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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 66, Iss. 7 — Jul. 1, 1976
  • pp: 717–723

Local structure of movement parallax of the plane

J. J. Koenderink and A. J. van Doorn  »View Author Affiliations

JOSA, Vol. 66, Issue 7, pp. 717-723 (1976)

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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

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

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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).
  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).
  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).
  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).
  15. J. B. J. Riemersma, K. W. Mess, and J. A. Michon, Institute for Perception TNO, Report No. IZF 1972-C7 (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.

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