Abstract

A lightness scale is derived from a theoretical estimate of the probability distribution of image intensities for natural scenes. The derived image intensity distribution considers three factors: reflectance; surface orientation and illumination; and surface texture (or roughness). The convolution of the effects of these three factors yields the theoretical probability distribution of image intensities. A useful lightness scale should be the integral of this probability density function, for then equal intervals along the scale are equally probable and carry equal information. The result is a scale similar to that used in photography or by the nervous system as its transfer function.

© 1982 Optical Society of America

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References

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  1. D. B. Judd, G. Wysecki, Color in Business, Science and Industry (Wiley, New York, 1975).
  2. G. T. Fechner, Elemente der Psychophysik (Bretkopf und Hartel, Leipzig, 1860).
  3. S. Stevens, Science 133, 80 (1961).
    [CrossRef] [PubMed]
  4. W. A. van de Grind, J. J. Koederink, H. A. A. Landman, M. A. Bouman, Kybernetik 8, 105 (1971).
    [CrossRef] [PubMed]
  5. M. Treisman, Percept. Psychophys. 1, 203 (1966).
  6. D. M. MacKay, Science 139, 1213 (1963).
    [CrossRef]
  7. H. L. Resnikoff, J. Math. Biol. 2, 265 (1975).
    [CrossRef]
  8. D. C. Marr, Vision: A Computational Investigation into the Human Representation and Processing of Visual Information (Freeman, San Francisco, 1982).
  9. H. A. T. Keitz, Light Calculations and Measurements (St. Martins Press, New York, 1971).
  10. B. K. P. Horn, R. W. Sjoberg, Appl. Opt. 18, 1770 (1979).
    [CrossRef] [PubMed]
  11. H. L. F. Helmholtz, Treatise on Physiological Optics, translated by J. P. Sonthall, (Dover, New York, 1910).
  12. E. H. Land, J. J. McCann, J. Opt. Soc. Am. 61, 1 (1971).
    [CrossRef] [PubMed]
  13. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).
  14. L. Brillouin, Science and Information Theory (Academic, New York, 1962).
  15. G. K. Zipf, Human Behavior and the Principal of Least Effort (Addison-Wesley, Reading, Mass., 1949).
  16. W. Richards, Tech. Eng. News 48, 11 (1967).
  17. P. Kubelka, F. Munk, Z. Tech. Phys. 12, 593 (1931).
  18. E. L. Krinov, Spectral Reflectance Properties of Natural Formations, translated by G. Belkov (NRC Canada, Technical Translation 439, 1971).
  19. H. R. Davidson, H. Hemmendinger, J. Opt. Soc. Am. 56, 1102 (1966).
    [CrossRef]
  20. G. Wyszecki, W. S. Stiles, Color Science (Wiley, New York, 1967).
  21. K. E. Torrance, E. M. Sparrow, J. Opt. Soc. Am. 57, 1105 (1967).
    [CrossRef]
  22. L. A. Jones, H. R. Condit, J. Opt. Soc. Am. 39, 94 (1949);C. N. Nelson, in Theory of Photographic Process, C. E. K. Mees, T. H. James, Eds. (Macmillan, New York, 1966), Chap. 22.
    [CrossRef] [PubMed]
  23. G. Priest, K. S. Gibson, H. J. McNicholas, “An Examination of the Munsell Color System,” Natl. Bur. Stand. Tech. Paper 167 (1920).
  24. K. J. Naka, W. A. H. Rushton, J. Physiol. 185, 536 (1966).
    [PubMed]
  25. R. A. Norman, F. S. Werblin, J. Gen. Physiol. 63, 37 (1974).
    [CrossRef]
  26. D. C. Hood, M. A. Finkelstein, E. Buckingham, Vis. Res. 19, 401 (1979).
    [CrossRef]
  27. S. Laughlin, Z. Naturforsch. 36, 910 (1981).

1981

S. Laughlin, Z. Naturforsch. 36, 910 (1981).

1979

D. C. Hood, M. A. Finkelstein, E. Buckingham, Vis. Res. 19, 401 (1979).
[CrossRef]

B. K. P. Horn, R. W. Sjoberg, Appl. Opt. 18, 1770 (1979).
[CrossRef] [PubMed]

1975

H. L. Resnikoff, J. Math. Biol. 2, 265 (1975).
[CrossRef]

1974

R. A. Norman, F. S. Werblin, J. Gen. Physiol. 63, 37 (1974).
[CrossRef]

1971

W. A. van de Grind, J. J. Koederink, H. A. A. Landman, M. A. Bouman, Kybernetik 8, 105 (1971).
[CrossRef] [PubMed]

E. H. Land, J. J. McCann, J. Opt. Soc. Am. 61, 1 (1971).
[CrossRef] [PubMed]

1967

1966

H. R. Davidson, H. Hemmendinger, J. Opt. Soc. Am. 56, 1102 (1966).
[CrossRef]

K. J. Naka, W. A. H. Rushton, J. Physiol. 185, 536 (1966).
[PubMed]

M. Treisman, Percept. Psychophys. 1, 203 (1966).

1963

D. M. MacKay, Science 139, 1213 (1963).
[CrossRef]

1961

S. Stevens, Science 133, 80 (1961).
[CrossRef] [PubMed]

1949

1931

P. Kubelka, F. Munk, Z. Tech. Phys. 12, 593 (1931).

1920

G. Priest, K. S. Gibson, H. J. McNicholas, “An Examination of the Munsell Color System,” Natl. Bur. Stand. Tech. Paper 167 (1920).

Bouman, M. A.

W. A. van de Grind, J. J. Koederink, H. A. A. Landman, M. A. Bouman, Kybernetik 8, 105 (1971).
[CrossRef] [PubMed]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).

Brillouin, L.

L. Brillouin, Science and Information Theory (Academic, New York, 1962).

Buckingham, E.

D. C. Hood, M. A. Finkelstein, E. Buckingham, Vis. Res. 19, 401 (1979).
[CrossRef]

Condit, H. R.

Davidson, H. R.

Fechner, G. T.

G. T. Fechner, Elemente der Psychophysik (Bretkopf und Hartel, Leipzig, 1860).

Finkelstein, M. A.

D. C. Hood, M. A. Finkelstein, E. Buckingham, Vis. Res. 19, 401 (1979).
[CrossRef]

Gibson, K. S.

G. Priest, K. S. Gibson, H. J. McNicholas, “An Examination of the Munsell Color System,” Natl. Bur. Stand. Tech. Paper 167 (1920).

Helmholtz, H. L. F.

H. L. F. Helmholtz, Treatise on Physiological Optics, translated by J. P. Sonthall, (Dover, New York, 1910).

Hemmendinger, H.

Hood, D. C.

D. C. Hood, M. A. Finkelstein, E. Buckingham, Vis. Res. 19, 401 (1979).
[CrossRef]

Horn, B. K. P.

Jones, L. A.

Judd, D. B.

D. B. Judd, G. Wysecki, Color in Business, Science and Industry (Wiley, New York, 1975).

Keitz, H. A. T.

H. A. T. Keitz, Light Calculations and Measurements (St. Martins Press, New York, 1971).

Koederink, J. J.

W. A. van de Grind, J. J. Koederink, H. A. A. Landman, M. A. Bouman, Kybernetik 8, 105 (1971).
[CrossRef] [PubMed]

Krinov, E. L.

E. L. Krinov, Spectral Reflectance Properties of Natural Formations, translated by G. Belkov (NRC Canada, Technical Translation 439, 1971).

Kubelka, P.

P. Kubelka, F. Munk, Z. Tech. Phys. 12, 593 (1931).

Land, E. H.

Landman, H. A. A.

W. A. van de Grind, J. J. Koederink, H. A. A. Landman, M. A. Bouman, Kybernetik 8, 105 (1971).
[CrossRef] [PubMed]

Laughlin, S.

S. Laughlin, Z. Naturforsch. 36, 910 (1981).

MacKay, D. M.

D. M. MacKay, Science 139, 1213 (1963).
[CrossRef]

Marr, D. C.

D. C. Marr, Vision: A Computational Investigation into the Human Representation and Processing of Visual Information (Freeman, San Francisco, 1982).

McCann, J. J.

McNicholas, H. J.

G. Priest, K. S. Gibson, H. J. McNicholas, “An Examination of the Munsell Color System,” Natl. Bur. Stand. Tech. Paper 167 (1920).

Munk, F.

P. Kubelka, F. Munk, Z. Tech. Phys. 12, 593 (1931).

Naka, K. J.

K. J. Naka, W. A. H. Rushton, J. Physiol. 185, 536 (1966).
[PubMed]

Norman, R. A.

R. A. Norman, F. S. Werblin, J. Gen. Physiol. 63, 37 (1974).
[CrossRef]

Priest, G.

G. Priest, K. S. Gibson, H. J. McNicholas, “An Examination of the Munsell Color System,” Natl. Bur. Stand. Tech. Paper 167 (1920).

Resnikoff, H. L.

H. L. Resnikoff, J. Math. Biol. 2, 265 (1975).
[CrossRef]

Richards, W.

W. Richards, Tech. Eng. News 48, 11 (1967).

Rushton, W. A. H.

K. J. Naka, W. A. H. Rushton, J. Physiol. 185, 536 (1966).
[PubMed]

Sjoberg, R. W.

Sparrow, E. M.

Stevens, S.

S. Stevens, Science 133, 80 (1961).
[CrossRef] [PubMed]

Stiles, W. S.

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

Torrance, K. E.

Treisman, M.

M. Treisman, Percept. Psychophys. 1, 203 (1966).

van de Grind, W. A.

W. A. van de Grind, J. J. Koederink, H. A. A. Landman, M. A. Bouman, Kybernetik 8, 105 (1971).
[CrossRef] [PubMed]

Werblin, F. S.

R. A. Norman, F. S. Werblin, J. Gen. Physiol. 63, 37 (1974).
[CrossRef]

Wysecki, G.

D. B. Judd, G. Wysecki, Color in Business, Science and Industry (Wiley, New York, 1975).

Wyszecki, G.

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

Zipf, G. K.

G. K. Zipf, Human Behavior and the Principal of Least Effort (Addison-Wesley, Reading, Mass., 1949).

Appl. Opt.

J. Gen. Physiol.

R. A. Norman, F. S. Werblin, J. Gen. Physiol. 63, 37 (1974).
[CrossRef]

J. Math. Biol.

H. L. Resnikoff, J. Math. Biol. 2, 265 (1975).
[CrossRef]

J. Opt. Soc. Am.

J. Physiol.

K. J. Naka, W. A. H. Rushton, J. Physiol. 185, 536 (1966).
[PubMed]

Kybernetik

W. A. van de Grind, J. J. Koederink, H. A. A. Landman, M. A. Bouman, Kybernetik 8, 105 (1971).
[CrossRef] [PubMed]

Natl. Bur. Stand. Tech. Paper

G. Priest, K. S. Gibson, H. J. McNicholas, “An Examination of the Munsell Color System,” Natl. Bur. Stand. Tech. Paper 167 (1920).

Percept. Psychophys.

M. Treisman, Percept. Psychophys. 1, 203 (1966).

Science

D. M. MacKay, Science 139, 1213 (1963).
[CrossRef]

S. Stevens, Science 133, 80 (1961).
[CrossRef] [PubMed]

Tech. Eng. News

W. Richards, Tech. Eng. News 48, 11 (1967).

Vis. Res.

D. C. Hood, M. A. Finkelstein, E. Buckingham, Vis. Res. 19, 401 (1979).
[CrossRef]

Z. Naturforsch.

S. Laughlin, Z. Naturforsch. 36, 910 (1981).

Z. Tech. Phys.

P. Kubelka, F. Munk, Z. Tech. Phys. 12, 593 (1931).

Other

E. L. Krinov, Spectral Reflectance Properties of Natural Formations, translated by G. Belkov (NRC Canada, Technical Translation 439, 1971).

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

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).

L. Brillouin, Science and Information Theory (Academic, New York, 1962).

G. K. Zipf, Human Behavior and the Principal of Least Effort (Addison-Wesley, Reading, Mass., 1949).

D. B. Judd, G. Wysecki, Color in Business, Science and Industry (Wiley, New York, 1975).

G. T. Fechner, Elemente der Psychophysik (Bretkopf und Hartel, Leipzig, 1860).

D. C. Marr, Vision: A Computational Investigation into the Human Representation and Processing of Visual Information (Freeman, San Francisco, 1982).

H. A. T. Keitz, Light Calculations and Measurements (St. Martins Press, New York, 1971).

H. L. F. Helmholtz, Treatise on Physiological Optics, translated by J. P. Sonthall, (Dover, New York, 1910).

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Figures (17)

Fig. 1
Fig. 1

Envelope characterizes a possible distribution of image intensities. Ideal sampling would require the measurements to be taken so that each interval has an equal area under the curve and hence is equally likely.

Fig. 2
Fig. 2

Lightness scales constructed by integration of various theoretical density functions for image intensities. The ordinate is in units of standard deviation, and thus a Gaussian probability distribution of (log) image intensities, such as in Fig. 1, will yield a straight line. A more plausible basis for a lightness scale is the Kulbelka-Munk theory of reflectance, which yields the K-M Theory curve. This result begins to approximate a common scale (Munsell) shown at left.

Fig. 3
Fig. 3

A comparison of empirical and theoretical probability distributions of reflectance. The histogram is an empirical distribution of the reflectances of natural materials taken from a compilation by Krinov.18 The smooth curve is a theoretical probability distribution based upon an extension of the Kubelka-Munk theory.

Fig. 4
Fig. 4

A hemisphere illuminated by an overhead source lying along L at 90° to the viewer's line of regard V. As before, N is the surface normal, and J is the flux emitted in the viewer's direction. The inset defines the slant angle σ. Orthographic projections and Lambertian surfaces are assumed.

Fig. 5
Fig. 5

Probability distribution functions of intensity for various amounts of an extended source, which, together with an overhead point source, illuminate a uniform distribution of surface orientations. Point source alone: E = 0; 20% extended illumination: E = 0.2; extended illumination alone: E = 1.

Fig. 6
Fig. 6

(A) Probability distribution of intensity values for a surface textured by cylinders and illuminated by a hemispheric source such as the sky. Each curve is for a different separation of the cylinders, with the gap S measured in radius units. (B) Distribution of intensity values for a surface pebbled by abutting spheres, illuminated by an extended overhead source such as the sky (see Fig. 17 for a comparison of the model with an image intensity distribution taken from a natural object).

Fig. 7
Fig. 7

Probability distribution of (log) intensity values for a surface pebbled with closely packed spheres T. The components T1 and T2 describe the density functions for image intensities arising from the surface or the gap between the spheres, respectively. The dashed curve, labeled J, is the joint intensity distribution function for the pebbled texture and an overhead illuminant.

Fig. 8
Fig. 8

Three-factor probability density function for image intensities (solid line) resulting from the convolution of intensity variations introduced by reflectance, surface orientation, and surface roughness (texture). The ogive is the integral of this function and is the theoretical lightness scale. The pluses describe the Naka-Rushton neural transfer function.

Fig. 9
Fig. 9

Distribution of the range of luminances found by Jones and Condit22 in 121 outdoor scenes is given by open circles. The solid curve is a distribution of the range predicted from the theoretical distribution of image intensities. The dashed curve is the prediction based on the best log–normal approximation to the theoretical image intensity distribution.

Fig. 10
Fig. 10

Two theoretical distributions of reflectance. The solid curve assumes a scatter-to-absorption ratio α of 0.5 and the same distribution as in text Fig. 3 but on a logI axis. The dashed curve assumes α = 1. Although the two means are clearly different, the standard deviations are almost the same.

Fig. 11
Fig. 11

Effect of variations of the two parameters α, β on the mean and standard deviation of the probability density function of reflectances as derived from the Kubelka-Munk theory. The practical range of α and β is indicated by the dashed rectangle. Note that over this range, the standard deviation of the distribution does not change appreciably.

Fig. 12
Fig. 12

Upper solid line shows the intensity distribution function for a single overhead point source illuminating a uniform distribution of Lambertian surfaces. If extended illumination is also present, the intensities are shifted to the right as indicated by the dashed line where 25% additional illumination everywhere is added as an example. When this new distribution is normalized so the maximum intensity is 1, the lower curve results, which is the expected image intensity distribution for an illuminant consisting of an overhead point source and plus an extended light source. These functions are also replotted in Fig. 5 on a logI intensity axis.

Fig. 13
Fig. 13

Cross-sectional view of three cylinders seen on end beneath a hemisphere of uniform illumination such as the sky.

Fig. 14
Fig. 14

Intensity profiles along a surface comprised of cylinders: (A) cylinders touch one another; (B) separation is 0.5 radius units; (C) 1.0 units; (D) 2 radius units. Note that the profile along the cylinder surface is relatively independent of gap size.

Fig. 15
Fig. 15

Overhead view of a portion of a surface pebbled with identical Lambertian spheres. The intensity profiles of the loci A, B, and C as shown in Fig. 16.

Fig. 16
Fig. 16

Three intensity profiles along the abutting spheres as shown in Fig. 15.

Fig. 17
Fig. 17

Comparison between an empirically measured image intensity distribution for a roughly textured surface and the pebble surface model. The smooth curve is the intensity histogram for the rhododendron bush shown in the top portion of the figure. The pebble surface prediction is the dashed line taken from Fig. 6(B) (courtesy of D. D. Hoffman).

Tables (2)

Tables Icon

Table I Summary of the Expected Means and Variances of the Separate Major Factors that Create Intensity Variations in an Image

Tables Icon

Table II “Luminance Values in an Outdoor Scene” (from Jones and Condit)22

Equations (38)

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I ( λ ) = ρ ( λ ) E ( λ ) ( NL ) R ( σ , θ ) ,
p 2 = 1 2 π a b exp ½ ( log I / I avg ) 2 d ( log I ) ,
H = i N p i log p i ,
ρ = 1 + C α ( 1 C ) C α ( 1 C ) [ 1 + 2 ( 1 C ) C α ] 1 / 2
pdf ( ρ ) = pdf ( C ) d ( C ) d ( ρ ) .
pdf ( C ) = C β ,
pdf ( ρ ) = 2 α β + 1 ( 1 ρ ) 2 β ( 1 ρ 2 ) [ α + ( 2 2 α ) ρ + ρ 2 ] β + 2 ,
pdf ( I ) = ( 1 I 2 ) ,
pdf ( I ) = I 2 ( 1 I 2 ) 1 / 3 .
V / V * = I e / ( I e + σ e ) ,
ρ = 1 + K / S ( K 2 / S 2 + 2 K / S ) 1 / 2 .
K / S = C K P / ( 1 C ) S S .
K / S = C α / ( 1 C ) .
ρ = 1 + C α ( 1 C ) C α ( 1 C ) [ 1 + 2 ( 1 C ) C α ] 1 / 2
J θ = R F ( N θ L ) = R F cos θ ,
J θ , σ = J θ ( N V ) = J θ cos σ .
I θ , σ = J θ , σ / cos σ = J θ ,
I θ , σ = R F cos θ = I θ .
A θ = ϕ = 0 π / 2 π sin θ cos σ d ϕ ,
cos σ = sin θ cos ϕ .
A θ = π sin 2 θ .
sin θ = ( 1 cos 2 θ ) 1 / 2 = [ 1 ( I θ 2 R F ) ] 1 / 2 .
pdf ( I ) = ( 1 I 2 ) .
T reflected = ρ = 0 1 pdf ( ρ ) ρ d ρ .
T incident = ρ = 0 1 pdf ( ρ ) d ρ .
I = E I max + ( 1 E ) I .
I = ( I E 1 ) / ( 1 E ) .
pdf ( I ) = 1 ( I E 1 E ) 2
pdf ( I ) = 1 ( 1 E ) [ 1 ( I E 1 E ) 2 ]
I ( y ) = ( π θ ϕ ) / π 0 y 1 ,
θ = arcsin ( y ) ,
ϕ = arctan ( 1 / B ) arctan ( cos θ / A ) ,
B = ( A 2 + cos 2 θ 1 ) 1 / 2 .
I ( y ) = ( π 2 arctan 1 A 2 arctan 1 y ) / π 1 < y < ( S + 1 ) ,
I ( y ) = 1 ( 3 / 4 ) S 1 < y < ( S + 1 ) .
pdf ( I ) = p ( y ) d y / d I ,
pdf ( I S ) = I 2 ( 1 I 2 ) 1 / 3
p ( I S ) = I 1 I log 1 1 I , 0 I 0.5 ,

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