Abstract

Our analysis of color constancy in a companion paper [ J. Opt. Soc. Am A 10, 2148 ( 1993)] provided an algorithm that lets one test how well linear color constancy schemes work. Here we present the results of applying the algorithm to a large parametric class of color constancy problems involving bilinear models that relate photoreceptoral spectral sensitivities, surface reflectance functions, and illuminant spectral power distributions. These results, supported by simulation and further analysis, provide a detailed classification of two-stage linear methods for recovering the spectral properties of reflectances and illuminants from reflected lights.

© 1993 Optical Society of America

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References

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    [CrossRef]
  2. 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]
  3. M. D’Zmura, “Color constancy: surface color from changing illumination,” J. Opt. Soc. Am. A 9, 490–493 (1992).
    [CrossRef]
  4. M. D’Zmura, G. Iverson, “Color structure from chromatic motion,” in Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 51.
  5. 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 (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).
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  19. G. Iverson, M. D’Zmura, “Criteria for color constancy in trichromatic bilinear models,” UC Irvine Institute for Mathematical Behavioral Sciences Tech. Rep. 93-18 (University of California, Irvine, Irvine, Calif., 1993).
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  27. D. B. Judd, “Hue, saturation and lightness of surface colors with chromatic illumination,”J. Opt. Soc. Am. 30, 2–32 (1940).
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  30. D. Brainard, B. A. Wandell, “An analysis of the retinex theory of color vision,” J. Opt. Soc. Am. A 3, 1651–1661 (1986).
    [CrossRef] [PubMed]
  31. D. I. A. MacLeod, “Receptoral constraints on colour appearance,” in Central and Peripheral Mechanisms of Colour Vision, D. Ottoson, S. Zeki, eds. (Macmillan, London, 1985), pp. 103–116.
  32. M. D’Zmura, P. Lennie, “Mechanisms of color constancy,” J. Opt. Soc. Am. A 3, 1662–1672 (1986).
    [CrossRef]
  33. M. Fairchild, P. Lennie, “Chromatic adaptation to natural and incandescent illuminants,” Vision Res. 32, 2077–2085 (1992).
    [CrossRef] [PubMed]
  34. S. Ahn, D. I. A. MacLeod, “Adaptation in the chromatic and luminance channels,” Invest. Ophthalmol. Vis. Sci. Suppl. 31, 109 (1990).
  35. E. H. Adelson, J. Bergen, “Spatiotemporal energy models for the perception of motion,” J. Opt. Soc. Am. A 2, 284–299 (1985).
    [CrossRef] [PubMed]
  36. E. H. Adelson, J. Bergen, “The plenoptic function and the elements of early vision,” in Computational Models of Visual Processing, M. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass., 1991), pp. 3–20.
  37. C. R. Michael, “Color vision mechanisms in monkey striate cortex: dual-opponent cells with concentric receptive fields,” J. Neurophysiol. 41, 572–588 (1978).
    [PubMed]
  38. P. Lennie, J. Krauskopf, G. Sclar, “Chromatic mechanisms in striate cortex of macaque,” J. Neurosci. 10, 649–669 (1990).
    [PubMed]
  39. D. Y. Ts’o, C. D. Gilbert, “The organization of chromatic and spatial interactions in the primate striate cortex,” J. Neurosci. 8, 1712–1727 (1988).
  40. M. S. Livingstone, D. H. Hubel, “Anatomy and physiology of a color system in primate primary visual cortex,” J. Neurosci. 4, 309–356 (1984).
    [PubMed]
  41. S. M. Zeki, “Colour coding in the cerebral cortex: the reaction of cells in monkey visual cortex to wavelengths and colours,” Neuroscience 9, 741–765 (1983).
    [CrossRef] [PubMed]
  42. S. M. Zeki, “Colour coding in the cerebral cortex: the response of wavelength-selective and colour-coded cells in monkey visual cortex to changes in wavelength composition,” Neuroscience 9, 767–781 (1983).
    [CrossRef] [PubMed]
  43. H. M. Wild, S. R. Butler, D. Carden, J. J. Kulikowski, “Primate cortical area V4 important for colour constancy but not wavelength discrimination,” Nature 313, 133–135 (1985).
    [CrossRef]
  44. C. A. Heywood, A. Cowey, “On the role of cortical area V4 in the discrimination of hue and pattern in macaque monkeys,” J. Neurosci. 7, 2601–2617 (1987).
    [PubMed]
  45. P. E. Haenny, P. H. Schiller, “State dependent activity in monkey visual cortex—I. Single cell activity in V1 and V4 on visual tasks,” Exp. Brain Res. 69, 225–244 (1988).
    [CrossRef]
  46. S. J. Schein, R. Desimone, “Spectral properties of V4 neurons in macaque,” J. Neurosci. 10, 3369–3389 (1990).
    [PubMed]

1993 (1)

1992 (2)

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

M. Fairchild, P. Lennie, “Chromatic adaptation to natural and incandescent illuminants,” Vision Res. 32, 2077–2085 (1992).
[CrossRef] [PubMed]

1991 (1)

1990 (4)

M. Tsukada, Y. Ohta, “An approach to color constancy using multiple images,” Proc. Third Int. Conf. Comput. Vis. 3, 385–393 (1990).

S. Ahn, D. I. A. MacLeod, “Adaptation in the chromatic and luminance channels,” Invest. Ophthalmol. Vis. Sci. Suppl. 31, 109 (1990).

P. Lennie, J. Krauskopf, G. Sclar, “Chromatic mechanisms in striate cortex of macaque,” J. Neurosci. 10, 649–669 (1990).
[PubMed]

S. J. Schein, R. Desimone, “Spectral properties of V4 neurons in macaque,” J. Neurosci. 10, 3369–3389 (1990).
[PubMed]

1989 (1)

1988 (2)

P. E. Haenny, P. H. Schiller, “State dependent activity in monkey visual cortex—I. Single cell activity in V1 and V4 on visual tasks,” Exp. Brain Res. 69, 225–244 (1988).
[CrossRef]

D. Y. Ts’o, C. D. Gilbert, “The organization of chromatic and spatial interactions in the primate striate cortex,” J. Neurosci. 8, 1712–1727 (1988).

1987 (1)

C. A. Heywood, A. Cowey, “On the role of cortical area V4 in the discrimination of hue and pattern in macaque monkeys,” J. Neurosci. 7, 2601–2617 (1987).
[PubMed]

1986 (6)

1985 (3)

1984 (1)

M. S. Livingstone, D. H. Hubel, “Anatomy and physiology of a color system in primate primary visual cortex,” J. Neurosci. 4, 309–356 (1984).
[PubMed]

1983 (3)

S. M. Zeki, “Colour coding in the cerebral cortex: the reaction of cells in monkey visual cortex to wavelengths and colours,” Neuroscience 9, 741–765 (1983).
[CrossRef] [PubMed]

S. M. Zeki, “Colour coding in the cerebral cortex: the response of wavelength-selective and colour-coded cells in monkey visual cortex to changes in wavelength composition,” Neuroscience 9, 767–781 (1983).
[CrossRef] [PubMed]

E. H. Land, “Recent advances in retinex theory and some implications for cortical computations: color vision and the natural image,” Proc. Natl. Acad. Sci. USA 80, 5163–5169 (1983).
[CrossRef] [PubMed]

1980 (1)

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

1978 (2)

C. R. Michael, “Color vision mechanisms in monkey striate cortex: dual-opponent cells with concentric receptive fields,” J. Neurophysiol. 41, 572–588 (1978).
[PubMed]

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

1976 (1)

J. J. McCann, S. P. McKee, T. H. Taylor, “Quantitative studies in retinex theory. A comparison between theoretical predictions and observer responses to the ‘color Mondrian’ experiments,” Vision Res. 16, 445–458 (1976).
[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)

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]

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

1962 (1)

E. H. Land, N. W. Daw, “Colors seen in a flash of light,” Proc. Natl. Acad. Sci. USA 48, 1000–1008 (1962).
[CrossRef] [PubMed]

1955 (1)

1940 (1)

Adelson, E. H.

E. H. Adelson, J. Bergen, “Spatiotemporal energy models for the perception of motion,” J. Opt. Soc. Am. A 2, 284–299 (1985).
[CrossRef] [PubMed]

E. H. Adelson, J. Bergen, “The plenoptic function and the elements of early vision,” in Computational Models of Visual Processing, M. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass., 1991), pp. 3–20.

Ahn, S.

S. Ahn, D. I. A. MacLeod, “Adaptation in the chromatic and luminance channels,” Invest. Ophthalmol. Vis. Sci. Suppl. 31, 109 (1990).

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 (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).

Arend, L.

Arend, L. E.

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 (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).

Benzschawel, T.

Bergen, J.

E. H. Adelson, J. Bergen, “Spatiotemporal energy models for the perception of motion,” J. Opt. Soc. Am. A 2, 284–299 (1985).
[CrossRef] [PubMed]

E. H. Adelson, J. Bergen, “The plenoptic function and the elements of early vision,” in Computational Models of Visual Processing, M. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass., 1991), pp. 3–20.

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 (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).

Brainard, D.

D. Brainard, B. A. Wandell, “An analysis of the retinex theory of color vision,” J. Opt. Soc. Am. A 3, 1651–1661 (1986).
[CrossRef] [PubMed]

D. Brainard, B. A. Wandell, “A bilinear model of the illuminant’s effect on color appearance,” in Computational Models of Visual Processing, M. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass., 1991), pp. 171–186.

Brill, M. H.

Buchsbaum, G.

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

Butler, S. R.

H. M. Wild, S. R. Butler, D. Carden, J. J. Kulikowski, “Primate cortical area V4 important for colour constancy but not wavelength discrimination,” Nature 313, 133–135 (1985).
[CrossRef]

Carden, D.

H. M. Wild, S. R. Butler, D. Carden, J. J. Kulikowski, “Primate cortical area V4 important for colour constancy but not wavelength discrimination,” Nature 313, 133–135 (1985).
[CrossRef]

Cohen, J.

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

Cowey, A.

C. A. Heywood, A. Cowey, “On the role of cortical area V4 in the discrimination of hue and pattern in macaque monkeys,” J. Neurosci. 7, 2601–2617 (1987).
[PubMed]

D’Zmura, M.

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

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

G. Iverson, M. D’Zmura, “Criteria for color constancy in trichromatic bilinear models,” UC Irvine Institute for Mathematical Behavioral Sciences Tech. Rep. 93-18 (University of California, Irvine, Irvine, Calif., 1993).

M. D’Zmura, G. Iverson, “Color structure from chromatic motion,” in Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 51.

Daw, N. W.

E. H. Land, N. W. Daw, “Colors seen in a flash of light,” Proc. Natl. Acad. Sci. USA 48, 1000–1008 (1962).
[CrossRef] [PubMed]

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 (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).

Desimone, R.

S. J. Schein, R. Desimone, “Spectral properties of V4 neurons in macaque,” J. Neurosci. 10, 3369–3389 (1990).
[PubMed]

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 (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 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 (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).

Fairchild, M.

M. Fairchild, P. Lennie, “Chromatic adaptation to natural and incandescent illuminants,” Vision Res. 32, 2077–2085 (1992).
[CrossRef] [PubMed]

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).

Gilbert, C. D.

D. Y. Ts’o, C. D. Gilbert, “The organization of chromatic and spatial interactions in the primate striate cortex,” J. Neurosci. 8, 1712–1727 (1988).

Goldstein, R.

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 (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).

Haenny, P. E.

P. E. Haenny, P. H. Schiller, “State dependent activity in monkey visual cortex—I. Single cell activity in V1 and V4 on visual tasks,” Exp. Brain Res. 69, 225–244 (1988).
[CrossRef]

Hallikainen, J.

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 (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).

Heywood, C. A.

C. A. Heywood, A. Cowey, “On the role of cortical area V4 in the discrimination of hue and pattern in macaque monkeys,” J. Neurosci. 7, 2601–2617 (1987).
[PubMed]

Hubel, D. H.

M. S. Livingstone, D. H. Hubel, “Anatomy and physiology of a color system in primate primary visual cortex,” J. Neurosci. 4, 309–356 (1984).
[PubMed]

Hurvich, L. M.

Iverson, G.

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 structure from chromatic motion,” in Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 51.

G. Iverson, M. D’Zmura, “Criteria for color constancy in trichromatic bilinear models,” UC Irvine Institute for Mathematical Behavioral Sciences Tech. Rep. 93-18 (University of California, Irvine, Irvine, Calif., 1993).

Jaaskelainen, T.

Jameson, D.

Judd, D. B.

Krauskopf, J.

P. Lennie, J. Krauskopf, G. Sclar, “Chromatic mechanisms in striate cortex of macaque,” J. Neurosci. 10, 649–669 (1990).
[PubMed]

Kulikowski, J. J.

H. M. Wild, S. R. Butler, D. Carden, J. J. Kulikowski, “Primate cortical area V4 important for colour constancy but not wavelength discrimination,” Nature 313, 133–135 (1985).
[CrossRef]

Land, E. H.

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

E. H. Land, “Recent advances in retinex theory and some implications for cortical computations: color vision and the natural image,” Proc. Natl. Acad. Sci. USA 80, 5163–5169 (1983).
[CrossRef] [PubMed]

E. H. Land, N. W. Daw, “Colors seen in a flash of light,” Proc. Natl. Acad. Sci. USA 48, 1000–1008 (1962).
[CrossRef] [PubMed]

Lennie, P.

M. Fairchild, P. Lennie, “Chromatic adaptation to natural and incandescent illuminants,” Vision Res. 32, 2077–2085 (1992).
[CrossRef] [PubMed]

P. Lennie, J. Krauskopf, G. Sclar, “Chromatic mechanisms in striate cortex of macaque,” J. Neurosci. 10, 649–669 (1990).
[PubMed]

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

Livingstone, M. S.

M. S. Livingstone, D. H. Hubel, “Anatomy and physiology of a color system in primate primary visual cortex,” J. Neurosci. 4, 309–356 (1984).
[PubMed]

MacAdam, D. L.

MacLeod, D. I. A.

S. Ahn, D. I. A. MacLeod, “Adaptation in the chromatic and luminance channels,” Invest. Ophthalmol. Vis. Sci. Suppl. 31, 109 (1990).

D. I. A. MacLeod, “Receptoral constraints on colour appearance,” in Central and Peripheral Mechanisms of Colour Vision, D. Ottoson, S. Zeki, eds. (Macmillan, London, 1985), pp. 103–116.

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]

B. A. Wandell, L. T. Maloney, “Color imaging process,” U.S. patent4,648,051 (March3, 1987).

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

McCann, J. J.

J. J. McCann, S. P. McKee, T. H. Taylor, “Quantitative studies in retinex theory. A comparison between theoretical predictions and observer responses to the ‘color Mondrian’ experiments,” Vision Res. 16, 445–458 (1976).
[CrossRef]

McKee, S. P.

J. J. McCann, S. P. McKee, T. H. Taylor, “Quantitative studies in retinex theory. A comparison between theoretical predictions and observer responses to the ‘color Mondrian’ experiments,” Vision Res. 16, 445–458 (1976).
[CrossRef]

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 (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).

Michael, C. R.

C. R. Michael, “Color vision mechanisms in monkey striate cortex: dual-opponent cells with concentric receptive fields,” J. Neurophysiol. 41, 572–588 (1978).
[PubMed]

Ohta, Y.

M. Tsukada, Y. Ohta, “An approach to color constancy using multiple images,” Proc. Third Int. Conf. Comput. Vis. 3, 385–393 (1990).

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 (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 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]

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).

Reeves, A.

Schein, S. J.

S. J. Schein, R. Desimone, “Spectral properties of V4 neurons in macaque,” J. Neurosci. 10, 3369–3389 (1990).
[PubMed]

Schiller, P. H.

P. E. Haenny, P. H. Schiller, “State dependent activity in monkey visual cortex—I. Single cell activity in V1 and V4 on visual tasks,” Exp. Brain Res. 69, 225–244 (1988).
[CrossRef]

Schirillo, J.

Sclar, G.

P. Lennie, J. Krauskopf, G. Sclar, “Chromatic mechanisms in striate cortex of macaque,” J. Neurosci. 10, 649–669 (1990).
[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.

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Figures (7)

Fig. 1
Fig. 1

Results for dichromatic bilinear models: problems involving square bilinear model matrices (p = m = 2). The horizontal axis marks the dimension n of the reflectance model, which is taken equal to the number s of surfaces, while the vertical axis marks the number of views. The solid lines divide cases that satisfy the necessary condition svpsn + vm − 1 [inequality (1)]. 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. Points that lie beneath and to the right of the solid lines, where Q < D − 1, fail the feasibility condition [inequality (1)] and so represent problems for which unique recovery is impossible. The dotted lines divide cases that satisfy the necessary condition for the test provided by the model check algorithm to be performed, namely, that E ≥ U [Eqs. (2) and (3)]. The pair E/U is shown directly beneath each point. In problems where mv, such checks provide necessary and sufficient tests of whether particular bilinear models with the problem’s parameters provide perfect recovery algorithms. The X marks the problem (p m n v s) = (2 2 2 2 2), for which recovery fails totally.1 The circled point marks the problem (2 2 3 2 3), for which successful model checks show that there are perfect two-stage linear recovery algorithms with these problem parameters. By transposition [entailments (d) and (l) of Table 2], the result for each problem (p m n v s) also represents the result for the transposed problem (p n m s v), and the transposed parameters are indicated in parentheses at the top of the diagram and along its axes. See text for further discussion.

Fig. 2
Fig. 2

Spectra of model check matrices for exemplary dichromatic bilinear models. Plotted on a log axis are the ordered singular values of the model check matrices for exemplary bilinear models that combine the Smith–Pokorny protanope,8 the CIE daylight basis for illumination,9,10 and the Fourier basis for reflectance (see Table 1). The model check matrix for the exemplary model with parameters (p m n v s) = (2 2 2 2 2) has a kernel of dimension one and fails the check; the matrix for the exemplary model with parameters (2 2 3 2 3) has full rank and passes the check. See text for discussion.

Fig. 3
Fig. 3

Results for trichromatic bilinear models. A, The case of square bilinear model matrices (p = m = 3 or p = n = 3); B, the case of rectangular bilinear model matrices (p = 3, m = 2 or p = 3, n = 2). The triangles mark problems that are shown by analysis to provide perfect recovery algorithms. The squares mark problems that are shown by successful simulation of recovery to provide, at worst, imperfect recovery. As in Fig. 1, the circles mark problems that are shown by the model check algorithm to support perfect recovery procedures. See the caption for Fig. 1 and text for further details.

Fig. 4
Fig. 4

Spectra of model check matrices for exemplary trichromatic bilinear models. A, The case of square bilinear model matrices (p = m = 3). The Smith–Pokorny trichromat8 and the CIE daylight basis9,10 were used in combination with Fourier reflectance models of dimension n ranging from two through eight. Plotted on a log axis are the ordered singular values of the model check matrices for these exemplary bilinear models. The spectra are shown, from bottom to top, in order of increasing view and, for a particular choice of view, in order of increasing dimension n for surface reflectance. We have scaled the spectra to stagger the maximal singular values along the vertical axis at half log unit intervals. The parameters of the exemplary models whose spectra are shown are, from bottom to top, (3 3 2 2 2), (3 3 3 2 3), (3 3 3 3 3), (3 3 4 3 4), (3 3 5 3 5), (3 3 6 3 6), (3 3 7 3 7), and (3 3 8 3 8). The model with parameters (3 3 2 2 2) provides a matrix with a kernel of dimension three and fails the model check; with the possible exception of the model with parameters (3 3 8 3 8), the remaining models provide matrices of full rank and so pass the model check. B, The case of rectangular bilinear model matrices (p = 3, m = 2). The Smith–Pokorny trichromat8 and the first two CIE daylight basis functions9,10 were used in combination with Fourier reflectance models of dimension n ranging from two through five. The spectra of the model check matrices for these exemplary bilinear models are ordered and staggered as in A; the parameters are, from bottom to top, (3 2 2 1 2), (3 2 2 2 2), (3 2 3 2 3), (3 2 4 2 4), and (3 2 5 2 5). All provide matrices of full rank and so pass the model check. See text for further discussion.

Fig. 5
Fig. 5

Possible failure of bilinear models with parameters (3 3 2 1 2) or (3 3 2 2 2). See text for further discussion.

Fig. 6
Fig. 6

Results for tetrachromatic bilinear models. A, Problems that involve square bilinear model matrices (p = m = 4 or p = n = 4); B, C, problems that involve rectangular bilinear model matrices (B: p = 4, m = 3 or p = 4, n = 3; C: p = 4, m = 2 or p = 4, n = 2); D, problems that involve the recovery of greater than eight reflectance descriptors per surface (or greater than eight descriptors per illuminant). See the captions for Figs. 1 and 3 and text for further discussion.

Fig. 7
Fig. 7

Spectra of model check matrices for exemplary tetrachromatic bilinear models. The Smith–Pokorny8 trichromat and the rod V λ photoreceptoral sensitivities, together with the CIE daylight basis,9,10 were used in combination with Fourier reflectance models of dimension n ranging from two through fifteen for formation of exemplary bilinear models. Plotted on a log axis are the ordered singular values of the model check matrices for these exemplary bilinear models. The spectra are shown, from bottom to top, in order of increasing view, and for a particular choice of view, in order of increasing dimension n for surface reflectance. We have scaled the spectra to stagger the maximal singular values along the vertical axis at half log unit intervals. A, The case of square bilinear model matrices (p = m = 4, n ≤ 8; see Fig. 6A). The parameters of the exemplary models whose spectra are shown are, from bottom to top, v = 2: (4 4 2 2 2), (4 4 3 2 3), and (4 4 4 2 4); v = 3: (4 4 2 3 2), (4 4 3 3 3), (4 4 4 3 4), (4 4 5 3 5), (4 4 6 3 6), (4 4 7 3 7), and (4 4 8 3 8); v = 4: (4 4 2 4 2), (4 4 3 4 3), (4 4 4 4 4), (4 4 5 4 5), (4 4 6 4 6), (4 4 7 4 7), and (4 4 8 4 8). Note that all the models that use two views fail the model check. B, The case of rectangular bilinear model matrices (p = 4, m = 3, n ≤ 8; see Fig. 6B). The spectra of the model check matrices for the exemplary bilinear models are ordered and staggered as in A; the parameters are, from bottom to top, v = 1: (4 3 2 1 2); v = 2: (4 3 2 2 2), (4 3 3 2 3), (4 3 4 2 4), (4 3 5 2 5), and (4 3 6 2 6); v = 3: (4 3 2 3 2), (4 3 3 3 3), (4 3 4 3 4), (4 3 5 3 5), (4 3 6 3 6), (4 3 7 3 7), and (4 3 8 3 8). The exemplary models provide matrices of full rank and so pass the model check, with the exception of the model with parameters (4 3 6 2 6). C, The case of rectangular bilinear model matrices (p = 4, m = 2, n ≤ 8; see Fig. 6C). The spectra of the model check matrices for the exemplary bilinear models are ordered and staggered as in A; the parameters are, from bottom to top, v = 1: (4 2 2 1 2), (4 2 3 1 3); v = 2: (4 2 2 2 2), (4 2 3 2 3), (4 2 4 2 4), (4 2 5 2 5), (4 2 6 2 6), and (4 2 7 2 7). All provide matrices of full rank and so pass the model check. D, Problems that involve the recovery of greater than eight reflectance descriptors per surface (see Fig. 6D). From bottom to top, the parameters of the exemplary bilinear models whose spectra are exhibited are (4 3 n 3 s): (4 3 9 3 9), (4 3 10 3 10), and (4 3 11 3 11); (4 4 n 4 s): (4 4 9 4 9), (4 4 10 4 10), (4 4 11 4 11), (4 4 12 4 12), (4 4 13 4 13), (4 4 14 4 14), and (4 4 15 4 15). See text for further discussion.

Tables (2)

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Table 1 Tested Bilinear Model Components

Equations (12)

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s v p s n + v m - 1.
Number of Equations = ( p m v , n = s ) = ( n p - m m - v + 1 ) ( m m - v + 1 ) ,
Number of Unknowns ( p m v , n = s ) = = ( n 2 + m - v - 1 m - v + 1 ) .
z = ( e 11 I + e 12 Γ 12 ) a = ( e 21 Γ 21 + e 22 I ) a .
a = α ɛ 1 + β ɛ 2 ,
( e 11 - e 22 ) ( α ɛ 1 + β ɛ 2 ) + e 12 ( α λ 1 ɛ 1 + β λ 2 ɛ 2 ) - e 21 ( α ɛ 1 / λ 1 + β ɛ 2 / λ 2 ) = 0.
α [ ( e 11 - e 22 ) + λ 1 e 12 - e 21 / λ 1 ] = β [ ( e 11 - e 22 ) + λ 2 e 12 - e 21 / λ 2 ] = 0 ,
( e 11 - e 22 ) + λ 1 e 12 - e 21 / λ 1 = 0 ( e 11 - e 22 ) + λ 2 e 12 - e 21 / λ 2 = 0.
e 12 = c , e 21 = - c λ 1 λ 2 , e 11 - e 22 = - c ( λ 1 + λ 2 ) ,
Γ 12 ɛ = λ ɛ .
a = ɛ .
e 33 = e 11 + λ e 12 , e 22 = e 11 + λ e 12 - e 21 / λ , e 31 = - λ e 32 .

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