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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Stephen A. Burns
  • Vol. 25, Iss. 9 — Sep. 1, 2008
  • pp: 2357–2369

Cyclopean geometry of binocular vision

Miles Hansard and Radu Horaud  »View Author Affiliations

JOSA A, Vol. 25, Issue 9, pp. 2357-2369 (2008)

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The geometry of binocular projection is analyzed in relation to the primate visual system. An oculomotor parameterization that includes the classical vergence and version angles is defined. It is shown that the epipolar geometry of the system is constrained by binocular coordination of the eyes. A local model of the scene is adopted in which depth is measured relative to a plane containing the fixation point. These constructions lead to an explicit parameterization of the binocular disparity field involving the gaze angles as well as the scene structure. The representation of visual direction and depth is discussed with reference to the relevant psychophysical and neurophysiological literature.

© 2008 Optical Society of America

OCIS Codes
(330.1400) Vision, color, and visual optics : Vision - binocular and stereopsis
(330.2210) Vision, color, and visual optics : Vision - eye movements

ToC Category:
Vision, Color, and Visual Optics

Original Manuscript: March 18, 2008
Revised Manuscript: June 26, 2008
Manuscript Accepted: June 27, 2008
Published: August 22, 2008

Virtual Issues
Vol. 3, Iss. 11 Virtual Journal for Biomedical Optics

Miles Hansard and Radu Horaud, "Cyclopean geometry of binocular vision," J. Opt. Soc. Am. A 25, 2357-2369 (2008)

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  1. J. J. Koenderink and A. J. van Doorn, “Affine structure from motion,” J. Opt. Soc. Am. A 8, 377-385 (1991). [CrossRef] [PubMed]
  2. O. Faugeras, “Stratification of three-dimensional vision: projective, affine, and metric representations,” J. Opt. Soc. Am. A 12, 465-484 (1995). [CrossRef]
  3. Q.-T. Luong and T. Viéville, “Canonical representations for the geometries of multiple projective views,” Comput. Vis. Image Underst. 64, 193-229 (1996). [CrossRef]
  4. E. Brenner and W. J. M. van Damme, “Judging distance from ocular convergence,” Vision Res. 38, 493-498 (1998). [CrossRef] [PubMed]
  5. R. H. S. Carpenter, Movements of the Eyes (Pion, 1988).
  6. G. L. Walls, “The evolutionary history of eye movements,” Vision Res. 2, 69-80 (1962). [CrossRef]
  7. C. Rashbass and G. Westheimer, “Disjunctive eye movements,” J. Physiol. (London) 159, 339-360 (1961).
  8. D. J. Fleet, H. Wagner, and D. J. Heeger, “Neural encoding of binocular disparity: Energy models, position shifts and phase shifts,” Vision Res. 36, 1839-1857 (1996). [CrossRef] [PubMed]
  9. S. J. D. Prince, B. G. Cumming, and A. J. Parker, “Range and mechanism of encoding of horizontal disparity in macaque V1,” J. Neurophysiol. 87, 209-221 (2002). [PubMed]
  10. J. D. Pettigrew, “Binocular visual processing in the owl's telencephalon,” Proc. R. Soc. London, Ser. B 204, 435-454 (1979). [CrossRef]
  11. B. Julesz, Foundations of Cyclopean Perception (U. Chicago Press, 1971).
  12. J. D. Pettigrew, “Evolution of binocular vision,” in Visual Neuroscience, J.D.Pettigrew, K.J.Sanderson, and W.R.Levick, eds. 208-22 (Cambridge U. Press, 1986).
  13. J. M. Foley, “Binocular distance perception,” Psychol. Rev. 87, 411-434 (1980). [CrossRef] [PubMed]
  14. D. Tweed, “Visual-motor optimization in binocular control,” Vision Res. 37, 1939-1951 (1997). [CrossRef] [PubMed]
  15. J. C. A. Read and B. G. Cumming, “Understanding the cortical specialization for horizontal disparity,” Neural Comput. 16, 1983-2020 (2004). [CrossRef] [PubMed]
  16. H. L. F. von Helmholtz, Treatise on Physiological Optics, Vol. 3, 3rd ed., 1910 (Optical Society of America, 1925), translated by J. P. C. Southall.
  17. E. Hering, The Theory of Binocular Vision (Englemann, 1868), edited and translated, B.Bridgeman and L.Stark (Plenum Press, 1977).
  18. C. Blakemore, “The range and scope of binocular depth discrimination in man,” J. Physiol. (London) 211, 599-622 (1970).
  19. B. G. Cumming and A. J. Parker, “Binocular neurons in V1 of awake monkeys are selective for absolute, not relative disparity,” J. Neurosci. 19, 5602-5618 (1999). [PubMed]
  20. H. C. Longuet-Higgins, “A computer algorithm for reconstructing a scene from two projections,” Nature (London) 293, 133-135 (1981). [CrossRef]
  21. K. N. Ogle, Researches in Binocular Vision (W. B. Saunders, 1950).
  22. J. R. Bergen, P. Anandan, K. J. Hanna, and R. Hingorani, “Hierarchical model-based motion estimation,” in Proceedings of the European Conference on Computer Vision (Springer-Verlag, 1992), pp. 237-252.
  23. A. Glennerster, S. P. McKee, and M. D. Birch, “Evidence for surface-based processing of binocular disparity,” Curr. Biol. 12, 825-828 (2002). [CrossRef] [PubMed]
  24. J. J. Koenderink, A. J. van Doorn, A. M. L. Kappers, and J. T. Todd, “Ambiguity and the 'mental eye' in pictorial relief,” Perception 30, 431-448 (2001). [CrossRef] [PubMed]
  25. B. Julesz, “Cyclopean perception and neurophysiology,” Invest. Ophthalmol. 11, 540-548 (1972). [PubMed]
  26. J. Gårding, J. Porrill, J. E. W. Mayhew, and J. P. Frisby, “Stereopsis, vertical disparity and relief transformations,” Vision Res. 35, 703-722 (1995). [CrossRef] [PubMed]
  27. J. J. Koenderink and A. J. van Doorn, “Geometry of binocular vision and a model for stereopsis,” Biol. Cybern. 21, 29-35 (1976). [CrossRef] [PubMed]
  28. J. J. Koenderink and A. J. van Doorn, “Second-order optic flow,” J. Opt. Soc. Am. A 9, 530-538 (1992). [CrossRef]
  29. B. Rogers and R. Cagenello, “Disparity curvature and the perception of three-dimensional surfaces,” Nature (London) 339, 135-137 (1989). [CrossRef]
  30. A. J. Noest, R. Van Ee, and A. V. van den Berg, “Direct extraction of curvature-based metric shape from stereo by view-modulated receptive fields,” Biol. Cybern. 95, 455-486 (2006). [CrossRef] [PubMed]
  31. J. E. W. Mayhew, “The interpretation of stereo-disparity information: The computation of surface orientation and depth,” Perception 11, 387-403 (1982). [CrossRef] [PubMed]
  32. J. E. W. Mayhew and H. C. Longuet-Higgins, “A computational model of binocular depth perception,” Nature (London) 297, 376-379 (1982). [CrossRef]
  33. D. Weinshall, “Qualitative depth from stereo, with applications,” Comput. Vis. Graph. Image Process. 49, 222-241 (1990). [CrossRef]
  34. B. Gillam, D. Chambers, and B. Lawergren, “The role of vertical disparity in the scaling of stereoscopic depth perception: An empirical and theoretical study,” Percept. Psychophys. 44, 473-484 (1988). [CrossRef] [PubMed]
  35. B. J. Rogers and M. F. Bradshaw, “Vertical disparities, differential perspective and binocular stereopsis,” Nature (London) 361, 253-255 (1993). [CrossRef]
  36. B. T. Backus, M. S. Banks, R. van Ee, and J. A. Crowell, “Horizontal and vertical disparity, eye position, and stereoscopic slant perception,” Vision Res. 39, 1143-1170 (1999). [CrossRef] [PubMed]
  37. E. M. Berends and C. J. Erkelens, “Strength of depth effects induced by three types of vertical disparity,” Vision Res. 41, 37-45 (2001). [CrossRef] [PubMed]
  38. C. J. Erkelens and R. van Ee, “A computational model of depth perception based on headcentric disparity,” Vision Res. 38, 2999-3018 (1998). [CrossRef] [PubMed]
  39. S. Maybank, Theory of Reconstruction from Image Motion (Springer-Verlag, 1993). [CrossRef]
  40. M. Armstrong, A. Zisserman, and R. I. Hartley, “Self-calibration from image triplets,” in Proceedings of the European Conference on Computer Vision (Springer-Verlag, 1996), Vol. 1, pp. 3-16.
  41. A. Shashua and N. Navab, “Relative affine structure: Canonical model for 3-D from 2-D geometry and applications,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 873-883 (1996). [CrossRef]
  42. R. I. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision (Cambridge U. Press, 2000).
  43. K. Hepp, “Oculomotor control: Listing's law and all that,” Curr. Opin. Neurobiol. 4, 862-868 (1994). [CrossRef] [PubMed]
  44. D. Mok, A. Ro, W. Cadera, J. D. Crawford, and T. Vilis, “Rotation of Listing's plane during vergence,” Vision Res. 32, 2055-2064 (1992). [CrossRef] [PubMed]
  45. L. J. Van Rijn and A. V. Van den Berg, “Binocular eye orientation during fixations: Listing's law extended to include eye vergence,” Vision Res. 33, 691-708 (1993). [CrossRef] [PubMed]
  46. C. W. Tyler, “The horopter and binocular fusion,” in Vision and Visual Disorders, Vol. 9, Binocular Vision, D.Regan, ed. (MacMillan, 1991), pp. 19-37.
  47. M. L. Cooper and J. D. Pettigrew, “A neurophysiological determination of the vertical horopter in the cat and owl,” J. Comp. Neurol. 184, 1-25 (1979). [CrossRef] [PubMed]
  48. M. J. Brooks, L. de Agapito, D. Q. Huynh, and L. Baumela, “Towards robust metric reconstruction via a dynamic uncalibrated stereo head,” Image and Vision Computing 16, 989-1002 (1998). [CrossRef]
  49. L. E. Mays, “Neural control of vergence eye movements: Convergence and divergence neurons in midbrain,” J. Neurophysiol. 51, 1091-1108 (1984). [PubMed]
  50. H. Collewijn, C. J. Erkelens, and R. M. Steinman, “Trajectories of the human binocular fixation point during conjugate and non-conjugate gaze-shifts,” Vision Res. 37, 1049-1069 (1997). [CrossRef] [PubMed]
  51. W. Zhou and W. M. King, “Binocular eye movements not coordinated during REM sleep,” Exp. Brain Res. 117, 153-160 (1997). [CrossRef]
  52. D. Marr and T. Poggio, “Cooperative computation of stereo disparity,” Science 194, 283-287 (1976). [CrossRef] [PubMed]
  53. A. Anzai, I. Ohzawa, and R. D. Freeman, “Neural mechanisms for encoding binocular disparity: Receptive field position versus phase,” J. Neurophysiol. 82, 874-890 (1999). [PubMed]
  54. S. J. Maybank and P. F. Sturm, “MDL, collineations and the fundamental matrix,” in Proceedings of the 10th British Machine Vision Conference (The British Machine Vision Association and Society for Pattern Recognition, 1999), pp. 53-62.
  55. G. F. Poggio and B. Fischer, “Binocular interaction and depth sensitivity in striate and prestriate cortex of behaving rhesus monkeys,” J. Neurophysiol. 40, 1392-1407 (1977). [PubMed]
  56. B. G. Cumming and G. C. DeAngelis, “The physiology of stereopsis,” Annu. Rev. Neurosci. 24, 203-238 (2001). [CrossRef] [PubMed]
  57. G. Westheimer, “Cooperative neural processes involved in stereoscopic acuity,” Exp. Brain Res. 36, 585-597 (1979). [CrossRef] [PubMed]
  58. G. J. Mitchison and S. P. McKee, “Interpolation in stereoscopic matching,” Nature (London) 315, 402-404 (1985). [CrossRef]
  59. H. Ono and A. P. Mapp, “A re-statement and modification of Wells-Hering's laws of visual direction,” Perception 24, 237-252 (1995). [CrossRef] [PubMed]

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