Abstract

For the egocentric orientation of observers moving with respect to a plane (e.g., pilots and automobile drivers), the movement parallax field provides the main cue. The parallax field is split into a lamellar and a solenoidal part, and it is shown that the solenoidal part is purely propriospecific. For instance, it can be shown that this component can be completely canceled by an appropriate eye movement. Thus all exterospecific information is contained in the lamellar part, and this part is completely determined by the divergence of the parallax field. Thus the measure of expansion of the visual field as a function of direction of gaze is sufficient to provide all information available for egocentric orientation. It is further shown that the widely used focus of expansion, as introduced by Gibson, is not invariant against eye movements and does not (in general) correspond to extrema of the divergence.

© 1981 Optical Society of America

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References

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  1. J. J. Gibson, P. Olum, and F. Rosenblatt, “Parallax and perspective during aircraft landings,” Am. J. Psychol. 68, 372–385 (1958).
    [Crossref]
  2. D. A. Gordon, “Static and dynamic visual fields in human space perception,” J. Opt. Soc. Am. 55, 1296–1303 (1965).
    [Crossref] [PubMed]
  3. J. J. Koenderink and A. J. van Doorn, “Local structure of movement parallax of the plane,” J. Opt. Soc. Am. 66, 717–723 (1976).
    [Crossref]
  4. J. J. Gibson, The Perception of the Visual World (Houghton Mifflin, Boston, Mass., 1950).
  5. G. A. Brecher and et al., “Relation of optical and labyrinthean orientation,” Opt. Acta 19, 467–471 (1972).
    [Crossref]
  6. D. Regan, K. Beverley, and M. Cynader, “The visual perception of motion in depth,” Sci. Am. 241(1), 122–133 (1979).
    [Crossref]
  7. Ref. 4, Fig. 56.
  8. J. J. Koenderink and A. J. van Doorn, “Invariant properties of the motion parallax field due to the movement of rigid bodies relative to an observer,” Opt. Acta 22, 773–791 (1975).
    [Crossref]
  9. F. Hoyle, The Black Cloud (Penguin, Harmondsworth, Middlesex, 1960).
  10. D. N. Lee, “A theory of visual control of braking based on information about time-to-collision,” Percept. 5, 437–459 (1976).
    [Crossref]

1979 (1)

D. Regan, K. Beverley, and M. Cynader, “The visual perception of motion in depth,” Sci. Am. 241(1), 122–133 (1979).
[Crossref]

1976 (2)

D. N. Lee, “A theory of visual control of braking based on information about time-to-collision,” Percept. 5, 437–459 (1976).
[Crossref]

J. J. Koenderink and A. J. van Doorn, “Local structure of movement parallax of the plane,” J. Opt. Soc. Am. 66, 717–723 (1976).
[Crossref]

1975 (1)

J. J. Koenderink and A. J. van Doorn, “Invariant properties of the motion parallax field due to the movement of rigid bodies relative to an observer,” Opt. Acta 22, 773–791 (1975).
[Crossref]

1972 (1)

G. A. Brecher and et al., “Relation of optical and labyrinthean orientation,” Opt. Acta 19, 467–471 (1972).
[Crossref]

1965 (1)

1958 (1)

J. J. Gibson, P. Olum, and F. Rosenblatt, “Parallax and perspective during aircraft landings,” Am. J. Psychol. 68, 372–385 (1958).
[Crossref]

Beverley, K.

D. Regan, K. Beverley, and M. Cynader, “The visual perception of motion in depth,” Sci. Am. 241(1), 122–133 (1979).
[Crossref]

Brecher, G. A.

G. A. Brecher and et al., “Relation of optical and labyrinthean orientation,” Opt. Acta 19, 467–471 (1972).
[Crossref]

Cynader, M.

D. Regan, K. Beverley, and M. Cynader, “The visual perception of motion in depth,” Sci. Am. 241(1), 122–133 (1979).
[Crossref]

Gibson, J. J.

J. J. Gibson, P. Olum, and F. Rosenblatt, “Parallax and perspective during aircraft landings,” Am. J. Psychol. 68, 372–385 (1958).
[Crossref]

J. J. Gibson, The Perception of the Visual World (Houghton Mifflin, Boston, Mass., 1950).

Gordon, D. A.

Hoyle, F.

F. Hoyle, The Black Cloud (Penguin, Harmondsworth, Middlesex, 1960).

Koenderink, J. J.

J. J. Koenderink and A. J. van Doorn, “Local structure of movement parallax of the plane,” J. Opt. Soc. Am. 66, 717–723 (1976).
[Crossref]

J. J. Koenderink and A. J. van Doorn, “Invariant properties of the motion parallax field due to the movement of rigid bodies relative to an observer,” Opt. Acta 22, 773–791 (1975).
[Crossref]

Lee, D. N.

D. N. Lee, “A theory of visual control of braking based on information about time-to-collision,” Percept. 5, 437–459 (1976).
[Crossref]

Olum, P.

J. J. Gibson, P. Olum, and F. Rosenblatt, “Parallax and perspective during aircraft landings,” Am. J. Psychol. 68, 372–385 (1958).
[Crossref]

Regan, D.

D. Regan, K. Beverley, and M. Cynader, “The visual perception of motion in depth,” Sci. Am. 241(1), 122–133 (1979).
[Crossref]

Rosenblatt, F.

J. J. Gibson, P. Olum, and F. Rosenblatt, “Parallax and perspective during aircraft landings,” Am. J. Psychol. 68, 372–385 (1958).
[Crossref]

van Doorn, A. J.

J. J. Koenderink and A. J. van Doorn, “Local structure of movement parallax of the plane,” J. Opt. Soc. Am. 66, 717–723 (1976).
[Crossref]

J. J. Koenderink and A. J. van Doorn, “Invariant properties of the motion parallax field due to the movement of rigid bodies relative to an observer,” Opt. Acta 22, 773–791 (1975).
[Crossref]

Am. J. Psychol. (1)

J. J. Gibson, P. Olum, and F. Rosenblatt, “Parallax and perspective during aircraft landings,” Am. J. Psychol. 68, 372–385 (1958).
[Crossref]

J. Opt. Soc. Am. (2)

Opt. Acta (2)

G. A. Brecher and et al., “Relation of optical and labyrinthean orientation,” Opt. Acta 19, 467–471 (1972).
[Crossref]

J. J. Koenderink and A. J. van Doorn, “Invariant properties of the motion parallax field due to the movement of rigid bodies relative to an observer,” Opt. Acta 22, 773–791 (1975).
[Crossref]

Percept. (1)

D. N. Lee, “A theory of visual control of braking based on information about time-to-collision,” Percept. 5, 437–459 (1976).
[Crossref]

Sci. Am. (1)

D. Regan, K. Beverley, and M. Cynader, “The visual perception of motion in depth,” Sci. Am. 241(1), 122–133 (1979).
[Crossref]

Other (3)

Ref. 4, Fig. 56.

F. Hoyle, The Black Cloud (Penguin, Harmondsworth, Middlesex, 1960).

J. J. Gibson, The Perception of the Visual World (Houghton Mifflin, Boston, Mass., 1950).

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Figures (6)

Fig. 1
Fig. 1

The observer moves with velocity v in a direction parallel to the ground plane at a distance h above the plane. He looks at two line pieces, l1, l2, in direction θ. Line piece l1 is perpendicular to v; line piece l2 is parallel to v.

Fig. 2
Fig. 2

The observer moves with unit velocity (v = 1) at a unit distance (h = 1) from the plane. At time t = 0 he looks at a small square directly beneath him. As time progresses the square shrinks in apparent size, but the sides parallel to v shrink faster than the sides perpendicular to v. Thus the figure represents the deformations suffered by the image of a floor tile in the visual field of a man who walks over the pavement.

Fig. 3
Fig. 3

Coordinate system employed in the text: O, observer; R, projection of observer on ground plane, OR = h; OQ, direction of movement, velocity v under angle α; P, general point of the plane in direction r, or (θ, ϕ); ex, ey, ez, Cartesian coordinates. Distance OP is denoted by ρ in the text. In Figs. 46 points on the horizon (θ = π/2) are denoted as follows: A, anterior direction (ϕ = 0); S, direction to the left (ϕ = π/2); P, posterior direction (ϕ = π); D, direction to the right (ϕ = 3π/2).

Fig. 4
Fig. 4

The solenoidal component of the flow field for α ≠ 0. The field of view is in stereographic projection. (This has the virtue that more than a hemisphere can be depicted.) The upper limit of the figure is the horizon. A is the anterior direction (ϕ = 0), D is the direction to the right [ϕ = −(π/2)], and S is the direction to the left [ϕ = (π/2)] of the observer. (a) The anterior direction is the center of the stereographic projection. (b) The same case, but here the direction to the left (θ = 0, ϕ = π/2) is the center of projection.

Fig. 5
Fig. 5

The flow field for α = (π/2)(v = 0). The movement is parallel to the plane in the anterior (A) direction. The flow field has both a lamellar and a solenoidal component. (The solenoidal component looks like Fig. 4, the lamellar component like Fig. 6.) The Gibsonean foci of expansion are at the right ahead (A) and in the backward direction (P). The projection is stereographic. (a) The center of projection is the anterior direction. (b) The center of projection is the direction to the left (θ = 0, ϕ = π/2).

Fig. 6
Fig. 6

The lamellar component of the flow field for α = (π/2)(v = 0). This is the flow field of Fig. 5 with the solenoidal component (Fig. 4) removed. (Thus the field in Fig. 5 is the sum of the solenoidal field depicted in Fig. 4 and the lamellar field depicted in Fig. 6.) Note that the Gibsonean foci of expansion have disappeared: at those foci the expansion actually vanishes. Instead there appear two nodes, one of expansion at (θ = π/4, ϕ = 0) [visible in Figs. 6(a) and 6(b)] and one of contraction at (θ = π/4, ϕ = π) [visible only in Fig. 6(b)]. The nodes are nondegenerate for the reason shown in Fig. 2. The projection is again stereographic. (a) Anterior direction is the center of projection. (b) Direction to the left (θ = 0, ϕ = π/2) is the center of projection.

Equations (23)

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α ˙ 1 α 1 = - θ ˙ × tan θ ;             α ˙ α 2 = - 2 θ ˙ × tan θ .
Ω ˙ Ω = α ˙ 1 α 1 + α ˙ 2 α 2 = - 3 θ ˙ × tan θ .
Ω ˙ Ω = 3 2 v h × sin 2 θ .
D = v cos θ h [ ( cos α sin θ - sin α cos θ cos ϕ ) e θ + sin α sin ϕ e ϕ ] ;
· D = v 2 h ( 3 sin α cos ϕ sin 2 θ + 3 cos α cos 2 θ + cos α ) ,
× D = - v h sin α sin θ sin ϕ .
D = Φ + J Ψ ,
Δ Φ = · D , Δ Ψ = × D .
ρ ( r ) = v 2 C 1 [ 1 - ( r · s ) 2 ]
Φ = - v 4 h ( sin α cos ϕ sin 2 θ + cos α cos 2 θ ) ,
Ψ = v 2 h sin α sin θ sin ϕ .
Φ ( r ) = - v 2 ρ ( e x · r ) - h v 2 ρ 2 ,
Ψ ( r ) = v ( e y · r ) 2 h .
ω = v 2 h .
θ ˙ = - v h cos 2 θ = - v 2 h = ω .
D = ( - v 2 h ρ 2 ) = 1 / 2 v h sin 2 θ · e θ .
D = - v 2 · e x · r ρ = - v 2 h ( cos ϕ cos 2 θ · e θ - sin ϕ cos θ e ϕ )
θ = π 2 , ϕ = π 2 ; θ = π 2 , ϕ = - π 2 , θ = π 4 , ϕ = 0 ; θ = π 4 , ϕ = π .
θ = π 4 + ξ ,             ϕ = η 2
θ = π 2 + ξ ,             ϕ = π 2 + η
D = - v 2 h [ ( sin α cos ϕ cos 2 θ - cos α sin 2 θ ) e θ - sin α sin ϕ cos θ e ϕ ] .
( Ω ˙ Ω ) exp + ( Ω ˙ Ω ) contr = 1 τ ,
3 [ ( Ω ˙ Ω ) exp + ( Ω ˙ Ω ) contr ] ( Ω ˙ Ω ) exp - ( Ω ˙ Ω ) contr = cos α .