Abstract

If two different surfaces look the same when viewed under a particular light source, then they are called metamers. We show mathematically how one can solve for the whole set of physically realizable natural surface reflectances that relate to the same tristimulus, the metamer set. Our analysis is based on very general linear models of reflectances, coupled with constraints that reflectances should adhere to (e.g., positivity and boundedness). We show that we can recover metamer sets for linear models of an arbitrary high dimension. To illustrate our new algorithm, we provide an example of calculating the metamer set and its manifestation as a mismatch region. Given a single XYZ observed under illuminant D65, we can examine the set of XYZs that would be possible under illuminant A.

© 2005 Optical Society of America

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References

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  1. G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982).
  2. G. Wyszecki, “Evaluation of metameric colors,” J. Opt. Soc. Am. 48, 451–454 (1958).
    [CrossRef]
  3. J. B. Cohen, W. E. Kappauf, “Metameric color stimuli, fundamental metamers, and Wyszecki’s metameric blacks,” Am. J. Psychol. 95, 537–564 (1982).
    [CrossRef] [PubMed]
  4. E. H. Land, “The retinex theory of color vision,” Sci. Am. 237, 108–128 (1977).
    [CrossRef] [PubMed]
  5. B. Smith, C. Spiekermann, R. Sember, “Numerical methods for colorimetric calculations: sampling density requirements,” Color Res. Appl. 17, 394–401 (1992).
    [CrossRef]
  6. E. L. Krinov, “Spectral reflectance properties of natural formations,” Tech. Transl. TT-439 (National Research Council of Canada, Ottawa, 1947).
  7. J. B. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychon. Sci. 1, 369–370 (1964).
    [CrossRef]
  8. L. T. Maloney, “Evaluation of linear models of surface spectral reflectance with small numbers of parameters,” J. Opt. Soc. Am. A 3, 1673–1683 (1986).
    [CrossRef] [PubMed]
  9. J. P. S. Parkkinen, J. Hallikanen, T. Jaaskelainen, “Characteristic spectra of Munsell colors,” J. Opt. Soc. Am. A 6, 318–322 (1989).
    [CrossRef]
  10. S. Westland, J. Shaw, H. Owens, “Color statistics of natural and man-made surfaces,” Remote Sens. Rev. 20, 50–55 (2000).
  11. L. T. Maloney, B. A. Wandell, “Color constancy: a method for recovering surface spectral reflectance,” J. Opt. Soc. Am. A 3, 29–33 (1986).
    [CrossRef] [PubMed]
  12. J. B. Cohen, W. E. Kappauf, “Color mixture and fundamental metamers: theory, algebra, geometry, application,” Am. J. Psychol. 98, 171–259 (1985).
    [CrossRef]
  13. J. B. Fraleigh, R. A. Beauregard, Linear Algebra (Addison-Wesley, Reading, Mass., 1990).
  14. S. A. Burns, J. B. Cohen, E. N. Kuznetsov, “Multiple metamers: preserving color matches under diverse illuminants,” Color Res. Appl. 14, 16–22 (1989).
    [CrossRef]
  15. F. J. M. Schmitt, “A method for the treatment of metamerism in colorimetry,” J. Opt. Soc. Am. 66, 601–608 (1976).
    [CrossRef] [PubMed]
  16. K. Takahama, Y. Nayatani, “New method for generating metameric stimuli of object colors,” J. Opt. Soc. Am. 62, 1516–1520 (1972).
    [CrossRef]
  17. N. Ohta, “Generating metameric object colors,” J. Opt. Soc. Am. 65, 1081–1082 (1975).
    [CrossRef]
  18. R. S. Berns, F. W. Billmeyer, R. S. Sacher, “Methods for generating spectral reflectance functions leading to color-constant properties,” Color Res. Appl. 10, 73–83 (1985).
    [CrossRef]
  19. J. M. Zoido, F. Carreño, E. Bernabeu, “Improved linear programming method to generate metameric spectral distributions,” Appl. Opt. 34, 1938–1943 (1995).
    [CrossRef] [PubMed]
  20. J. M. Speigle, D. H. Brainard, “Luminosity thresholds: effects of test chromaticity and ambient illumination,” J. Opt. Soc. Am. A 13, 436–451 (1996).
    [CrossRef]
  21. H. J. Trussell, “Applications of set-theoretic methods to color systems,” Color Res. Appl. 16, 31–41 (1991).
    [CrossRef]
  22. D. H. Marimont, B. A. Wandell, “Linear models of surface and illuminant spectra,” J. Opt. Soc. Am. A 9, 1905–1913 (1992).
    [CrossRef] [PubMed]
  23. T. Katriňák, M. Galvanec, E. Gedeonová, J. Smítal, Algebra a Teoretická Aritmetika (Alfa, Bratislava, Czechoslovakia, 1985).
  24. P. Morovic, “Metamer Sets,” Ph.D. thesis (University of East Anglia, Norwich, UK, 2002).
  25. G. H. Golub, C. F. van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, Baltimore, Md., 1996).
  26. F. P. Preparata, M. I. Shamos, Computational Geometry—An Introduction, 2nd ed. (Springer-Verlag, New York, 1988).
  27. C. B. Barber, D. D. Dobkin, H. Huhdanpaa, “The Quickhull algorithm for convex hulls,” ACM Trans. Math. Softw. 22, 469–483 (1996).
    [CrossRef]
  28. V. Chvátal, Linear Programing (Freeman, New York, 1983).
  29. N. Ohta, G. Wyszecki, “Theoretical chromaticity mismatch limits of metamers viewed under different illuminants,” J. Opt. Soc. Am. 65, 327–333 (1975).
    [CrossRef]
  30. J. A. Worthey, “Calculation of metameric reflectances,” Color Res. Appl. 13, 76–84 (1988).
    [CrossRef]
  31. W. A. Thornton, “Matching lights, metamers, and human visual response,” J. Color Appearance 2, 23–29 (1973).
  32. N. Ohta, G. Wyszecki, “Location of the nodes of metameric color stimuli,” Color Res. Appl. 2, 183–186 (1977).
    [CrossRef]
  33. N. Ohta, “Intersections of spectral curves of metameric colors,” Color Res. Appl. 12, 85–87 (1987).
    [CrossRef]
  34. G. D. Finlayson, P. Morovic, “Metamer crossovers of infinite metamer sets,” in Proceedings of the IS&T/SID Eighth Color Imaging Conference (The Society of Imaging Science and Technology, Springfield, Va. 2000), pp. 13–17..
  35. W. S. Stiles, G. Wyszecki, “Counting metameric object colors,” J. Opt. Soc. Am. 52, 313–328 (1962).
    [CrossRef]
  36. G. D. Finlayson, P. Morovic, “Metamer constrained colour correction,” J. Imaging Sci. Technol. 44, 295–300 (2000).

2000 (2)

S. Westland, J. Shaw, H. Owens, “Color statistics of natural and man-made surfaces,” Remote Sens. Rev. 20, 50–55 (2000).

G. D. Finlayson, P. Morovic, “Metamer constrained colour correction,” J. Imaging Sci. Technol. 44, 295–300 (2000).

1996 (2)

C. B. Barber, D. D. Dobkin, H. Huhdanpaa, “The Quickhull algorithm for convex hulls,” ACM Trans. Math. Softw. 22, 469–483 (1996).
[CrossRef]

J. M. Speigle, D. H. Brainard, “Luminosity thresholds: effects of test chromaticity and ambient illumination,” J. Opt. Soc. Am. A 13, 436–451 (1996).
[CrossRef]

1995 (1)

1992 (2)

B. Smith, C. Spiekermann, R. Sember, “Numerical methods for colorimetric calculations: sampling density requirements,” Color Res. Appl. 17, 394–401 (1992).
[CrossRef]

D. H. Marimont, B. A. Wandell, “Linear models of surface and illuminant spectra,” J. Opt. Soc. Am. A 9, 1905–1913 (1992).
[CrossRef] [PubMed]

1991 (1)

H. J. Trussell, “Applications of set-theoretic methods to color systems,” Color Res. Appl. 16, 31–41 (1991).
[CrossRef]

1989 (2)

S. A. Burns, J. B. Cohen, E. N. Kuznetsov, “Multiple metamers: preserving color matches under diverse illuminants,” Color Res. Appl. 14, 16–22 (1989).
[CrossRef]

J. P. S. Parkkinen, J. Hallikanen, T. Jaaskelainen, “Characteristic spectra of Munsell colors,” J. Opt. Soc. Am. A 6, 318–322 (1989).
[CrossRef]

1988 (1)

J. A. Worthey, “Calculation of metameric reflectances,” Color Res. Appl. 13, 76–84 (1988).
[CrossRef]

1987 (1)

N. Ohta, “Intersections of spectral curves of metameric colors,” Color Res. Appl. 12, 85–87 (1987).
[CrossRef]

1986 (2)

1985 (2)

J. B. Cohen, W. E. Kappauf, “Color mixture and fundamental metamers: theory, algebra, geometry, application,” Am. J. Psychol. 98, 171–259 (1985).
[CrossRef]

R. S. Berns, F. W. Billmeyer, R. S. Sacher, “Methods for generating spectral reflectance functions leading to color-constant properties,” Color Res. Appl. 10, 73–83 (1985).
[CrossRef]

1982 (1)

J. B. Cohen, W. E. Kappauf, “Metameric color stimuli, fundamental metamers, and Wyszecki’s metameric blacks,” Am. J. Psychol. 95, 537–564 (1982).
[CrossRef] [PubMed]

1977 (2)

E. H. Land, “The retinex theory of color vision,” Sci. Am. 237, 108–128 (1977).
[CrossRef] [PubMed]

N. Ohta, G. Wyszecki, “Location of the nodes of metameric color stimuli,” Color Res. Appl. 2, 183–186 (1977).
[CrossRef]

1976 (1)

1975 (2)

1973 (1)

W. A. Thornton, “Matching lights, metamers, and human visual response,” J. Color Appearance 2, 23–29 (1973).

1972 (1)

1964 (1)

J. B. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychon. Sci. 1, 369–370 (1964).
[CrossRef]

1962 (1)

1958 (1)

Barber, C. B.

C. B. Barber, D. D. Dobkin, H. Huhdanpaa, “The Quickhull algorithm for convex hulls,” ACM Trans. Math. Softw. 22, 469–483 (1996).
[CrossRef]

Beauregard, R. A.

J. B. Fraleigh, R. A. Beauregard, Linear Algebra (Addison-Wesley, Reading, Mass., 1990).

Bernabeu, E.

Berns, R. S.

R. S. Berns, F. W. Billmeyer, R. S. Sacher, “Methods for generating spectral reflectance functions leading to color-constant properties,” Color Res. Appl. 10, 73–83 (1985).
[CrossRef]

Billmeyer, F. W.

R. S. Berns, F. W. Billmeyer, R. S. Sacher, “Methods for generating spectral reflectance functions leading to color-constant properties,” Color Res. Appl. 10, 73–83 (1985).
[CrossRef]

Brainard, D. H.

Burns, S. A.

S. A. Burns, J. B. Cohen, E. N. Kuznetsov, “Multiple metamers: preserving color matches under diverse illuminants,” Color Res. Appl. 14, 16–22 (1989).
[CrossRef]

Carreño, F.

Chvátal, V.

V. Chvátal, Linear Programing (Freeman, New York, 1983).

Cohen, J. B.

S. A. Burns, J. B. Cohen, E. N. Kuznetsov, “Multiple metamers: preserving color matches under diverse illuminants,” Color Res. Appl. 14, 16–22 (1989).
[CrossRef]

J. B. Cohen, W. E. Kappauf, “Color mixture and fundamental metamers: theory, algebra, geometry, application,” Am. J. Psychol. 98, 171–259 (1985).
[CrossRef]

J. B. Cohen, W. E. Kappauf, “Metameric color stimuli, fundamental metamers, and Wyszecki’s metameric blacks,” Am. J. Psychol. 95, 537–564 (1982).
[CrossRef] [PubMed]

J. B. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychon. Sci. 1, 369–370 (1964).
[CrossRef]

Dobkin, D. D.

C. B. Barber, D. D. Dobkin, H. Huhdanpaa, “The Quickhull algorithm for convex hulls,” ACM Trans. Math. Softw. 22, 469–483 (1996).
[CrossRef]

Finlayson, G. D.

G. D. Finlayson, P. Morovic, “Metamer constrained colour correction,” J. Imaging Sci. Technol. 44, 295–300 (2000).

G. D. Finlayson, P. Morovic, “Metamer crossovers of infinite metamer sets,” in Proceedings of the IS&T/SID Eighth Color Imaging Conference (The Society of Imaging Science and Technology, Springfield, Va. 2000), pp. 13–17..

Fraleigh, J. B.

J. B. Fraleigh, R. A. Beauregard, Linear Algebra (Addison-Wesley, Reading, Mass., 1990).

Galvanec, M.

T. Katriňák, M. Galvanec, E. Gedeonová, J. Smítal, Algebra a Teoretická Aritmetika (Alfa, Bratislava, Czechoslovakia, 1985).

Gedeonová, E.

T. Katriňák, M. Galvanec, E. Gedeonová, J. Smítal, Algebra a Teoretická Aritmetika (Alfa, Bratislava, Czechoslovakia, 1985).

Golub, G. H.

G. H. Golub, C. F. van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, Baltimore, Md., 1996).

Hallikanen, J.

Huhdanpaa, H.

C. B. Barber, D. D. Dobkin, H. Huhdanpaa, “The Quickhull algorithm for convex hulls,” ACM Trans. Math. Softw. 22, 469–483 (1996).
[CrossRef]

Jaaskelainen, T.

Kappauf, W. E.

J. B. Cohen, W. E. Kappauf, “Color mixture and fundamental metamers: theory, algebra, geometry, application,” Am. J. Psychol. 98, 171–259 (1985).
[CrossRef]

J. B. Cohen, W. E. Kappauf, “Metameric color stimuli, fundamental metamers, and Wyszecki’s metameric blacks,” Am. J. Psychol. 95, 537–564 (1982).
[CrossRef] [PubMed]

Katrinák, T.

T. Katriňák, M. Galvanec, E. Gedeonová, J. Smítal, Algebra a Teoretická Aritmetika (Alfa, Bratislava, Czechoslovakia, 1985).

Krinov, E. L.

E. L. Krinov, “Spectral reflectance properties of natural formations,” Tech. Transl. TT-439 (National Research Council of Canada, Ottawa, 1947).

Kuznetsov, E. N.

S. A. Burns, J. B. Cohen, E. N. Kuznetsov, “Multiple metamers: preserving color matches under diverse illuminants,” Color Res. Appl. 14, 16–22 (1989).
[CrossRef]

Land, E. H.

E. H. Land, “The retinex theory of color vision,” Sci. Am. 237, 108–128 (1977).
[CrossRef] [PubMed]

Maloney, L. T.

Marimont, D. H.

Morovic, P.

G. D. Finlayson, P. Morovic, “Metamer constrained colour correction,” J. Imaging Sci. Technol. 44, 295–300 (2000).

G. D. Finlayson, P. Morovic, “Metamer crossovers of infinite metamer sets,” in Proceedings of the IS&T/SID Eighth Color Imaging Conference (The Society of Imaging Science and Technology, Springfield, Va. 2000), pp. 13–17..

P. Morovic, “Metamer Sets,” Ph.D. thesis (University of East Anglia, Norwich, UK, 2002).

Nayatani, Y.

Ohta, N.

N. Ohta, “Intersections of spectral curves of metameric colors,” Color Res. Appl. 12, 85–87 (1987).
[CrossRef]

N. Ohta, G. Wyszecki, “Location of the nodes of metameric color stimuli,” Color Res. Appl. 2, 183–186 (1977).
[CrossRef]

N. Ohta, G. Wyszecki, “Theoretical chromaticity mismatch limits of metamers viewed under different illuminants,” J. Opt. Soc. Am. 65, 327–333 (1975).
[CrossRef]

N. Ohta, “Generating metameric object colors,” J. Opt. Soc. Am. 65, 1081–1082 (1975).
[CrossRef]

Owens, H.

S. Westland, J. Shaw, H. Owens, “Color statistics of natural and man-made surfaces,” Remote Sens. Rev. 20, 50–55 (2000).

Parkkinen, J. P. S.

Preparata, F. P.

F. P. Preparata, M. I. Shamos, Computational Geometry—An Introduction, 2nd ed. (Springer-Verlag, New York, 1988).

Sacher, R. S.

R. S. Berns, F. W. Billmeyer, R. S. Sacher, “Methods for generating spectral reflectance functions leading to color-constant properties,” Color Res. Appl. 10, 73–83 (1985).
[CrossRef]

Schmitt, F. J. M.

Sember, R.

B. Smith, C. Spiekermann, R. Sember, “Numerical methods for colorimetric calculations: sampling density requirements,” Color Res. Appl. 17, 394–401 (1992).
[CrossRef]

Shamos, M. I.

F. P. Preparata, M. I. Shamos, Computational Geometry—An Introduction, 2nd ed. (Springer-Verlag, New York, 1988).

Shaw, J.

S. Westland, J. Shaw, H. Owens, “Color statistics of natural and man-made surfaces,” Remote Sens. Rev. 20, 50–55 (2000).

Smítal, J.

T. Katriňák, M. Galvanec, E. Gedeonová, J. Smítal, Algebra a Teoretická Aritmetika (Alfa, Bratislava, Czechoslovakia, 1985).

Smith, B.

B. Smith, C. Spiekermann, R. Sember, “Numerical methods for colorimetric calculations: sampling density requirements,” Color Res. Appl. 17, 394–401 (1992).
[CrossRef]

Speigle, J. M.

Spiekermann, C.

B. Smith, C. Spiekermann, R. Sember, “Numerical methods for colorimetric calculations: sampling density requirements,” Color Res. Appl. 17, 394–401 (1992).
[CrossRef]

Stiles, W. S.

W. S. Stiles, G. Wyszecki, “Counting metameric object colors,” J. Opt. Soc. Am. 52, 313–328 (1962).
[CrossRef]

G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982).

Takahama, K.

Thornton, W. A.

W. A. Thornton, “Matching lights, metamers, and human visual response,” J. Color Appearance 2, 23–29 (1973).

Trussell, H. J.

H. J. Trussell, “Applications of set-theoretic methods to color systems,” Color Res. Appl. 16, 31–41 (1991).
[CrossRef]

van Loan, C. F.

G. H. Golub, C. F. van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, Baltimore, Md., 1996).

Wandell, B. A.

Westland, S.

S. Westland, J. Shaw, H. Owens, “Color statistics of natural and man-made surfaces,” Remote Sens. Rev. 20, 50–55 (2000).

Worthey, J. A.

J. A. Worthey, “Calculation of metameric reflectances,” Color Res. Appl. 13, 76–84 (1988).
[CrossRef]

Wyszecki, G.

Zoido, J. M.

ACM Trans. Math. Softw. (1)

C. B. Barber, D. D. Dobkin, H. Huhdanpaa, “The Quickhull algorithm for convex hulls,” ACM Trans. Math. Softw. 22, 469–483 (1996).
[CrossRef]

Am. J. Psychol. (2)

J. B. Cohen, W. E. Kappauf, “Metameric color stimuli, fundamental metamers, and Wyszecki’s metameric blacks,” Am. J. Psychol. 95, 537–564 (1982).
[CrossRef] [PubMed]

J. B. Cohen, W. E. Kappauf, “Color mixture and fundamental metamers: theory, algebra, geometry, application,” Am. J. Psychol. 98, 171–259 (1985).
[CrossRef]

Appl. Opt. (1)

Color Res. Appl. (7)

H. J. Trussell, “Applications of set-theoretic methods to color systems,” Color Res. Appl. 16, 31–41 (1991).
[CrossRef]

R. S. Berns, F. W. Billmeyer, R. S. Sacher, “Methods for generating spectral reflectance functions leading to color-constant properties,” Color Res. Appl. 10, 73–83 (1985).
[CrossRef]

J. A. Worthey, “Calculation of metameric reflectances,” Color Res. Appl. 13, 76–84 (1988).
[CrossRef]

N. Ohta, G. Wyszecki, “Location of the nodes of metameric color stimuli,” Color Res. Appl. 2, 183–186 (1977).
[CrossRef]

N. Ohta, “Intersections of spectral curves of metameric colors,” Color Res. Appl. 12, 85–87 (1987).
[CrossRef]

S. A. Burns, J. B. Cohen, E. N. Kuznetsov, “Multiple metamers: preserving color matches under diverse illuminants,” Color Res. Appl. 14, 16–22 (1989).
[CrossRef]

B. Smith, C. Spiekermann, R. Sember, “Numerical methods for colorimetric calculations: sampling density requirements,” Color Res. Appl. 17, 394–401 (1992).
[CrossRef]

J. Color Appearance (1)

W. A. Thornton, “Matching lights, metamers, and human visual response,” J. Color Appearance 2, 23–29 (1973).

J. Imaging Sci. Technol. (1)

G. D. Finlayson, P. Morovic, “Metamer constrained colour correction,” J. Imaging Sci. Technol. 44, 295–300 (2000).

J. Opt. Soc. Am. (6)

J. Opt. Soc. Am. A (5)

Psychon. Sci. (1)

J. B. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychon. Sci. 1, 369–370 (1964).
[CrossRef]

Remote Sens. Rev. (1)

S. Westland, J. Shaw, H. Owens, “Color statistics of natural and man-made surfaces,” Remote Sens. Rev. 20, 50–55 (2000).

Sci. Am. (1)

E. H. Land, “The retinex theory of color vision,” Sci. Am. 237, 108–128 (1977).
[CrossRef] [PubMed]

Other (9)

G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982).

E. L. Krinov, “Spectral reflectance properties of natural formations,” Tech. Transl. TT-439 (National Research Council of Canada, Ottawa, 1947).

J. B. Fraleigh, R. A. Beauregard, Linear Algebra (Addison-Wesley, Reading, Mass., 1990).

V. Chvátal, Linear Programing (Freeman, New York, 1983).

T. Katriňák, M. Galvanec, E. Gedeonová, J. Smítal, Algebra a Teoretická Aritmetika (Alfa, Bratislava, Czechoslovakia, 1985).

P. Morovic, “Metamer Sets,” Ph.D. thesis (University of East Anglia, Norwich, UK, 2002).

G. H. Golub, C. F. van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, Baltimore, Md., 1996).

F. P. Preparata, M. I. Shamos, Computational Geometry—An Introduction, 2nd ed. (Springer-Verlag, New York, 1988).

G. D. Finlayson, P. Morovic, “Metamer crossovers of infinite metamer sets,” in Proceedings of the IS&T/SID Eighth Color Imaging Conference (The Society of Imaging Science and Technology, Springfield, Va. 2000), pp. 13–17..

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Figures (6)

Fig. 1
Fig. 1

Spectral sensitivities of the CIE 1931 color-matching functions (left) and the spectral power distribution of CIE illuminants D65 and A (right).

Fig. 2
Fig. 2

Macbeth ColorChecker Chart spectral reflectances (left) and its first eight characteristic vectors (right).

Fig. 3
Fig. 3

Particular solutions for patch numbers 13 and 19 (dashed curve, fundamental; solid curve, 3D) and the metameric black basis.

Fig. 4
Fig. 4

Reflectances in M ( χ ̱ ) , where χ ̱ corresponds on the left to the blue patch and to the white patch on the right.

Fig. 5
Fig. 5

CIE x y chromaticity mismatch regions corresponding to metamer sets under CIE illuminant D65 projected to CIE illuminant A. Top left to bottom right: by use of a five- to eight-dimensional linear model representation of surface reflectance.

Fig. 6
Fig. 6

Progressive growth of a mismatch region, as a function of the linear model dimension. 5D–10D, five to ten dimensions.

Equations (43)

Equations on this page are rendered with MathJax. Learn more.

R ( λ ) = R x ( λ ) + R 0 ( λ ) ,
χ = ω R ( λ ) E ( λ ) X ( λ ) d λ ,
χ = ω R x ( λ ) E ( λ ) X ( λ ) d λ ,
0 = ω R 0 ( λ ) E ( λ ) X ( λ ) d λ .
χ 1 = ω R ( λ ) E ( λ ) X 1 ( λ ) d λ ,
χ 2 = ω R ( λ ) E ( λ ) X 2 ( λ ) d λ ,
χ 3 = ω R ( λ ) E ( λ ) X 3 ( λ ) d λ ,
χ ̱ = ω R ( λ ) E ( λ ) X ̱ ( λ ) d λ ,
χ ̱ = X T D ( e ̱ ) r ̱ ,
r ̱ B σ ̱ ,
Λ = X T D ( e ̱ ) B ,
χ ̱ = Λ σ ̱ .
β ̱ 1 = [ 1 , 0 , 0 , , a 1 , , b 1 , , c 1 , , 0 ] ,
β ̱ 2 = [ 0 , 1 , 0 , , a 2 , , b 2 , , c 2 , , 0 ] ,
β ̱ q 3 = [ 0 , 0 , 0 , , a q 3 , , b q 3 , , c q 3 , , 1 ] ,
Q = Λ T ( Λ Λ T ) 1 Λ .
r ̱ = Q r ̱ + Q r ̱ .
min R i * ( λ ) ω [ R i ( λ ) R i * ( λ ) ] 2 d λ ,
ω R i * ( λ ) E ( λ ) X ̱ ( λ ) d λ = χ ̱ * .
min r ̱ f ( r ̱ ) ,
A r ̱ b ̱ .
χ ̱ = Λ σ ̱ ,
σ ̱ = σ ̱ x + σ ̱ o ,
χ ̱ = Λ σ ̱ x ,
0 ̱ = Λ σ ̱ o .
M ( χ ̱ ) = { σ ̱ σ ̱ = σ ̱ x + σ ̱ o : Λ σ ̱ = χ ̱ } .
σ ̱ x = Λ T ( Λ Λ T ) 1 χ ̱ .
σ ̱ x = Λ 1 χ ̱ .
σ ̱ o = Ξ α ̱ ,
σ ̱ = σ ̱ x + Ξ α ̱ .
P = { σ ̱ B σ ̱ 0 ̱ B σ ̱ 1 ̱ } ,
N = { σ ̱ w ̱ 0 ̱ : σ ̱ = S w ̱ i = 1 n w i = 1 } .
M ( χ ̱ ) R .
B σ ̱ 0 ̱ ,
B ( σ ̱ x + Ξ α ̱ ) 0 ̱ ,
B Ξ α ̱ B σ ̱ x .
B Ξ α ̱ 1 ̱ B σ ̱ x ,
A σ ̱ b ̱ .
A ( σ ̱ x + Ξ α ̱ ) b ̱ ,
A Ξ α ̱ b ̱ A σ ̱ x .
Ξ α ̱ s σ ̱ min σ ̱ x ,
Ξ α ̱ σ ̱ max σ ̱ x .

Metrics