Abstract

We evaluate three link functions (square root, logit, and copula) and Matérn kernel in the kernel-based estimation of reflectance spectra of the Munsell Matte collection in the 400–700 nm region. We estimate reflectance spectra from RGB camera responses in case of real and simulated responses and show that a combination of link function and a kernel regression model with a Matérn kernel decreases spectral errors when compared to a Gaussian mixture model or kernel regression with the Gaussian kernel. Matérn kernel produces performance similar to the thin plate spline model, but does not require a parametric polynomial part in the model.

© 2013 Optical Society of America

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References

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    [CrossRef]
  12. N. Shimano, “Recovery of spectral reflectances of objects being imaged without prior knowledge,” IEEE Trans. Image Process. 15, 1848–1856 (2006).
    [CrossRef]
  13. M. Yamaguchi, H. Haneishi, and N. Ohyama, “Beyond red-green-blue (RGB): spectrum-based color imaging technology,” J. Imaging Sci. Technol. 52, 10201 (2008).
    [CrossRef]
  14. P. Urban, M. R. Rosen, and R. S. Berns, “Spectral image reconstruction using an edge preserving spatio-spectral Wiener estimation,” J. Opt. Soc. Am. A 26, 1865–1875 (2009).
    [CrossRef]
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    [CrossRef]
  16. G. Wahba, Spline Models for Observational Data, SIAM CBMS-NSF Regional Conference Series in Applied Mathematics (SIAM, 1990), Vol. 59.
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    [CrossRef]
  22. W.-F. Zhang and D.-Q. Dai, “Spectral reflectance estimation from camera responses by support vector regression and a composite model,” J. Opt. Soc. Am. A 25, 2286–2296 (2008).
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  23. J. M. Dicarlo and B. A. Wandell, “Spectral estimation theory: beyond linear but before Bayesian,” J. Opt. Soc. Am. A 20, 1261–1270 (2003).
    [CrossRef]

2009

2008

2007

2006

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

2004

2003

2002

2000

H. Haneishi, T. Hasegawa, A. Hosoi, Y. Yokoyama, N. Tsumura, and Y. Miyake, “System design for accurately estimating the reflectance spectra of art paintings,” Appl. Opt. 39, 6621–6632 (2000).
[CrossRef]

F. Imai, R. S. Berns, and D. Tzeng, “A comparative analysis of spectral reflectance estimation in various spaces using a trichromatic camera system,” J. Imaging Sci. Technol. 44, 280–287 (2000).

1989

1976

Attewell, D.

D. Attewell and B. J. Roland, “The distribution of reflectances within the visual environment,” Vision Res. 47, 548–554 (2007).
[CrossRef]

Berns, R. S.

P. Urban, M. R. Rosen, and R. S. Berns, “Spectral image reconstruction using an edge preserving spatio-spectral Wiener estimation,” J. Opt. Soc. Am. A 26, 1865–1875 (2009).
[CrossRef]

F. Imai, R. S. Berns, and D. Tzeng, “A comparative analysis of spectral reflectance estimation in various spaces using a trichromatic camera system,” J. Imaging Sci. Technol. 44, 280–287 (2000).

R. S. Berns, “Color-accurate image archives using spectral imaging,” in Scientific Examination of Art: Modern Techniques in Conservation and Analysis (National Academy, 2005), pp. 105–119.

Cucker, F.

F. Cucker and S. Smale, “On the mathematical foundations of learning,” Bulletin of the American Mathematical Society 39, 1–49 (2002).
[CrossRef]

Dai, D.-Q.

Dicarlo, J. M.

Friedman, J.

T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference and Prediction (Springer-Verlag, 2001).

Hallikainen, J.

Haneishi, H.

M. Yamaguchi, H. Haneishi, and N. Ohyama, “Beyond red-green-blue (RGB): spectrum-based color imaging technology,” J. Imaging Sci. Technol. 52, 10201 (2008).
[CrossRef]

H. Haneishi, T. Hasegawa, A. Hosoi, Y. Yokoyama, N. Tsumura, and Y. Miyake, “System design for accurately estimating the reflectance spectra of art paintings,” Appl. Opt. 39, 6621–6632 (2000).
[CrossRef]

Hasegawa, T.

Hastie, T.

T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference and Prediction (Springer-Verlag, 2001).

Hauta-Kasari, M.

Heikkinen, V.

Hernández-Andrés, J.

Hosoi, A.

Imai, F.

F. Imai, R. S. Berns, and D. Tzeng, “A comparative analysis of spectral reflectance estimation in various spaces using a trichromatic camera system,” J. Imaging Sci. Technol. 44, 280–287 (2000).

Jääskeläinen, T.

Jetsu, T.

Lee, S. D.

Lenz, R.

Mancill, C. E.

Miyake, Y.

Murakami, Y.

Nieves, J. L.

Obi, T.

Ohyama, N.

M. Yamaguchi, H. Haneishi, and N. Ohyama, “Beyond red-green-blue (RGB): spectrum-based color imaging technology,” J. Imaging Sci. Technol. 52, 10201 (2008).
[CrossRef]

Y. Murakami, T. Obi, M. Yamaguchi, and N. Ohyama, “Nonlinear estimation of spectral reflectance on Gaussian mixture distribution for color image reproduction,” Appl. Opt. 41, 4840–4847 (2002).
[CrossRef]

Parkkinen, J.

Pratt, W. K.

Rao, C. R.

C. R. Rao, Linear Statistical Inference and Its Applications, 2nd ed. (Wiley, 1973).

Rasmussen, C. E.

C. E. Rasmussen and C. Williams, Gaussian Processes for Machine Learning (MIT, 2006).

Roland, B. J.

D. Attewell and B. J. Roland, “The distribution of reflectances within the visual environment,” Vision Res. 47, 548–554 (2007).
[CrossRef]

Romero, J.

Rosen, M. R.

Shimano, N.

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

Smale, S.

F. Cucker and S. Smale, “On the mathematical foundations of learning,” Bulletin of the American Mathematical Society 39, 1–49 (2002).
[CrossRef]

Tibshirani, R.

T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference and Prediction (Springer-Verlag, 2001).

Tsumura, N.

Tzeng, D.

F. Imai, R. S. Berns, and D. Tzeng, “A comparative analysis of spectral reflectance estimation in various spaces using a trichromatic camera system,” J. Imaging Sci. Technol. 44, 280–287 (2000).

Urban, P.

Valero, E. M.

Wahba, G.

G. Wahba, Spline Models for Observational Data, SIAM CBMS-NSF Regional Conference Series in Applied Mathematics (SIAM, 1990), Vol. 59.

Wandell, B. A.

Williams, C.

C. E. Rasmussen and C. Williams, Gaussian Processes for Machine Learning (MIT, 2006).

Yamaguchi, M.

M. Yamaguchi, H. Haneishi, and N. Ohyama, “Beyond red-green-blue (RGB): spectrum-based color imaging technology,” J. Imaging Sci. Technol. 52, 10201 (2008).
[CrossRef]

Y. Murakami, T. Obi, M. Yamaguchi, and N. Ohyama, “Nonlinear estimation of spectral reflectance on Gaussian mixture distribution for color image reproduction,” Appl. Opt. 41, 4840–4847 (2002).
[CrossRef]

Yokoyama, Y.

Zhang, W.-F.

Appl. Opt.

Bulletin of the American Mathematical Society

F. Cucker and S. Smale, “On the mathematical foundations of learning,” Bulletin of the American Mathematical Society 39, 1–49 (2002).
[CrossRef]

IEEE Trans. Image Process.

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

J. Imaging Sci. Technol.

M. Yamaguchi, H. Haneishi, and N. Ohyama, “Beyond red-green-blue (RGB): spectrum-based color imaging technology,” J. Imaging Sci. Technol. 52, 10201 (2008).
[CrossRef]

F. Imai, R. S. Berns, and D. Tzeng, “A comparative analysis of spectral reflectance estimation in various spaces using a trichromatic camera system,” J. Imaging Sci. Technol. 44, 280–287 (2000).

J. Opt. Soc. Am. A

Vision Res.

D. Attewell and B. J. Roland, “The distribution of reflectances within the visual environment,” Vision Res. 47, 548–554 (2007).
[CrossRef]

Other

R. S. Berns, “Color-accurate image archives using spectral imaging,” in Scientific Examination of Art: Modern Techniques in Conservation and Analysis (National Academy, 2005), pp. 105–119.

University of Eastern Finland Color Group., “Spectral Database,” http://uef.fi/spectral .

C. R. Rao, Linear Statistical Inference and Its Applications, 2nd ed. (Wiley, 1973).

V. Heikkinen, Kernel methods for estimation and classification of data from spectral imaging, Ph.D. Thesis (School of Computing, University of Eastern Finland, 2011).

G. Wahba, Spline Models for Observational Data, SIAM CBMS-NSF Regional Conference Series in Applied Mathematics (SIAM, 1990), Vol. 59.

C. E. Rasmussen and C. Williams, Gaussian Processes for Machine Learning (MIT, 2006).

T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference and Prediction (Springer-Verlag, 2001).

E. M. Valero, Y. Hu, T. Eckhard, J. L. Nieves, J. Romero, M. Schnitzlein, and D. Nowack, “Comparative performance analysis of spectral estimation algorithms and computational optimization of a multispectral imaging system for print inspection,” Color Res. Appl. Available online (2012), http://onlinelibrary.wiley.com/doi/10.1002/col.21763/abstract .

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Figures (5)

Fig. 1.
Fig. 1.

Left: Location of wavelength-wise errors for the GMM and for a kernel model with a link function (a result corresponding to one data randomization in Experiment 1) with respect to the sensor sensitivity properties. Red curves represent relative sensitivity of CIE x¯,y¯, z¯ 1931 sensor system. Right: Wavelength-wise errors for the Gaussian and Matérn kernel and TPS (Experiment 1).

Fig. 2.
Fig. 2.

Left: Wavelength-wise errors for the Gaussian kernel and for the Matérn kernel models with link functions (a result corresponding to one data randomization in Experiment 1). Right: Wavelength-wise errors for the Gaussian kernel and the Matérn kernel models with link functions (a result corresponding to one data randomization in Experiment 2).

Fig. 3.
Fig. 3.

First row (image on left): Sample corresponding to largest PD difference between the Gaussian kernel model (0.03987) and the Matérn kernel model with link function (0.00831) in Experiment 1 (simulated data). First row (image on right): Sample corresponding to the largest RMSE difference between the Gaussian kernel model (0.0261) and the Matérn kernel model with link function (0.0083) in Experiment 1 (simulated data). Second row (image on left): Sample corresponding to largest PD difference between the Gaussian kernel model (0.01043) and the Matérn kernel model with link function (0.01758) in Experiment 2 (real data). Second row (image on right): Sample corresponding to the largest RMSE difference between the Gaussian kernel model (0.0352) and the Matérn kernel model with link function (0.0179) in Experiment 2 (real data).

Fig. 4.
Fig. 4.

Error values for individual test samples corresponding to Experiment 1 (simulated data) and tabulated results. Values are ordered according to (Matérn kernel+logit) results [maximum PD for Gaussian (0.0399) is excluded from the image].

Fig. 5.
Fig. 5.

Error values for individual test samples corresponding to Experiment 2 (real data) and tabulated results. Values are ordered according to (Matérn kernel+logit) results.

Tables (3)

Tables Icon

Table 1. Overview over Training and Test Setsa

Tables Icon

Table 2. Spectral Error Valuesa

Tables Icon

Table 3. Color Error Values, Corresponding to DeltaE2000 Errors under D65, A, and F11 Illuminanta

Equations (40)

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z=Wq,
S={(x1,q1),,(xm,qm)}Rk×Rn
[u1u2]N([μ1μ2],[Σ11Σ12Σ21Σ22]),
u2|u1N(μ2+Σ21Σ111(u1μ1),Σ22Σ21Σ111Σ12).
q^=ΣqWT(WΣqWT+σe2Ik)1x.
p(q)=i=1mvipi(q),
pi(q)=1(2π)n/2σqnexp(12σq2qqi2),i=1,,m.
[xq]=N([Wqiqi],[σq2WWT+σe2Ikσq2Wσq2WTσq2In]).
q|xN(q˜i(x),Σ)
q˜i(x)=WT(WWT+γIk)1(xWqi)+qi,
Σ=σq2(InWT(WWT+γIk)1W),
x|qN(Wq,σe2Ik),
pi(x,q)=p(x|q)pi(q).
p(x|q)p(q)=i=1mvipi(q|x)pi(x),
p(x)=i=1mvipi(x),
p(q|x)=i=1mpi(q|x)pi(x)i=1mpi(x).
p˜i(x)=pi(x)/i=1mpi(x),
pi(x)exp(1σq2(xWqi)T(WWT+γIk)1(xWqi)).
q^=E[q|x]=i=1mp˜i(x)E[qi|x]=i=1mp˜i(x)q˜i(x),
q^=(QTWT(WWT+γIk)1WQT)p˜(x)+WT(WWT+γIk)1x,
V[fi]=j=1m(qjifi(xj))2+ηfiH2,
fi(x)=aiTk(x),
argminARm×nV(A)=j=1mqjATk(xj)2+ηTr(ATKA),
A^=(K+ηIm)1Q
q^=A^Tk(x)=QT(K+ηIm)1k(x).
κ1(x,v)=exp(xv22ς2),
κ2(x,v)=(1+3xvς)exp(3xvς),
κ3(x,v)=cd,kxv2dk,cd,k=Γ(k/2d)22dπk/2(d1)!,
argminBRm×n,CRN×nV(B,C)=j=1mqjBTk(xj)CTΨ(xj)2+ηTr(BTKB)s.t.Ψ(X)TB=0,
q^=B^Tk(x)+C^TΨ(x),
q˜=logit(q)=log(q1q),
q=exp(q˜)1+exp(q˜).
q˜=Φ1(F(q))=Φ1(F1(q1),F2(q2),,Fn(qn))T,
q˜i=Φ1(Fi(qi))=Φ1(Γ(ai+bi)Γ(ai)Γ(bi)0qitai1(1t)bi1dt),i=1,,n,
qi=Fi1(Φ(q˜i)),i=1,,n,
RMSE(q,q^)=qq^2/n,
PD(q,q^)=1qTq^qq^.
MSE(i)=j=1l(qjiq^ji)2/l,i=1,,n,
Q=[q1q2ql]T=[qji],j=1,,l,i=1,,n
Q^=[q^1q^2q^l]T=[q^ji],j=1,,l,i=1,,n.

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