Abstract

The theory of statistical fluctuations has been applied to fitting Munsell data for nearly neutral specimens. This leads to a form of hyperbolic cosine law for lightness value expressed as a function of luminous reflectance. An inverse relationship can be readily obtained in terms of standard mathematical functions. Account is taken of the effects of adaptation.

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References

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  1. M. A. Bouman, J. J. Vos, and P. L. Walraven, J. Opt. Soc.Am. 53, 121 (1963).
    [CrossRef] [PubMed]
  2. S. M. Newhall, D. Nickerson, and D. B. Judd, J. Opt. Soc. Am. 33, 385 (1943).
    [CrossRef]
  3. K. L. Kelly, K. S. Gibson, and D. Nickerson, J. Opt. Soc. Am. 33, 355 (1943).
    [CrossRef]
  4. In lieu of the exponential distribution of times between spikes used by Bouman et al, it is possible to use the more-general probability distribution for which dp = u(t - t0)F{n¯(t - t0)n¯} d t, where u (t) is the Heaviside unit function and ∫-∞ d p = 1. Apart from numerical factors that can be absorbed into k and ƒ, this leads to essentially the same forms of expressions as those given here.
  5. S. M. Newhall, J. Opt. Soc. Am. 30, 617 (1940).
    [CrossRef]
  6. J. A. Nelder and R. Mead, Comput. J. 7, 308 (1965).
    [CrossRef]
  7. P. Moon and D. E. Spencer, J. Opt. Soc. Am. 33, 270 (1943).
    [CrossRef]
  8. Colorimetry, Publication CIE No. 15 (1971), p. 14.
  9. E. A. Trabka, Vision Res. 8, 113 (1968).
    [CrossRef] [PubMed]
  10. E. A. Trabka, J. Opt. Soc. Am. 59, 345 (1969).
    [CrossRef] [PubMed]
  11. W. S. Stiles, Proc. Phys. Soc. Lond. 58, 41 (1945).
    [CrossRef]
  12. G. Wyszecki and W. S. Stiles, Color Science (Wiley, New York, 1967), p. 578.

1969 (1)

E. A. Trabka, J. Opt. Soc. Am. 59, 345 (1969).
[CrossRef] [PubMed]

1968 (1)

E. A. Trabka, Vision Res. 8, 113 (1968).
[CrossRef] [PubMed]

1965 (1)

J. A. Nelder and R. Mead, Comput. J. 7, 308 (1965).
[CrossRef]

1963 (1)

M. A. Bouman, J. J. Vos, and P. L. Walraven, J. Opt. Soc.Am. 53, 121 (1963).
[CrossRef] [PubMed]

1945 (1)

W. S. Stiles, Proc. Phys. Soc. Lond. 58, 41 (1945).
[CrossRef]

1943 (3)

P. Moon and D. E. Spencer, J. Opt. Soc. Am. 33, 270 (1943).
[CrossRef]

S. M. Newhall, D. Nickerson, and D. B. Judd, J. Opt. Soc. Am. 33, 385 (1943).
[CrossRef]

K. L. Kelly, K. S. Gibson, and D. Nickerson, J. Opt. Soc. Am. 33, 355 (1943).
[CrossRef]

1940 (1)

S. M. Newhall, J. Opt. Soc. Am. 30, 617 (1940).
[CrossRef]

Bouman, M. A.

M. A. Bouman, J. J. Vos, and P. L. Walraven, J. Opt. Soc.Am. 53, 121 (1963).
[CrossRef] [PubMed]

Gibson, K. S.

K. L. Kelly, K. S. Gibson, and D. Nickerson, J. Opt. Soc. Am. 33, 355 (1943).
[CrossRef]

Judd, D. B.

S. M. Newhall, D. Nickerson, and D. B. Judd, J. Opt. Soc. Am. 33, 385 (1943).
[CrossRef]

Kelly, K. L.

K. L. Kelly, K. S. Gibson, and D. Nickerson, J. Opt. Soc. Am. 33, 355 (1943).
[CrossRef]

Mead, R.

J. A. Nelder and R. Mead, Comput. J. 7, 308 (1965).
[CrossRef]

Moon, P.

P. Moon and D. E. Spencer, J. Opt. Soc. Am. 33, 270 (1943).
[CrossRef]

Nelder, J. A.

J. A. Nelder and R. Mead, Comput. J. 7, 308 (1965).
[CrossRef]

Newhall, S. M.

S. M. Newhall, D. Nickerson, and D. B. Judd, J. Opt. Soc. Am. 33, 385 (1943).
[CrossRef]

S. M. Newhall, J. Opt. Soc. Am. 30, 617 (1940).
[CrossRef]

Nickerson, D.

S. M. Newhall, D. Nickerson, and D. B. Judd, J. Opt. Soc. Am. 33, 385 (1943).
[CrossRef]

K. L. Kelly, K. S. Gibson, and D. Nickerson, J. Opt. Soc. Am. 33, 355 (1943).
[CrossRef]

Spencer, D. E.

P. Moon and D. E. Spencer, J. Opt. Soc. Am. 33, 270 (1943).
[CrossRef]

Stiles, W. S.

W. S. Stiles, Proc. Phys. Soc. Lond. 58, 41 (1945).
[CrossRef]

G. Wyszecki and W. S. Stiles, Color Science (Wiley, New York, 1967), p. 578.

Trabka, E. A.

E. A. Trabka, J. Opt. Soc. Am. 59, 345 (1969).
[CrossRef] [PubMed]

E. A. Trabka, Vision Res. 8, 113 (1968).
[CrossRef] [PubMed]

Vos, J. J.

M. A. Bouman, J. J. Vos, and P. L. Walraven, J. Opt. Soc.Am. 53, 121 (1963).
[CrossRef] [PubMed]

Walraven, P. L.

M. A. Bouman, J. J. Vos, and P. L. Walraven, J. Opt. Soc.Am. 53, 121 (1963).
[CrossRef] [PubMed]

Wyszecki, G.

G. Wyszecki and W. S. Stiles, Color Science (Wiley, New York, 1967), p. 578.

Other (12)

M. A. Bouman, J. J. Vos, and P. L. Walraven, J. Opt. Soc.Am. 53, 121 (1963).
[CrossRef] [PubMed]

S. M. Newhall, D. Nickerson, and D. B. Judd, J. Opt. Soc. Am. 33, 385 (1943).
[CrossRef]

K. L. Kelly, K. S. Gibson, and D. Nickerson, J. Opt. Soc. Am. 33, 355 (1943).
[CrossRef]

In lieu of the exponential distribution of times between spikes used by Bouman et al, it is possible to use the more-general probability distribution for which dp = u(t - t0)F{n¯(t - t0)n¯} d t, where u (t) is the Heaviside unit function and ∫-∞ d p = 1. Apart from numerical factors that can be absorbed into k and ƒ, this leads to essentially the same forms of expressions as those given here.

S. M. Newhall, J. Opt. Soc. Am. 30, 617 (1940).
[CrossRef]

J. A. Nelder and R. Mead, Comput. J. 7, 308 (1965).
[CrossRef]

P. Moon and D. E. Spencer, J. Opt. Soc. Am. 33, 270 (1943).
[CrossRef]

Colorimetry, Publication CIE No. 15 (1971), p. 14.

E. A. Trabka, Vision Res. 8, 113 (1968).
[CrossRef] [PubMed]

E. A. Trabka, J. Opt. Soc. Am. 59, 345 (1969).
[CrossRef] [PubMed]

W. S. Stiles, Proc. Phys. Soc. Lond. 58, 41 (1945).
[CrossRef]

G. Wyszecki and W. S. Stiles, Color Science (Wiley, New York, 1967), p. 578.

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Figures (2)

Fig. 1
Fig. 1

Least-squares fit of Eq. (8) to Munsell data for neutral samples by varying b. The continuous line shows a plot of the Munsell quintic function V. The solid points correspond to Eq. (10), which is the optimized form of Eq. (8). Original data for neutral Munsell samples, from p. 373 of Ref. 3, are shown by enclosed crosses.

Fig. 2
Fig. 2

By various choices of b, fits have been obtained to estimates of lightness value for Munsell neutral samples seen against a black background and a white background, in addition to the fit of Fig. 1. For a background of reflectance Y0, values of b correspond approximately to 1/b = 5.59 + 1.45 Y0. The uppermost plot is for the black background, with 1/a = 3.34, 1/b = 11.39, and Y0 = 4%. The lowest plot is for the white background, with 1/a = 8.44, 1/b = 130.2, and Y0 = 85%. Data are shown by encircled points and theoretical values by crosses. For the original Munsell data, the reflectance of the grey background was 19.77% (see text).

Equations (13)

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n ¯ = f I 0 Y / ( 1 + I 0 Y 0 ) .
ν ¯ = T A / ( t 0 + 1 / n ¯ ) .
σ ν ¯ = { T A / ( t 0 + 1 / n ¯ ) 3 } 1 2 / n ¯ .
( T A t 0 ) 1 2 δ Y [ Y { Y + ( 1 / I 0 + Y 0 ) / f t 0 } ] 1 2 = ± k .
d Y / { Y ( Y + 2 / b ) } 1 2 = a d V ,
a = C ( t 0 / T A ) 1 2
b = 2 f t 0 / ( 1 / I 0 + Y 0 ) .
a V = cosh 1 ( 1 + b Y ) .
V = 10 cosh 1 ( 1 + b Y ) / cosh 1 ( 1 + 102.56 b ) .
b Y = cosh ( a V ) 1 , with 10 a = cosh 1 ( 1 + 102.56 b ) .
b = 1 / ( 5.59 + 1.45 Y 0 )
Y = 34.26 { cosh ( V / 4.850 ) 1.0 } ,
V = 4.850 cosh 1 ( 1 + Y / 34.26 ) .