Abstract

When successive ridges of distant mountains are seen, observers often report that, near the ridge where the brightness changes abruptly, the upper part of the nearer ridge appears darker than at its lower portions. Similarly, they report that the base of the more distant mountain seems brighter adjacent to the nearer ridge than on its upper portions. The explanation of this phenomenon, known as the step contrast effect, is a special case of Mach bands. It is usually attributed to a visual illusion involving lateral inhibition in the eye, which is most apparent in the vicinity of step brightness changes. Using analytic techniques and numerical integrations to simulate the airlight-induced brightness distributions of such scenes, we show that in many cases the perceived brightness distribution is qualitatively similar to the true brightness distribution and thus is not a visual illusion.

© 1991 Optical Society of America

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References

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  1. M. Minnaert, The Nature of Light and Colour in the Open Air (Bell, London, 1939; reprinted by Dover, New York, 1954), pp. 130–131.
  2. A. Fiorentino, M. Jeanne, G. Toraldo di Francia, “Mesures photometriques visuelles sur un champ á gradient d’éclairement variable,” Opt. Acta 1, 192–193 (1955).
    [Crossref]
  3. A. Fiorentino, “Mach band phenomena,” in Handbook of Sensory Physiology, VII/4, D. Jameson, L. Hurvich, eds. (Springer-Verlag, Berlin, 1972), pp. 188–201.
    [Crossref]
  4. H. Harms, E. Aulhorn, “Studien ueber den Grenzkontrast. I. Mitteilung, Ein neues Grenzphänomen,” Arch. Ophthalmol. (Graefes) 157, 3–23 (1955).
    [Crossref]
  5. F. Ratliff, K. H. Hartline, “The response of Limulus optic nerve fibers to patterns of illumination on the receptor mosaic,” J. Gen. Physiol. 42, 1241–1255 (1959).
    [Crossref] [PubMed]
  6. R. B. Barlow, “Inhibitory fields in the Limulus lateral eye,” thesis (Rockefeller University, New York, N.Y.1967).
  7. F. Ratliff, “Contour and contrast,” J. Am. Phil. Soc. 115(2), 150–163 (Apr.1971).
  8. D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London Ser. B 207, 187–217 (1980).
    [Crossref]
  9. F. Ratliff, Mach Bands: Quantitative Studies on Neural Networks in Retina (Holden-Day, New York, 1965).
  10. T. N. Cornsweet, Visual Perception (Academic, New York, 1970), pp. 342–364.
  11. H. Neuberger, Introduction to Physical Meteorology (Mineral Industries Extension Services, State College, Pa., 1951), pp. 245–247.
  12. W. E. K. Middleton, Vision Through the Atmosphere (U. Toronto Press, Toronto, 1952), p. 61.
  13. T. H. James, G. C. Higgins, Fundamentals of Photographic Theory (Morgan & Morgan, Hastings-on-Hudson, N.Y., 1968).
  14. Scientific Imaging with Kodak Films, and Plates, P-315 (Eastman Kodak Co., Rochester, N.Y., 1987).
  15. J. Stock, W. A. Williams, “Photographic photometry,” in Astronomical Techniques, W. A. Hiltner, ed. (U. Chicago Press, Chicago, Ill., 1960), pp. 386–388.

1980 (1)

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London Ser. B 207, 187–217 (1980).
[Crossref]

1971 (1)

F. Ratliff, “Contour and contrast,” J. Am. Phil. Soc. 115(2), 150–163 (Apr.1971).

1959 (1)

F. Ratliff, K. H. Hartline, “The response of Limulus optic nerve fibers to patterns of illumination on the receptor mosaic,” J. Gen. Physiol. 42, 1241–1255 (1959).
[Crossref] [PubMed]

1955 (2)

A. Fiorentino, M. Jeanne, G. Toraldo di Francia, “Mesures photometriques visuelles sur un champ á gradient d’éclairement variable,” Opt. Acta 1, 192–193 (1955).
[Crossref]

H. Harms, E. Aulhorn, “Studien ueber den Grenzkontrast. I. Mitteilung, Ein neues Grenzphänomen,” Arch. Ophthalmol. (Graefes) 157, 3–23 (1955).
[Crossref]

Aulhorn, E.

H. Harms, E. Aulhorn, “Studien ueber den Grenzkontrast. I. Mitteilung, Ein neues Grenzphänomen,” Arch. Ophthalmol. (Graefes) 157, 3–23 (1955).
[Crossref]

Barlow, R. B.

R. B. Barlow, “Inhibitory fields in the Limulus lateral eye,” thesis (Rockefeller University, New York, N.Y.1967).

Cornsweet, T. N.

T. N. Cornsweet, Visual Perception (Academic, New York, 1970), pp. 342–364.

Fiorentino, A.

A. Fiorentino, M. Jeanne, G. Toraldo di Francia, “Mesures photometriques visuelles sur un champ á gradient d’éclairement variable,” Opt. Acta 1, 192–193 (1955).
[Crossref]

A. Fiorentino, “Mach band phenomena,” in Handbook of Sensory Physiology, VII/4, D. Jameson, L. Hurvich, eds. (Springer-Verlag, Berlin, 1972), pp. 188–201.
[Crossref]

Harms, H.

H. Harms, E. Aulhorn, “Studien ueber den Grenzkontrast. I. Mitteilung, Ein neues Grenzphänomen,” Arch. Ophthalmol. (Graefes) 157, 3–23 (1955).
[Crossref]

Hartline, K. H.

F. Ratliff, K. H. Hartline, “The response of Limulus optic nerve fibers to patterns of illumination on the receptor mosaic,” J. Gen. Physiol. 42, 1241–1255 (1959).
[Crossref] [PubMed]

Higgins, G. C.

T. H. James, G. C. Higgins, Fundamentals of Photographic Theory (Morgan & Morgan, Hastings-on-Hudson, N.Y., 1968).

Hildreth, E.

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London Ser. B 207, 187–217 (1980).
[Crossref]

James, T. H.

T. H. James, G. C. Higgins, Fundamentals of Photographic Theory (Morgan & Morgan, Hastings-on-Hudson, N.Y., 1968).

Jeanne, M.

A. Fiorentino, M. Jeanne, G. Toraldo di Francia, “Mesures photometriques visuelles sur un champ á gradient d’éclairement variable,” Opt. Acta 1, 192–193 (1955).
[Crossref]

Marr, D.

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London Ser. B 207, 187–217 (1980).
[Crossref]

Middleton, W. E. K.

W. E. K. Middleton, Vision Through the Atmosphere (U. Toronto Press, Toronto, 1952), p. 61.

Minnaert, M.

M. Minnaert, The Nature of Light and Colour in the Open Air (Bell, London, 1939; reprinted by Dover, New York, 1954), pp. 130–131.

Neuberger, H.

H. Neuberger, Introduction to Physical Meteorology (Mineral Industries Extension Services, State College, Pa., 1951), pp. 245–247.

Ratliff, F.

F. Ratliff, “Contour and contrast,” J. Am. Phil. Soc. 115(2), 150–163 (Apr.1971).

F. Ratliff, K. H. Hartline, “The response of Limulus optic nerve fibers to patterns of illumination on the receptor mosaic,” J. Gen. Physiol. 42, 1241–1255 (1959).
[Crossref] [PubMed]

F. Ratliff, Mach Bands: Quantitative Studies on Neural Networks in Retina (Holden-Day, New York, 1965).

Stock, J.

J. Stock, W. A. Williams, “Photographic photometry,” in Astronomical Techniques, W. A. Hiltner, ed. (U. Chicago Press, Chicago, Ill., 1960), pp. 386–388.

Toraldo di Francia, G.

A. Fiorentino, M. Jeanne, G. Toraldo di Francia, “Mesures photometriques visuelles sur un champ á gradient d’éclairement variable,” Opt. Acta 1, 192–193 (1955).
[Crossref]

Williams, W. A.

J. Stock, W. A. Williams, “Photographic photometry,” in Astronomical Techniques, W. A. Hiltner, ed. (U. Chicago Press, Chicago, Ill., 1960), pp. 386–388.

Arch. Ophthalmol. (Graefes) (1)

H. Harms, E. Aulhorn, “Studien ueber den Grenzkontrast. I. Mitteilung, Ein neues Grenzphänomen,” Arch. Ophthalmol. (Graefes) 157, 3–23 (1955).
[Crossref]

J. Am. Phil. Soc. (1)

F. Ratliff, “Contour and contrast,” J. Am. Phil. Soc. 115(2), 150–163 (Apr.1971).

J. Gen. Physiol. (1)

F. Ratliff, K. H. Hartline, “The response of Limulus optic nerve fibers to patterns of illumination on the receptor mosaic,” J. Gen. Physiol. 42, 1241–1255 (1959).
[Crossref] [PubMed]

Opt. Acta (1)

A. Fiorentino, M. Jeanne, G. Toraldo di Francia, “Mesures photometriques visuelles sur un champ á gradient d’éclairement variable,” Opt. Acta 1, 192–193 (1955).
[Crossref]

Proc. R. Soc. London Ser. B (1)

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London Ser. B 207, 187–217 (1980).
[Crossref]

Other (10)

F. Ratliff, Mach Bands: Quantitative Studies on Neural Networks in Retina (Holden-Day, New York, 1965).

T. N. Cornsweet, Visual Perception (Academic, New York, 1970), pp. 342–364.

H. Neuberger, Introduction to Physical Meteorology (Mineral Industries Extension Services, State College, Pa., 1951), pp. 245–247.

W. E. K. Middleton, Vision Through the Atmosphere (U. Toronto Press, Toronto, 1952), p. 61.

T. H. James, G. C. Higgins, Fundamentals of Photographic Theory (Morgan & Morgan, Hastings-on-Hudson, N.Y., 1968).

Scientific Imaging with Kodak Films, and Plates, P-315 (Eastman Kodak Co., Rochester, N.Y., 1987).

J. Stock, W. A. Williams, “Photographic photometry,” in Astronomical Techniques, W. A. Hiltner, ed. (U. Chicago Press, Chicago, Ill., 1960), pp. 386–388.

A. Fiorentino, “Mach band phenomena,” in Handbook of Sensory Physiology, VII/4, D. Jameson, L. Hurvich, eds. (Springer-Verlag, Berlin, 1972), pp. 188–201.
[Crossref]

R. B. Barlow, “Inhibitory fields in the Limulus lateral eye,” thesis (Rockefeller University, New York, N.Y.1967).

M. Minnaert, The Nature of Light and Colour in the Open Air (Bell, London, 1939; reprinted by Dover, New York, 1954), pp. 130–131.

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Figures (8)

Fig. 1
Fig. 1

Bands of uniform brightness. Note the bright and dark bands at the step brightness changes.

Fig. 2
Fig. 2

True and perceived brightness distribution across a step in Fig. 1. The response function (lower curve) has negative values that produce lateral inhibition.

Fig. 3
Fig. 3

Schematic diagram of distant mountain ridges used to model the viewing circumstances.

Fig. 4
Fig. 4

Relative brightness I(α) versus altitude α for various scattering scale heights Ho for the geometry shown in Fig. 5.

Fig. 5
Fig. 5

Vertical brightness gradients dI/dα of the brightness distributions I(α) shown in Fig. 6.

Fig. 6
Fig. 6

Foreshortening can cause a uniform distribution of dark trees on light-colored hills to show dark bands on the ridges.

Fig. 7
Fig. 7

Adjacency effects in photographic emulsions; reprinted from Ref. 13.

Fig. 8
Fig. 8

Density tracing across a step brightness change photographed with conventional black-and-white film. This is an example of adjacency effects in photographic emulsions, reprinted from Ref. 13.

Equations (15)

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

I = K 0 R n ( z ) d r = K 0 R n 0 exp ( - z / H o ) d r ,
I ( α ) = - n 0 H o exp ( - h o / H o ) { exp [ - ( R / H o ) sin α ] - 1 } / sin α .
I ( α ) = - n 0 H o exp ( - h o / H o ) [ - ( R / H o ) sin α ] / sin α ,
I ( α ) = n 0 exp ( - h o / H o ) R .
exp [ - ( R / H o ) sin α ] - 1 ~ exp [ - ( R / H o ) sin α ] ,
I = - n 0 H o exp ( - h o / H o ) exp ( - R sin α / H o ) / sin α .
R ( α ) = [ ( h o - A ) / ( tan β - tan α ) ] sec 2 α ,
d R / d α = R [ 1 + sec 2 α / ( tan β - tan α ) ] ,
H o = R cos α tan α + R C sin α tan α ,
C = tan α + [ sec 2 α / ( tan β - tan α ) + cos α ] .
H o = R tan α ( 1 + C sin α ) .
H o = R tan α = R sin α
H o = h o .
H o = h m .
I = i n ( r i ) Δ r i ,

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