D. L. MacAdam, Color Measurement; Theme and Variations (Springer-Verlag, Berlin, 1981), p. 128.
J. N. Lythgoe, The Ecology of Vision (Clarendon, Oxford, 1979), p. 31.
H. Yilmaz, “Color vision and a new approach to color perception,” in Biological Prototypes and Synthetic Systems (Plenum, New York, 1962), pp. 126–141.
J. M. Tenenbaum, Ph.D. dissertation (Stanford University, Stanford, Calif., 1971).
P. Sallstrom, “Colour and physics: some remarks concerning the physical aspects of human colour vision,” Inst. Phys. Rep. no. 73-09 (Institute of Physics, University of Stockholm, 1973).
E. L. Krinov, Spectral Reflectance Properties of Natural Formations (National Research Council of Canada, Ottawa, technical translation: TT-439, 1947). Krinov reports 370 surface spectral reflectances. The 337 analyzed were those that had complete and legible data available from 400 to 650 nm.
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.
Regression on the first three characteristic vectors derived from the Munsell data directly produced similar results.
K. Nassau, The Physics and Chemistry of Color: The Fifteen Causes of Color (Wiley, New York, 1983).
H. Suzuki, Electronic Absorption Spectra and Geometry of Organic Molecules; An Application of Molecular Orbital Theory (Academic, New York, 1967), p. 79.
W. Kauzmann, Quantum Chemistry (Academic, New York, 1957), pp. 582 and 669.
W. Kauzmann, ibid., pp. 669–670.
P. Pringsheim, Fluorescence and Phosphorescence (Interscience, New York, 1949).
T. H. Wonnacott, R. J. Wonnacott, Regression: A Second Course in Statistics (Wiley, New York, 1981).
G. Wyszecki, W. S. Stiles, Color Science; Concepts and Methods, Quantitative Data and Formulas, 2nd ed. (Wiley, New York, 1982), p. 256.
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.
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.
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.
See D. A. Belsey, E. Kuh, R. W. Welsch, Regression Diagnostics; Identifying Influential Data and Sources of Collinearity (Wiley, New York, 1980).
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.
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).
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).
L. T. Maloney, “Computational approaches to color constancy,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1984).