Abstract

In this work, we evaluate the conditionally positive definite logarithmic kernel in kernel-based estimation of reflectance spectra. Reflectance spectra are estimated from responses of a 12-channel multispectral imaging system. We demonstrate the performance of the logarithmic kernel in comparison with the linear and Gaussian kernel using simulated and measured camera responses for the Pantone and HKS color charts. Especially, we focus on the estimation model evaluations in case the selection of model parameters is optimized using a cross-validation technique. In experiments, it was found that the Gaussian and logarithmic kernel outperformed the linear kernel in almost all evaluation cases (training set size, response channel number) for both sets. Furthermore, the spectral and color estimation accuracies of the Gaussian and logarithmic kernel were found to be similar in several evaluation cases for real and simulated responses. However, results suggest that for a relatively small training set size, the accuracy of the logarithmic kernel can be markedly lower when compared to the Gaussian kernel. Further it was found from our data that the parameter of the logarithmic kernel could be fixed, which simplified the use of this kernel when compared with the Gaussian kernel.

© 2014 Optical Society of America

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References

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  1. B. Schölkopf, A. Smola, Learning with Kernels (MIT, 2002).
  2. J. Shawe-Taylor, N. Cristianini, Kernel Methods for Pattern Analysis (Cambridge University, 2004).
  3. M. G. Genton, “Classes of kernels for machine learning: a statistics perspective,” J. Mach. Learn. Res. 2, 299–312 (2002).
  4. V. Heikkinen, T. Jetsu, J. Parkkinen, M. Hauta-Kasari, T. Jääskeläinen, 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]
  5. V. Heikkinen, R. Lenz, T. Jetsu, J. Parkkinen, M. Hauta-Kasari, T. Jääskeläinen, “Evaluation and unification of some methods for estimating reflectance spectra from RGB images,” J. Opt. Soc. Am. A 25, 2444–2458 (2008).
    [CrossRef]
  6. T. Eckhard, E. M. Valero, J. Hernández-Andrés, “A comparative analysis of spectral estimation approaches applied to print inspection,” in Proceedings of the Annual German Colour Group Meeting (IDD TU Darmstadt, GRIS TU Darmstadt, IGD Fraunhofer, 2012), pp. 13–24.
  7. W.-F. Zhang, P. Yang, D.-Q. Dai, A. Nehorai, “Reflectance estimation using local regression methods,” Lect. Notes Comput. Sci. 7367, 116–122 (2012).
    [CrossRef]
  8. V. Heikkinen, “Kernel methods for estimation and classification of data from spectral imaging,” Dissertations in Forestry and Natural Sciences (University of Eastern Finland, 2011).
  9. S. Boughorbel, J.-P. Tarel, N. Boujemaa, “Conditionally positive definite kernels for svm based image recognition,” in Proceedings of IEEE Conference on Multimedia and Expo (IEEE, 2005), pp. 113–116.
  10. C. Berg, J. Christensen, P. Ressel, Harmonic Analysis on Semigroups Theory of Positive Definite and Related Functions (Springer, 1984).
  11. B. Schöllkopf, “The kernel trick for distances,” in Proceedings of Advances in Neural Information Processing Systems, T. K. Leen, ed. (MIT, 2001), pp. 301–307.
  12. M. Schnitzlein, M. Hund, “Chromasens GmbH,” http://www.chromasens.de/en .
  13. R. G. Lyons, Understanding Digital Signal Processing (Prentice Hall, 2004).
  14. T. Eckhard, “Color imaging web page of the University of Granada,” http://www.ugr.es/local/colorimg/suppl_docs/log_kernel.html .
  15. F. H. Imai, M. R. Rosen, R. S. Berns, “Comparative study of metrics for spectral match quality,” in CGIV, First European Conference on Colour in Graphics, Imaging and Vision (Society for Imaging Science and Technology, 2002), pp. 492–496.
  16. J. A. S. Viggiano, “Metrics for evaluating spectral matches: a quantitative comparison,” in Second European Conference on Colour Graphics, Imaging and Vision (Society for Imaging Science and Technology, 2004), pp. 286–291.
  17. J. Romero, A. García-Beltrań, J. Hernández-Andrés, “Linear bases for representation of natural and artificial illuminants,” J. Opt. Soc. Am. A 14, 1007–1014 (1997).
    [CrossRef]
  18. G. Wyszecki, W. S. Stiles, Color Science, 2nd ed. (Wiley, 1982).
  19. “Improvement to industrial color-difference evaluation,” CIE Technical Report, (CIE, 2001).
  20. J. Schanda, Colorimetry: Understanding the CIE System (Wiley, 2007).

2012 (1)

W.-F. Zhang, P. Yang, D.-Q. Dai, A. Nehorai, “Reflectance estimation using local regression methods,” Lect. Notes Comput. Sci. 7367, 116–122 (2012).
[CrossRef]

2008 (1)

2007 (1)

2002 (1)

M. G. Genton, “Classes of kernels for machine learning: a statistics perspective,” J. Mach. Learn. Res. 2, 299–312 (2002).

1997 (1)

Berg, C.

C. Berg, J. Christensen, P. Ressel, Harmonic Analysis on Semigroups Theory of Positive Definite and Related Functions (Springer, 1984).

Berns, R. S.

F. H. Imai, M. R. Rosen, R. S. Berns, “Comparative study of metrics for spectral match quality,” in CGIV, First European Conference on Colour in Graphics, Imaging and Vision (Society for Imaging Science and Technology, 2002), pp. 492–496.

Boughorbel, S.

S. Boughorbel, J.-P. Tarel, N. Boujemaa, “Conditionally positive definite kernels for svm based image recognition,” in Proceedings of IEEE Conference on Multimedia and Expo (IEEE, 2005), pp. 113–116.

Boujemaa, N.

S. Boughorbel, J.-P. Tarel, N. Boujemaa, “Conditionally positive definite kernels for svm based image recognition,” in Proceedings of IEEE Conference on Multimedia and Expo (IEEE, 2005), pp. 113–116.

Christensen, J.

C. Berg, J. Christensen, P. Ressel, Harmonic Analysis on Semigroups Theory of Positive Definite and Related Functions (Springer, 1984).

Cristianini, N.

J. Shawe-Taylor, N. Cristianini, Kernel Methods for Pattern Analysis (Cambridge University, 2004).

Dai, D.-Q.

W.-F. Zhang, P. Yang, D.-Q. Dai, A. Nehorai, “Reflectance estimation using local regression methods,” Lect. Notes Comput. Sci. 7367, 116–122 (2012).
[CrossRef]

Eckhard, T.

T. Eckhard, E. M. Valero, J. Hernández-Andrés, “A comparative analysis of spectral estimation approaches applied to print inspection,” in Proceedings of the Annual German Colour Group Meeting (IDD TU Darmstadt, GRIS TU Darmstadt, IGD Fraunhofer, 2012), pp. 13–24.

García-Beltran, A.

Genton, M. G.

M. G. Genton, “Classes of kernels for machine learning: a statistics perspective,” J. Mach. Learn. Res. 2, 299–312 (2002).

Hauta-Kasari, M.

Heikkinen, V.

Hernández-Andrés, J.

J. Romero, A. García-Beltrań, J. Hernández-Andrés, “Linear bases for representation of natural and artificial illuminants,” J. Opt. Soc. Am. A 14, 1007–1014 (1997).
[CrossRef]

T. Eckhard, E. M. Valero, J. Hernández-Andrés, “A comparative analysis of spectral estimation approaches applied to print inspection,” in Proceedings of the Annual German Colour Group Meeting (IDD TU Darmstadt, GRIS TU Darmstadt, IGD Fraunhofer, 2012), pp. 13–24.

Imai, F. H.

F. H. Imai, M. R. Rosen, R. S. Berns, “Comparative study of metrics for spectral match quality,” in CGIV, First European Conference on Colour in Graphics, Imaging and Vision (Society for Imaging Science and Technology, 2002), pp. 492–496.

Jääskeläinen, T.

Jetsu, T.

Lee, S. D.

Lenz, R.

Lyons, R. G.

R. G. Lyons, Understanding Digital Signal Processing (Prentice Hall, 2004).

Nehorai, A.

W.-F. Zhang, P. Yang, D.-Q. Dai, A. Nehorai, “Reflectance estimation using local regression methods,” Lect. Notes Comput. Sci. 7367, 116–122 (2012).
[CrossRef]

Parkkinen, J.

Ressel, P.

C. Berg, J. Christensen, P. Ressel, Harmonic Analysis on Semigroups Theory of Positive Definite and Related Functions (Springer, 1984).

Romero, J.

Rosen, M. R.

F. H. Imai, M. R. Rosen, R. S. Berns, “Comparative study of metrics for spectral match quality,” in CGIV, First European Conference on Colour in Graphics, Imaging and Vision (Society for Imaging Science and Technology, 2002), pp. 492–496.

Schanda, J.

J. Schanda, Colorimetry: Understanding the CIE System (Wiley, 2007).

Schölkopf, B.

B. Schölkopf, A. Smola, Learning with Kernels (MIT, 2002).

Schöllkopf, B.

B. Schöllkopf, “The kernel trick for distances,” in Proceedings of Advances in Neural Information Processing Systems, T. K. Leen, ed. (MIT, 2001), pp. 301–307.

Shawe-Taylor, J.

J. Shawe-Taylor, N. Cristianini, Kernel Methods for Pattern Analysis (Cambridge University, 2004).

Smola, A.

B. Schölkopf, A. Smola, Learning with Kernels (MIT, 2002).

Stiles, W. S.

G. Wyszecki, W. S. Stiles, Color Science, 2nd ed. (Wiley, 1982).

Tarel, J.-P.

S. Boughorbel, J.-P. Tarel, N. Boujemaa, “Conditionally positive definite kernels for svm based image recognition,” in Proceedings of IEEE Conference on Multimedia and Expo (IEEE, 2005), pp. 113–116.

Valero, E. M.

T. Eckhard, E. M. Valero, J. Hernández-Andrés, “A comparative analysis of spectral estimation approaches applied to print inspection,” in Proceedings of the Annual German Colour Group Meeting (IDD TU Darmstadt, GRIS TU Darmstadt, IGD Fraunhofer, 2012), pp. 13–24.

Viggiano, J. A. S.

J. A. S. Viggiano, “Metrics for evaluating spectral matches: a quantitative comparison,” in Second European Conference on Colour Graphics, Imaging and Vision (Society for Imaging Science and Technology, 2004), pp. 286–291.

Wyszecki, G.

G. Wyszecki, W. S. Stiles, Color Science, 2nd ed. (Wiley, 1982).

Yang, P.

W.-F. Zhang, P. Yang, D.-Q. Dai, A. Nehorai, “Reflectance estimation using local regression methods,” Lect. Notes Comput. Sci. 7367, 116–122 (2012).
[CrossRef]

Zhang, W.-F.

W.-F. Zhang, P. Yang, D.-Q. Dai, A. Nehorai, “Reflectance estimation using local regression methods,” Lect. Notes Comput. Sci. 7367, 116–122 (2012).
[CrossRef]

J. Mach. Learn. Res. (1)

M. G. Genton, “Classes of kernels for machine learning: a statistics perspective,” J. Mach. Learn. Res. 2, 299–312 (2002).

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

Lect. Notes Comput. Sci. (1)

W.-F. Zhang, P. Yang, D.-Q. Dai, A. Nehorai, “Reflectance estimation using local regression methods,” Lect. Notes Comput. Sci. 7367, 116–122 (2012).
[CrossRef]

Other (15)

V. Heikkinen, “Kernel methods for estimation and classification of data from spectral imaging,” Dissertations in Forestry and Natural Sciences (University of Eastern Finland, 2011).

S. Boughorbel, J.-P. Tarel, N. Boujemaa, “Conditionally positive definite kernels for svm based image recognition,” in Proceedings of IEEE Conference on Multimedia and Expo (IEEE, 2005), pp. 113–116.

C. Berg, J. Christensen, P. Ressel, Harmonic Analysis on Semigroups Theory of Positive Definite and Related Functions (Springer, 1984).

B. Schöllkopf, “The kernel trick for distances,” in Proceedings of Advances in Neural Information Processing Systems, T. K. Leen, ed. (MIT, 2001), pp. 301–307.

M. Schnitzlein, M. Hund, “Chromasens GmbH,” http://www.chromasens.de/en .

R. G. Lyons, Understanding Digital Signal Processing (Prentice Hall, 2004).

T. Eckhard, “Color imaging web page of the University of Granada,” http://www.ugr.es/local/colorimg/suppl_docs/log_kernel.html .

F. H. Imai, M. R. Rosen, R. S. Berns, “Comparative study of metrics for spectral match quality,” in CGIV, First European Conference on Colour in Graphics, Imaging and Vision (Society for Imaging Science and Technology, 2002), pp. 492–496.

J. A. S. Viggiano, “Metrics for evaluating spectral matches: a quantitative comparison,” in Second European Conference on Colour Graphics, Imaging and Vision (Society for Imaging Science and Technology, 2004), pp. 286–291.

B. Schölkopf, A. Smola, Learning with Kernels (MIT, 2002).

J. Shawe-Taylor, N. Cristianini, Kernel Methods for Pattern Analysis (Cambridge University, 2004).

T. Eckhard, E. M. Valero, J. Hernández-Andrés, “A comparative analysis of spectral estimation approaches applied to print inspection,” in Proceedings of the Annual German Colour Group Meeting (IDD TU Darmstadt, GRIS TU Darmstadt, IGD Fraunhofer, 2012), pp. 13–24.

G. Wyszecki, W. S. Stiles, Color Science, 2nd ed. (Wiley, 1982).

“Improvement to industrial color-difference evaluation,” CIE Technical Report, (CIE, 2001).

J. Schanda, Colorimetry: Understanding the CIE System (Wiley, 2007).

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Figures (9)

Fig. 1.
Fig. 1.

CIE-L*a*b* coordinates for the Pantone (blue circle markers) and HKS dataset (red diamond markers).

Fig. 2.
Fig. 2.

Schematic illustration of the working principle of the 12-channel line-scan camera truePIXA. The RGB channel of the line sensor in combination with the four filters allow acquisition of 12 camera responses per image pixel at once.

Fig. 3.
Fig. 3.

Illustration of the system responsivities [sensor spectral responsivity plus illumination in arbitrary units (AU)]. Blue corresponds to the three-channel system, blue+green to the six-channel, blue+green+red to the 12 channel system used for experiments in Section 5.E.

Fig. 4.
Fig. 4.

Illustration of the mean RMSE (over 10 folds of parameter optimization) over the parameter search space for Pantone 12C measured camera response data and the Gaussian kernel. The optimal parameter selection is illustrated by a green square. The parameter search space has been cropped for illustration purposes.

Fig. 5.
Fig. 5.

Illustration of the mean RMSE (over 10 folds of parameter optimization) over the parameter search space for Pantone 12C measured camera response data and the logarithmic kernel. The optimal parameter selection is illustrated by a green square. The parameter search space has been cropped for illustration purposes.

Fig. 6.
Fig. 6.

Illustration of the mean RMSE (over 10 folds of parameter optimization) over the parameter search space for Pantone 12C measured camera response data and varying number of training samples for the logarithmic kernel. The optimal parameters (indicated as a green square) are β 1 4 = 2 and λ 1 4 = 10 4 .

Fig. 7.
Fig. 7.

Analysis of the influence of the number of training samples on the estimation performance (RMSE) for the logarithmic, Gaussian, and linear kernel function. Camera responses were simulated noiseless for the Pantone dataset.

Fig. 8.
Fig. 8.

Analysis of the influence of the number of training samples on the estimation performance (RMSE) for the logarithmic, Gaussian, and linear kernel function and measured camera responses of the Pantone dataset.

Fig. 9.
Fig. 9.

Estimation results for the Pantone dataset and measured camera responses: sample reflectance with lowest (left) and highest (middle) RMSE error for the logarithmic kernel and corresponding estimated spectra of the linear and Gaussian kernel. The plot on the right illustrates the sample corresponding to the 2nd highest RMSE for the logarithmic kernel.

Tables (3)

Tables Icon

Table 1. Spectral Estimation Results for Measured Data of the Pantone and HKS Dataset and the 12-Channel Acquisition System, Logarithmic, Gaussian, and Linear Kernel

Tables Icon

Table 2. Spectral Estimation Results for Simulated Data of the Pantone and HKS Dataset and 12-Channel Acquisition System, Logarithmic, Gaussian, and Linear Kernel

Tables Icon

Table 3. Comparison of the Number of Channels for Measured Data of the Pantone Dataset, the Logarithmic, Gaussian, and Linear Kernel

Equations (13)

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x = W r + b ,
argmin F ( i = 1 l r i F T Φ ( x i ) 2 + λ F F 2 ) ,
argmin A ( i = 1 l r i A T k ( x i ) 2 + λ Tr ( A T K A ) ) ,
( K + λ I l ) A = R T ,
r ˜ = A T k ( x ) = R ( K + λ I l ) 1 k ( x ) .
k ( x i , x j ) = ( x i T x j ) d , with d = 1 ,
k ( x i , x j ) = exp ( x i x j 2 2 σ 2 ) ,
arg min { A , a } ( i = 1 l r i A T k ( x i ) a 2 + λ Tr ( A T K A ) ) s.t. 1 T A = 0 T
[ K + λ I l 1 1 T 0 ] [ A a T ] = [ R T 0 T ] ,
r ˜ = A T k ( x ) + a .
k ( x i , x j ) = log ( 1 + x i x j β ) , with 0 < β 2 ,
RMSE = 1 m r r ˜ 2 .
d p = 1 r T r ˜ r r ˜ .

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