Abstract

The problem of color constancy may be solved if we can recover the physical properties of illuminants and surfaces from photosensor responses. We consider this problem within the framework of Bayesian decision theory. First, we model the relation among illuminants, surfaces, and photosensor responses. Second, we construct prior distributions that describe the probability that particular illuminants and surfaces exist in the world. Given a set of photosensor responses, we can then use Bayes’s rule to compute the posterior distribution for the illuminants and the surfaces in the scene. There are two widely used methods for obtaining a single best estimate from a posterior distribution. These are maximum a posteriori (MAP) and minimum mean-squared-error (MMSE) estimation. We argue that neither is appropriate for perception problems. We describe a new estimator, which we call the maximum local mass (MLM) estimate, that integrates local probability density. The new method uses an optimality criterion that is appropriate for perception tasks: It finds the most probable approximately correct answer. For the case of low observation noise, we provide an efficient approximation. We develop the MLM estimator for the color-constancy problem in which flat matte surfaces are uniformly illuminated. In simulations we show that the MLM method performs better than the MAP estimator and better than a number of standard color-constancy algorithms. We note conditions under which even the optimal estimator produces poor estimates: when the spectral properties of the surfaces in the scene are biased.

© 1997 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. O. Berger, Statistical Decision Theory and Bayesian Analysis (Springer-Verlag, New York, 1985).
  2. G. E. P. Box, G. C. Tiao, Bayesian Inference in Statistical Analysis (Wiley, New York, 1973).
  3. D. Knill, W. Richards, eds., Perception as Bayesian Inference (Cambridge U. Press, Cambridge, 1996).
  4. Preliminary reports of our work may be found in Brainard and Freeman 5 and Freeman and Brainard.6 See Trussell and Vrhel7,8 and D’Zmura et al.9 for related statistical approaches to color constancy.
  5. D. H. Brainard, W. T. Freeman, “Bayesian method for recovering surface and illuminant properties from photoreceptor responses,” in Human Vision, Visual Processing, and Display V, B. E. Rogowitz, J. P. Allebach, eds., Proc. SPIE2179, 364–376 (1994).
    [CrossRef]
  6. W. T. Freeman, D. H. Brainard, “Bayesian decision theory, the maximum local mass estimate, and color constancy,” in Proceedings of the 5th International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1995), pp. 210–217.
  7. H. J. Trussell, M. J. Vrhel, “Estimation of illumination for color correction,” Proceedings of the International Conference in Acoustics, Speech, and Signal Processing (IEEE, New York, 1991), pp. 2513–2516.
  8. M. J. Vrhel, H. J. Trussell, “Color correction using principal components,” Color Res. Appl. 17, 329–337 (1992).
    [CrossRef]
  9. M. D’Zmura, G. Iverson, B. Singer, “Probabilistic color constancy,” in Geometric Representations of Perceptual Phenomena: Papers in Honor of Tarow Indow’s 70th Birthday, R. D. Luce, M. D’Zmura, D. Hoffman, G. Iverson, A. K. Romney, eds. (Erlbaum, Hillsdale, N. J., 1995), pp. 187–202.
  10. R. M. Evans, The Perception of Color (Wiley, New York, 1974).
  11. 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]
  12. M. D. Fairchild, P. Lennie, “Chromatic adaptation to natural and incandescent illuminants,” Vision Res. 32, 2077–2085 (1992).
    [CrossRef] [PubMed]
  13. D. H. Brainard, B. A. Wandell, E.-J. Chichilnisky, “Color constancy: from physics to appearance,” Curr. Dir. Psychol. Sci. 2, 165–170 (1993).
  14. L. E. Arend, “How much does illuminant color affect unattributed colors?” J. Opt. Soc. Am. A 10, 2134–2147 (1993).
    [CrossRef]
  15. D. H. Brainard, J. M. Speigle, “Achromatic loci measured under realistic viewing conditions,” Invest. Ophthalmol. Visual Sci. Suppl. 35, 1328 (1994).
  16. D. A. Forsyth, “A novel algorithm for color constancy,” Int. J. Comput. Vision 5, 5–36 (1990).
    [CrossRef]
  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. E. H. Land, J. J. McCann, “Lightness and retinex theory,” J. Opt. Soc. Am. 61, 1–11 (1971).
    [CrossRef] [PubMed]
  19. E. H. Land, “Recent advances in retinex theory,” Vision Res. 26, 7–21 (1986).
    [CrossRef] [PubMed]
  20. 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]
  21. W. Menke, Geophysical Data Analysis: Discrete Inverse Theory (Academic, San Diego, 1989).
  22. J. J. McCann, “Psychophysical experiments in search of adaptation and the gray world,” in Proceedings of the 47th Annual Conference on Imaging Science and Technology (The Society for Imaging Science and Technology, Springfield, Va., 1994), pp. 397–401.
  23. J. J. McCann, J. A. Hall, E. H. Land, “Color Mondrian experiments: the study of average spectral distributions,” J. Opt. Soc. Am. 67, 1380 (1977).
  24. L. E. Arend, A. Reeves, “Simultaneous color constancy,” J. Opt. Soc. Am. A 3, 1743–1751 (1986).
    [CrossRef] [PubMed]
  25. L. E. Arend, A. Reeves, J. Schirillo, R. Goldstein, “Simultaneous color constancy: papers with diverse Munsell values,” J. Opt. Soc. Am. A 8, 661–672 (1991).
    [CrossRef] [PubMed]
  26. A. Valberg, B. Lange-Malecki, “Mondrian complexity does not improve ‘color constancy’,” Invest. Ophthalmol. Visual Sci. Suppl. 28, 92 (1987).
  27. D. H. Brainard, B. A. Wandell, “Asymmetric color-matching: how color appearance depends on the illuminant,” J. Opt. Soc. Am. A 9, 1433–1448 (1992).
    [CrossRef] [PubMed]
  28. K. H. Bauml, “Illuminant changes under different surface collections: examining some principles of color appearance,” J. Opt. Soc. Am. A 12, 261–271 (1995).
    [CrossRef]
  29. K. H. Bauml, “Color appearance: effects of illuminant changes under different surface collections,” J. Opt. Soc. Am. A 11, 531–542 (1994).
    [CrossRef]
  30. G. Buchsbaum, “A spatial processor model for object colour perception,” J. Franklin Inst. 310, 1–26 (1980).
    [CrossRef]
  31. L. T. Maloney, B. A. Wandell, “Color constancy: a method for recovering surface spectral reflectances,” J. Opt. Soc. Am. A 3, 29–33 (1986).
    [CrossRef] [PubMed]
  32. 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]
  33. 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]
  34. G. D. Finlayson, “Color constancy in diagonal chromaticity space,” in Proceedings of the 5th International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1995), pp. 218–223.
  35. B. K. P. Horn, B. G. Schunk, “Determining optical flow,” Artif. Intell. 17, 185–203 (1981).
    [CrossRef]
  36. J. J. Koenderink, A. J. van Doorn, “Affine structure from motion,” J. Opt. Soc. Am. A 8, 377–385 (1991).
    [CrossRef] [PubMed]
  37. T. Marill, “Emulating the human interpretation of line-drawings as three-dimensional objects,” Int. J. Comput. Vision 6, 147–161 (1991).
    [CrossRef]
  38. Y. G. Leclerc, M. A. Fischler, “Line drawings as 3D wire frames,” Int. J. Comput. Vision 9, 113–136 (1992).
    [CrossRef]
  39. P. Sinha, E. H. Adelson, “Recovering reflectance and illumination in a world of painted polyhedra,” in Proceedings of the 4th International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1993), pp. 156–163.
  40. E. Saund, T. P. Moran, “Perceptual organization in an interactive sketch editing application,” in Proceedings of the 5th International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1995), pp. 597–604.
  41. D. H. Brainard, “Colorimetry,” in Handbook of Optics: Volume 1. Fundamentals, Techniques, and Design, M. Bass, ed. (McGraw-Hill, New York, 1995), pp. 26.1–26.54.
  42. D. B. Judd, D. L. MacAdam, G. W. Wyszecki, “Spectral distribution of typical daylight as a function of correlated color temperature,” J. Opt. Soc. Am. 54, 1031–1040 (1964).
    [CrossRef]
  43. 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]
  44. T. Jaaskelainen, J. Parkkinen, S. Toyooka, “A vector-subspace model for color representation,” J. Opt. Soc. Am. A 7, 725–730 (1990).
    [CrossRef]
  45. D. H. Brainard, B. A. Wandell, “Analysis of the retinex theory of color vision,” J. Opt. Soc. Am. A 3, 1651–1661 (1986).
    [CrossRef] [PubMed]
  46. A. L. Gilchrist, “When does perceived lightness depend on perceived spatial arrangements?” Percept. Psychophys. 28, 527–538 (1980).
    [CrossRef] [PubMed]
  47. A. L. Gilchrist, “Lightness contrast and failures of constancy: a common explanation,” Percept. Psychophys. 43, 415–424 (1988).
    [CrossRef] [PubMed]
  48. R. O. Brown, “Saturation and color constancy,” in Advances in Color Vision, Vol. 4 of 1992 OSA Technical Digest Series (Optical Society of America, 1992), pp. 110–111.
  49. A. B. Poirson, B. A. Wandell, “Appearance of colored patterns—pattern color separability,” J. Opt. Soc. Am. A 10, 2458–2470 (1993).
    [CrossRef]
  50. B. Singer, M. D’Zmura, “Color contrast induction,” Vision Res. 34, 3111–3126 (1994).
    [CrossRef] [PubMed]
  51. B. Singer, M. D’Zmura, “Contrast gain control—a bilinear model for chromatic selectivity,” J. Opt. Soc. Am. A 12, 667–685 (1995).
    [CrossRef]
  52. J. W. Jenness, S. K. Shevell, “Color appearance with sparse chromatic context,” Vision Res. 35, 797–805 (1995).
    [CrossRef] [PubMed]
  53. B. A. Wandell, “The synthesis and analysis of color images,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-9, 2–13 (1987).
    [CrossRef]
  54. We use the notation p( ) to denote different probability-density functions. The particular function in any context is indicated by the argument.
  55. T. W. Hungerford, Algebra (Springer-Verlag, New York, 1974).
  56. D. H. Marimont, B. A. Wandell, “Linear models of surface and illuminant spectra,” J. Opt. Soc. Am. A 9, 1905–1913 (1992).
    [CrossRef] [PubMed]
  57. J. B. Tenenbaum, W. T. Freeman, “Separable mixture models: separating style and content,” in Advances in Neural Information Processing Systems9, M. C. Mozer, M. I. Jordan, T. Petsche, eds. (MIT Press, Cambridge, Mass., to be published).
  58. J. J. Koenderink, A. J. van Doorn, “The generic bilinear calibration–estimation problem,” Int. J. Comput. Vision (to be published).
  59. M. S. Landy, J. A. Movshon, eds., Computational Models of Visual Processing (MIT Press, Cambridge, Mass., 1991).
  60. S. Geman, D. Geman, “Stochastic relaxation, Gibbs distribution, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721–741 (1984).
    [CrossRef]
  61. R. Szeliski, Bayesian Modeling of Uncertainty in Low-Level Vision (Kluwer, Boston, 1989).
  62. T. Poggio, V. Torre, C. Koch, “Computational vision and regularization theory,” Nature (London) 317, 314–319 (1985).
    [CrossRef]
  63. D. Terzopoulos, “Regularization of inverse problems involving discontinuities,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-8, 413–424 (1986).
    [CrossRef]
  64. A. P. Pentland, “Automatic extraction of deformable part models,” Int. J. Comput. Vision 4, 107–126 (1990).
    [CrossRef]
  65. Y. G. Leclerc, “Constructing simple stable descriptions for image partitioning,” Int. J. Comput. Vision 3, 73–102 (1989).
    [CrossRef]
  66. A. Gelb, Applied Optimal Estimation (MIT Press, Cambridge, Mass., 1974).
  67. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984).
  68. K. L. Kelley, K. S. Gibson, D. Nickerson, “Tristimulus specification of the Munsell Book of Color from spectrophotometric measurements,” J. Opt. Soc. Am. 33, 355–376 (1943).
    [CrossRef]
  69. D. Nickerson, “Spectrophotometric data for a collection of Munsell samples,” (U.S. Department of Agriculture, Washington, D.C., 1957; available from Munsell Color Company, Baltimore, Md.).
  70. Although the Munsell papers are a man-made collection of surfaces, analyses of natural surfaces43,44 suggest that these are described by similar linear models.
  71. E. Schrodinger, “Theorie der pigmente von grosster leuchtkraft,” Ann. Phys. (Leipzig) 62, 603–622 (1920), as discussed in Ref. 72.
    [CrossRef]
  72. G. Wyszecki, “Color appearance,” in Handbook of Perception and Human Performance, K. R. Boff, L. Kaufman, J. P. Thomas, eds. (Wiley, New York, 1986), pp. 9.1–9.56.
  73. S. Rosch, “Die Kennzeichnung der Farben,” Phys. Z. 29, 83–91 (1928), as discussed in Ref. 72.
  74. D. L. MacAdam, “The theory of the maximum visual efficiency of colored materials,” J. Opt. Soc. Am. 25, 249–252 (1935).
    [CrossRef]
  75. M. R. Pointer, “The gamut of real surface colours,” Color Res. Appl. 5, 145–155 (1980).
    [CrossRef]
  76. M. Richter, K. Witt, “The story of the DIN color system,” Color Res. Appl. 11, 138–145 (1986).
    [CrossRef]
  77. G. Wyszecki, W. S. Stiles, Color Science—Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982).
  78. 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]
  79. D. A. Forsyth, “A novel approach to colour constancy,” in Proceedings of the International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1988), pp. 9–18.
  80. CIE, Colorimetry, 2nd ed. (Bureau Central de la CIE, Paris, 1986).
  81. P. DeMarco, J. Pokorny, V. C. Smith, “Full-spectrum cone sensitivity functions for X-chromosome-linked anomalous trichromats”, J. Opt. Soc. Am. A 9, 1465–1476 (1992).
    [CrossRef] [PubMed]
  82. A. Grace, Optimization Toolbox for Use with MATLAB—User’s Guide (MathWorks, Natick, Mass., 1990).
  83. H. Fuchs, “Eine experimentelle undersuchung zur farbkonstanz,” unpublished Ph.D. dissertation (University of Regensburg, Regensburg, Germany, 1992).
  84. M. S. Drew, B. V. Funt, “Variational approach to interreflection in color images”, J. Opt. Soc. Am. A 9, 1255–1265 (1992).
    [CrossRef]
  85. H. Lee, “Method for computing the scene-illuminant chromaticity from specular highlights,” J. Opt. Soc. Am. A 3, 1694–1699 (1986).
    [CrossRef] [PubMed]
  86. M. D’Zmura, P. Lennie, “Mechanisms of color constancy,” J. Opt. Soc. Am. A 3, 1662–1672 (1986).
    [CrossRef] [PubMed]
  87. L. T. Moloney, Department of Psychology, New York University, New York, N.Y. (personal communication, 1995).
  88. A. L. Yuille, H. H. Bulthoff, “Bayesian decision theory and psychophysics,” in Visual Perception: Computation and Psychophysics, D. Knill, W. Richards, eds. (Cambridge U. Press, Cambridge, 1995), pp. 123–161.
  89. R. N. Shepard, “Toward a universal law of generalization for psychological science,” Science 237, 1317–1323 (1987).
    [CrossRef] [PubMed]
  90. A. Blake, A. Zisserman, Visual Reconstruction (MIT Press, Cambridge, Mass., 1987).
  91. P. Meer, D. Mintz, A. Rosenfeld, “Robust regression methods for computer vision: a review,” Int. J. Comput. Vision 6, 59–70 (1991).
    [CrossRef]
  92. M. J. Black, P. Anandan, “A framework for the robust estimation of optical flow,” in Proceedings of the 4th International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1993), pp. 231–236.
  93. W. T. Freeman, “The generic viewpoint assumption in a framework for visual perception,” Nature (London) 368, 542–545 (1994).
    [CrossRef]
  94. W. T. Freeman, “Exploiting the generic viewpoint assumption,” Int. J. Comput. Vision 20, 243–261 (1996).
    [CrossRef]
  95. K. R. K. Nielsen, B. A. Wandell, “Discrete analysis of spatial sensitivity models,” J. Opt. Soc. Am. A 5, 743–755 (1988).
    [CrossRef] [PubMed]
  96. N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1986).
  97. P. S. Laplace, Theorie Analytique des Probabilities (Courcier, Paris, 1812).
  98. R. A. Fisher, Statistical Methods and Scientific Inference, 2nd ed. (Hafner, Oliver and Boyd, Edinburgh, 1959).
  99. H. Jeffreys, Theory of Probability (Clarendon, Oxford, 1961).
  100. G. E. P. Box, G. C. Tiao, “A Bayesian approach to the importance of assumptions applied to the comparison of variances,” Biometrika 51, 153–167 (1964).
  101. D. V. Lindley, Bayesian Statistics, a Review (Society for Industrial and Applied Mathematics, Philadelphia, 1971).
  102. S. F. Gull, “Bayesian inductive inference and maximum entropy,” in Maximum-Entropy and Bayesian Methods in Science and Engineering, G. J. Erickson, C. R. Smith, eds. (Kluwer, Boston, 1988), pp. 53–74.
  103. J. Skilling, “Classic maximum entropy,” in Maximum Entropy and Bayesian Methods, Cambridge, J. Skilling, ed. (Kluwer, Dordrecht, The Netherlands, 1989), pp. 45–52.
  104. D. J. C. MacKay, “Bayesian interpolation,” Neural Comput. 4, 415–447 (1992).
    [CrossRef]

1996 (2)

1995 (3)

1994 (4)

B. Singer, M. D’Zmura, “Color contrast induction,” Vision Res. 34, 3111–3126 (1994).
[CrossRef] [PubMed]

K. H. Bauml, “Color appearance: effects of illuminant changes under different surface collections,” J. Opt. Soc. Am. A 11, 531–542 (1994).
[CrossRef]

D. H. Brainard, J. M. Speigle, “Achromatic loci measured under realistic viewing conditions,” Invest. Ophthalmol. Visual Sci. Suppl. 35, 1328 (1994).

W. T. Freeman, “The generic viewpoint assumption in a framework for visual perception,” Nature (London) 368, 542–545 (1994).
[CrossRef]

1993 (5)

1992 (8)

1991 (4)

P. Meer, D. Mintz, A. Rosenfeld, “Robust regression methods for computer vision: a review,” Int. J. Comput. Vision 6, 59–70 (1991).
[CrossRef]

L. E. Arend, A. Reeves, J. Schirillo, R. Goldstein, “Simultaneous color constancy: papers with diverse Munsell values,” J. Opt. Soc. Am. A 8, 661–672 (1991).
[CrossRef] [PubMed]

J. J. Koenderink, A. J. van Doorn, “Affine structure from motion,” J. Opt. Soc. Am. A 8, 377–385 (1991).
[CrossRef] [PubMed]

T. Marill, “Emulating the human interpretation of line-drawings as three-dimensional objects,” Int. J. Comput. Vision 6, 147–161 (1991).
[CrossRef]

1990 (3)

T. Jaaskelainen, J. Parkkinen, S. Toyooka, “A vector-subspace model for color representation,” J. Opt. Soc. Am. A 7, 725–730 (1990).
[CrossRef]

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

A. P. Pentland, “Automatic extraction of deformable part models,” Int. J. Comput. Vision 4, 107–126 (1990).
[CrossRef]

1989 (2)

Y. G. Leclerc, “Constructing simple stable descriptions for image partitioning,” Int. J. Comput. Vision 3, 73–102 (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]

1988 (2)

A. L. Gilchrist, “Lightness contrast and failures of constancy: a common explanation,” Percept. Psychophys. 43, 415–424 (1988).
[CrossRef] [PubMed]

K. R. K. Nielsen, B. A. Wandell, “Discrete analysis of spatial sensitivity models,” J. Opt. Soc. Am. A 5, 743–755 (1988).
[CrossRef] [PubMed]

1987 (3)

R. N. Shepard, “Toward a universal law of generalization for psychological science,” Science 237, 1317–1323 (1987).
[CrossRef] [PubMed]

B. A. Wandell, “The synthesis and analysis of color images,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-9, 2–13 (1987).
[CrossRef]

A. Valberg, B. Lange-Malecki, “Mondrian complexity does not improve ‘color constancy’,” Invest. Ophthalmol. Visual Sci. Suppl. 28, 92 (1987).

1986 (9)

1985 (1)

T. Poggio, V. Torre, C. Koch, “Computational vision and regularization theory,” Nature (London) 317, 314–319 (1985).
[CrossRef]

1984 (1)

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distribution, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721–741 (1984).
[CrossRef]

1983 (1)

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]

1981 (1)

B. K. P. Horn, B. G. Schunk, “Determining optical flow,” Artif. Intell. 17, 185–203 (1981).
[CrossRef]

1980 (3)

A. L. Gilchrist, “When does perceived lightness depend on perceived spatial arrangements?” Percept. Psychophys. 28, 527–538 (1980).
[CrossRef] [PubMed]

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

M. R. Pointer, “The gamut of real surface colours,” Color Res. Appl. 5, 145–155 (1980).
[CrossRef]

1977 (1)

J. J. McCann, J. A. Hall, E. H. Land, “Color Mondrian experiments: the study of average spectral distributions,” J. Opt. Soc. Am. 67, 1380 (1977).

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]

1971 (1)

1964 (2)

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

G. E. P. Box, G. C. Tiao, “A Bayesian approach to the importance of assumptions applied to the comparison of variances,” Biometrika 51, 153–167 (1964).

1943 (1)

1935 (1)

1928 (1)

S. Rosch, “Die Kennzeichnung der Farben,” Phys. Z. 29, 83–91 (1928), as discussed in Ref. 72.

1920 (1)

E. Schrodinger, “Theorie der pigmente von grosster leuchtkraft,” Ann. Phys. (Leipzig) 62, 603–622 (1920), as discussed in Ref. 72.
[CrossRef]

Adelson, E. H.

P. Sinha, E. H. Adelson, “Recovering reflectance and illumination in a world of painted polyhedra,” in Proceedings of the 4th International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1993), pp. 156–163.

Anandan, P.

M. J. Black, P. Anandan, “A framework for the robust estimation of optical flow,” in Proceedings of the 4th International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1993), pp. 231–236.

Arend, L. E.

Bauml, K. H.

Berger, T. O.

T. O. Berger, Statistical Decision Theory and Bayesian Analysis (Springer-Verlag, New York, 1985).

Black, M. J.

M. J. Black, P. Anandan, “A framework for the robust estimation of optical flow,” in Proceedings of the 4th International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1993), pp. 231–236.

Blake, A.

A. Blake, A. Zisserman, Visual Reconstruction (MIT Press, Cambridge, Mass., 1987).

Bleistein, N.

N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1986).

Box, G. E. P.

G. E. P. Box, G. C. Tiao, “A Bayesian approach to the importance of assumptions applied to the comparison of variances,” Biometrika 51, 153–167 (1964).

G. E. P. Box, G. C. Tiao, Bayesian Inference in Statistical Analysis (Wiley, New York, 1973).

Brainard, D. H.

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]

D. H. Brainard, J. M. Speigle, “Achromatic loci measured under realistic viewing conditions,” Invest. Ophthalmol. Visual Sci. Suppl. 35, 1328 (1994).

D. H. Brainard, B. A. Wandell, E.-J. Chichilnisky, “Color constancy: from physics to appearance,” Curr. Dir. Psychol. Sci. 2, 165–170 (1993).

D. H. Brainard, B. A. Wandell, “Asymmetric color-matching: how color appearance depends on the illuminant,” J. Opt. Soc. Am. A 9, 1433–1448 (1992).
[CrossRef] [PubMed]

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

D. H. Brainard, W. T. Freeman, “Bayesian method for recovering surface and illuminant properties from photoreceptor responses,” in Human Vision, Visual Processing, and Display V, B. E. Rogowitz, J. P. Allebach, eds., Proc. SPIE2179, 364–376 (1994).
[CrossRef]

W. T. Freeman, D. H. Brainard, “Bayesian decision theory, the maximum local mass estimate, and color constancy,” in Proceedings of the 5th International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1995), pp. 210–217.

D. H. Brainard, “Colorimetry,” in Handbook of Optics: Volume 1. Fundamentals, Techniques, and Design, M. Bass, ed. (McGraw-Hill, New York, 1995), pp. 26.1–26.54.

Brown, R. O.

R. O. Brown, “Saturation and color constancy,” in Advances in Color Vision, Vol. 4 of 1992 OSA Technical Digest Series (Optical Society of America, 1992), pp. 110–111.

Buchsbaum, G.

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

Bulthoff, H. H.

A. L. Yuille, H. H. Bulthoff, “Bayesian decision theory and psychophysics,” in Visual Perception: Computation and Psychophysics, D. Knill, W. Richards, eds. (Cambridge U. Press, Cambridge, 1995), pp. 123–161.

Chichilnisky, E.-J.

D. H. Brainard, B. A. Wandell, E.-J. Chichilnisky, “Color constancy: from physics to appearance,” Curr. Dir. Psychol. Sci. 2, 165–170 (1993).

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.

DeMarco, P.

Drew, M. S.

Evans, R. M.

R. M. Evans, The Perception of Color (Wiley, New York, 1974).

Fairchild, M. D.

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

Finlayson, G. D.

G. D. Finlayson, “Color constancy in diagonal chromaticity space,” in Proceedings of the 5th International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1995), pp. 218–223.

Fischler, M. A.

Y. G. Leclerc, M. A. Fischler, “Line drawings as 3D wire frames,” Int. J. Comput. Vision 9, 113–136 (1992).
[CrossRef]

Fisher, R. A.

R. A. Fisher, Statistical Methods and Scientific Inference, 2nd ed. (Hafner, Oliver and Boyd, Edinburgh, 1959).

Forsyth, D. A.

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

D. A. Forsyth, “A novel approach to colour constancy,” in Proceedings of the International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1988), pp. 9–18.

Freeman, W. T.

W. T. Freeman, “Exploiting the generic viewpoint assumption,” Int. J. Comput. Vision 20, 243–261 (1996).
[CrossRef]

W. T. Freeman, “The generic viewpoint assumption in a framework for visual perception,” Nature (London) 368, 542–545 (1994).
[CrossRef]

J. B. Tenenbaum, W. T. Freeman, “Separable mixture models: separating style and content,” in Advances in Neural Information Processing Systems9, M. C. Mozer, M. I. Jordan, T. Petsche, eds. (MIT Press, Cambridge, Mass., to be published).

D. H. Brainard, W. T. Freeman, “Bayesian method for recovering surface and illuminant properties from photoreceptor responses,” in Human Vision, Visual Processing, and Display V, B. E. Rogowitz, J. P. Allebach, eds., Proc. SPIE2179, 364–376 (1994).
[CrossRef]

W. T. Freeman, D. H. Brainard, “Bayesian decision theory, the maximum local mass estimate, and color constancy,” in Proceedings of the 5th International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1995), pp. 210–217.

Fuchs, H.

H. Fuchs, “Eine experimentelle undersuchung zur farbkonstanz,” unpublished Ph.D. dissertation (University of Regensburg, Regensburg, Germany, 1992).

Funt, B. V.

Gelb, A.

A. Gelb, Applied Optimal Estimation (MIT Press, Cambridge, Mass., 1974).

Geman, D.

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distribution, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721–741 (1984).
[CrossRef]

Geman, S.

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distribution, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721–741 (1984).
[CrossRef]

Gibson, K. S.

Gilchrist, A. L.

A. L. Gilchrist, “Lightness contrast and failures of constancy: a common explanation,” Percept. Psychophys. 43, 415–424 (1988).
[CrossRef] [PubMed]

A. L. Gilchrist, “When does perceived lightness depend on perceived spatial arrangements?” Percept. Psychophys. 28, 527–538 (1980).
[CrossRef] [PubMed]

Goldstein, R.

Grace, A.

A. Grace, Optimization Toolbox for Use with MATLAB—User’s Guide (MathWorks, Natick, Mass., 1990).

Gull, S. F.

S. F. Gull, “Bayesian inductive inference and maximum entropy,” in Maximum-Entropy and Bayesian Methods in Science and Engineering, G. J. Erickson, C. R. Smith, eds. (Kluwer, Boston, 1988), pp. 53–74.

Hall, J. A.

J. J. McCann, J. A. Hall, E. H. Land, “Color Mondrian experiments: the study of average spectral distributions,” J. Opt. Soc. Am. 67, 1380 (1977).

Handelsman, R. A.

N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1986).

Horn, B. K. P.

B. K. P. Horn, B. G. Schunk, “Determining optical flow,” Artif. Intell. 17, 185–203 (1981).
[CrossRef]

Hungerford, T. W.

T. W. Hungerford, Algebra (Springer-Verlag, New York, 1974).

Iverson, G.

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, B. Singer, “Probabilistic color constancy,” in Geometric Representations of Perceptual Phenomena: Papers in Honor of Tarow Indow’s 70th Birthday, R. D. Luce, M. D’Zmura, D. Hoffman, G. Iverson, A. K. Romney, eds. (Erlbaum, Hillsdale, N. J., 1995), pp. 187–202.

Jaaskelainen, T.

Jeffreys, H.

H. Jeffreys, Theory of Probability (Clarendon, Oxford, 1961).

Jenness, J. W.

J. W. Jenness, S. K. Shevell, “Color appearance with sparse chromatic context,” Vision Res. 35, 797–805 (1995).
[CrossRef] [PubMed]

Judd, D. B.

Kelley, K. L.

Koch, C.

T. Poggio, V. Torre, C. Koch, “Computational vision and regularization theory,” Nature (London) 317, 314–319 (1985).
[CrossRef]

Koenderink, J. J.

J. J. Koenderink, A. J. van Doorn, “Affine structure from motion,” J. Opt. Soc. Am. A 8, 377–385 (1991).
[CrossRef] [PubMed]

J. J. Koenderink, A. J. van Doorn, “The generic bilinear calibration–estimation problem,” Int. J. Comput. Vision (to be published).

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]

J. J. McCann, J. A. Hall, E. H. Land, “Color Mondrian experiments: the study of average spectral distributions,” J. Opt. Soc. Am. 67, 1380 (1977).

E. H. Land, J. J. McCann, “Lightness and retinex theory,” J. Opt. Soc. Am. 61, 1–11 (1971).
[CrossRef] [PubMed]

Lange-Malecki, B.

A. Valberg, B. Lange-Malecki, “Mondrian complexity does not improve ‘color constancy’,” Invest. Ophthalmol. Visual Sci. Suppl. 28, 92 (1987).

Laplace, P. S.

P. S. Laplace, Theorie Analytique des Probabilities (Courcier, Paris, 1812).

Leclerc, Y. G.

Y. G. Leclerc, M. A. Fischler, “Line drawings as 3D wire frames,” Int. J. Comput. Vision 9, 113–136 (1992).
[CrossRef]

Y. G. Leclerc, “Constructing simple stable descriptions for image partitioning,” Int. J. Comput. Vision 3, 73–102 (1989).
[CrossRef]

Lee, H.

Lennie, P.

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

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

Lindley, D. V.

D. V. Lindley, Bayesian Statistics, a Review (Society for Industrial and Applied Mathematics, Philadelphia, 1971).

MacAdam, D. L.

MacKay, D. J. C.

D. J. C. MacKay, “Bayesian interpolation,” Neural Comput. 4, 415–447 (1992).
[CrossRef]

Maloney, L. T.

Marill, T.

T. Marill, “Emulating the human interpretation of line-drawings as three-dimensional objects,” Int. J. Comput. Vision 6, 147–161 (1991).
[CrossRef]

Marimont, D. H.

McCann, J. J.

J. J. McCann, J. A. Hall, E. H. Land, “Color Mondrian experiments: the study of average spectral distributions,” J. Opt. Soc. Am. 67, 1380 (1977).

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]

E. H. Land, J. J. McCann, “Lightness and retinex theory,” J. Opt. Soc. Am. 61, 1–11 (1971).
[CrossRef] [PubMed]

J. J. McCann, “Psychophysical experiments in search of adaptation and the gray world,” in Proceedings of the 47th Annual Conference on Imaging Science and Technology (The Society for Imaging Science and Technology, Springfield, Va., 1994), pp. 397–401.

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]

Meer, P.

P. Meer, D. Mintz, A. Rosenfeld, “Robust regression methods for computer vision: a review,” Int. J. Comput. Vision 6, 59–70 (1991).
[CrossRef]

Menke, W.

W. Menke, Geophysical Data Analysis: Discrete Inverse Theory (Academic, San Diego, 1989).

Mintz, D.

P. Meer, D. Mintz, A. Rosenfeld, “Robust regression methods for computer vision: a review,” Int. J. Comput. Vision 6, 59–70 (1991).
[CrossRef]

Moloney, L. T.

L. T. Moloney, Department of Psychology, New York University, New York, N.Y. (personal communication, 1995).

Moran, T. P.

E. Saund, T. P. Moran, “Perceptual organization in an interactive sketch editing application,” in Proceedings of the 5th International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1995), pp. 597–604.

Nickerson, D.

K. L. Kelley, K. S. Gibson, D. Nickerson, “Tristimulus specification of the Munsell Book of Color from spectrophotometric measurements,” J. Opt. Soc. Am. 33, 355–376 (1943).
[CrossRef]

D. Nickerson, “Spectrophotometric data for a collection of Munsell samples,” (U.S. Department of Agriculture, Washington, D.C., 1957; available from Munsell Color Company, Baltimore, Md.).

Nielsen, K. R. K.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984).

Parkkinen, J.

Pentland, A. P.

A. P. Pentland, “Automatic extraction of deformable part models,” Int. J. Comput. Vision 4, 107–126 (1990).
[CrossRef]

Poggio, T.

T. Poggio, V. Torre, C. Koch, “Computational vision and regularization theory,” Nature (London) 317, 314–319 (1985).
[CrossRef]

Pointer, M. R.

M. R. Pointer, “The gamut of real surface colours,” Color Res. Appl. 5, 145–155 (1980).
[CrossRef]

Poirson, A. B.

Pokorny, J.

Reeves, A.

Richter, M.

M. Richter, K. Witt, “The story of the DIN color system,” Color Res. Appl. 11, 138–145 (1986).
[CrossRef]

Rosch, S.

S. Rosch, “Die Kennzeichnung der Farben,” Phys. Z. 29, 83–91 (1928), as discussed in Ref. 72.

Rosenfeld, A.

P. Meer, D. Mintz, A. Rosenfeld, “Robust regression methods for computer vision: a review,” Int. J. Comput. Vision 6, 59–70 (1991).
[CrossRef]

Saund, E.

E. Saund, T. P. Moran, “Perceptual organization in an interactive sketch editing application,” in Proceedings of the 5th International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1995), pp. 597–604.

Schirillo, J.

Schrodinger, E.

E. Schrodinger, “Theorie der pigmente von grosster leuchtkraft,” Ann. Phys. (Leipzig) 62, 603–622 (1920), as discussed in Ref. 72.
[CrossRef]

Schunk, B. G.

B. K. P. Horn, B. G. Schunk, “Determining optical flow,” Artif. Intell. 17, 185–203 (1981).
[CrossRef]

Shepard, R. N.

R. N. Shepard, “Toward a universal law of generalization for psychological science,” Science 237, 1317–1323 (1987).
[CrossRef] [PubMed]

Shevell, S. K.

J. W. Jenness, S. K. Shevell, “Color appearance with sparse chromatic context,” Vision Res. 35, 797–805 (1995).
[CrossRef] [PubMed]

Singer, B.

B. Singer, M. D’Zmura, “Contrast gain control—a bilinear model for chromatic selectivity,” J. Opt. Soc. Am. A 12, 667–685 (1995).
[CrossRef]

B. Singer, M. D’Zmura, “Color contrast induction,” Vision Res. 34, 3111–3126 (1994).
[CrossRef] [PubMed]

M. D’Zmura, G. Iverson, B. Singer, “Probabilistic color constancy,” in Geometric Representations of Perceptual Phenomena: Papers in Honor of Tarow Indow’s 70th Birthday, R. D. Luce, M. D’Zmura, D. Hoffman, G. Iverson, A. K. Romney, eds. (Erlbaum, Hillsdale, N. J., 1995), pp. 187–202.

Sinha, P.

P. Sinha, E. H. Adelson, “Recovering reflectance and illumination in a world of painted polyhedra,” in Proceedings of the 4th International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1993), pp. 156–163.

Skilling, J.

J. Skilling, “Classic maximum entropy,” in Maximum Entropy and Bayesian Methods, Cambridge, J. Skilling, ed. (Kluwer, Dordrecht, The Netherlands, 1989), pp. 45–52.

Smith, V. C.

Speigle, J. M.

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]

D. H. Brainard, J. M. Speigle, “Achromatic loci measured under realistic viewing conditions,” Invest. Ophthalmol. Visual Sci. Suppl. 35, 1328 (1994).

Stiles, W. S.

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

Szeliski, R.

R. Szeliski, Bayesian Modeling of Uncertainty in Low-Level Vision (Kluwer, Boston, 1989).

Taylor, T. H.

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]

Tenenbaum, J. B.

J. B. Tenenbaum, W. T. Freeman, “Separable mixture models: separating style and content,” in Advances in Neural Information Processing Systems9, M. C. Mozer, M. I. Jordan, T. Petsche, eds. (MIT Press, Cambridge, Mass., to be published).

Terzopoulos, D.

D. Terzopoulos, “Regularization of inverse problems involving discontinuities,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-8, 413–424 (1986).
[CrossRef]

Tiao, G. C.

G. E. P. Box, G. C. Tiao, “A Bayesian approach to the importance of assumptions applied to the comparison of variances,” Biometrika 51, 153–167 (1964).

G. E. P. Box, G. C. Tiao, Bayesian Inference in Statistical Analysis (Wiley, New York, 1973).

Torre, V.

T. Poggio, V. Torre, C. Koch, “Computational vision and regularization theory,” Nature (London) 317, 314–319 (1985).
[CrossRef]

Toyooka, S.

Trussell, H. J.

M. J. Vrhel, H. J. Trussell, “Color correction using principal components,” Color Res. Appl. 17, 329–337 (1992).
[CrossRef]

H. J. Trussell, M. J. Vrhel, “Estimation of illumination for color correction,” Proceedings of the International Conference in Acoustics, Speech, and Signal Processing (IEEE, New York, 1991), pp. 2513–2516.

Valberg, A.

A. Valberg, B. Lange-Malecki, “Mondrian complexity does not improve ‘color constancy’,” Invest. Ophthalmol. Visual Sci. Suppl. 28, 92 (1987).

van Doorn, A. J.

J. J. Koenderink, A. J. van Doorn, “Affine structure from motion,” J. Opt. Soc. Am. A 8, 377–385 (1991).
[CrossRef] [PubMed]

J. J. Koenderink, A. J. van Doorn, “The generic bilinear calibration–estimation problem,” Int. J. Comput. Vision (to be published).

Vrhel, M. J.

M. J. Vrhel, H. J. Trussell, “Color correction using principal components,” Color Res. Appl. 17, 329–337 (1992).
[CrossRef]

H. J. Trussell, M. J. Vrhel, “Estimation of illumination for color correction,” Proceedings of the International Conference in Acoustics, Speech, and Signal Processing (IEEE, New York, 1991), pp. 2513–2516.

Wandell, B. A.

Witt, K.

M. Richter, K. Witt, “The story of the DIN color system,” Color Res. Appl. 11, 138–145 (1986).
[CrossRef]

Wyszecki, G.

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

G. Wyszecki, “Color appearance,” in Handbook of Perception and Human Performance, K. R. Boff, L. Kaufman, J. P. Thomas, eds. (Wiley, New York, 1986), pp. 9.1–9.56.

Wyszecki, G. W.

Yuille, A. L.

A. L. Yuille, H. H. Bulthoff, “Bayesian decision theory and psychophysics,” in Visual Perception: Computation and Psychophysics, D. Knill, W. Richards, eds. (Cambridge U. Press, Cambridge, 1995), pp. 123–161.

Zisserman, A.

A. Blake, A. Zisserman, Visual Reconstruction (MIT Press, Cambridge, Mass., 1987).

Ann. Phys. (Leipzig) (1)

E. Schrodinger, “Theorie der pigmente von grosster leuchtkraft,” Ann. Phys. (Leipzig) 62, 603–622 (1920), as discussed in Ref. 72.
[CrossRef]

Artif. Intell. (1)

B. K. P. Horn, B. G. Schunk, “Determining optical flow,” Artif. Intell. 17, 185–203 (1981).
[CrossRef]

Biometrika (1)

G. E. P. Box, G. C. Tiao, “A Bayesian approach to the importance of assumptions applied to the comparison of variances,” Biometrika 51, 153–167 (1964).

Color Res. Appl. (3)

M. J. Vrhel, H. J. Trussell, “Color correction using principal components,” Color Res. Appl. 17, 329–337 (1992).
[CrossRef]

M. R. Pointer, “The gamut of real surface colours,” Color Res. Appl. 5, 145–155 (1980).
[CrossRef]

M. Richter, K. Witt, “The story of the DIN color system,” Color Res. Appl. 11, 138–145 (1986).
[CrossRef]

Curr. Dir. Psychol. Sci. (1)

D. H. Brainard, B. A. Wandell, E.-J. Chichilnisky, “Color constancy: from physics to appearance,” Curr. Dir. Psychol. Sci. 2, 165–170 (1993).

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]

IEEE Trans. Pattern Anal. Mach. Intell. (3)

D. Terzopoulos, “Regularization of inverse problems involving discontinuities,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-8, 413–424 (1986).
[CrossRef]

B. A. Wandell, “The synthesis and analysis of color images,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-9, 2–13 (1987).
[CrossRef]

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distribution, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721–741 (1984).
[CrossRef]

Int. J. Comput. Vision (1)

T. Marill, “Emulating the human interpretation of line-drawings as three-dimensional objects,” Int. J. Comput. Vision 6, 147–161 (1991).
[CrossRef]

Int. J. Comput. Vision (6)

Y. G. Leclerc, M. A. Fischler, “Line drawings as 3D wire frames,” Int. J. Comput. Vision 9, 113–136 (1992).
[CrossRef]

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

A. P. Pentland, “Automatic extraction of deformable part models,” Int. J. Comput. Vision 4, 107–126 (1990).
[CrossRef]

Y. G. Leclerc, “Constructing simple stable descriptions for image partitioning,” Int. J. Comput. Vision 3, 73–102 (1989).
[CrossRef]

W. T. Freeman, “Exploiting the generic viewpoint assumption,” Int. J. Comput. Vision 20, 243–261 (1996).
[CrossRef]

P. Meer, D. Mintz, A. Rosenfeld, “Robust regression methods for computer vision: a review,” Int. J. Comput. Vision 6, 59–70 (1991).
[CrossRef]

Invest. Ophthalmol. Visual Sci. Suppl. (2)

D. H. Brainard, J. M. Speigle, “Achromatic loci measured under realistic viewing conditions,” Invest. Ophthalmol. Visual Sci. Suppl. 35, 1328 (1994).

A. Valberg, B. Lange-Malecki, “Mondrian complexity does not improve ‘color constancy’,” Invest. Ophthalmol. Visual Sci. Suppl. 28, 92 (1987).

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. (5)

J. Opt. Soc. Am. A (22)

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

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

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

K. R. K. Nielsen, B. A. Wandell, “Discrete analysis of spatial sensitivity models,” J. Opt. Soc. Am. A 5, 743–755 (1988).
[CrossRef] [PubMed]

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]

T. Jaaskelainen, J. Parkkinen, S. Toyooka, “A vector-subspace model for color representation,” J. Opt. Soc. Am. A 7, 725–730 (1990).
[CrossRef]

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

A. B. Poirson, B. A. Wandell, “Appearance of colored patterns—pattern color separability,” J. Opt. Soc. Am. A 10, 2458–2470 (1993).
[CrossRef]

B. Singer, M. D’Zmura, “Contrast gain control—a bilinear model for chromatic selectivity,” J. Opt. Soc. Am. A 12, 667–685 (1995).
[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]

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]

P. DeMarco, J. Pokorny, V. C. Smith, “Full-spectrum cone sensitivity functions for X-chromosome-linked anomalous trichromats”, J. Opt. Soc. Am. A 9, 1465–1476 (1992).
[CrossRef] [PubMed]

L. E. Arend, “How much does illuminant color affect unattributed colors?” J. Opt. Soc. Am. A 10, 2134–2147 (1993).
[CrossRef]

L. E. Arend, A. Reeves, “Simultaneous color constancy,” J. Opt. Soc. Am. A 3, 1743–1751 (1986).
[CrossRef] [PubMed]

L. E. Arend, A. Reeves, J. Schirillo, R. Goldstein, “Simultaneous color constancy: papers with diverse Munsell values,” J. Opt. Soc. Am. A 8, 661–672 (1991).
[CrossRef] [PubMed]

J. J. Koenderink, A. J. van Doorn, “Affine structure from motion,” J. Opt. Soc. Am. A 8, 377–385 (1991).
[CrossRef] [PubMed]

L. T. Maloney, B. A. Wandell, “Color constancy: a method for recovering surface spectral reflectances,” J. Opt. Soc. Am. A 3, 29–33 (1986).
[CrossRef] [PubMed]

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]

D. H. Brainard, B. A. Wandell, “Asymmetric color-matching: how color appearance depends on the illuminant,” J. Opt. Soc. Am. A 9, 1433–1448 (1992).
[CrossRef] [PubMed]

K. H. Bauml, “Illuminant changes under different surface collections: examining some principles of color appearance,” J. Opt. Soc. Am. A 12, 261–271 (1995).
[CrossRef]

K. H. Bauml, “Color appearance: effects of illuminant changes under different surface collections,” J. Opt. Soc. Am. A 11, 531–542 (1994).
[CrossRef]

Nature (London) (2)

T. Poggio, V. Torre, C. Koch, “Computational vision and regularization theory,” Nature (London) 317, 314–319 (1985).
[CrossRef]

W. T. Freeman, “The generic viewpoint assumption in a framework for visual perception,” Nature (London) 368, 542–545 (1994).
[CrossRef]

Neural Comput. (1)

D. J. C. MacKay, “Bayesian interpolation,” Neural Comput. 4, 415–447 (1992).
[CrossRef]

Percept. Psychophys. (2)

A. L. Gilchrist, “When does perceived lightness depend on perceived spatial arrangements?” Percept. Psychophys. 28, 527–538 (1980).
[CrossRef] [PubMed]

A. L. Gilchrist, “Lightness contrast and failures of constancy: a common explanation,” Percept. Psychophys. 43, 415–424 (1988).
[CrossRef] [PubMed]

Phys. Z. (1)

S. Rosch, “Die Kennzeichnung der Farben,” Phys. Z. 29, 83–91 (1928), as discussed in Ref. 72.

Proc. Natl. Acad. Sci. USA (1)

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]

Science (1)

R. N. Shepard, “Toward a universal law of generalization for psychological science,” Science 237, 1317–1323 (1987).
[CrossRef] [PubMed]

Vision Res. (5)

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

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]

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

J. W. Jenness, S. K. Shevell, “Color appearance with sparse chromatic context,” Vision Res. 35, 797–805 (1995).
[CrossRef] [PubMed]

B. Singer, M. D’Zmura, “Color contrast induction,” Vision Res. 34, 3111–3126 (1994).
[CrossRef] [PubMed]

Other (43)

R. O. Brown, “Saturation and color constancy,” in Advances in Color Vision, Vol. 4 of 1992 OSA Technical Digest Series (Optical Society of America, 1992), pp. 110–111.

J. B. Tenenbaum, W. T. Freeman, “Separable mixture models: separating style and content,” in Advances in Neural Information Processing Systems9, M. C. Mozer, M. I. Jordan, T. Petsche, eds. (MIT Press, Cambridge, Mass., to be published).

J. J. Koenderink, A. J. van Doorn, “The generic bilinear calibration–estimation problem,” Int. J. Comput. Vision (to be published).

M. S. Landy, J. A. Movshon, eds., Computational Models of Visual Processing (MIT Press, Cambridge, Mass., 1991).

R. Szeliski, Bayesian Modeling of Uncertainty in Low-Level Vision (Kluwer, Boston, 1989).

We use the notation p( ) to denote different probability-density functions. The particular function in any context is indicated by the argument.

T. W. Hungerford, Algebra (Springer-Verlag, New York, 1974).

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

A. Grace, Optimization Toolbox for Use with MATLAB—User’s Guide (MathWorks, Natick, Mass., 1990).

H. Fuchs, “Eine experimentelle undersuchung zur farbkonstanz,” unpublished Ph.D. dissertation (University of Regensburg, Regensburg, Germany, 1992).

D. A. Forsyth, “A novel approach to colour constancy,” in Proceedings of the International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1988), pp. 9–18.

CIE, Colorimetry, 2nd ed. (Bureau Central de la CIE, Paris, 1986).

D. Nickerson, “Spectrophotometric data for a collection of Munsell samples,” (U.S. Department of Agriculture, Washington, D.C., 1957; available from Munsell Color Company, Baltimore, Md.).

Although the Munsell papers are a man-made collection of surfaces, analyses of natural surfaces43,44 suggest that these are described by similar linear models.

G. Wyszecki, “Color appearance,” in Handbook of Perception and Human Performance, K. R. Boff, L. Kaufman, J. P. Thomas, eds. (Wiley, New York, 1986), pp. 9.1–9.56.

A. Gelb, Applied Optimal Estimation (MIT Press, Cambridge, Mass., 1974).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984).

M. D’Zmura, G. Iverson, B. Singer, “Probabilistic color constancy,” in Geometric Representations of Perceptual Phenomena: Papers in Honor of Tarow Indow’s 70th Birthday, R. D. Luce, M. D’Zmura, D. Hoffman, G. Iverson, A. K. Romney, eds. (Erlbaum, Hillsdale, N. J., 1995), pp. 187–202.

R. M. Evans, The Perception of Color (Wiley, New York, 1974).

T. O. Berger, Statistical Decision Theory and Bayesian Analysis (Springer-Verlag, New York, 1985).

G. E. P. Box, G. C. Tiao, Bayesian Inference in Statistical Analysis (Wiley, New York, 1973).

D. Knill, W. Richards, eds., Perception as Bayesian Inference (Cambridge U. Press, Cambridge, 1996).

Preliminary reports of our work may be found in Brainard and Freeman 5 and Freeman and Brainard.6 See Trussell and Vrhel7,8 and D’Zmura et al.9 for related statistical approaches to color constancy.

D. H. Brainard, W. T. Freeman, “Bayesian method for recovering surface and illuminant properties from photoreceptor responses,” in Human Vision, Visual Processing, and Display V, B. E. Rogowitz, J. P. Allebach, eds., Proc. SPIE2179, 364–376 (1994).
[CrossRef]

W. T. Freeman, D. H. Brainard, “Bayesian decision theory, the maximum local mass estimate, and color constancy,” in Proceedings of the 5th International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1995), pp. 210–217.

H. J. Trussell, M. J. Vrhel, “Estimation of illumination for color correction,” Proceedings of the International Conference in Acoustics, Speech, and Signal Processing (IEEE, New York, 1991), pp. 2513–2516.

W. Menke, Geophysical Data Analysis: Discrete Inverse Theory (Academic, San Diego, 1989).

J. J. McCann, “Psychophysical experiments in search of adaptation and the gray world,” in Proceedings of the 47th Annual Conference on Imaging Science and Technology (The Society for Imaging Science and Technology, Springfield, Va., 1994), pp. 397–401.

G. D. Finlayson, “Color constancy in diagonal chromaticity space,” in Proceedings of the 5th International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1995), pp. 218–223.

P. Sinha, E. H. Adelson, “Recovering reflectance and illumination in a world of painted polyhedra,” in Proceedings of the 4th International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1993), pp. 156–163.

E. Saund, T. P. Moran, “Perceptual organization in an interactive sketch editing application,” in Proceedings of the 5th International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1995), pp. 597–604.

D. H. Brainard, “Colorimetry,” in Handbook of Optics: Volume 1. Fundamentals, Techniques, and Design, M. Bass, ed. (McGraw-Hill, New York, 1995), pp. 26.1–26.54.

A. Blake, A. Zisserman, Visual Reconstruction (MIT Press, Cambridge, Mass., 1987).

M. J. Black, P. Anandan, “A framework for the robust estimation of optical flow,” in Proceedings of the 4th International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1993), pp. 231–236.

L. T. Moloney, Department of Psychology, New York University, New York, N.Y. (personal communication, 1995).

A. L. Yuille, H. H. Bulthoff, “Bayesian decision theory and psychophysics,” in Visual Perception: Computation and Psychophysics, D. Knill, W. Richards, eds. (Cambridge U. Press, Cambridge, 1995), pp. 123–161.

N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1986).

P. S. Laplace, Theorie Analytique des Probabilities (Courcier, Paris, 1812).

R. A. Fisher, Statistical Methods and Scientific Inference, 2nd ed. (Hafner, Oliver and Boyd, Edinburgh, 1959).

H. Jeffreys, Theory of Probability (Clarendon, Oxford, 1961).

D. V. Lindley, Bayesian Statistics, a Review (Society for Industrial and Applied Mathematics, Philadelphia, 1971).

S. F. Gull, “Bayesian inductive inference and maximum entropy,” in Maximum-Entropy and Bayesian Methods in Science and Engineering, G. J. Erickson, C. R. Smith, eds. (Kluwer, Boston, 1988), pp. 53–74.

J. Skilling, “Classic maximum entropy,” in Maximum Entropy and Bayesian Methods, Cambridge, J. Skilling, ed. (Kluwer, Dordrecht, The Netherlands, 1989), pp. 45–52.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Image formation for a simple geometry. The image is formed when light from an illuminant reflects off a collection of surfaces to the imaging device. We assume that the illuminant is diffuse and spatially uniform, so that it may be characterized by a single spectral power distribution. We assume that each surface is flat and matte, so that it may be characterized by a single spectral reflectance function. The spectral power distribution of the light reaching the observer from each surface is given as the wavelength-by-wavelength product of the illuminant spectral power distribution and the surface reflectance function. At each location this light is sampled by a small number of types of photosensors. For the case of human vision (as shown), these are the long-, medium-, and short-wavelength (LMS) sensitive cones. For a typical digital camera, these are the red, green, and blue (RGB) sensors.

Fig. 2
Fig. 2

Bayesian analysis of the product example: (a) posterior probability for the observed data ab=1 for Gaussian observation noise of variance σ2=0.18 and uniform prior probabilities over the plotted region, (b) cross section through the posterior at two different locations. Note the different thicknesses of the ridge; some local regions have more probability mass than others, even though the entire ridge has a constant maximum height.

Fig. 3
Fig. 3

Product example loss functions and corresponding expected losses. (a)–(c): Three loss functions. The plots show the penalty for guessing parameter values offset from the actual value, taken to be the plot center. Each loss function is shift invariant. (a) Minus delta function loss, implicit in MAP estimation. The penalty is constant except for the correct estimate. (b) Squared-error loss (a parabola), implicit in MMSE estimation. Very inaccurate estimates can carry inordinate influence. (c) Local mass loss function. Nearly correct estimates are rewarded, while all others carry nearly equal penalty. (d)–(f): Corresponding expected loss for the product example. Note that the plots are inverted, so that increasing height on the plots represents decreasing loss. This convention allows better visualization of the location of minimum expected loss. (d) The expected loss for the MAP estimator is minus the posterior probability. There is no unique point of minimum expected loss. (e) Expected loss for the MMSE estimator. The MMSE estimate (1.3, 1.3) does not lie along the ridge of solutions to ab=1. (f) Expected loss for the MLM estimator. The local mass loss favors the point (1.0, 1.0), where the ridge of high probability is widest. There is the most probability mass in the local neighborhood of this estimate.

Fig. 4
Fig. 4

Distribution of surface weights. The histograms show the distribution of linear model weights derived from the measurements of Kelly et al.68 and Nickerson.69 Each histogram corresponds to one basis vector. The solid curves show the fit of a truncated trivariate normal distribution to the weights.

Fig. 5
Fig. 5

Basic algorithm performance. Each plot illustrates performance for one of the algorithms. To generate each panel, we fixed the simulated illuminant and repeated the simulation for 15 different sets of eight randomly drawn surfaces. The solid curve in each plot shows the simulated illuminant. The dashed curves show the individual estimates of this illuminant produced by each algorithm. The simulated illuminant was a CIE 6500-K daylight, which is the mode of the illuminant prior. The surfaces were drawn at random from the surface prior. The six algorithms compared in this figure are (a) MLM, (b) Realizability 1, (c) Gray World, (d) MAP, (e) Realizability 2, and (f) Subspace.

Fig. 6
Fig. 6

Performance of the MLM algorithm for two illuminants that differ from the prior mode. To generate each panel, we fixed the simulated illuminant and repeated the simulation for 15 different sets of eight randomly drawn surfaces (no surface bias). The solid curve in each plot shows the simulated illuminant. The dashed curves show the individual estimates of this illuminant produced by the MLM algorithm. (a) Results for the simulated 4000-K daylight, (b) results for the simulated 15,000-K daylight.

Fig. 7
Fig. 7

Performance of MLM and Gray World algorithms when the surfaces are drawn from biased distributions. (a) Performance of the MLM algorithm for the simulated 6500-K daylight when the surfaces were drawn from the “blue” distribution specified by Table 2, (c) performance of the Gray World algorithm for the same conditions. In each case we fixed the simulated illuminant and repeated the simulation for 15 different sets of eight randomly drawn surfaces. Plots (b) and (d) highlight the difference between the performances of the two algorithms. Plot (b) compares the mean estimate of the MLM algorithm for the simulated 6500-K daylight when the surfaces were drawn from the blue distribution with the mean estimate of the Gray World algorithm for the same conditions. Plot (d) shows the same comparison when the surfaces were drawn from the “yellow” distribution.

Fig. 8
Fig. 8

Summary of algorithm performance for a variety of simulation conditions. The figure shows the mean fractional RMSE computed in the spectral domain for each algorithm for a variety of simulation conditions. (a) Error plotted as a function of the seven simulated illuminants listed in Table 1. For these simulations there was no bias in the distribution of surfaces. (b) Error plotted as a function of the 11 surface distributions listed in Table 2. For these simulations the 6500-K simulated illuminant was used.

Fig. 9
Fig. 9

Product example with an asymmetric loss function. (a) Loss function; (b) corresponding expected loss, computed according to expression (A5). As in Fig. 3, decreasing loss is plotted upward for visual clarity. The loss function aspect ratio is 0.14 to 1.0. The optimal parameter estimate is (0.37, 2.7), which differs from the results obtained for the symmetric local mass loss function (Fig. 3).

Fig. 10
Fig. 10

Expected utility (minus expected loss) derived for the case of monochromatic lightness constancy. The surfaces and the illuminant were assumed to have a uniform prior probability. To make the plot, we calculated with eight surfaces, the maximum observation set to 10, and the mean squared observation set to 33. Land and McCann’s white hypothesis (call the brightest surface white or equivalently choose the dimmest possible illuminant consistent with physical realizability) is the MLM estimate here.

Tables (2)

Tables Icon

Table 1 Simulated Illuminants a

Tables Icon

Table 2 Mean Simulated Surfaces a

Equations (36)

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

rj=Rcj=R(e . *sj).
p(x|y)=p(y|x)p(x)p(y)=Cp(y|x)p(x).
L¯(x˜|y)=xL(x˜,x)p(x|y)dx.
rj=L(e)sj=L(sj)e,
rj=M(we)wsj=M(wsj)we.
r=N(we)ws=N(ws)we.
p(y|x)=12πσ2 exp-|y-f(x)|22σ2.
p(x|y)=C exp|1-ab|22(0.18),0<a, b<40otherwise,
L(x˜, x)=-δ(x˜-x).
L(x˜, x)=|x˜-x|2.
L(x˜, x)=-exp[-|KL-1/2(x˜-x)|2],
p(wsj)=N(usj,Ksj),Bswsjrealizable0,otherwise.
p(we)=N(ue,Ke),Bewerealizable0,otherwise.
p(ws)=j=1Nsp(wsj).
p(we, ws)=p(we)p(ws).
L(x˜, x)=-exp{-|KL-1/2[g(x˜)-g(x)]|2}.
L¯(x˜|y)=C  (likelihood)(priors)(lossfunction)dx=-C  exp-τ2 |Kn-1/2[y-f(x)]|2p(x)×exp-τ2 |KL-1/2(x-x˜)|2dx.
I(τ)= exp[-τϕ(x)]g(x)dx,
I(τ)exp[-τϕ(x0)]{|det[ϕxx(x0)]|}1/2 2πτn/2g(x0),
ϕ(x)=12|Kn-1/2[y-f(x)]|2+12|KL-1/2(x-x˜)|2.
L¯(x˜|y)-C exp(-τ{12|Kn-1/2[y-f(x0)]|2+ 12|KL-1/2(x0-x˜)|2}) p(x0){|det[ϕxx(x0)]|}1/2.
[ϕxx(x0)]ij=fiTKn-1fj-[y-f(x0)]TKn-1fij+[KL-1]ij,
fi=f(x)xix=x0andfij=2f(x)xixjx=x0.
x=wswe,x0=ws0we0.
f(ws, we)/wsi=N(we)i,
f(ws, we)/wei=N(ws)i,
ϕxx(x0)=NT(we0)NT(ws0)Kn-1(N(we0)N(ws0))+KL-1,
x=x1x2xNxI,
y=f(x)=x1x2xNxI.
fTKn-1f=1σn2 xI200xIx10xI20xIx200xI2xIxNxIx1xIx2xIxNi=1Nx12,
ϕxx=1σn2xI2+σn2/σL200xIx10xI2+σn2/σL20xIx200xI2+σn2/σL2xIxNxIx1xIx2xIxNi=1Nxi2+σn2/σL2.
det[ϕxx(x)]=i=1N xi2+σn2σL2xI2+σn2σL2N-j=1N(xIxj)2xI2+σn2σL2N-1=σn2σL2 xI2+σn2σL2N-1i=1N xi2+xI2+σn2σL2.
det[ϕxx(x)]σn2σL2 (xI2)N-1i=1N xi2+xI2σn2σL2 (xI2)N-2i=1N yi2+xI4.
L¯(x|y)-Cp(x)xIN-2i=1N yi2+xI41/2,
p(xi)=1ifxiisbetween0and10otherwise
pyixI=1ifyi/xIisbetween0and10otherwise.

Metrics