Pontil, and T. Poggio, “Regularization networks and support vector machines,” Adv. Comput. Math. 13, 1-50 (2000).
[CrossRef]

Fairchild, M. D.

M. D. Fairchild, Color Appearance Models (Addison-Wesley, 1998).

Fan, R. E.

R. E. Fan, P. H. Chen, and C. J. Lin, “Working set selection using second order information for training SVM,” J. Mach. Learn. Res. 6, 1889-1918 (2005).

Farsiu, S.

H. Takeda, S. Farsiu, and P. Milanfar, “Kernel regression for image processing and reconstruction,” IEEE Trans. Image Process. 16, 349-366 (2007).
[CrossRef] [PubMed]

Feng, H. Q.

X. Wang, A. Li, Z. H. Jiang, and H. Q. Feng, “Missing value estimation for DNA microarray gene expression data by support vector regression imputation and orthogonal coding scheme,” BMC Bioinf. 7, 32 (2006).
[CrossRef]

Haneishi, H.

Hardeberg, J.

Hardeberg, J. Y.

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

J. Y. Hardeberg, “Filter selection for multispectral color image acquistion,” J. Imaging Sci. Technol. 48, 105-110 (2004).

J. Y. Hardeberg, Acquisition and Reproduction of Color Images--Colorimetric and Multispectral Approaches (Dissertation.com, 2001).

A. Mansouri, T. Silwa, J. Y. Hardeberg, and Y. Voisin, “Spectral reflectance reconstruction using wavelet basis decomposition,” in Proceedings of the 9th International Symposium on Multispectral Color Science and Application (MCS 07) (2007), pp. 149-155.

Hasegawa, T.

Hauta-Kasari, M.

V. Heikkinen, T. Jetsu, J. Parkkinen, M. Hauta-Kasari, T. Jaaskelainen, and S. D. Lee, “Regularized learning framework in the estimation of reflectance spectra from camera responses,” J. Opt. Soc. Am. A 24, 2673-2683 (2007).
[CrossRef]

P. Stigell, K. Miyata, and M. Hauta-Kasari, “Wiener estimation method in estimation of spectral reflectance from RGB images,” Pattern Recogn. Image Anal. 15, 327-329 (2005).

T. Jetsu, V. Heikkinen, J. Parkkinen, M. Hauta-Kasari, B. Martinkauppi, S. D. Lee, H. W. Ok, and C. Y. Kim, “Color calibration of digital camera using polynomial transformation,” in CGIV, Third European Conference on Color in Graphics, Imaging and Vision (IS&T, 2006), pp. 163-166.

Heikkinen, V.

V. Heikkinen, T. Jetsu, J. Parkkinen, M. Hauta-Kasari, T. Jaaskelainen, and S. D. Lee, “Regularized learning framework in the estimation of reflectance spectra from camera responses,” J. Opt. Soc. Am. A 24, 2673-2683 (2007).
[CrossRef]

T. Jetsu, V. Heikkinen, J. Parkkinen, M. Hauta-Kasari, B. Martinkauppi, S. D. Lee, H. W. Ok, and C. Y. Kim, “Color calibration of digital camera using polynomial transformation,” in CGIV, Third European Conference on Color in Graphics, Imaging and Vision (IS&T, 2006), pp. 163-166.

Hironaga, M.

Hong, G.

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

Hosoi, A.

Jaaskelainen, T.

Jetsu, T.

V. Heikkinen, T. Jetsu, J. Parkkinen, M. Hauta-Kasari, T. Jaaskelainen, and S. D. Lee, “Regularized learning framework in the estimation of reflectance spectra from camera responses,” J. Opt. Soc. Am. A 24, 2673-2683 (2007).
[CrossRef]

T. Jetsu, V. Heikkinen, J. Parkkinen, M. Hauta-Kasari, B. Martinkauppi, S. D. Lee, H. W. Ok, and C. Y. Kim, “Color calibration of digital camera using polynomial transformation,” in CGIV, Third European Conference on Color in Graphics, Imaging and Vision (IS&T, 2006), pp. 163-166.

Jiang, Z. H.

X. Wang, A. Li, Z. H. Jiang, and H. Q. Feng, “Missing value estimation for DNA microarray gene expression data by support vector regression imputation and orthogonal coding scheme,” BMC Bioinf. 7, 32 (2006).
[CrossRef]

Joachims, T.

T. Joachims, “Making large-scale SVM learning practical,” in Advances in Kernel Methods-Support Vector Learning (MIT Press, 1999), pp. 169-184.

Kim, C. Y.

T. Jetsu, V. Heikkinen, J. Parkkinen, M. Hauta-Kasari, B. Martinkauppi, S. D. Lee, H. W. Ok, and C. Y. Kim, “Color calibration of digital camera using polynomial transformation,” in CGIV, Third European Conference on Color in Graphics, Imaging and Vision (IS&T, 2006), pp. 163-166.

Lee, S. D.

V. Heikkinen, T. Jetsu, J. Parkkinen, M. Hauta-Kasari, T. Jaaskelainen, and S. D. Lee, “Regularized learning framework in the estimation of reflectance spectra from camera responses,” J. Opt. Soc. Am. A 24, 2673-2683 (2007).
[CrossRef]

T. Jetsu, V. Heikkinen, J. Parkkinen, M. Hauta-Kasari, B. Martinkauppi, S. D. Lee, H. W. Ok, and C. Y. Kim, “Color calibration of digital camera using polynomial transformation,” in CGIV, Third European Conference on Color in Graphics, Imaging and Vision (IS&T, 2006), pp. 163-166.

Li, A.

X. Wang, A. Li, Z. H. Jiang, and H. Q. Feng, “Missing value estimation for DNA microarray gene expression data by support vector regression imputation and orthogonal coding scheme,” BMC Bioinf. 7, 32 (2006).
[CrossRef]

Li, C.

Lin, C. J.

R. E. Fan, P. H. Chen, and C. J. Lin, “Working set selection using second order information for training SVM,” J. Mach. Learn. Res. 6, 1889-1918 (2005).

C. C. Chang and C. J. Lin, “LIBSVM: a library for support vector machines,” 2001. Software available at: http://www.csie.ntu.edu.tw/~cjlin/libsvm.

Luo, M. R.

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

Maloney, L. T.

Mansouri, A.

A. Mansouri, T. Silwa, J. Y. Hardeberg, and Y. Voisin, “Spectral reflectance reconstruction using wavelet basis decomposition,” in Proceedings of the 9th International Symposium on Multispectral Color Science and Application (MCS 07) (2007), pp. 149-155.

Martinkauppi, B.

T. Jetsu, V. Heikkinen, J. Parkkinen, M. Hauta-Kasari, B. Martinkauppi, S. D. Lee, H. W. Ok, and C. Y. Kim, “Color calibration of digital camera using polynomial transformation,” in CGIV, Third European Conference on Color in Graphics, Imaging and Vision (IS&T, 2006), pp. 163-166.

Milanfar, P.

H. Takeda, S. Farsiu, and P. Milanfar, “Kernel regression for image processing and reconstruction,” IEEE Trans. Image Process. 16, 349-366 (2007).
[CrossRef] [PubMed]

Miyake, Y.

Miyata, K.

P. Stigell, K. Miyata, and M. Hauta-Kasari, “Wiener estimation method in estimation of spectral reflectance from RGB images,” Pattern Recogn. Image Anal. 15, 327-329 (2005).

Ok, H. W.

T. Jetsu, V. Heikkinen, J. Parkkinen, M. Hauta-Kasari, B. Martinkauppi, S. D. Lee, H. W. Ok, and C. Y. Kim, “Color calibration of digital camera using polynomial transformation,” in CGIV, Third European Conference on Color in Graphics, Imaging and Vision (IS&T, 2006), pp. 163-166.

Parkkinen, J.

V. Heikkinen, T. Jetsu, J. Parkkinen, M. Hauta-Kasari, T. Jaaskelainen, and S. D. Lee, “Regularized learning framework in the estimation of reflectance spectra from camera responses,” J. Opt. Soc. Am. A 24, 2673-2683 (2007).
[CrossRef]

T. Jetsu, V. Heikkinen, J. Parkkinen, M. Hauta-Kasari, B. Martinkauppi, S. D. Lee, H. W. Ok, and C. Y. Kim, “Color calibration of digital camera using polynomial transformation,” in CGIV, Third European Conference on Color in Graphics, Imaging and Vision (IS&T, 2006), pp. 163-166.

Platt, J. C.

J. C. Platt, “Fast training of support vector machines using sequential minimal optimization,” in Advances in Kernel Methods-Support Vector Learning (MIT Press, 1999), pp. 185-208.

Poggio, T.

T. Evgeniou, M. Pontil, and T. Poggio, “Regularization networks and support vector machines,” Adv. Comput. Math. 13, 1-50 (2000).
[CrossRef]

M. Bertero, T. Poggio, and V. Torre, “Ill-posed problems in early vision,” Proc. IEEE 76, 869-889 (1988).
[CrossRef]

Pontil, M.

T. Evgeniou, M. Pontil, and T. Poggio, “Regularization networks and support vector machines,” Adv. Comput. Math. 13, 1-50 (2000).
[CrossRef]

Rakotomamonjy, A.

A. Rakotomamonjy and S. Canu, “Frames, reproducing kernels, regularization and learning,” J. Mach. Learn. Res. 6, 1485-1515 (2005).

Rhodes, P. A.

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

Ripamonti, C.

T. L. V. Cheung, S. Westland, D. R. Connah, and C. Ripamonti, “A comparative study of the characterization of color cameras by means of neural networks and polynomial transforms,” J. Coloration Technol. 120, 19-25 (2004).
[CrossRef]

Schettini, R.

R. Schettini and S. Zuffi, “A computational strategy exploiting genetic algorithms to recover color surface reflectance functions,” Neural Comput. Appl. 16, 69-79 (2007).

Schölkopf, B.

A. Smola and B. Schölkopf, “A tutorial on support vector regression,” Stat. Comput. 14, 199-222 (2004).
[CrossRef]

Shao, S. J.

Shawe-Taylor, J.

J. Shawe-Taylor and N. Cristianini, Kernel Methods for Pattern Analysis (Cambridge U. Press, 2004).
[CrossRef]

Shen, H. L.

Shimano, N.

N. Shimano, K. Terai, and M. Hironaga, “Recovery of spectral reflectances of objects being imaged by multispectral cameras,” J. Opt. Soc. Am. A 24, 3211-3219 (2007).
[CrossRef]

N. Shimano, “Recovery of spectral reflectances of objects being imaged without prior knowledge,” IEEE Trans. Image Process. 15, 1848-1856 (2006).
[CrossRef] [PubMed]

Silwa, T.

A. Mansouri, T. Silwa, J. Y. Hardeberg, and Y. Voisin, “Spectral reflectance reconstruction using wavelet basis decomposition,” in Proceedings of the 9th International Symposium on Multispectral Color Science and Application (MCS 07) (2007), pp. 149-155.

Smola, A.

A. Smola and B. Schölkopf, “A tutorial on support vector regression,” Stat. Comput. 14, 199-222 (2004).
[CrossRef]

Stigell, P.

P. Stigell, K. Miyata, and M. Hauta-Kasari, “Wiener estimation method in estimation of spectral reflectance from RGB images,” Pattern Recogn. Image Anal. 15, 327-329 (2005).

Takeda, H.

H. Takeda, S. Farsiu, and P. Milanfar, “Kernel regression for image processing and reconstruction,” IEEE Trans. Image Process. 16, 349-366 (2007).
[CrossRef] [PubMed]

Tay, F. E. H.

L. J. Cao and F. E. H. Tay, “Support vector machine with adaptive parameters in financial time series forecasting,” IEEE Trans. Neural Netw. 14, 1506-1518 (2003).
[CrossRef]

Terai, K.

Tikhonov, A. N.

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (W. H. Winston, 1977).

Torre, V.

M. Bertero, T. Poggio, and V. Torre, “Ill-posed problems in early vision,” Proc. IEEE 76, 869-889 (1988).
[CrossRef]

Tsumura, N.

Vapnik, V. N.

V. N. Vapnik, The Nature of Statistical Learning Theory (Springer-Verlag, 1995).

V. N. Vapnik, Statistical Learning Theory (Wiley, 1998).

Voisin, Y.

A. Mansouri, T. Silwa, J. Y. Hardeberg, and Y. Voisin, “Spectral reflectance reconstruction using wavelet basis decomposition,” in Proceedings of the 9th International Symposium on Multispectral Color Science and Application (MCS 07) (2007), pp. 149-155.

Wandell, B. A.

Wang, X.

X. Wang, A. Li, Z. H. Jiang, and H. Q. Feng, “Missing value estimation for DNA microarray gene expression data by support vector regression imputation and orthogonal coding scheme,” BMC Bioinf. 7, 32 (2006).
[CrossRef]

Westland, S.

V. Cheung, S. Westland, C. Li, J. Hardeberg, and D. Connah, “Characterization of trichromatic color cameras by using a new multispectral imaging technique,” J. Opt. Soc. Am. A 22, 1231-1240 (2005).
[CrossRef]

T. L. V. Cheung, S. Westland, D. R. Connah, and C. Ripamonti, “A comparative study of the characterization of color cameras by means of neural networks and polynomial transforms,” J. Coloration Technol. 120, 19-25 (2004).
[CrossRef]

Xin, J. H.

Yan, H.

W. F. Zhang, D. Q. Dai, and H. Yan, “On a new class of framelet kernels for support vector regression and regularization networks,” in Advances in Knowledge Discovery and Data Mining (Springer, 2007), Vol. 4426, pp. 355-366.
[CrossRef]

Yokoyama, Y.

Zhang, W. F.

W. F. Zhang, D. Q. Dai, and H. Yan, “On a new class of framelet kernels for support vector regression and regularization networks,” in Advances in Knowledge Discovery and Data Mining (Springer, 2007), Vol. 4426, pp. 355-366.
[CrossRef]

Zuffi, S.

R. Schettini and S. Zuffi, “A computational strategy exploiting genetic algorithms to recover color surface reflectance functions,” Neural Comput. Appl. 16, 69-79 (2007).

Adv. Comput. Math. (1)

T. Evgeniou, M. Pontil, and T. Poggio, “Regularization networks and support vector machines,” Adv. Comput. Math. 13, 1-50 (2000).
[CrossRef]

Appl. Opt. (2)

BMC Bioinf. (1)

X. Wang, A. Li, Z. H. Jiang, and H. Q. Feng, “Missing value estimation for DNA microarray gene expression data by support vector regression imputation and orthogonal coding scheme,” BMC Bioinf. 7, 32 (2006).
[CrossRef]

Color Res. Appl. (1)

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

IEEE Trans. Image Process. (2)

N. Shimano, “Recovery of spectral reflectances of objects being imaged without prior knowledge,” IEEE Trans. Image Process. 15, 1848-1856 (2006).
[CrossRef] [PubMed]

H. Takeda, S. Farsiu, and P. Milanfar, “Kernel regression for image processing and reconstruction,” IEEE Trans. Image Process. 16, 349-366 (2007).
[CrossRef] [PubMed]

IEEE Trans. Neural Netw. (1)

L. J. Cao and F. E. H. Tay, “Support vector machine with adaptive parameters in financial time series forecasting,” IEEE Trans. Neural Netw. 14, 1506-1518 (2003).
[CrossRef]

J. Coloration Technol. (1)

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Figures (5)

Fig. 1
Fig. 1

Spectral sensitivities of the camera system used in the experiments.

Fig. 2
Fig. 2

Spectral power distribution of CIE illuminant D65.

Fig. 3
Fig. 3

Mean RMSE errors plotted versus the number of training samples when SNR = .

Fig. 4
Fig. 4

Mean RMSE errors plotted versus the number of training samples. when SNR = 40 .

Fig. 5
Fig. 5

Mean RMSE errors plotted versus the number of training samples. when SNR = 30 .

Tables (3)

Tables Icon

Table 1 Training, Validation, and Test Sets in the Experiment

Tables Icon

Table 2 Estimation Errors of Different Methods on the Test Set a

Tables Icon

Table 3 Estimation Errors of Different Methods on the Test Set a

Equations (34)

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P i = λ R ( λ ) E ( λ ) Q i ( λ ) d λ + e i ,
x = S L y + e ,
y ̂ = K r M T ( M K r M T + K e ) 1 x ,
K r = E { y y T } , K e = E { e e T }
y i = f i ( x ) + ϵ i , i = 1 , , n
L ( f , S ) = v X w 2 ,
w ̃ = ( X T X ) 1 X T v ,
f ( x ) = w , Φ p k ( x ) ,
w ̃ = ( P T P ) 1 P T v .
L ( f , S ) = v P w 2 + γ w 2 ,
w ̃ = ( P T P + γ I ) 1 P T v ,
min f H K H [ f ] = i = 1 m ( v i f ( x i ) ) 2 + γ f K 2 ,
f ( x ) = i = 1 m c i K ( x i , x ) .
c = ( K + γ I ) 1 v ,
K i j = K ( x i , x j ) .
K ( x , z ) = exp ( x z 2 2 σ 2 )
K ( x , z ) = ( x , z + 1 ) k ,
K ( x , z ) = x z 2 2 m n ,
y i = f i ( R , G , B ) , i = 1 , , n
r = f ( R , G , B , λ ) .
D = { ( ( R i , G i , B i , λ j ) T , R i ( λ j ) ) , i = 1 , , m , j = 1 , , n } ,
R ( λ ) = f ( R 0 , G 0 , B 0 , λ ) .
( f ( R 0 , G 0 , B 0 , λ 1 ) , , f ( R 0 , G 0 , B 0 , λ n ) ) T .
f ( R , G , B , λ i ) = f i ( R , G , B ) , i = 1 , , n .
x = ( R , G , B , λ ) T
D = { ( x i , r i ) } i = 1 l ,
min f H K H [ f ] = i = 1 l r i f ( x i ) ϵ + γ f K 2 ,
x ϵ = { 0 if x < ϵ x ϵ otherwise } ,
f ( x ) = i = 1 l c i K ( x i , x ) + b ,
min α , α * W α , α * = ϵ i = 1 l ( α i + α i * ) i = 1 l r i ( α i α i * ) + 1 2 i , j = 1 l ( α i α i * ) ( α j α j * ) K ( x i , x j ) ,
i = 1 l ( α i α i * ) = 0 , 0 α i , α i * 1 2 γ , i = 1 , , l .
SNR = 10 log 10 ( Tr ( M K r M T ) σ 2 ) .
RMSE = λ ( R p ( λ ) R o ( λ ) ) 2 n ,
Δ E a b = ( Δ L ) 2 + ( Δ a ) 2 + ( Δ b ) 2 .

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