Abstract

The movement parallax field due to the translation of an observer relative to a plane surface is studied in an infinitesimal neighborhood of a visual direction. The parallax field is decomposed into elementary transformations: a translation, a rigid rotation, a similarity, and a deformation. A topologically invariant classification based on critical-point analysis is also obtained. It is shown that the field is either that of a node or that of a saddle point. Numerical results for a general case are offered as illustration. We discuss the relevance of the local, as opposed to the global structure of the parallax field for visual perception and the visual space sense.

© 1976 Optical Society of America

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References

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  1. J. J. Gibson, The Perception of the Visual World (Riverside, Cambridge, England, 1950).
  2. D. A. Gordon, J. Opt. Soc. Am. 55, 1296 (1965).
    [CrossRef] [PubMed]
  3. H. L. F. von Helmholtz, Treatise on Physiological Optics. Vol III, translated by J. P. C. Southall (Optical Society of America, Rochester, 1924–1925).
  4. M. H. Pirenne, Optics, Painting and Photography (University Press, Cambridge, England, 1970).
  5. K. Nakayama and J. M. Loomis, Perception 3, 63 (1974).
    [CrossRef]
  6. J. J. Gibson, P. Olum, and F. Rosenblatt, Am. J. Psych. 68, 372 (1958).
  7. Gibson (Ref. 1) uses “gradient” in an intuitive way; Gordon (Ref. 2) talks of “parallax curl.” Neither gives these terms an unambiguous mathematical meaning. The rudiments of “parallax curl” were already discussed by Euclid.
  8. This is the name given by Gibson (Ref. 1) to the manifold of visual directions.
  9. S. Lefschetz, Differential Equations: Geometric Theory (Interscience, New York, 1957).
  10. T. C. D. Whiteside and G. D. Samuel, Nature 225, 94 (1970).
    [CrossRef] [PubMed]
  11. Gibson (Ref. 1) stresses the importance of “gradients” of the optical flow, hence of the local structure. Gordon’s (Ref. 2) “positional velocity field” is a structured entity, but this is not explicit in his formulation. Nakayama and Loomis (Ref. 5) also stress the fact that the velocity field is continuous, but their “convexity detectors” detect discontinuities, and they make no further use of the structure.
  12. The intuitive notion of “slant” is probably best represented by the vector (1/ρ) ∇ρ. This vector specifies the direction of the surface normal relative to the visual direction, irrespective of the absolute distance. From Eqs. (19)–(21) we have def D = ||vang|| · || ∇ (1/ρ)||. This, together with the direction of the deformation axes, gives us ∇ (1/ρ).
  13. The deformation describes the nonconformal component of the perspective transformations. The action of the deformation component is to alter the mutual angle relations of image detail; conversely the deformation field can easily be found from the spatiotemporal change of the mutual orientation of image details.
  14. D. H. Hubel and T. N. Wiesel, J. Neurophysiol. 28, 229 (1965).
    [PubMed]
  15. J. B. J. Riemersma, K. W. Mess, and J. A. Michon, Institute for Perception TNO, (Soesterberg, 1972).
  16. Committee on Vision, Assembly of Behavioral and Social Sciences, National Research Council, Visual Elements in Flight Simulation, Report of Working Group 34 (National Academy of Sciences, Washington, D. C., 1975). We quote the following. “It may prove useful to consider various dimensions of the visual world and their possible significance in the design of a satisfactory visual simulation,” (p. 8); “We have woefully little information on the response characteristics of the visual system for discrimination of movement on the basis of which to determine what sorts of approximation can be made without any loss in the realism of the display,” (p. 13). In our opinion a theoretical description of the stimulus is a prerequisite of any experimental approach to solve these questions.

1974 (1)

K. Nakayama and J. M. Loomis, Perception 3, 63 (1974).
[CrossRef]

1970 (1)

T. C. D. Whiteside and G. D. Samuel, Nature 225, 94 (1970).
[CrossRef] [PubMed]

1965 (2)

D. H. Hubel and T. N. Wiesel, J. Neurophysiol. 28, 229 (1965).
[PubMed]

D. A. Gordon, J. Opt. Soc. Am. 55, 1296 (1965).
[CrossRef] [PubMed]

1958 (1)

J. J. Gibson, P. Olum, and F. Rosenblatt, Am. J. Psych. 68, 372 (1958).

Gibson, J. J.

J. J. Gibson, P. Olum, and F. Rosenblatt, Am. J. Psych. 68, 372 (1958).

J. J. Gibson, The Perception of the Visual World (Riverside, Cambridge, England, 1950).

Gordon, D. A.

Hubel, D. H.

D. H. Hubel and T. N. Wiesel, J. Neurophysiol. 28, 229 (1965).
[PubMed]

Lefschetz, S.

S. Lefschetz, Differential Equations: Geometric Theory (Interscience, New York, 1957).

Loomis, J. M.

K. Nakayama and J. M. Loomis, Perception 3, 63 (1974).
[CrossRef]

Mess, K. W.

J. B. J. Riemersma, K. W. Mess, and J. A. Michon, Institute for Perception TNO, (Soesterberg, 1972).

Michon, J. A.

J. B. J. Riemersma, K. W. Mess, and J. A. Michon, Institute for Perception TNO, (Soesterberg, 1972).

Nakayama, K.

K. Nakayama and J. M. Loomis, Perception 3, 63 (1974).
[CrossRef]

Olum, P.

J. J. Gibson, P. Olum, and F. Rosenblatt, Am. J. Psych. 68, 372 (1958).

Pirenne, M. H.

M. H. Pirenne, Optics, Painting and Photography (University Press, Cambridge, England, 1970).

Riemersma, J. B. J.

J. B. J. Riemersma, K. W. Mess, and J. A. Michon, Institute for Perception TNO, (Soesterberg, 1972).

Rosenblatt, F.

J. J. Gibson, P. Olum, and F. Rosenblatt, Am. J. Psych. 68, 372 (1958).

Samuel, G. D.

T. C. D. Whiteside and G. D. Samuel, Nature 225, 94 (1970).
[CrossRef] [PubMed]

von Helmholtz, H. L. F.

H. L. F. von Helmholtz, Treatise on Physiological Optics. Vol III, translated by J. P. C. Southall (Optical Society of America, Rochester, 1924–1925).

Whiteside, T. C. D.

T. C. D. Whiteside and G. D. Samuel, Nature 225, 94 (1970).
[CrossRef] [PubMed]

Wiesel, T. N.

D. H. Hubel and T. N. Wiesel, J. Neurophysiol. 28, 229 (1965).
[PubMed]

Am. J. Psych. (1)

J. J. Gibson, P. Olum, and F. Rosenblatt, Am. J. Psych. 68, 372 (1958).

J. Neurophysiol. (1)

D. H. Hubel and T. N. Wiesel, J. Neurophysiol. 28, 229 (1965).
[PubMed]

J. Opt. Soc. Am. (1)

Nature (1)

T. C. D. Whiteside and G. D. Samuel, Nature 225, 94 (1970).
[CrossRef] [PubMed]

Perception (1)

K. Nakayama and J. M. Loomis, Perception 3, 63 (1974).
[CrossRef]

Other (11)

J. J. Gibson, The Perception of the Visual World (Riverside, Cambridge, England, 1950).

H. L. F. von Helmholtz, Treatise on Physiological Optics. Vol III, translated by J. P. C. Southall (Optical Society of America, Rochester, 1924–1925).

M. H. Pirenne, Optics, Painting and Photography (University Press, Cambridge, England, 1970).

Gibson (Ref. 1) uses “gradient” in an intuitive way; Gordon (Ref. 2) talks of “parallax curl.” Neither gives these terms an unambiguous mathematical meaning. The rudiments of “parallax curl” were already discussed by Euclid.

This is the name given by Gibson (Ref. 1) to the manifold of visual directions.

S. Lefschetz, Differential Equations: Geometric Theory (Interscience, New York, 1957).

Gibson (Ref. 1) stresses the importance of “gradients” of the optical flow, hence of the local structure. Gordon’s (Ref. 2) “positional velocity field” is a structured entity, but this is not explicit in his formulation. Nakayama and Loomis (Ref. 5) also stress the fact that the velocity field is continuous, but their “convexity detectors” detect discontinuities, and they make no further use of the structure.

The intuitive notion of “slant” is probably best represented by the vector (1/ρ) ∇ρ. This vector specifies the direction of the surface normal relative to the visual direction, irrespective of the absolute distance. From Eqs. (19)–(21) we have def D = ||vang|| · || ∇ (1/ρ)||. This, together with the direction of the deformation axes, gives us ∇ (1/ρ).

The deformation describes the nonconformal component of the perspective transformations. The action of the deformation component is to alter the mutual angle relations of image detail; conversely the deformation field can easily be found from the spatiotemporal change of the mutual orientation of image details.

J. B. J. Riemersma, K. W. Mess, and J. A. Michon, Institute for Perception TNO, (Soesterberg, 1972).

Committee on Vision, Assembly of Behavioral and Social Sciences, National Research Council, Visual Elements in Flight Simulation, Report of Working Group 34 (National Academy of Sciences, Washington, D. C., 1975). We quote the following. “It may prove useful to consider various dimensions of the visual world and their possible significance in the design of a satisfactory visual simulation,” (p. 8); “We have woefully little information on the response characteristics of the visual system for discrimination of movement on the basis of which to determine what sorts of approximation can be made without any loss in the realism of the display,” (p. 13). In our opinion a theoretical description of the stimulus is a prerequisite of any experimental approach to solve these questions.

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Figures (8)

FIG. 1
FIG. 1

To the left, the parallax field at a stable node. (At an unstable node the arrows point away from the origin.) This is the general case. Degenerate cases may occur for which the field is somewhat simpler. The directions of the characteristic vectors of the gradient are dotted. To the right, the parallax field at a saddle point. The general case is depicted. For a pure deformation field (div D = curl D = 0) the characteristic directions (axes of contraction and expansion; dotted lines) are perpendicular to each other.

FIG. 2
FIG. 2

Part of the plane p in central projection. B is the point of p nearest to the observer. The vertical axis through B is the meridian φ = 0°, 180°. The meridian φ = 0° carries a scale for the angle θ in degrees (because of the central projection this scale is nonlinear). The circle η and the line ζ correspond with Eqs. (24) and (25). The circle ξ is the locus of zero divergence. The curl is zero on the meridian φ = 0°, 180°. The observer is headed towards point A (θ = α = 60°, φ = 0°). Extrema of the divergence are at θ = 30°, φ = 0°, and at θ = 60°, φ = 180° (point D). Extrema of the curl are at θ = 90°, φ = 90° (point E), and at θ = 90°, φ = 270° (point F). The deformation is everywhere positive. It is zero at point A and B. Extrema of the magnitude of D are at θ = 60°, φ = 0° (||D|| = 0), and at θ = 15°, φ = 180° (point C).

FIG. 3
FIG. 3

An impression of the distortion of details in the visual field due to the angular velocity field generated by the translation of an observer relative to a plane surface. This figure (in contradistinction to Figs. 2 and 48) is in stereographic projection, so that the whole plane is depicted. At t = 0 a square grid was superimposed on the stereographic projection. This grid was permitted to be carried along with the optical flow for a short period. Because of the finite period we used an exact numerical calculation, not the first order approximation given in the text. The horizon is a circle, touching the edges of the figure at all four sides. The point of the plane at the shortest distance to the observer is the center of the figure. We took α = 60°. The various transformations (congruences, similarities, and deformations) discussed in the the text are clearly perceived in this figure because the stereographic projection is conformal.

FIG. 4
FIG. 4

Contour lines of the value of ||D||, generated by the same field as that depicted in Fig. 3. Central projection was used. The point of the plane at the shortest distance to the observer is at the center of the figure. Figures 2 and 4 are on the same scale. Contour lines were generated at equal intervals. Locations of the extrema of ||D|| are depicted in Fig. 2.

FIG. 5
FIG. 5

Contour lines of the value of the divergence, generated by the same field as that depicted in Fig. 3. Central projection was used. The point of the plane at the shortest distance to the observer is at the center of the figure. Figures 2 and 5 are on the same scale. Contour lines were generated at equal intervals. Locations of the contour of zero divergence, and of the extrema are depicted in Fig. 2.

FIG. 6
FIG. 6

Contour lines of the value of the curl, generated by the same field as that depicted in Fig. 3. Central projection was used. The point of the plane at the shortest distance to the observer is at the center of the figure. Figures 2 and 6 are on the same scale. Contour lines were generated at equal intervals. Locations of the contour of zero curl, and of the extrema are depicted in Fig. 2.

FIG. 7
FIG. 7

Contour lines of the value of the deformation, generated by the same field as that depicted in Fig. 3. Central projection was used. The point of the plane at the shortest distance to the observer is at the center of the figure. Figures 2 and 7 are on the same scale. Contour lines were generated at equal intervals. Locations of the points with zero deformation are depicted in Fig. 2.

FIG. 8
FIG. 8

The directions of the axes of expansion (closed curves) and contraction (open curves) of the deformation component of the field depicted in Fig. 3. The scale of this figure and the location of some characteristic points are given in Fig. 2. The axes of expansion and contraction are everywhere perpendicular to each other. This is not immediately apparent from the figure because the central projection is not conformal.

Equations (39)

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ρ = ( x 2 + y 2 + z 2 ) 1 / 2 ,
θ = arctan [ ( x 2 + y 2 ) 1 / 2 / z ] ,
φ = arctan ( y / x ) ,
x = ρ cos φ sin θ ,
y = ρ sin φ sin θ ,
z = ρ cos θ .
ρ ( θ , φ ) = z 0 / cos θ .
r ( θ , φ ) = ( cos φ sin θ , sin φ sin θ , cos θ )
ρ ( θ , φ ) = z 0 / cos θ .
v = ( v sin α , 0 , v cos α ) .
P ( t ) = P ( θ ( t ) , φ ( t ) ) = P ( θ , φ ) - t v ( sin α , 0 , cos α ) = [ x ( t ) , y ( t ) , z ( t ) ] ,
r ( θ ( t ) , φ ( t ) ) of P ( t ) .
D ( r ) = r t | t = 0 ,
r t = r θ d θ d t + r φ d φ d t .
d θ d t | t = 0 = θ x d x d t | t = 0 + θ y d y d t | t = 0 + θ z d z d t | t = 0 = cos 2 θ cos φ ( - v sin α ) z 0 + 0 + ( - sin θ cos θ ) ( - v cos α ) z 0 ,
d φ d t | t = 0 = φ x d x d t | t = 0 + φ y d y d t | t = 0 + φ z d z d t | t = 0 = ( - sin φ cos θ z 0 sin θ ) ( - v sin α ) + ( φ y ) · 0 + 0. ( d z d t ) .
D ( r ) = ( v cos θ z 0 ( cos α sin θ - sin α cos θ cos φ ) ) r θ + ( v cos θ z 0 ( sin α sin φ ) ) ( 1 sin θ r φ ) = D θ e θ + D φ e φ ,
d r d s = d θ d s r θ + d φ d s r φ ,
( d θ d s sin θ d φ d s ) .
M ( r ; d r d s ) = d d s D ( r ( s ) ) s = 0
d d s D ( r ( s ) ) = d d s ( D θ ) e θ + D θ d d s ( e θ ) + d d s ( D φ ) e φ + D φ d d s ( e φ ) = ( D θ θ d θ d s + D θ φ d φ d s ) e θ + D θ ( e θ ; θ d θ d s + e θ ; φ d φ d s ) + ( D φ θ d θ d s + D φ φ d φ d s ) e φ + D φ ( e φ ; θ d θ d s + e φ ; φ d φ d s ) .
d d s D ( r ( s ) ) = ( D θ θ d θ d s + D θ φ d φ d s - cos θ D φ d φ d s ) e θ + ( D φ θ d θ d s + D φ φ d φ d s + D θ cos θ d φ d s ) e φ = ( D θ θ 1 sin θ ( D θ φ - cos θ D φ ) D φ θ 1 sin θ ( D φ φ + cos θ D θ ) ) ( d θ d s sin θ d φ d s ) = D r d r d s .
D r = ( v z 0 ( cos α cos 2 θ + sin α cos φ sin 2 θ ) 0 - v z 0 ( sin θ sin α sin φ ) 1 2 v z 0 ( cos α cos 2 θ + sin α cos φ sin 2 θ + cos α ) ) .
A = ( a b c d ) ,
1 2 ( 0 b - c - b + c 0 )
1 2 ( 2 a b + c b + c 2 d ) ,
1 2 ( a + d 0 0 a + d )
1 2 ( a - d b + c b + c - a + d )
D r = 1 2 curlD ( 0 - 1 1 0 ) + 1 2 divD ( 1 0 0 1 ) + 1 2 defD S ( D ) ,
S ( D ) = Q ( D ) - 1 ( 1 0 0 - 1 )             Q ( D ) .
divD = v 2 z 0 [ 3 sin α cos φ sin 2 θ + cos α ( 3 cos 2 θ + 1 ) ] ,
curlD = - v z 0 ( sin α sin θ sin φ ) ,
defD = v z 0 sin θ × [ ( sin α cos θ cos φ - cos α sin θ ) 2 + sin 2 α sin 2 φ ] 1 / 2 .
( 1 ρ ) = θ ( 1 ρ ) e θ + 1 sin θ φ ( 1 ρ ) e φ = - ( sin θ z 0 ) e θ
v ang = ( v , e θ ) e θ + ( v , e φ ) e φ
x = z 0 · tan θ · cos φ ,
y = z 0 · tan θ · sin φ .
( x - z 0 · tan α ) 2 + y 2 = ( z 0 / cos α ) 2 ,
x = - z 0 · tan ( 90 ° - α ) .