Abstract

We examine conditions under which the spectral properties of lights and surfaces may be recovered by a trichromatic visual system that uses bilinear models. We derive criteria for perfect recovery, formulated in terms of invariant properties of model matrices, for situations in which either two or three lights are shone sequentially on a set of surfaces.

© 1994 Optical Society of America

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References

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  1. M. D’Zmura, “Color constancy: surface color from changing illumination,”J. Opt. Soc. Am. A 9, 490–493 (1992).
    [CrossRef]
  2. M. D’Zmura, G. Iverson, “Color constancy. I. Basic theory of two-stage linear recovery of spectral descriptions for lights and surfaces,” J. Opt. Soc. Am. A 10, 2148–2165 (1993).
    [CrossRef]
  3. M. D’Zmura, G. Iverson, “Color constancy. II. Results for two-stage linear recovery of spectral descriptions for lights and surfaces,” J. Opt. Soc. Am. A 10, 2166–2180 (1993).
    [CrossRef]
  4. D. H. Marimont, B. A. Wandell, “Linear models of surface and illuminant spectra,” J. Opt. Soc. Am. A 9, 1905–1913 (1992).
    [CrossRef] [PubMed]
  5. G. J. Iverson, M. D’Zmura, “Criteria for color constancy in trichromatic bilinear models,” presented at the Twenty-Fifth Annual Mathematical Psychology Meeting, Stanford University, Palo Alto, Calif., August 1992.
  6. J. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychonom. Sci. 1, 369–370 (1964).
  7. J. P. S. Parkkinen, J. Hallikainen, T. Jaaskelainen, “Characteristic spectra of Munsell colors,” J. Opt. Soc. Am. A 6, 318–322 (1989).
    [CrossRef]
  8. D. B. Judd, D. L. MacAdam, G. Wyszecki, “Spectral distribution of typical daylight as a function of correlated color temperature,” J. Opt. Soc. Am. 54, 1031–1040 (1964).
    [CrossRef]
  9. E. R. Dixon, “Spectral distribution of Australian daylight,”J. Opt. Soc. Am. 68, 437–450 (1978).
    [CrossRef]
  10. 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]
  11. Note that our notion of decomposability is often referred to in the linear algebra literature as reducibility, e.g., P. R. Halmos, Finite-Dimensional Vector Spaces, 2nd ed. (Van Nostrand, Princeton, N.J., 1958),and S. K. Berberian, Linear Algebra (Oxford U. Press, New York, 1992).Our usage agrees with terminology used in the theory of group representations, e.g., M. Hamermesh, Group Theory and Its Application to Physical Problems (Addison-Wesley, Reading, Mass., 1962).
  12. S. Lang, Linear Algebra, 2nd ed. (Addison-Wesley, Reading, Mass., 1971).
  13. E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, A. DuCroz, S. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen, lapack User’s Guide (SIAM, Philadelphia, 1992).
  14. V. C. Smith, J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
    [CrossRef] [PubMed]
  15. L. M. Hurvich, D. Jameson, “Some quantitative aspects of an opponent-colors theory. II. Brightness, saturation, and hue in normal and dichromatic vision,”J. Opt. Soc. Am. 45, 602–616 (1955).
    [CrossRef] [PubMed]
  16. G. Wyszecki, W. S. Stiles, Color Science. Concepts and Methods, Quantitative Data and Formulas, 2nd ed. (Wiley, New York, 1982).

1993 (2)

1992 (2)

1989 (1)

1986 (1)

1978 (1)

1975 (1)

V. C. Smith, J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
[CrossRef] [PubMed]

1964 (2)

J. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychonom. Sci. 1, 369–370 (1964).

D. B. Judd, D. L. MacAdam, G. Wyszecki, “Spectral distribution of typical daylight as a function of correlated color temperature,” J. Opt. Soc. Am. 54, 1031–1040 (1964).
[CrossRef]

1955 (1)

Anderson, E.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, A. DuCroz, S. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen, lapack User’s Guide (SIAM, Philadelphia, 1992).

Bai, Z.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, A. DuCroz, S. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen, lapack User’s Guide (SIAM, Philadelphia, 1992).

Bischof, C.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, A. DuCroz, S. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen, lapack User’s Guide (SIAM, Philadelphia, 1992).

Cohen, J.

J. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychonom. Sci. 1, 369–370 (1964).

D’Zmura, M.

Demmel, J.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, A. DuCroz, S. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen, lapack User’s Guide (SIAM, Philadelphia, 1992).

Dixon, E. R.

Dongarra, J.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, A. DuCroz, S. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen, lapack User’s Guide (SIAM, Philadelphia, 1992).

DuCroz, A.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, A. DuCroz, S. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen, lapack User’s Guide (SIAM, Philadelphia, 1992).

Greenbaum, S.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, A. DuCroz, S. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen, lapack User’s Guide (SIAM, Philadelphia, 1992).

Hallikainen, J.

Halmos, P. R.

Note that our notion of decomposability is often referred to in the linear algebra literature as reducibility, e.g., P. R. Halmos, Finite-Dimensional Vector Spaces, 2nd ed. (Van Nostrand, Princeton, N.J., 1958),and S. K. Berberian, Linear Algebra (Oxford U. Press, New York, 1992).Our usage agrees with terminology used in the theory of group representations, e.g., M. Hamermesh, Group Theory and Its Application to Physical Problems (Addison-Wesley, Reading, Mass., 1962).

Hammarling, S.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, A. DuCroz, S. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen, lapack User’s Guide (SIAM, Philadelphia, 1992).

Hurvich, L. M.

Iverson, G.

Iverson, G. J.

G. J. Iverson, M. D’Zmura, “Criteria for color constancy in trichromatic bilinear models,” presented at the Twenty-Fifth Annual Mathematical Psychology Meeting, Stanford University, Palo Alto, Calif., August 1992.

Jaaskelainen, T.

Jameson, D.

Judd, D. B.

Lang, S.

S. Lang, Linear Algebra, 2nd ed. (Addison-Wesley, Reading, Mass., 1971).

MacAdam, D. L.

Maloney, L. T.

Marimont, D. H.

McKenney, A.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, A. DuCroz, S. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen, lapack User’s Guide (SIAM, Philadelphia, 1992).

Ostrouchov, S.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, A. DuCroz, S. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen, lapack User’s Guide (SIAM, Philadelphia, 1992).

Parkkinen, J. P. S.

Pokorny, J.

V. C. Smith, J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
[CrossRef] [PubMed]

Smith, V. C.

V. C. Smith, J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
[CrossRef] [PubMed]

Sorensen, D.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, A. DuCroz, S. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen, lapack User’s Guide (SIAM, Philadelphia, 1992).

Stiles, W. S.

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

Wandell, B. A.

Wyszecki, G.

D. B. Judd, D. L. MacAdam, G. Wyszecki, “Spectral distribution of typical daylight as a function of correlated color temperature,” J. Opt. Soc. Am. 54, 1031–1040 (1964).
[CrossRef]

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

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (6)

Psychonom. Sci. (1)

J. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychonom. Sci. 1, 369–370 (1964).

Vision Res. (1)

V. C. Smith, J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
[CrossRef] [PubMed]

Other (5)

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

G. J. Iverson, M. D’Zmura, “Criteria for color constancy in trichromatic bilinear models,” presented at the Twenty-Fifth Annual Mathematical Psychology Meeting, Stanford University, Palo Alto, Calif., August 1992.

Note that our notion of decomposability is often referred to in the linear algebra literature as reducibility, e.g., P. R. Halmos, Finite-Dimensional Vector Spaces, 2nd ed. (Van Nostrand, Princeton, N.J., 1958),and S. K. Berberian, Linear Algebra (Oxford U. Press, New York, 1992).Our usage agrees with terminology used in the theory of group representations, e.g., M. Hamermesh, Group Theory and Its Application to Physical Problems (Addison-Wesley, Reading, Mass., 1962).

S. Lang, Linear Algebra, 2nd ed. (Addison-Wesley, Reading, Mass., 1971).

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, A. DuCroz, S. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen, lapack User’s Guide (SIAM, Philadelphia, 1992).

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Tables (1)

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Table 1 Tested Bilinear Model Components

Equations (40)

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R ( λ ) = j = 1 3 r j R j ( λ ) ,
r j = R ( λ ) R j ( λ ) d λ .
A ( λ ) = i = 1 3 a i A i ( λ ) ,
a i = A ( λ ) A i ( λ ) d λ .
b j i k = R j ( λ ) A i ( λ ) Q k ( λ ) d λ ,
q k = j = 1 3 i = 1 3 r j b j i k a i for k = 1 , 2 , 3 .
q t w k = j = 1 3 i = 1 3 r t j b j i k a i w for t = 1 , 2 , 3 , w = 1 , , υ , k = 1 , 2 , 3 .
Δ k = R β k A , k = 1 , 2 , 3 .
Δ k = R β k A = S β k z , k = 1 , 2 , 3 .
β k 1 S 1 R β k A = z , k = 1 , 2 , 3 .
( β k 1 E β k β 1 1 E β 1 ) A = 0 , k = 2 , 3 .
[ G k 1 , E ] β 1 A = 0 , k = 2 , 3 ,
E = c I ,
[ x x 0 x x 0 x x x ] , [ x 0 0 x x x x x x ] ,
[ x 0 0 0 x x 0 x x ] .
G 21 = [ λ 1 0 0 0 λ 2 0 0 0 λ 3 ] ,
[ G 21 , E ] = 0 = [ G 31 , E ] .
[ G 21 , E ] = [ 0 ( λ 1 λ 2 ) e 12 ( λ 1 λ 3 ) e 13 ( λ 2 λ 1 ) e 21 0 ( λ 2 λ 3 ) e 23 ( λ 3 λ 1 ) e 31 ( λ 3 λ 2 ) e 32 0 ] ,
[ G 31 , E ] = [ 0 ( e 22 e 11 ) g 12 ( e 33 e 11 ) g 13 ( e 11 e 22 ) g 21 0 ( e 33 e 22 ) g 23 ( e 11 e 33 ) g 31 ( e 22 e 33 ) g 32 0 ] ,
( a ) e 11 = e 22 = e 33
( b 1 ) e 11 e 22 and e 11 e 33 , or ( b 2 ) e 11 e 22 and e 22 e 33 , or ( b 3 ) e 11 e 33 and e 22 e 33 .
E = [ e 11 0 0 0 e 22 0 0 0 e 22 ]
[ G 21 , E ] = z α T ,
[ G 31 , E ] = z * α T ,
[ G 21 , E ] = [ z 1 α 1 z 1 α 2 z 1 α 3 z 2 α 1 z 2 α 2 z 2 α 3 z 3 α 1 z 3 α 2 z 3 α 3 ] ,
E = [ e 11 0 0 0 e 22 0 e 31 e 32 e 33 ] ,
[ G 21 , E ] = [ 0 0 0 0 0 0 z 3 α 1 z 3 α 2 0 ] = [ 0 0 0 0 0 0 ( λ 3 λ 1 ) e 31 ( λ 3 λ 2 ) e 32 0 ] .
[ G 31 , E ] = [ z 1 * α 1 z 1 * α 2 0 z 2 * α 1 z 2 * α 2 0 z 3 * α 1 z 3 * α 2 0 ] .
[ G 31 , E ] z = z * α T z = 0 .
[ G 31 , E ] = [ e 31 g 13 e 32 g 13 0 e 31 g 23 e 32 g 23 0 e 31 ( g 33 g 11 ) e 32 g 21 e 32 ( g 33 g 22 ) e 31 g 12 e 31 g 13 + e 32 g 23 ] = [ z 1 * α 1 z 1 * α 2 0 z 2 * α 1 z 2 * α 2 0 z 3 * α 1 z 3 * α 2 0 ] .
E = [ e 11 0 0 e 21 e 22 0 e 31 0 e 33 ] .
[ G 21 , E ] = [ 0 0 0 z 2 0 0 z 3 0 0 ] .
[ G 31 , E ] = [ e 21 g 12 + e 31 g 13 ( e 22 e 11 ) g 12 ( e 33 e 11 ) g 13 x e 21 g 12 e 21 g 13 + ( e 33 e 22 ) g 23 x e 31 g 12 + ( e 22 e 33 ) g 32 e 31 g 13 ]
= [ z 1 * 0 0 z 2 * 0 0 z 3 * 0 0 ] ,
G 31 = [ g 11 0 0 g 21 g 22 g 23 g 31 g 32 g 33 ] .
E = [ e 11 0 0 0 e 33 0 0 0 e 33 ] .
[ 0 0 0 x 0 0 x 0 0 ] .
G 31 = [ g 11 g 12 0 g 21 g 22 0 g 31 g 32 g 33 ] .
E = [ e 11 0 0 0 e 11 0 0 0 e 33 ] .
[ G 31 , E ] = [ 0 0 0 0 0 0 g 31 ( e 33 e 11 ) g 32 ( e 33 e 11 ) 0 ] .

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