Abstract

Spectral calibration of digital cameras based on the spectral data of commercially available calibration charts is an ill-conditioned problem that has an infinite number of solutions. We introduce a method to estimate the sensor's spectral sensitivity function based on metamers. For a given patch on the calibration chart we construct numerical metamers by computing convex linear combinations of spectra from calibration chips with lower and higher sensor response values. The difference between the measured reflectance spectrum and the numerical metamer lies in the null space of the sensor. For each measured spectrum we use this procedure to compute a collection of color signals that lie in the null space of the sensor. For a collection of such spaces we compute the robust principal components, and we obtain an estimate of the sensor by computing the common null space spanned by these vectors. Our approach has a number of advantages over standard techniques: It is robust to outliers and is not dominated by larger response values, and it offers the ability to evaluate the goodness of the solution where it is possible to show that the solution is optimal, given the data, if the calculated range is one dimensional.

© 2006 Optical Society of America

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References

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  1. A. Alsam and G. D. Finlayson, 'Recovering spectral sensitivities with uncertainty,' in Proceedings of the First European Conference on Color in Graphics, Imaging and Vision (CG1V 2002) (Society for Imaging Science and Technology, 2002), pp. 22-26.
  2. G. Sharma and H. Trussell, 'Characterization of scanner sensitivity,' in Proceedings of the IS&T/SID Color Imaging Conference (Society for Imaging Science and Technology/Society for Information Display, 1993), pp. 103-107.
  3. G. Finlayson, S. Hordley, and P. Hubel, 'Recovering device sensitivities with quadratic programming,' in Proceedings of the IS&T/SID Sixth Color Imaging Conference: Color Science, Systems and Applications (Society for Imaging Science and Technology/Society for Information Display, 1998), pp. 90-95.
  4. B. Dyas, 'Robust sensor response characterization,' in Proceedings of the IS&T/SID Eighth Color Imaging Conference (Society for Imaging Science and Technology/Society for Information Display, 2000), pp. 144-148.
  5. K. Barnard and B. Funt, 'Camera characterization for color research,' Color Res. Appl. 27, 153-164 (2002).
    [CrossRef]
  6. K. Barnard and B. Funt, 'Camera calibration for colour vision research,' in Human Vision and Electronic Imaging IV, B.E. Roqowitz and T.N. Pappas, eds., Proc. SPIE 3644, 576-585 (1999).
  7. B. Smith, C. Spiekermann, and R. Sember, 'Numerical methods for colorimetric calculations: Sampling density requirements,' Color Res. Appl. 17, 394-401 (1992).
    [CrossRef]
  8. P. Dimitri Bertsekas, Nonlinear Programming, 2nd ed. (Athena Scientific, 1999).
  9. R. D. Fierro, G. H. Golub, P. C. Hansen, and D. P. O'Leary, 'Regularization by truncated total least squares,' SIAM J. Sci. Comput. (USA) 18, 1223-1241 (1997).
    [CrossRef]
  10. F. Konig and P. Herzog, 'Spectral calibration using linear programming,' in Color Imaging, Device Independent Color, Color Hard Copy, and Graphic Arts V, R.Eschbach and G.G.Marcu, eds., Proc. SPIE 3963, 36-56 (2000).
  11. A. Alsam, 'Optimising spectral calibration,' Ph.D. thesis (University of East Anglia, 2004).
  12. G. H. Golub and C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, 1989).
  13. P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, SIAM Monographs on Mathematical Modeling and Computation (Society for Industrial and Applied Mathematics, 1998).
    [CrossRef]
  14. L. T. Maloney and B. A. Wandell, 'Color constancy: a method for recording surface spectral reflectance,' J. Opt. Soc. Am. A 3, 29-33 (1986).
    [CrossRef] [PubMed]
  15. J. Parkkinen, J. Hallikainen, and T. Jaaskelainen, 'Characteristic spectra of Munsell colors,' J. Opt. Soc. Am. A 6, 318-322 (1989).
    [CrossRef]
  16. P. C. Hansen, 'Analysis of discrete ill-posed problems by means of the l-curve,' SIAM Rev. 34, 561-580 (1992).
    [CrossRef]
  17. P. C. Hansen, 'The l-curve and its use in the numerical treatment of inverse problems,' in Computational Inverse Problems in Electrocardiology (WIT Press, Southampton, 2001), pp. 119-142.
  18. A. Alsam and G. Finlayson, 'Metamer sets without spectral calibration,' in Proceedings of the 13th IS&T/SID Color Imaging Conference (Society for Imaging Science and Technology/Society for Information Display, 2005), pp. 104-108.
  19. M. Hubert, P. J. Rousseeuw, and S. Verboven, 'A fast method for robust principal components with applications to chemometrics,' Chemom. Intell. Lab. Syst. 60, 101-111 (2002).
    [CrossRef]
  20. P. L. Vora and H. J. Trussell, 'Measure of goodness of a set of color scanning filters,' J. Opt. Soc. Am. A 10, 1499-1508 (1993).
    [CrossRef]

2002 (2)

K. Barnard and B. Funt, 'Camera characterization for color research,' Color Res. Appl. 27, 153-164 (2002).
[CrossRef]

M. Hubert, P. J. Rousseeuw, and S. Verboven, 'A fast method for robust principal components with applications to chemometrics,' Chemom. Intell. Lab. Syst. 60, 101-111 (2002).
[CrossRef]

1997 (1)

R. D. Fierro, G. H. Golub, P. C. Hansen, and D. P. O'Leary, 'Regularization by truncated total least squares,' SIAM J. Sci. Comput. (USA) 18, 1223-1241 (1997).
[CrossRef]

1993 (1)

1992 (2)

P. C. Hansen, 'Analysis of discrete ill-posed problems by means of the l-curve,' SIAM Rev. 34, 561-580 (1992).
[CrossRef]

B. Smith, C. Spiekermann, and R. Sember, 'Numerical methods for colorimetric calculations: Sampling density requirements,' Color Res. Appl. 17, 394-401 (1992).
[CrossRef]

1989 (1)

1986 (1)

Alsam, A.

A. Alsam and G. D. Finlayson, 'Recovering spectral sensitivities with uncertainty,' in Proceedings of the First European Conference on Color in Graphics, Imaging and Vision (CG1V 2002) (Society for Imaging Science and Technology, 2002), pp. 22-26.

A. Alsam, 'Optimising spectral calibration,' Ph.D. thesis (University of East Anglia, 2004).

A. Alsam and G. Finlayson, 'Metamer sets without spectral calibration,' in Proceedings of the 13th IS&T/SID Color Imaging Conference (Society for Imaging Science and Technology/Society for Information Display, 2005), pp. 104-108.

Barnard, K.

K. Barnard and B. Funt, 'Camera characterization for color research,' Color Res. Appl. 27, 153-164 (2002).
[CrossRef]

K. Barnard and B. Funt, 'Camera calibration for colour vision research,' in Human Vision and Electronic Imaging IV, B.E. Roqowitz and T.N. Pappas, eds., Proc. SPIE 3644, 576-585 (1999).

Bertsekas, P. Dimitri

P. Dimitri Bertsekas, Nonlinear Programming, 2nd ed. (Athena Scientific, 1999).

Dyas, B.

B. Dyas, 'Robust sensor response characterization,' in Proceedings of the IS&T/SID Eighth Color Imaging Conference (Society for Imaging Science and Technology/Society for Information Display, 2000), pp. 144-148.

Fierro, R. D.

R. D. Fierro, G. H. Golub, P. C. Hansen, and D. P. O'Leary, 'Regularization by truncated total least squares,' SIAM J. Sci. Comput. (USA) 18, 1223-1241 (1997).
[CrossRef]

Finlayson, G.

G. Finlayson, S. Hordley, and P. Hubel, 'Recovering device sensitivities with quadratic programming,' in Proceedings of the IS&T/SID Sixth Color Imaging Conference: Color Science, Systems and Applications (Society for Imaging Science and Technology/Society for Information Display, 1998), pp. 90-95.

A. Alsam and G. Finlayson, 'Metamer sets without spectral calibration,' in Proceedings of the 13th IS&T/SID Color Imaging Conference (Society for Imaging Science and Technology/Society for Information Display, 2005), pp. 104-108.

Finlayson, G. D.

A. Alsam and G. D. Finlayson, 'Recovering spectral sensitivities with uncertainty,' in Proceedings of the First European Conference on Color in Graphics, Imaging and Vision (CG1V 2002) (Society for Imaging Science and Technology, 2002), pp. 22-26.

Funt, B.

K. Barnard and B. Funt, 'Camera characterization for color research,' Color Res. Appl. 27, 153-164 (2002).
[CrossRef]

K. Barnard and B. Funt, 'Camera calibration for colour vision research,' in Human Vision and Electronic Imaging IV, B.E. Roqowitz and T.N. Pappas, eds., Proc. SPIE 3644, 576-585 (1999).

Golub, G. H.

R. D. Fierro, G. H. Golub, P. C. Hansen, and D. P. O'Leary, 'Regularization by truncated total least squares,' SIAM J. Sci. Comput. (USA) 18, 1223-1241 (1997).
[CrossRef]

G. H. Golub and C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, 1989).

Hallikainen, J.

Hansen, P. C.

R. D. Fierro, G. H. Golub, P. C. Hansen, and D. P. O'Leary, 'Regularization by truncated total least squares,' SIAM J. Sci. Comput. (USA) 18, 1223-1241 (1997).
[CrossRef]

P. C. Hansen, 'Analysis of discrete ill-posed problems by means of the l-curve,' SIAM Rev. 34, 561-580 (1992).
[CrossRef]

P. C. Hansen, 'The l-curve and its use in the numerical treatment of inverse problems,' in Computational Inverse Problems in Electrocardiology (WIT Press, Southampton, 2001), pp. 119-142.

P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, SIAM Monographs on Mathematical Modeling and Computation (Society for Industrial and Applied Mathematics, 1998).
[CrossRef]

Herzog, P.

F. Konig and P. Herzog, 'Spectral calibration using linear programming,' in Color Imaging, Device Independent Color, Color Hard Copy, and Graphic Arts V, R.Eschbach and G.G.Marcu, eds., Proc. SPIE 3963, 36-56 (2000).

Hordley, S.

G. Finlayson, S. Hordley, and P. Hubel, 'Recovering device sensitivities with quadratic programming,' in Proceedings of the IS&T/SID Sixth Color Imaging Conference: Color Science, Systems and Applications (Society for Imaging Science and Technology/Society for Information Display, 1998), pp. 90-95.

Hubel, P.

G. Finlayson, S. Hordley, and P. Hubel, 'Recovering device sensitivities with quadratic programming,' in Proceedings of the IS&T/SID Sixth Color Imaging Conference: Color Science, Systems and Applications (Society for Imaging Science and Technology/Society for Information Display, 1998), pp. 90-95.

Hubert, M.

M. Hubert, P. J. Rousseeuw, and S. Verboven, 'A fast method for robust principal components with applications to chemometrics,' Chemom. Intell. Lab. Syst. 60, 101-111 (2002).
[CrossRef]

Jaaskelainen, T.

Konig, F.

F. Konig and P. Herzog, 'Spectral calibration using linear programming,' in Color Imaging, Device Independent Color, Color Hard Copy, and Graphic Arts V, R.Eschbach and G.G.Marcu, eds., Proc. SPIE 3963, 36-56 (2000).

Maloney, L. T.

O'Leary, D. P.

R. D. Fierro, G. H. Golub, P. C. Hansen, and D. P. O'Leary, 'Regularization by truncated total least squares,' SIAM J. Sci. Comput. (USA) 18, 1223-1241 (1997).
[CrossRef]

Parkkinen, J.

Rousseeuw, P. J.

M. Hubert, P. J. Rousseeuw, and S. Verboven, 'A fast method for robust principal components with applications to chemometrics,' Chemom. Intell. Lab. Syst. 60, 101-111 (2002).
[CrossRef]

Sember, R.

B. Smith, C. Spiekermann, and R. Sember, 'Numerical methods for colorimetric calculations: Sampling density requirements,' Color Res. Appl. 17, 394-401 (1992).
[CrossRef]

Sharma, G.

G. Sharma and H. Trussell, 'Characterization of scanner sensitivity,' in Proceedings of the IS&T/SID Color Imaging Conference (Society for Imaging Science and Technology/Society for Information Display, 1993), pp. 103-107.

Smith, B.

B. Smith, C. Spiekermann, and R. Sember, 'Numerical methods for colorimetric calculations: Sampling density requirements,' Color Res. Appl. 17, 394-401 (1992).
[CrossRef]

Spiekermann, C.

B. Smith, C. Spiekermann, and R. Sember, 'Numerical methods for colorimetric calculations: Sampling density requirements,' Color Res. Appl. 17, 394-401 (1992).
[CrossRef]

Trussell, H.

G. Sharma and H. Trussell, 'Characterization of scanner sensitivity,' in Proceedings of the IS&T/SID Color Imaging Conference (Society for Imaging Science and Technology/Society for Information Display, 1993), pp. 103-107.

Trussell, H. J.

van Loan, C. F.

G. H. Golub and C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, 1989).

Verboven, S.

M. Hubert, P. J. Rousseeuw, and S. Verboven, 'A fast method for robust principal components with applications to chemometrics,' Chemom. Intell. Lab. Syst. 60, 101-111 (2002).
[CrossRef]

Vora, P. L.

Wandell, B. A.

Chemom. Intell. Lab. Syst. (1)

M. Hubert, P. J. Rousseeuw, and S. Verboven, 'A fast method for robust principal components with applications to chemometrics,' Chemom. Intell. Lab. Syst. 60, 101-111 (2002).
[CrossRef]

Color Res. Appl. (2)

K. Barnard and B. Funt, 'Camera characterization for color research,' Color Res. Appl. 27, 153-164 (2002).
[CrossRef]

B. Smith, C. Spiekermann, and R. Sember, 'Numerical methods for colorimetric calculations: Sampling density requirements,' Color Res. Appl. 17, 394-401 (1992).
[CrossRef]

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

SIAM J. Sci. Comput. (USA) (1)

R. D. Fierro, G. H. Golub, P. C. Hansen, and D. P. O'Leary, 'Regularization by truncated total least squares,' SIAM J. Sci. Comput. (USA) 18, 1223-1241 (1997).
[CrossRef]

SIAM Rev. (1)

P. C. Hansen, 'Analysis of discrete ill-posed problems by means of the l-curve,' SIAM Rev. 34, 561-580 (1992).
[CrossRef]

Other (12)

P. C. Hansen, 'The l-curve and its use in the numerical treatment of inverse problems,' in Computational Inverse Problems in Electrocardiology (WIT Press, Southampton, 2001), pp. 119-142.

A. Alsam and G. Finlayson, 'Metamer sets without spectral calibration,' in Proceedings of the 13th IS&T/SID Color Imaging Conference (Society for Imaging Science and Technology/Society for Information Display, 2005), pp. 104-108.

F. Konig and P. Herzog, 'Spectral calibration using linear programming,' in Color Imaging, Device Independent Color, Color Hard Copy, and Graphic Arts V, R.Eschbach and G.G.Marcu, eds., Proc. SPIE 3963, 36-56 (2000).

A. Alsam, 'Optimising spectral calibration,' Ph.D. thesis (University of East Anglia, 2004).

G. H. Golub and C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, 1989).

P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, SIAM Monographs on Mathematical Modeling and Computation (Society for Industrial and Applied Mathematics, 1998).
[CrossRef]

P. Dimitri Bertsekas, Nonlinear Programming, 2nd ed. (Athena Scientific, 1999).

K. Barnard and B. Funt, 'Camera calibration for colour vision research,' in Human Vision and Electronic Imaging IV, B.E. Roqowitz and T.N. Pappas, eds., Proc. SPIE 3644, 576-585 (1999).

A. Alsam and G. D. Finlayson, 'Recovering spectral sensitivities with uncertainty,' in Proceedings of the First European Conference on Color in Graphics, Imaging and Vision (CG1V 2002) (Society for Imaging Science and Technology, 2002), pp. 22-26.

G. Sharma and H. Trussell, 'Characterization of scanner sensitivity,' in Proceedings of the IS&T/SID Color Imaging Conference (Society for Imaging Science and Technology/Society for Information Display, 1993), pp. 103-107.

G. Finlayson, S. Hordley, and P. Hubel, 'Recovering device sensitivities with quadratic programming,' in Proceedings of the IS&T/SID Sixth Color Imaging Conference: Color Science, Systems and Applications (Society for Imaging Science and Technology/Society for Information Display, 1998), pp. 90-95.

B. Dyas, 'Robust sensor response characterization,' in Proceedings of the IS&T/SID Eighth Color Imaging Conference (Society for Imaging Science and Technology/Society for Information Display, 2000), pp. 144-148.

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Figures (4)

Fig. 1
Fig. 1

Three-dimensional example of the metameric black planes, shown to be orthogonal to a single vector. The axes a 1 , a 2 , and a 3 are the three-dimensional axes of the input data.

Fig. 2
Fig. 2

Estimated spectral sensitivities of the Nikon D70 camera as a function of wavelength 380– 750 nm . Estimation methods used the proposed MB (circles), TR (diamonds), and TSVD (asterisks).

Fig. 3
Fig. 3

Estimates of the red (solid curve), green (dashed curve), and blue (dotted curve).

Fig. 4
Fig. 4

Estimated spectral sensitivities (dashed curves) of the MegaVision camera as a function of wavelength 400– 700 nm . Actual sensitivities are shown by solid curves.

Tables (3)

Tables Icon

Table 1 Absolute Error between Measured and Estimated Responses for the Red, Green, and Blue Channels of the Nikon D70 Camera: Esser Calibration Chart a

Tables Icon

Table 2 Absolute Error between Measured and Estimated Responses for the Red, Green, and Blue Chanels of the Nikon D70 Camera: Macbeth Color Checker a

Tables Icon

Table 3 Absolute Error between Measured and Estimated Responses for the Red, Green, and Blue Channels of the MegaVision Camera: Macbeth Color Checker a

Equations (26)

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Y Υ = Y Υ = AX .
ρ = ω E ( λ ) S ( λ ) X ( λ ) d λ .
ρ = ω A ( λ ) X ( λ ) d λ .
e = ( E ( λ 1 ) , E ( λ 2 ) , , E ( λ n ) ) T ,
s = ( S ( λ 1 ) , S ( λ 2 ) , , S ( λ n ) ) T ,
x = ( X ( λ 1 ) , X ( λ 2 ) , , X ( λ n ) ) T ,
ρ = j = 1 n e j s j x j Δ λ ,
ρ = s T diag ( e ) x ,
A = ( diag ( e ) S ) T .
y = A x + ϵ ,
y ϵ = A x ,
A = U Σ V T ,
x = i = 1 n v i u i T σ i y ϵ .
x = i = 1 r v i u i T σ i y ϵ ,
x = i = 1 n v i u i T σ i + δ 2 y ϵ ,
x = ( A T A + I δ 2 ) 1 A y ϵ ,
min r A x y ϵ
subject to B x t ,
AU T = [ A 1 0 ] ,
AX = ( AU T ) ( UX ) = [ A 1 0 ] [ X 1 X 2 ] = A 1 X 1 .
γ = y m y ( μ ) y ( ν ) y ( μ ) .
P 0 = U Z n U Z n T .
A = A AP 0 .
A = U A D A V A T .
g = d A 1 i = 1 n d A × 100 ,
A E = a i x ̃ y i ,

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