Abstract

The problem of inferring local surface orientation from changing images is studied computationally by deriving conditions under which the motion information is sufficient for an information-processing system, biological or otherwise, to infer unique descriptions of the local surface orientation. The analysis is based on a shape-from-motion proposition, which states that, given the first spatial derivatives of the orthographically projected velocity and acceleration fields of a rigidly rotating regular surface, then the angular velocity and the surface normal at each visible point on that surface are uniquely determined up to a reflection. The proof proceeds in two steps. First it is shown that surface tilt and one component of the angular velocity are uniquely determined by the first spatial derivatives of the velocity field. Then it is shown that surface slant and the remaining two components of the angular velocity are uniquely determined if the first spatial derivatives of the acceleration field are also available.

© 1982 Optical Society of America

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  1. W. Miles, "Movement interpretations of the silhouette of a revolving fan," Am. J. Psychol 43, 392–505 (1931).
  2. W. Metzger, "Tiefenerscheinungen in optischen Bewegungsfeldern," Psychol. Forsch. 20, 195–260 (1934).
  3. H. Wallach and D. O'Connell, "The kinetic depth effect," J. Exp. Psychol. 45, 205–217 (1953).
  4. J. Gibson and E. Gibson, "Continuous perspective transformations and the perception of rigid motion," J. Exp. Psychol. 54, 129–138 (1957).
  5. K. von Fieandt and J. Gibson, "The sensitivity of the eye to two kinds of continuous transformations of a shadow pattern," J. Exp. Psychol. 57, 344–347 (1959).
  6. E. Gibson, J. Gibson, O. Smith, and H. Flock, "Motion parallax as a determinant of perceived depth," J. Exp. Psychol. 8, 40–51 (1959).
  7. B. White and G. Mueser, "Accuracy in reconstructing the arrangement of elements generating kinetic depth displays," J. Exp. Psychol. 60, 1–11 (1960).
  8. B. F. Green, "Figure coherence in the kinetic depth effect," J. Exp. Psychol. 62, 272–282 (1961).
  9. M. Braunstein, "Depth perception in rotation dot patterns; effects of numerosity and perspective," J. Exp. Psychol. 64, 415–420 (1962).
  10. J. Gibson, "Research on the visual perception of motion and change," in Readings in the Study of Visually Perceived Motion, I. Spigel, ed. (Harper and Row, New York, 1965), pp. 125–146.
  11. G. Johannson, "Visual motion perception," Sci. Am. 232(6),76–88 (1975).
  12. S. Ullman, The Interpretation of Visual Motion (MIT Press, Cambridge, Mass., 1979).
  13. This eliminates single-frame information, such as texture gradients, from being a plausible explanation of this ability.
  14. G. Johansson, "Perception of motion and changing form," Scand. J. Psychol. 5, 181–208 (1964).
  15. C. Hay, "Optical motions and space perception-an extension of Gibson's analysis," Psychol. Rev. 73, 550–565 (1966).
  16. D. Hoffman and B. Flinchbaugh, "The interpretation of biological motion," Biol. Cybernet. 42, 195–204 (1982); also MIT AI Memo 608 (1980).
  17. Thus the goal is to provide a computational theory in the sense of D. Marr and T. Poggio, "From understanding computation to understanding neural circuitry," Neurol. Res. Prog. Bull. 15, 470–488 (1977).
  18. H. Longuet-Higgins and K. Prazdny, "The interpretation of a moving retinal image," Proc. R. Soc. Lond. B, 208, 385–397 (1980).
  19. J. Gibson, The Perception of the Visual World (Houghton Mifflin, Boston, Mass., 1950).
  20. K. Nakayama and J. Loomis, "Optical velocity patterns, velocity-sensitive neurons, and space perception: a hypothesis," Percept. 3, 63–80 (1974).
  21. J. Koenderink and A. 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).
  22. J. Koenderink and A. van Dorn, "Local structure of movement parallax of the plane," J. Opt. Soc. Am. 66, 717–723 (1976).
  23. J. Koenderink and A. van Doorn, "The internal representation of solid shape with respect to vision," Biol. Cybernet. 32, 211–216 (1979).
  24. J. Koenderink and A. van Doorn, "Exterospecific component of the motion parallax field," J. Opt. Soc. Am. 71, 953–957 (1981).
  25. Actually the surface normal is expressed in terms of zx and zy in Eqs. (8)–(11). The transformations to slant σ and tilt τ are given in Fig. 2.
  26. K. Stevens, "Representing and analyzing surface orientation," in Artificial Intelligence: An MIT Perspective, P. Winston and R. Brown, eds. (MIT Press, Cambridge, Mass. 1979), pp. 104–125.
  27. F. Attneave, "Representation of Physical space," in Coding Processes in Human Memory, A. W. Melton and E. Martino, eds. (Wiley, New York, 1972), pp. 283–306.

1982 (1)

D. Hoffman and B. Flinchbaugh, "The interpretation of biological motion," Biol. Cybernet. 42, 195–204 (1982); also MIT AI Memo 608 (1980).

1981 (1)

1980 (1)

H. Longuet-Higgins and K. Prazdny, "The interpretation of a moving retinal image," Proc. R. Soc. Lond. B, 208, 385–397 (1980).

1979 (1)

J. Koenderink and A. van Doorn, "The internal representation of solid shape with respect to vision," Biol. Cybernet. 32, 211–216 (1979).

1977 (1)

Thus the goal is to provide a computational theory in the sense of D. Marr and T. Poggio, "From understanding computation to understanding neural circuitry," Neurol. Res. Prog. Bull. 15, 470–488 (1977).

1976 (1)

1975 (2)

J. Koenderink and A. 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).

G. Johannson, "Visual motion perception," Sci. Am. 232(6),76–88 (1975).

1974 (1)

K. Nakayama and J. Loomis, "Optical velocity patterns, velocity-sensitive neurons, and space perception: a hypothesis," Percept. 3, 63–80 (1974).

1966 (1)

C. Hay, "Optical motions and space perception-an extension of Gibson's analysis," Psychol. Rev. 73, 550–565 (1966).

1964 (1)

G. Johansson, "Perception of motion and changing form," Scand. J. Psychol. 5, 181–208 (1964).

1962 (1)

M. Braunstein, "Depth perception in rotation dot patterns; effects of numerosity and perspective," J. Exp. Psychol. 64, 415–420 (1962).

1961 (1)

B. F. Green, "Figure coherence in the kinetic depth effect," J. Exp. Psychol. 62, 272–282 (1961).

1960 (1)

B. White and G. Mueser, "Accuracy in reconstructing the arrangement of elements generating kinetic depth displays," J. Exp. Psychol. 60, 1–11 (1960).

1959 (2)

K. von Fieandt and J. Gibson, "The sensitivity of the eye to two kinds of continuous transformations of a shadow pattern," J. Exp. Psychol. 57, 344–347 (1959).

E. Gibson, J. Gibson, O. Smith, and H. Flock, "Motion parallax as a determinant of perceived depth," J. Exp. Psychol. 8, 40–51 (1959).

1957 (1)

J. Gibson and E. Gibson, "Continuous perspective transformations and the perception of rigid motion," J. Exp. Psychol. 54, 129–138 (1957).

1953 (1)

H. Wallach and D. O'Connell, "The kinetic depth effect," J. Exp. Psychol. 45, 205–217 (1953).

1934 (1)

W. Metzger, "Tiefenerscheinungen in optischen Bewegungsfeldern," Psychol. Forsch. 20, 195–260 (1934).

1931 (1)

W. Miles, "Movement interpretations of the silhouette of a revolving fan," Am. J. Psychol 43, 392–505 (1931).

Attneave, F.

F. Attneave, "Representation of Physical space," in Coding Processes in Human Memory, A. W. Melton and E. Martino, eds. (Wiley, New York, 1972), pp. 283–306.

Braunstein, M.

M. Braunstein, "Depth perception in rotation dot patterns; effects of numerosity and perspective," J. Exp. Psychol. 64, 415–420 (1962).

Flinchbaugh, B.

D. Hoffman and B. Flinchbaugh, "The interpretation of biological motion," Biol. Cybernet. 42, 195–204 (1982); also MIT AI Memo 608 (1980).

Flock, H.

E. Gibson, J. Gibson, O. Smith, and H. Flock, "Motion parallax as a determinant of perceived depth," J. Exp. Psychol. 8, 40–51 (1959).

Gibson, E.

E. Gibson, J. Gibson, O. Smith, and H. Flock, "Motion parallax as a determinant of perceived depth," J. Exp. Psychol. 8, 40–51 (1959).

J. Gibson and E. Gibson, "Continuous perspective transformations and the perception of rigid motion," J. Exp. Psychol. 54, 129–138 (1957).

Gibson, J.

K. von Fieandt and J. Gibson, "The sensitivity of the eye to two kinds of continuous transformations of a shadow pattern," J. Exp. Psychol. 57, 344–347 (1959).

E. Gibson, J. Gibson, O. Smith, and H. Flock, "Motion parallax as a determinant of perceived depth," J. Exp. Psychol. 8, 40–51 (1959).

J. Gibson and E. Gibson, "Continuous perspective transformations and the perception of rigid motion," J. Exp. Psychol. 54, 129–138 (1957).

J. Gibson, "Research on the visual perception of motion and change," in Readings in the Study of Visually Perceived Motion, I. Spigel, ed. (Harper and Row, New York, 1965), pp. 125–146.

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

Green, B. F.

B. F. Green, "Figure coherence in the kinetic depth effect," J. Exp. Psychol. 62, 272–282 (1961).

Hay, C.

C. Hay, "Optical motions and space perception-an extension of Gibson's analysis," Psychol. Rev. 73, 550–565 (1966).

Hoffman, D.

D. Hoffman and B. Flinchbaugh, "The interpretation of biological motion," Biol. Cybernet. 42, 195–204 (1982); also MIT AI Memo 608 (1980).

Johannson, G.

G. Johannson, "Visual motion perception," Sci. Am. 232(6),76–88 (1975).

Johansson, G.

G. Johansson, "Perception of motion and changing form," Scand. J. Psychol. 5, 181–208 (1964).

Koenderink, J.

J. Koenderink and A. van Doorn, "Exterospecific component of the motion parallax field," J. Opt. Soc. Am. 71, 953–957 (1981).

J. Koenderink and A. van Doorn, "The internal representation of solid shape with respect to vision," Biol. Cybernet. 32, 211–216 (1979).

J. Koenderink and A. van Dorn, "Local structure of movement parallax of the plane," J. Opt. Soc. Am. 66, 717–723 (1976).

J. Koenderink and A. 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).

Longuet-Higgins, H.

H. Longuet-Higgins and K. Prazdny, "The interpretation of a moving retinal image," Proc. R. Soc. Lond. B, 208, 385–397 (1980).

Loomis, J.

K. Nakayama and J. Loomis, "Optical velocity patterns, velocity-sensitive neurons, and space perception: a hypothesis," Percept. 3, 63–80 (1974).

Marr, D.

Thus the goal is to provide a computational theory in the sense of D. Marr and T. Poggio, "From understanding computation to understanding neural circuitry," Neurol. Res. Prog. Bull. 15, 470–488 (1977).

Metzger, W.

W. Metzger, "Tiefenerscheinungen in optischen Bewegungsfeldern," Psychol. Forsch. 20, 195–260 (1934).

Miles, W.

W. Miles, "Movement interpretations of the silhouette of a revolving fan," Am. J. Psychol 43, 392–505 (1931).

Mueser, G.

B. White and G. Mueser, "Accuracy in reconstructing the arrangement of elements generating kinetic depth displays," J. Exp. Psychol. 60, 1–11 (1960).

Nakayama, K.

K. Nakayama and J. Loomis, "Optical velocity patterns, velocity-sensitive neurons, and space perception: a hypothesis," Percept. 3, 63–80 (1974).

O’Connell, D.

H. Wallach and D. O'Connell, "The kinetic depth effect," J. Exp. Psychol. 45, 205–217 (1953).

Poggio, T.

Thus the goal is to provide a computational theory in the sense of D. Marr and T. Poggio, "From understanding computation to understanding neural circuitry," Neurol. Res. Prog. Bull. 15, 470–488 (1977).

Prazdny, K.

H. Longuet-Higgins and K. Prazdny, "The interpretation of a moving retinal image," Proc. R. Soc. Lond. B, 208, 385–397 (1980).

Smith, O.

E. Gibson, J. Gibson, O. Smith, and H. Flock, "Motion parallax as a determinant of perceived depth," J. Exp. Psychol. 8, 40–51 (1959).

Stevens, K.

K. Stevens, "Representing and analyzing surface orientation," in Artificial Intelligence: An MIT Perspective, P. Winston and R. Brown, eds. (MIT Press, Cambridge, Mass. 1979), pp. 104–125.

Ullman, S.

S. Ullman, The Interpretation of Visual Motion (MIT Press, Cambridge, Mass., 1979).

van Doorn, A.

J. Koenderink and A. van Doorn, "Exterospecific component of the motion parallax field," J. Opt. Soc. Am. 71, 953–957 (1981).

J. Koenderink and A. van Doorn, "The internal representation of solid shape with respect to vision," Biol. Cybernet. 32, 211–216 (1979).

J. Koenderink and A. 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).

van Dorn, A.

von Fieandt, K.

K. von Fieandt and J. Gibson, "The sensitivity of the eye to two kinds of continuous transformations of a shadow pattern," J. Exp. Psychol. 57, 344–347 (1959).

Wallach, H.

H. Wallach and D. O'Connell, "The kinetic depth effect," J. Exp. Psychol. 45, 205–217 (1953).

White, B.

B. White and G. Mueser, "Accuracy in reconstructing the arrangement of elements generating kinetic depth displays," J. Exp. Psychol. 60, 1–11 (1960).

Am. J. Psychol (1)

W. Miles, "Movement interpretations of the silhouette of a revolving fan," Am. J. Psychol 43, 392–505 (1931).

Biol. Cybernet. (2)

D. Hoffman and B. Flinchbaugh, "The interpretation of biological motion," Biol. Cybernet. 42, 195–204 (1982); also MIT AI Memo 608 (1980).

J. Koenderink and A. van Doorn, "The internal representation of solid shape with respect to vision," Biol. Cybernet. 32, 211–216 (1979).

J. Exp. Psychol. (7)

H. Wallach and D. O'Connell, "The kinetic depth effect," J. Exp. Psychol. 45, 205–217 (1953).

J. Gibson and E. Gibson, "Continuous perspective transformations and the perception of rigid motion," J. Exp. Psychol. 54, 129–138 (1957).

K. von Fieandt and J. Gibson, "The sensitivity of the eye to two kinds of continuous transformations of a shadow pattern," J. Exp. Psychol. 57, 344–347 (1959).

E. Gibson, J. Gibson, O. Smith, and H. Flock, "Motion parallax as a determinant of perceived depth," J. Exp. Psychol. 8, 40–51 (1959).

B. White and G. Mueser, "Accuracy in reconstructing the arrangement of elements generating kinetic depth displays," J. Exp. Psychol. 60, 1–11 (1960).

B. F. Green, "Figure coherence in the kinetic depth effect," J. Exp. Psychol. 62, 272–282 (1961).

M. Braunstein, "Depth perception in rotation dot patterns; effects of numerosity and perspective," J. Exp. Psychol. 64, 415–420 (1962).

J. Opt. Soc. Am. (2)

Neurol. Res. Prog. Bull. (1)

Thus the goal is to provide a computational theory in the sense of D. Marr and T. Poggio, "From understanding computation to understanding neural circuitry," Neurol. Res. Prog. Bull. 15, 470–488 (1977).

Opt. Acta (1)

J. Koenderink and A. 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).

Percept. (1)

K. Nakayama and J. Loomis, "Optical velocity patterns, velocity-sensitive neurons, and space perception: a hypothesis," Percept. 3, 63–80 (1974).

Proc. R. Soc. Lond. B (1)

H. Longuet-Higgins and K. Prazdny, "The interpretation of a moving retinal image," Proc. R. Soc. Lond. B, 208, 385–397 (1980).

Psychol. Forsch. (1)

W. Metzger, "Tiefenerscheinungen in optischen Bewegungsfeldern," Psychol. Forsch. 20, 195–260 (1934).

Psychol. Rev. (1)

C. Hay, "Optical motions and space perception-an extension of Gibson's analysis," Psychol. Rev. 73, 550–565 (1966).

Scand. J. Psychol. (1)

G. Johansson, "Perception of motion and changing form," Scand. J. Psychol. 5, 181–208 (1964).

Sci. Am. (1)

G. Johannson, "Visual motion perception," Sci. Am. 232(6),76–88 (1975).

Other (7)

S. Ullman, The Interpretation of Visual Motion (MIT Press, Cambridge, Mass., 1979).

This eliminates single-frame information, such as texture gradients, from being a plausible explanation of this ability.

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

J. Gibson, "Research on the visual perception of motion and change," in Readings in the Study of Visually Perceived Motion, I. Spigel, ed. (Harper and Row, New York, 1965), pp. 125–146.

Actually the surface normal is expressed in terms of zx and zy in Eqs. (8)–(11). The transformations to slant σ and tilt τ are given in Fig. 2.

K. Stevens, "Representing and analyzing surface orientation," in Artificial Intelligence: An MIT Perspective, P. Winston and R. Brown, eds. (MIT Press, Cambridge, Mass. 1979), pp. 104–125.

F. Attneave, "Representation of Physical space," in Coding Processes in Human Memory, A. W. Melton and E. Martino, eds. (Wiley, New York, 1972), pp. 283–306.

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