Abstract

We present a color-constancy algorithm that uses quantum-catch data from reflected lights to recover surface reflectance functions and illuminant spectral power distributions. The algorithm recovers both surface and light-source spectral properties simultaneously. The method works in all situations that were handled by the earlier two-stage algorithms of Maloney and Wandell [ J. Opt. Soc. Am. A 3, 29 ( 1986)] and D’Zmura and Iverson [ J. Opt. Soc. Am. A 9, 490 ( 1992); J. Opt. Soc. Am. A 10, 2148, 2166 ( 1993); J. Opt. Soc. Am. A 11, 1970 ( 1994)]. In addition, the method handles problems that lie outside the scope of earlier algorithms. Using this method, a trichromatic visual system can recover, when provided adequate information, spectral descriptions of arbitrarily high accuracy for lights and surfaces. We determine conditions under which bilinear models can be used to recover color properties uniquely with the new procedure, and we formulate an algorithm for checking whether a particular bilinear model provides perfect color constancy. This research extends our analysis of linear methods for color constancy begun earlier [ J. Opt. Soc. Am. A 10, 2148, 2166 ( 1993)].

© 1994 Optical Society of America

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References

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  1. 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]
  2. M. D’Zmura, “Color constancy: surface color from changing illumination,” J. Opt. Soc. Am. A 9, 490–493 (1992).
    [Crossref]
  3. 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]
  4. 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]
  5. G. Iverson, M. D’Zmura, “Criteria for color constancy in trichromatic bilinear models,” J. Opt. Soc. Am. A 11, 1970–1975 (1994).
    [Crossref]
  6. D. Katz, The World of Color, R. B. MacLeod, C. W. Fox, trans. (Kegan Paul, Trench, Trubner, London, 1935).
  7. J. Beck, Surface Color Perception (Cornell U. Press, Ithaca, N.Y., 1972).
  8. E. H. Land, “Recent advances in retinex theory,” Vision Res. 26, 7–21 (1986).
    [Crossref] [PubMed]
  9. P. Sallstrom, “Colour and physics: some remarks concerning the physical aspects of human colour vision,” University of Stockholm Institute of Physics Rep. 73–09 (University of Stockholm, Stockholm, 1973).
  10. M. H. Brill, “A device performing illuminant-invariant assessment of chromatic relations,”J. Theor. Biol. 71, 473–478 (1978).
    [Crossref] [PubMed]
  11. M. H. Brill, “Further features of the illuminant-invariant trichromatic photosensor,”J. Theor. Biol. 78, 305–308 (1979).
    [Crossref] [PubMed]
  12. G. Buchsbaum, “A spatial processor model for object colour perception,”J. Franklin Inst. 310, 1–26 (1980).
    [Crossref]
  13. L. T. Maloney, “Computational approaches to color constancy,” Stanford Applied Psychology Laboratory Tech. Rep. 1985–01 (Stanford University, Palo Alto, Calif., 1985).
  14. A. Hurlbert, “Formal connections between lightness algorithms,” J. Opt. Soc. Am. A 3, 1684–1693 (1986).
    [Crossref] [PubMed]
  15. M. D’Zmura, P. Lennie, “Mechanisms of color constancy,” J. Opt. Soc. Am. A 3, 1662–1672 (1986).
    [Crossref]
  16. H.-C. Lee, “Method for computing the scene-illuminant chromaticity from specular highlights,” J. Opt. Soc. Am. A 3, 1694–1699 (1986).
    [Crossref] [PubMed]
  17. D. H. Brainard, B. A. Wandell, W. B. Cowan, “Black light: how sensors filter spectral variation of the illuminant,”IEEE Trans. Biomed. Eng. 36, 140–149 (1989).
    [Crossref] [PubMed]
  18. J. Rubner, K. Schulten, “A regularized approach to color constancy,” Biol. Cybern. 61, 29–36 (1989).
    [Crossref] [PubMed]
  19. D. A. Forsyth, “A novel algorithm for color constancy,” Int. J. Comput. Vis. 5, 5–36 (1990).
    [Crossref]
  20. M. S. Drew, B. V. Funt, “Variational approach to interreflection in color images,” J. Opt. Soc. Am. A 9, 1255–1265 (1992).
    [Crossref]
  21. M. D’Zmura, G. Iverson, “Color constancy: feasibility and recovery,” Invest. Ophthalmol. Vis. Sci. Suppl. 34, 748 (1993).
  22. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C. The Art of Scientific Computing (Cambridge U. Press, New York, 1988).
  23. M. D’Zmura, G. Iverson, “Color constancy. III. General linear recovery of spectral descriptions for lights and surfaces,” Institute for Mathematical Behavioral Sciences Tech. Rep. MBS 93-38 (University of California, Irvine, Irvine, Calif., 1993).
  24. 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]
  25. 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]
  26. J. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychonom. Sci. 1, 369–370 (1964).
  27. E. R. Dixon, “Spectral distribution of Australian daylight,”J. Opt. Soc. Am. 68, 437–450 (1978).
    [Crossref]
  28. J. P. S. Parkkinen, J. Hallikainen, T. Jaaskelainen, “Characteristic spectra of Munsell colors,” J. Opt. Soc. Am. A 6, 318–322 (1989).
    [Crossref]
  29. L. M. Hurvich, D. Jameson, “Some quantitative aspects of an opponent-color theory. II. Brightness, saturation, and hue in normal and dichromatic vision,”J. Opt. Soc. Am. 45, 602–616 (1955).
    [Crossref] [PubMed]
  30. G. Wyszecki, W. S. Stiles, Color Science. Concepts and Methods, Quantitative Data and Formulas, 2nd ed. (Wiley, New York, 1982).
  31. A. Stockman, D. I. A. MacLeod, N. E. Johnson, “Spectral sensitivities of the human cones,” J. Opt. Soc. Am. A 10, 2491–2521 (1993).
    [Crossref]
  32. S. Lang, Linear Algebra, 2nd ed. (Addison-Wesley, Reading, Mass., 1971).

1994 (1)

1993 (4)

1992 (2)

1990 (1)

D. A. Forsyth, “A novel algorithm for color constancy,” Int. J. Comput. Vis. 5, 5–36 (1990).
[Crossref]

1989 (3)

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

D. H. Brainard, B. A. Wandell, W. B. Cowan, “Black light: how sensors filter spectral variation of the illuminant,”IEEE Trans. Biomed. Eng. 36, 140–149 (1989).
[Crossref] [PubMed]

J. Rubner, K. Schulten, “A regularized approach to color constancy,” Biol. Cybern. 61, 29–36 (1989).
[Crossref] [PubMed]

1986 (5)

1980 (1)

G. Buchsbaum, “A spatial processor model for object colour perception,”J. Franklin Inst. 310, 1–26 (1980).
[Crossref]

1979 (1)

M. H. Brill, “Further features of the illuminant-invariant trichromatic photosensor,”J. Theor. Biol. 78, 305–308 (1979).
[Crossref] [PubMed]

1978 (2)

M. H. Brill, “A device performing illuminant-invariant assessment of chromatic relations,”J. Theor. Biol. 71, 473–478 (1978).
[Crossref] [PubMed]

E. R. Dixon, “Spectral distribution of Australian daylight,”J. Opt. Soc. Am. 68, 437–450 (1978).
[Crossref]

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)

Beck, J.

J. Beck, Surface Color Perception (Cornell U. Press, Ithaca, N.Y., 1972).

Brainard, D. H.

D. H. Brainard, B. A. Wandell, W. B. Cowan, “Black light: how sensors filter spectral variation of the illuminant,”IEEE Trans. Biomed. Eng. 36, 140–149 (1989).
[Crossref] [PubMed]

Brill, M. H.

M. H. Brill, “Further features of the illuminant-invariant trichromatic photosensor,”J. Theor. Biol. 78, 305–308 (1979).
[Crossref] [PubMed]

M. H. Brill, “A device performing illuminant-invariant assessment of chromatic relations,”J. Theor. Biol. 71, 473–478 (1978).
[Crossref] [PubMed]

Buchsbaum, G.

G. Buchsbaum, “A spatial processor model for object colour perception,”J. Franklin Inst. 310, 1–26 (1980).
[Crossref]

Cohen, J.

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

Cowan, W. B.

D. H. Brainard, B. A. Wandell, W. B. Cowan, “Black light: how sensors filter spectral variation of the illuminant,”IEEE Trans. Biomed. Eng. 36, 140–149 (1989).
[Crossref] [PubMed]

D’Zmura, M.

Dixon, E. R.

Drew, M. S.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C. The Art of Scientific Computing (Cambridge U. Press, New York, 1988).

Forsyth, D. A.

D. A. Forsyth, “A novel algorithm for color constancy,” Int. J. Comput. Vis. 5, 5–36 (1990).
[Crossref]

Funt, B. V.

Hallikainen, J.

Hurlbert, A.

Hurvich, L. M.

Iverson, G.

G. Iverson, M. D’Zmura, “Criteria for color constancy in trichromatic bilinear models,” J. Opt. Soc. Am. A 11, 1970–1975 (1994).
[Crossref]

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]

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]

M. D’Zmura, G. Iverson, “Color constancy: feasibility and recovery,” Invest. Ophthalmol. Vis. Sci. Suppl. 34, 748 (1993).

M. D’Zmura, G. Iverson, “Color constancy. III. General linear recovery of spectral descriptions for lights and surfaces,” Institute for Mathematical Behavioral Sciences Tech. Rep. MBS 93-38 (University of California, Irvine, Irvine, Calif., 1993).

Jaaskelainen, T.

Jameson, D.

Johnson, N. E.

Judd, D. B.

Katz, D.

D. Katz, The World of Color, R. B. MacLeod, C. W. Fox, trans. (Kegan Paul, Trench, Trubner, London, 1935).

Land, E. H.

E. H. Land, “Recent advances in retinex theory,” Vision Res. 26, 7–21 (1986).
[Crossref] [PubMed]

Lang, S.

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

Lee, H.-C.

Lennie, P.

MacAdam, D. L.

MacLeod, D. I. A.

Maloney, L. T.

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]

L. T. Maloney, “Computational approaches to color constancy,” Stanford Applied Psychology Laboratory Tech. Rep. 1985–01 (Stanford University, Palo Alto, Calif., 1985).

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]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C. The Art of Scientific Computing (Cambridge U. Press, New York, 1988).

Rubner, J.

J. Rubner, K. Schulten, “A regularized approach to color constancy,” Biol. Cybern. 61, 29–36 (1989).
[Crossref] [PubMed]

Sallstrom, P.

P. Sallstrom, “Colour and physics: some remarks concerning the physical aspects of human colour vision,” University of Stockholm Institute of Physics Rep. 73–09 (University of Stockholm, Stockholm, 1973).

Schulten, K.

J. Rubner, K. Schulten, “A regularized approach to color constancy,” Biol. Cybern. 61, 29–36 (1989).
[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]

Stiles, W. S.

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

Stockman, A.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C. The Art of Scientific Computing (Cambridge U. Press, New York, 1988).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C. The Art of Scientific Computing (Cambridge U. Press, New York, 1988).

Wandell, B. A.

D. H. Brainard, B. A. Wandell, W. B. Cowan, “Black light: how sensors filter spectral variation of the illuminant,”IEEE Trans. Biomed. Eng. 36, 140–149 (1989).
[Crossref] [PubMed]

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]

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).

Biol. Cybern. (1)

J. Rubner, K. Schulten, “A regularized approach to color constancy,” Biol. Cybern. 61, 29–36 (1989).
[Crossref] [PubMed]

IEEE Trans. Biomed. Eng. (1)

D. H. Brainard, B. A. Wandell, W. B. Cowan, “Black light: how sensors filter spectral variation of the illuminant,”IEEE Trans. Biomed. Eng. 36, 140–149 (1989).
[Crossref] [PubMed]

Int. J. Comput. Vis. (1)

D. A. Forsyth, “A novel algorithm for color constancy,” Int. J. Comput. Vis. 5, 5–36 (1990).
[Crossref]

Invest. Ophthalmol. Vis. Sci. Suppl. (1)

M. D’Zmura, G. Iverson, “Color constancy: feasibility and recovery,” Invest. Ophthalmol. Vis. Sci. Suppl. 34, 748 (1993).

J. Franklin Inst. (1)

G. Buchsbaum, “A spatial processor model for object colour perception,”J. Franklin Inst. 310, 1–26 (1980).
[Crossref]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (11)

M. D’Zmura, “Color constancy: surface color from changing illumination,” J. Opt. Soc. Am. A 9, 490–493 (1992).
[Crossref]

G. Iverson, M. D’Zmura, “Criteria for color constancy in trichromatic bilinear models,” J. Opt. Soc. Am. A 11, 1970–1975 (1994).
[Crossref]

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]

M. D’Zmura, P. Lennie, “Mechanisms of color constancy,” J. Opt. Soc. Am. A 3, 1662–1672 (1986).
[Crossref]

A. Hurlbert, “Formal connections between lightness algorithms,” J. Opt. Soc. Am. A 3, 1684–1693 (1986).
[Crossref] [PubMed]

H.-C. Lee, “Method for computing the scene-illuminant chromaticity from specular highlights,” J. Opt. Soc. Am. A 3, 1694–1699 (1986).
[Crossref] [PubMed]

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

M. S. Drew, B. V. Funt, “Variational approach to interreflection in color images,” J. Opt. Soc. Am. A 9, 1255–1265 (1992).
[Crossref]

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]

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]

A. Stockman, D. I. A. MacLeod, N. E. Johnson, “Spectral sensitivities of the human cones,” J. Opt. Soc. Am. A 10, 2491–2521 (1993).
[Crossref]

J. Theor. Biol. (2)

M. H. Brill, “A device performing illuminant-invariant assessment of chromatic relations,”J. Theor. Biol. 71, 473–478 (1978).
[Crossref] [PubMed]

M. H. Brill, “Further features of the illuminant-invariant trichromatic photosensor,”J. Theor. Biol. 78, 305–308 (1979).
[Crossref] [PubMed]

Psychonom. Sci. (1)

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

Vision Res. (2)

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]

E. H. Land, “Recent advances in retinex theory,” Vision Res. 26, 7–21 (1986).
[Crossref] [PubMed]

Other (8)

P. Sallstrom, “Colour and physics: some remarks concerning the physical aspects of human colour vision,” University of Stockholm Institute of Physics Rep. 73–09 (University of Stockholm, Stockholm, 1973).

D. Katz, The World of Color, R. B. MacLeod, C. W. Fox, trans. (Kegan Paul, Trench, Trubner, London, 1935).

J. Beck, Surface Color Perception (Cornell U. Press, Ithaca, N.Y., 1972).

L. T. Maloney, “Computational approaches to color constancy,” Stanford Applied Psychology Laboratory Tech. Rep. 1985–01 (Stanford University, Palo Alto, Calif., 1985).

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

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

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C. The Art of Scientific Computing (Cambridge U. Press, New York, 1988).

M. D’Zmura, G. Iverson, “Color constancy. III. General linear recovery of spectral descriptions for lights and surfaces,” Institute for Mathematical Behavioral Sciences Tech. Rep. MBS 93-38 (University of California, Irvine, Irvine, Calif., 1993).

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Figures (5)

Fig. 1
Fig. 1

Results for dichromatic bilinear models: A, the case in which the illumination model has dimension three (p = 2, m = 3 or by transposition p = 2, n = 3); B, the case in which the illumination model has dimension four (p = 2, m = 4 or by transposition p = 2, n = 4). The format is like that used in Fig. 1 of Ref. 4. The horizontal axes mark the dimension n of the reflectance model, which is taken equal to the number s of surfaces, and the vertical axes mark the number v of views. The solid lines divide cases that satisfy the necessary conditions pnvn2 + vm − 1 [inequality (15)] and p + v > m [inequality (16)]. The number of quantum-catch data Q = svp and the number D = sn + vm of spectral descriptors to be recovered are indicated for each problem by the bracketed pair [Q/D] beneath the appropriate point. The dotted lines divide cases that satisfy the necessary condition for the sufficient test provided by the first nonlinear model-check algorithm, namely, that E1 ≥ U1 [Eqs. (A5) and (A11) of Ref. 23]. The pair E1/U1 is shown directly beneath each point. In cases where m = v (top rows), a necessary and sufficient model check examines the rank of a system of npm equations in n2 + m2 unknowns [Eq. (33)]; these two values are shown directly beneath such points. An X marks problems for which recovery fails totally. The circled points mark problems for which there are perfect recovery algorithms. Parameters for transposed problems are indicated in parentheses at the tops of the diagrams and along their axes.

Fig. 2
Fig. 2

Spectra of model check matrices for exemplary dichromatic bilinear models. Plotted are the ordered singular values of the model check matrices for exemplary bilinear models that combine the Smith–Pokorny protanope24 the CIE daylight basis for illumination,25,30 and the Fourier basis for reflectance (see Table 2). The bottom two graphs pertain to Fig. 1A (p = 2, m = 3) and are spectra for matrices of models with parameters (2 3 3 3 3) (bottom, with a kernel of dimension three) and (2 3 4 3 4) (second from bottom, with a kernel of dimension one). To pass this model check, a matrix must have a kernel of dimension one, so that the results indicate failure for (2 3 3 3 3) and success for (2 3 4 3 4). The top two spectra pertain to Fig. 1B (p = 2, m = 4) and are from model check matrices for models with parameters (2 4 4 4 4) (second to top), with a kernel of dimension four, so failing the check, and (2 4 5 4 5) (top), with a kernel of dimension one, so passing the check.

Fig. 3
Fig. 3

Results for trichromatic bilinear models: A, the case of four-dimensional models of illumination (p = 3, m = 4, or by transposition p = 3, n = 4); B, the case of five-dimensional models of illumination (p = 3, m = 5, or by transposition p = 3, n = 5); C, the case of six-dimensional models of illumination (p = 3, m = 6, or by transposition p = 3, n = 6). The circles mark problems that are shown by the model-check algorithm to support perfect recovery procedures. See the caption for Fig. 1 and the text for further details.

Fig. 4
Fig. 4

Spectra of model check matrices for exemplary trichromatic bilinear models. The Smith–Pokorny24 trichromat and the CIE daylight basis25,30 were used in combination with Fourier reflectance models. The spectra shown are, from bottom to top, for the problems (3 4 4 4 4), (3 4 5 4 5), (3 4 6 4 6), (3 5 5 5 5), (3 5 6 5 6), and (3 6 6 6 6). The spectra show that the corresponding model check matrices had one-dimensional kernels, so passing the check. See the text for further discussion.

Fig. 5
Fig. 5

Spectra of model check matrices for exemplary trichromatic bilinear models with parameters (3 c c c c) for 3 ≤ c ≤ 31, c odd. Plotted on a log axis are the ordered singular values of the model check matrices for these models. We scaled the spectra to stagger the maximal singular values along the vertical axis at half-log-unit intervals. The parameters of the models whose spectra are shown are, from bottom to top, (3 3 3 3 3), (3 5 5 5 5), (3 7 7 7 7), (3 9 9 9 9), (3 11 11 11 11), (3 13 13 13 13), (3 15 15 15 15), (3 17 17 17 17), (3 19 19 19 19), (3 21 21 21 21), (3 23 23 23 23), (3 25 25 2525), (3 27 27 27 27), (3 29 29 29 29), and (3 31 31 31 31). Each has a kernel of dimension one and so passes the necessary and sufficient model check. See the text for further discussion.

Tables (2)

Tables Icon

Table 1 List of Symbols

Tables Icon

Table 2 Tested Bilinear Model Components

Equations (47)

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

( B j ) k i = Q k ( λ ) A i ( λ ) R j ( λ ) d λ ,             j = 1 , , n ,
q t w k = j = 1 n i = 1 m r t j ( B j ) k i a w i .
d t w = j = 1 n r t j B j a w ,
C j = diag [ B j , , B j ] ,
d t = j = 1 n r t j C j a ,             t = 1 , , s .
P = R - 1 .
t = 1 n ρ j t d t - C j a = 0 ,             j = 1 , , n .
ρ j = [ ρ j 1 ρ j n ] T ,             j = 1 , , n ,
ρ = [ ρ 1 T ρ n T ] T .
δ = [ ρ a ] .
D = [ d 1 d n ] .
F = [ D 0 0 - C 1 0 D 0 - C 2 0 0 D - C n ] ,
F δ = 0 .
dim [ ker ( F ) ] = 1.
p n v n 2 + m v - 1.
p + v > m .
a j = 1 n ker ( C j ) ,
j = 1 n r j 0 C j a 0 = 0 .
j = 1 n r j 0 B j A 0 = 0 ,
e = ( p m - v + 1 ) ( m m - v + 1 ) .
r j 0 = 0 ,             j = 1 , , n .
u = ( n + m - v m - v + 1 ) .
p n v > p v + n 2 - 1 ,
p v ( n - 1 ) > ( n - 1 ) ( n + 1 ) .
p v > n + 1.
D t = j = 1 n r t j B j A ,             t = 1 , , n ,
D t = i = 1 n s t i B i Z = j = 1 n r t j B j A ,             t = 1 , , s = n .
B i Z = j = 1 n e i j B j A ,             i = 1 , , n ,
e i j = t = 1 n σ i t r t j ,
B i H = j = 1 n e i j B j ,             i = 1 , , n ,
H = ZA - 1 .
j = 1 n e 1 j B j - B 1 H = 0 , j = 1 n e n j B j - B n H = 0 .
L ω = 0 ,
ω = [ e 11 e 1 n e n 1 e n n             h 11 h 1 m h m 1 h m m ] T .
X = [ ( B 1 ) 11 ( B n ) 11 ( B 1 ) 1 m ( B n ) 1 m ( B 1 ) p 1 ( B n ) p 1 ( B 1 ) p m ( B n ) p m ] ,
Y i = [ [ - ( B i ) 11 - ( B i ) 1 m ] 0 m 0 [ - ( B i ) 11 - ( B i ) 1 m ] [ - ( B i ) p 1 - ( B i ) p m ] 0 m 0 [ - ( B i ) p 1 - ( B i ) p m ] ] .
L = [ X 0 Y 1 n 0 X Y n ] .
dim [ ker ( L ) ] = 1.
e 11 - e j j = 0 for j = 2 , , n , e i j = 0 for i j , h 11 - h j j = 0 for j = 2 , , m , h i j = 0 for i j , e 11 = h 11 .
Δ k = R β k A ,             k = 1 , 2 , 3 ,
R β k A = S β k Z ,             k = 1 , 2 , 3.
E β k = β k H ,             k = 1 , 2 , 3 ,
β k - 1 E β k = H ,             k = 1 , 2 , 3.
β k - 1 E β k = β l - 1 E β l ,             k , l = 1 , 2 , 3.
E G k l = G k l E ,             k , l = 1 , 2 , 3 ,
[ E , G 21 ] = 0 = [ E , G 31 ] ,
G 21 = [ μ 1 0 0 μ c ] .

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