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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 20, Iss. 7 — Jul. 1, 2003
  • pp: 1398–1406

Bayesian modeling of cue interaction: bistability in stereoscopic slant perception

Raymond van Ee, Wendy J. Adams, and Pascal Mamassian  »View Author Affiliations

JOSA A, Vol. 20, Issue 7, pp. 1398-1406 (2003)

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Our two eyes receive different views of a visual scene, and the resulting binocular disparities enable us to reconstruct its three-dimensional layout. However, the visual environment is also rich in monocular depth cues. We examined the resulting percept when observers view a scene in which there are large conflicts between the surface slant signaled by binocular disparities and the slant signaled by monocular perspective. For a range of disparity–perspective cue conflicts, many observers experience bistability: They are able to perceive two distinct slants and to flip between the two percepts in a controlled way. We present a Bayesian model that describes the quantitative aspects of perceived slant on the basis of the likelihoods of both perspective and disparity slant information combined with prior assumptions about the shape and orientation of objects in the scene. Our Bayesian approach can be regarded as an overarching framework that allows researchers to study all cue integration aspects—including perceptual decisions—in a unified manner.

© 2003 Optical Society of America

OCIS Codes
(330.0330) Vision, color, and visual optics : Vision, color, and visual optics
(330.1400) Vision, color, and visual optics : Vision - binocular and stereopsis
(330.4060) Vision, color, and visual optics : Vision modeling
(330.5510) Vision, color, and visual optics : Psychophysics
(330.7310) Vision, color, and visual optics : Vision

Original Manuscript: September 27, 2002
Revised Manuscript: February 26, 2003
Manuscript Accepted: February 26, 2003
Published: July 1, 2003

Raymond van Ee, Wendy J. Adams, and Pascal Mamassian, "Bayesian modeling of cue interaction: bistability in stereoscopic slant perception," J. Opt. Soc. Am. A 20, 1398-1406 (2003)

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  1. These assumptions are often unnoticed, and the prior knowledge is not something the observer needs to be aware of.2Bayesian theory provides a general framework that incorporates such assumptions.
  2. H. von Helmholtz, Handbuch der Physiologischen Optik (Voss, Hamburg, Germany, 1866), Vol. 3, Sec. 26.
  3. R. S. Allison, I. P. Howard, “Temporal dependencies in resolving monocular and binocular cue conflict in slant perception,” Vision Res. 40, 1869–1886 (2000). [CrossRef] [PubMed]
  4. R. S. Allison, I. P. Howard, “Stereopsis with persisting and dynamic textures,” Vision Res. 40, 3823–3827 (2000). [CrossRef] [PubMed]
  5. B. J. Gillam, “Perception of slant when perspective and stereopsis conflict: experiments with aniseikonic lenses,” J. Exp. Psychol. 78, 299–305 (1968). [CrossRef] [PubMed]
  6. B. J. Gillam, C. Ryan, “Perspective, orientation disparity, and anisotropy in stereoscopic slant perception,” Perception 21, 427–439 (1992). [CrossRef] [PubMed]
  7. C. Ryan, B. Gillam, “Cue conflict and stereoscopic surface slant about horizontal and vertical axes,” Perception 23, 645–658 (1994). [CrossRef] [PubMed]
  8. B. J. Gillam, M. L. Cook, “Perspective based on stereopsis and occlusion,” Psychol. Sci. 12, 424–429 (2001). [CrossRef] [PubMed]
  9. A. H. Smith, “Perceived slant as a function of stimulus contour and vertical dimension,” Percept. Mot. Skills 24, 167–173 (1967). [CrossRef]
  10. R. van Ee, M. S. Banks, B. T. Backus, “An analysis of binocular slant contrast,” Perception 28, 1121–1145 (1999). [CrossRef]
  11. M. S. Banks, B. T. Backus, “Extra-retinal and perspective cues cause the small range of the induced effect,” Vision Res. 38, 187–194 (1998). [CrossRef] [PubMed]
  12. W. M. Youngs, “The influence of perspective and disparity cues on the perception of slant,” Vision Res. 16, 79–82 (1976). [CrossRef] [PubMed]
  13. C. Wheatstone, “Contributions to the physiology of vision—part the first. On some remarkable and hitherto unobserved phenomena of binocular vision,” Philos. Trans. R. Soc. London 128, 371–394 (1838). [CrossRef]
  14. W. Schriever, “Experimentelle Studien über stereoskopisches Sehen,” Z. Psychol. Physiol. Sinnesorgane 96, 113–170 (1925).
  15. K. A. Stevens, M. Lees, A. Brookes, “Combining binocular and monocular curvature features,” Perception 20, 425–440 (1991). [CrossRef] [PubMed]
  16. H. Hill, V. Bruce, “Independent effects of lighting, orientation, and stereopsis on the hollow-face illusion,” Perception 22, 887–897 (1993). [CrossRef] [PubMed]
  17. R. van Ee, K. Hol, C. J. Erkelens, “Bistable stereoscopic percepts and depth cue combination,” Perception 30, S42 (2001).
  18. T. V. Papathomas, “Experiments on the role of painted cues in Hughes’s reverspectives,” Perception 31, 521–530 (2002). [CrossRef]
  19. See also other interesting contributions in Refs. 20-23.
  20. R. Gregory, The Intelligent Eye (Weidenfeld and Nicholson, London, 1970).
  21. J. Slyce, Patrick Hughes: Perverspective (Momentum, London, 1998).
  22. N. J. Wade, P. Hughes, “Fooling the eyes: trompe l’oeil and reverse perspective,” Perception 28, 1115–1119 (1999). [CrossRef]
  23. E. Mach, “Über die physiologische Wirkung räumlich verteilter Lichtreize,” Sitzungsber. d. Wiener Akad. 54, 3 (1866).
  24. R. van Ee, L. C. J. van Dam, C. J. Erkelens, “Bi-stability in perceived slant when binocular disparity and monocular perspective specify different slants,” J. Vision 2, 597–607 (2002). [CrossRef]
  25. Although eye movements play a role, the perceptual bistability seems to be predominantly central. We are currently measuring eye movements while subjects experience bistability in our grid stimuli. Our preliminary findings reveal that switching between the two percepts can occur by effort of will while subjects keep strict fixation. When eye movements are allowed, there is no clear correlation between perceptual flips and both eye movements and blinks.26
  26. L. C. J. van Dam, R. van Ee, “Bistability in stereoscopically perceived slant about a horizontal axis,” J. Vision (to be published) (Abstract Book VSS03).
  27. R. van Ee, W. Richards, “A planar and a volumetric test for stereoanomaly,” Perception 31, 51–64 (2002). [CrossRef] [PubMed]
  28. R. van Ee, C. J. Erkelens, “Temporal aspects of binocular slant perception,” Vision Res. 36, 43–51 (1996). [CrossRef] [PubMed]
  29. A sensible objection to this metrical slant-estimation method is that it is hard to interpret the data because a slant angle that is estimated at 35 deg in one trial might look like 40 deg in another trial. Previous work has demonstrated, however, that subjects have a relatively constant internal reference and that they do not regard this task as difficult. This estimation method has been used previously for real planes10and when subjects wore distorting lenses.30In addition, a similar metrical depth-estimation method was successfully used for volumetric stimuli.31
  30. W. J. Adams, M. S. Banks, R. van Ee, “Adaptation to three-dimensional distortions in human vision,” Nat. Neurosci. 4, 1063–1064 (2001). [CrossRef] [PubMed]
  31. R. van Ee, B. L. Anderson, “Motion direction, speed, and orientation in binocular matching,” Nature 410, 690–694 (2001). [CrossRef] [PubMed]
  32. Bayesian theory is a rich mathematical theory.33,34Massaro35and Clark and Yuille36made Bayesian theory accessible to speech perception and visual perception, respectively. See also excellent chapters in Refs. 37and 38and introductory tutorials in Refs. 39and 40on applications in visual cue integration.
  33. J. O. Berger, Statistical Decision Theory and Bayesian Analysis (Springer-Verlag, Berlin, 1985).
  34. T. Ferguson, Mathematical Statistics: a Decision Theoretic Approach (Academic, New York, 1967).
  35. D. W. Massaro, Speech Perception by Ear and Eye (Erlbaum, Hillsdale, N.J., 1987).
  36. J. J. Clark, A. L. Yuille, Data Fusion for Sensory Information Processing Systems (Kluwer Academic, Boston, 1990).
  37. L. T. Maloney, “Statistical decision theory and biological vision,” in Perception and the Physical World, D. Heyer, R. Mausfeld, eds. (Wiley, Chichester, UK, 2002).
  38. A. L. Yuille, H. H. Bülthoff, “Bayesian decision theory and psychophysics,” in Perception as Bayesian Inference, D. C. Knill, W. Richards, eds. (Cambridge U. Press, Cambridge, UK, 1996).
  39. D. C. Knill, D. Kersten, A. L. Yuille, “Introduction: a Bayesian formulation of visual perception,” in Perception as Bayesian Inference, D. C. Knill, W. Richards, eds. (Cambridge U. Press, Cambridge, UK, 1996).
  40. P. Mamassian, M. S. Landy, L. T. Maloney, “Bayesian modelling of visual perception,” in Probabilistic Models of the Brain, R. P. N. Rao, B. A. Olshausen, M. S. Lewicki, eds. (MIT, Cambridge, Mass., 2002).
  41. D. C. Knill, W. Richards, Perception as Bayesian Inference (Cambridge U. Press, Cambridge, UK, 1996).
  42. J. Porrill, J. P. Frisby, W. J. Adams, D. Buckley, “Robust and optimal use of information in stereo vision,” Nature 397, 63–66 (1999). [CrossRef] [PubMed]
  43. H. H. Bülthoff, H. A. Mallot, “Integration of stereo, shading and texture,” in AI and the Eye, A. Blake, T. Troscianko, eds. (Wiley, New York, 1990).
  44. W. T. Freeman, “The generic viewpoint assumption in a framework for visual perception,” Nature 368, 542–545 (1994). [CrossRef] [PubMed]
  45. W. T. Freeman, “The generic viewpoint assumption in a Bayesian framework,” in Perception as Bayesian Inference, D. C. Knill, W. Richards, eds. (Cambridge U. Press, Cambridge, UK, 1996).
  46. H. H. Bülthoff, A. L. Yuille, “Shape from X: psychophysics and computation,” in Sensor Fusion III: 3D Perception and Recognition, P. S. Schenker, ed., Proc. SPIE1383, 235–246 (1990). [CrossRef]
  47. H. H. Bülthoff, “Shape from X: psychophysics and computation,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds. (MIT, Cambridge, Mass., 1991).
  48. A. L. Yuille, D. Geiger, H. H. Bülthoff, “Stereo integration, mean field theory and psychophysics,” Network 2, 423–442 (1991). [CrossRef]
  49. D. Ascher, N. M. Grzywacz, “A Bayesian model for the measurement of visual velocity,” Vision Res. 40, 3427–3434 (2000). [CrossRef] [PubMed]
  50. M. A. Hogervorst, R. A. Eagle, “Biases in three-dimensional structure-from-motion arise from noise in the early visual system,” Proc. R. Soc. London Ser. B 265, 1587–1593 (1998). [CrossRef]
  51. L. L. Kontsevich, C. W. Tyler, “Bayesian adaptive estimation of psychometric slope and threshold,” Vision Res. 39, 2729–2737 (1999). [CrossRef] [PubMed]
  52. P. Mamassian, M. S. Landy, “Observer biases in the 3D interpretation of line drawings,” Vision Res. 38, 2817–2832 (1998). [CrossRef] [PubMed]
  53. P. Mamassian, M. S. Landy, “Interaction of visual prior constraints,” Vision Res. 41, 2653–2668 (2001). [CrossRef] [PubMed]
  54. J. C. A. Read, “A Bayesian model of stereopsis depth and motion direction discrimination,” Biol. Cybern. 86, 117–136 (2002). [CrossRef] [PubMed]
  55. It is of historical interest to note that Bayes died in 1761 and that an essay that Bayes wrote had been published by the Royal Society56two years after his death. Bayes’s theorem was originally developed to model human conscious judgments during the playing of games, but it has proven to be wrong for this purpose.37In modern vision science, Bayes’s work has been attached to the following equation: p(S|I)∝p(I|S)p(S).It therefore comes as a surprise that this equation is not present in Bayes’s essay. According to Dale,57Laplace’s58formulations have mistakenly been applied as those of Bayes. This is not to say that Bayes does not deserve the name for the theory.
  56. T. Bayes, “An essay towards solving a problem in the doctrine of chances,” Philos. Trans. R. Soc. London 53, 370–418 (1763). [CrossRef]
  57. A. I. Dale, “Bayes or Laplace? An examination of the origin and early applications of Bayes’ theorem,” Arch. Hist. Exact Sci. 27, 23–47 (1982).
  58. P. S. Laplace, Théorie Analytique des Probabilités (Courcier, Paris, 1812).
  59. To decide which of the peaks corresponds to the weak rectangularity mode and which of the peaks corresponds to the strong rectangularity mode, we compared the peaks in the expected gain distribution with the highest peaks in the individual posterior distributions. It is relatively straightforward to shift the bifurcation point by applying a different gain function, producing bifurcation points that perfectly fit the obtained data. However, the coefficient goodness of fit that we generally applied (see Table 1) becomes then slightly worse relative to the best fit of the model.
  60. D. Kersten, H. H. Bülthoff, B. L. Schwartz, K. J. Kurtz, “Interaction between transparency and SFM,” Neural Comput. 4, 573–589 (1992). [CrossRef]
  61. In the top left panel of Fig. 3the model prediction exceeds the disparity-specified slant. This overprediction is relatively easy to prevent, but it involves, to our mind, ad hocphysiological assumptions.
  62. M. L. Braunstein, G. J. Andersen, M. W. Rouse, J. S. Tittle, “Recovering viewer-centered depth from disparity, occlusion, and velocity gradients,” Percept. Psychophys. 40, 216–224 (1986). [CrossRef] [PubMed]
  63. H. H. Bülthoff, H. A. Mallot, “Integration of depth modules: stereo and shading,” J. Opt. Soc. Am. A 5, 1749–1758 (1988). [CrossRef] [PubMed]
  64. B. A. Dosher, G. Sperling, S. A. Wurst, “Tradeoffs between stereopsis and proximity luminance covariance as determinants of perceived 3D structure,” Vision Res. 26, 973–990 (1986). [CrossRef] [PubMed]
  65. B. J. Rogers, T. S. Collett, “The appearance of surfaces specified by motion parallax and binocular disparity,” Q. J. Exp. Psychol. A 41, 697–717 (1989). [CrossRef] [PubMed]
  66. J. Turner, M. L. Braunstein, G. J. Andersen, “Relationship between binocular disparity and motion parallax insurface detection,” Percept. Psychophys. 59, 370–380 (1997). [CrossRef] [PubMed]
  67. H. C. van der Meer, “Interrelation of the effects of binocular disparity and perspective cues on judgments of depth and height,” Percept. Psychophys. 29, 481–488 (1979). [CrossRef]
  68. C. Wheatstone, “The Bakerian lecture: contributions to the physiology of vision—part the second. On some remarkable and hitherto unobserved phenomena of binocular vision,” Philos. Trans. R. Soc. London 142, 1–17 (1852). [CrossRef]
  69. R. B. Freeman, “Theory of cues and the psychophysics of visual space perception,” Psychonom. Monogr. 3, 171–181 (1970).
  70. L. T. Maloney, M. S. Landy, “A statistical framework for robust fusion of depth information,” in Visual Communications and Image Processing IV, W. A. Pearlman, ed., Proc. SPIE1199, 1154–1163 (1989). [CrossRef]
  71. M. S. Landy, L. T. Maloney, E. B. Johnston, M. Young, “Measurement and modeling of depth cue combination: in defense of weak fusion,” Vision Res. 35, 389–412 (1995). [CrossRef] [PubMed]
  72. I. Fine, R. A. Jacobs, “Modeling the combination of motion, stereo, and vergence angle cues to visual depth,” Neural Comput. 11, 1297–1330 (1999). [CrossRef] [PubMed]
  73. T. Poggio, E. B. Gamble, J. J. Little, “Parallel integration of vision modules,” Science 242, 436–440 (1988). [CrossRef] [PubMed]
  74. R. van Ee, C. J. Erkelens, “Conscious selection of bi-stable 3D percepts described by neural population codes,” J. Vision 2, S549a (2002).

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