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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 3, Iss. 10 — Oct. 1, 1986
  • pp: 1673–1683

Evaluation of linear models of surface spectral reflectance with small numbers of parameters

Laurence T. Maloney  »View Author Affiliations

JOSA A, Vol. 3, Issue 10, pp. 1673-1683 (1986)

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Recent computational models of color vision demonstrate that it is possible to achieve exact color constancy over a limited range of lights and surfaces described by linear models. The success of these computational models hinges on whether any sizable range of surface spectral reflectances can be described by a linear model with about three parameters. In the first part of this paper, I analyze two large sets of empirical surface spectral reflectances and examine three conjectures concerning constraints on surface reflectance: (1) that empirical surface reflectances fall within a linear model with a small number of parameters, (2) that empirical surface reflectances fall within a linear model composed of band-limited functions with a small number of parameters, and (3) that the shape of the spectral-sensitivity curves of human vision enhance the fit between empirical surface reflectances and a linear model. I conclude that the first and second conjectures hold for the two sets of spectral reflectances analyzed but that the number of parameters required to model the spectral reflectances is five to seven, not three. A reanalysis of the empirical data that takes human visual sensitivity into account gives more promising results. The linear models derived provide excellent fits to the data with as few as three or four parameters, confirming the third conjecture. The results suggest that constraints on possible surface-reflectance functions and the “filtering” properties of the shapes of the spectral-sensitivity curves of photoreceptors can both contribute to color constancy. In the last part of the paper I derive the relation between the number of photoreceptor classes present in vision and the “filtering” properties of each class. The results of this analysis reverse a conclusion reached by Barlow: the “filtering” properties of human photoreceptors are consistent with a trichromatic visual system that is color constant.

© 1986 Optical Society of America

Original Manuscript: March 10, 1986
Manuscript Accepted: June 27, 1986
Published: October 1, 1986

Laurence T. Maloney, "Evaluation of linear models of surface spectral reflectance with small numbers of parameters," J. Opt. Soc. Am. A 3, 1673-1683 (1986)

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  19. The analyses reported below were repeated for the full range of Munsell data, 380 to 770 nm, inclusive. The analyses were also repeated after each surface spectral reflectance was transformed from equal-spaced wavelength measurements to equal-spaced wave-number measurements by 4-point Lagrangean interpolation as described by Barlow. The results for both these analyses are qualitatively similar to the results reported here and are not reported.
  20. Regression on the first three characteristic vectors derived from the Munsell data directly produced similar results.
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  27. W. Kauzmann, ibid., pp. 669–670.
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  31. Barlow performs his analysis with color signals plotted against wave number, not wavelength. To avoid confusion, I continue to use wavelength as the independent variable. A change to wave number does not substantially affect the following argument.
  32. The spectrum of the light is convolved with the spectrum of the surface reflectance. See R. N. Bracewell, The Fourier Transform and Its Applications., 2nd ed. (McGraw-Hill, New York, 1978), Chap. 3. It is easily shown that the highest frequency present in a convolution is the sum of the two highest frequencies present in the functions being convolved.
  33. Barlow uses terahertz rather than inverse centimeters. The conversion factor is 33.33. Barlow assumes that the width of the visible spectrum is W= 228 THz. Then, using the result of Slepian et al., N= 2WF becomes N= 0.456F, where N is the number of samples required to capture the essentially frequency limited functions with cutoff F in cycles/1000 THz. Note that Barlow’s computations use a different formula based on the Shannon–Whittaker Theorem that gives N= 2WF+ 1.
  34. See D. A. Belsey, E. Kuh, R. W. Welsch, Regression Diagnostics; Identifying Influential Data and Sources of Collinearity (Wiley, New York, 1980). [CrossRef]
  35. The analyses reported in the text were also carried out on the unnormalized data. Comparison of the fits obtained on normalized and unnormalized data showed that normalized data produced worse fits in all cases. In effect, the penalty associated with failing to fit surface reflectances with small entries is reduced by neglecting to normalize.
  36. Also known as principal-components analysis or the Karhunen–Loeve decomposition. See K. V. Mardia, J. T. Kent, J. M. Bibby, Multivariate Analysis (Academic, London, 1979). All characteristic vector analyses and multiple regressions were done in AT&T Bell Laboratories’ statistical language S. See R. A. Becker, J. M. Chambers, S: An Interactive Environment for Data Analysis and Graphics (Wadsworth, Belmont, Calif., 1984).
  37. See R. B. Blackman, J. W. Tukey, The Measurement of Power Spectra from the Point of View of Communications Engineering (Dover, New York; 1959); P. Bloomfield, Fourier Analysis of Time Series: An Introduction (Wiley, New York, 1976).

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