Abstract

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

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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).

2002 (5)

T. V. Papathomas, “Experiments on the role of painted cues in Hughes’s reverspectives,” Perception 31, 521–530 (2002).
[CrossRef]

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]

R. van Ee, W. Richards, “A planar and a volumetric test for stereoanomaly,” Perception 31, 51–64 (2002).
[CrossRef] [PubMed]

J. C. A. Read, “A Bayesian model of stereopsis depth and motion direction discrimination,” Biol. Cybern. 86, 117–136 (2002).
[CrossRef] [PubMed]

R. van Ee, C. J. Erkelens, “Conscious selection of bi-stable 3D percepts described by neural population codes,” J. Vision 2, S549a (2002).

2001 (5)

P. Mamassian, M. S. Landy, “Interaction of visual prior constraints,” Vision Res. 41, 2653–2668 (2001).
[CrossRef] [PubMed]

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]

R. van Ee, B. L. Anderson, “Motion direction, speed, and orientation in binocular matching,” Nature 410, 690–694 (2001).
[CrossRef] [PubMed]

R. van Ee, K. Hol, C. J. Erkelens, “Bistable stereoscopic percepts and depth cue combination,” Perception 30, S42 (2001).

B. J. Gillam, M. L. Cook, “Perspective based on stereopsis and occlusion,” Psychol. Sci. 12, 424–429 (2001).
[CrossRef] [PubMed]

2000 (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]

R. S. Allison, I. P. Howard, “Stereopsis with persisting and dynamic textures,” Vision Res. 40, 3823–3827 (2000).
[CrossRef] [PubMed]

D. Ascher, N. M. Grzywacz, “A Bayesian model for the measurement of visual velocity,” Vision Res. 40, 3427–3434 (2000).
[CrossRef] [PubMed]

1999 (5)

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]

L. L. Kontsevich, C. W. Tyler, “Bayesian adaptive estimation of psychometric slope and threshold,” Vision Res. 39, 2729–2737 (1999).
[CrossRef] [PubMed]

R. van Ee, M. S. Banks, B. T. Backus, “An analysis of binocular slant contrast,” Perception 28, 1121–1145 (1999).
[CrossRef]

N. J. Wade, P. Hughes, “Fooling the eyes: trompe l’oeil and reverse perspective,” Perception 28, 1115–1119 (1999).
[CrossRef]

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]

1998 (3)

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]

P. Mamassian, M. S. Landy, “Observer biases in the 3D interpretation of line drawings,” Vision Res. 38, 2817–2832 (1998).
[CrossRef] [PubMed]

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]

1997 (1)

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]

1996 (1)

R. van Ee, C. J. Erkelens, “Temporal aspects of binocular slant perception,” Vision Res. 36, 43–51 (1996).
[CrossRef] [PubMed]

1995 (1)

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]

1994 (2)

W. T. Freeman, “The generic viewpoint assumption in a framework for visual perception,” Nature 368, 542–545 (1994).
[CrossRef] [PubMed]

C. Ryan, B. Gillam, “Cue conflict and stereoscopic surface slant about horizontal and vertical axes,” Perception 23, 645–658 (1994).
[CrossRef] [PubMed]

1993 (1)

H. Hill, V. Bruce, “Independent effects of lighting, orientation, and stereopsis on the hollow-face illusion,” Perception 22, 887–897 (1993).
[CrossRef] [PubMed]

1992 (2)

B. J. Gillam, C. Ryan, “Perspective, orientation disparity, and anisotropy in stereoscopic slant perception,” Perception 21, 427–439 (1992).
[CrossRef] [PubMed]

D. Kersten, H. H. Bülthoff, B. L. Schwartz, K. J. Kurtz, “Interaction between transparency and SFM,” Neural Comput. 4, 573–589 (1992).
[CrossRef]

1991 (2)

A. L. Yuille, D. Geiger, H. H. Bülthoff, “Stereo integration, mean field theory and psychophysics,” Network 2, 423–442 (1991).
[CrossRef]

K. A. Stevens, M. Lees, A. Brookes, “Combining binocular and monocular curvature features,” Perception 20, 425–440 (1991).
[CrossRef] [PubMed]

1989 (1)

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]

1988 (2)

T. Poggio, E. B. Gamble, J. J. Little, “Parallel integration of vision modules,” Science 242, 436–440 (1988).
[CrossRef] [PubMed]

H. H. Bülthoff, H. A. Mallot, “Integration of depth modules: stereo and shading,” J. Opt. Soc. Am. A 5, 1749–1758 (1988).
[CrossRef] [PubMed]

1986 (2)

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]

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]

1982 (1)

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).

1979 (1)

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]

1976 (1)

W. M. Youngs, “The influence of perspective and disparity cues on the perception of slant,” Vision Res. 16, 79–82 (1976).
[CrossRef] [PubMed]

1970 (1)

R. B. Freeman, “Theory of cues and the psychophysics of visual space perception,” Psychonom. Monogr. 3, 171–181 (1970).

1968 (1)

B. J. Gillam, “Perception of slant when perspective and stereopsis conflict: experiments with aniseikonic lenses,” J. Exp. Psychol. 78, 299–305 (1968).
[CrossRef] [PubMed]

1967 (1)

A. H. Smith, “Perceived slant as a function of stimulus contour and vertical dimension,” Percept. Mot. Skills 24, 167–173 (1967).
[CrossRef]

1925 (1)

W. Schriever, “Experimentelle Studien über stereoskopisches Sehen,” Z. Psychol. Physiol. Sinnesorgane 96, 113–170 (1925).

1866 (1)

E. Mach, “Über die physiologische Wirkung räumlich verteilter Lichtreize,” Sitzungsber. d. Wiener Akad. 54, 3 (1866).

1852 (1)

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]

1838 (1)

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]

1763 (1)

T. Bayes, “An essay towards solving a problem in the doctrine of chances,” Philos. Trans. R. Soc. London 53, 370–418 (1763).
[CrossRef]

Adams, W. J.

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]

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]

Allison, R. S.

R. S. Allison, I. P. Howard, “Stereopsis with persisting and dynamic textures,” Vision Res. 40, 3823–3827 (2000).
[CrossRef] [PubMed]

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]

Andersen, G. J.

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]

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]

Anderson, B. L.

R. van Ee, B. L. Anderson, “Motion direction, speed, and orientation in binocular matching,” Nature 410, 690–694 (2001).
[CrossRef] [PubMed]

Ascher, D.

D. Ascher, N. M. Grzywacz, “A Bayesian model for the measurement of visual velocity,” Vision Res. 40, 3427–3434 (2000).
[CrossRef] [PubMed]

Backus, B. T.

R. van Ee, M. S. Banks, B. T. Backus, “An analysis of binocular slant contrast,” Perception 28, 1121–1145 (1999).
[CrossRef]

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]

Banks, M. S.

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]

R. van Ee, M. S. Banks, B. T. Backus, “An analysis of binocular slant contrast,” Perception 28, 1121–1145 (1999).
[CrossRef]

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]

Bayes, T.

T. Bayes, “An essay towards solving a problem in the doctrine of chances,” Philos. Trans. R. Soc. London 53, 370–418 (1763).
[CrossRef]

Berger, J. O.

J. O. Berger, Statistical Decision Theory and Bayesian Analysis (Springer-Verlag, Berlin, 1985).

Braunstein, M. L.

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]

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]

Brookes, A.

K. A. Stevens, M. Lees, A. Brookes, “Combining binocular and monocular curvature features,” Perception 20, 425–440 (1991).
[CrossRef] [PubMed]

Bruce, V.

H. Hill, V. Bruce, “Independent effects of lighting, orientation, and stereopsis on the hollow-face illusion,” Perception 22, 887–897 (1993).
[CrossRef] [PubMed]

Buckley, D.

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]

Bülthoff, H. H.

D. Kersten, H. H. Bülthoff, B. L. Schwartz, K. J. Kurtz, “Interaction between transparency and SFM,” Neural Comput. 4, 573–589 (1992).
[CrossRef]

A. L. Yuille, D. Geiger, H. H. Bülthoff, “Stereo integration, mean field theory and psychophysics,” Network 2, 423–442 (1991).
[CrossRef]

H. H. Bülthoff, H. A. Mallot, “Integration of depth modules: stereo and shading,” J. Opt. Soc. Am. A 5, 1749–1758 (1988).
[CrossRef] [PubMed]

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).

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]

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).

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).

Clark, J. J.

J. J. Clark, A. L. Yuille, Data Fusion for Sensory Information Processing Systems (Kluwer Academic, Boston, 1990).

Collett, T. S.

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]

Cook, M. L.

B. J. Gillam, M. L. Cook, “Perspective based on stereopsis and occlusion,” Psychol. Sci. 12, 424–429 (2001).
[CrossRef] [PubMed]

Dale, A. I.

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).

Dosher, B. A.

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]

Eagle, R. A.

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]

Erkelens, C. J.

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]

R. van Ee, C. J. Erkelens, “Conscious selection of bi-stable 3D percepts described by neural population codes,” J. Vision 2, S549a (2002).

R. van Ee, K. Hol, C. J. Erkelens, “Bistable stereoscopic percepts and depth cue combination,” Perception 30, S42 (2001).

R. van Ee, C. J. Erkelens, “Temporal aspects of binocular slant perception,” Vision Res. 36, 43–51 (1996).
[CrossRef] [PubMed]

Ferguson, T.

T. Ferguson, Mathematical Statistics: a Decision Theoretic Approach (Academic, New York, 1967).

Fine, I.

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]

Freeman, R. B.

R. B. Freeman, “Theory of cues and the psychophysics of visual space perception,” Psychonom. Monogr. 3, 171–181 (1970).

Freeman, W. T.

W. T. Freeman, “The generic viewpoint assumption in a framework for visual perception,” Nature 368, 542–545 (1994).
[CrossRef] [PubMed]

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).

Frisby, J. P.

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]

Gamble, E. B.

T. Poggio, E. B. Gamble, J. J. Little, “Parallel integration of vision modules,” Science 242, 436–440 (1988).
[CrossRef] [PubMed]

Geiger, D.

A. L. Yuille, D. Geiger, H. H. Bülthoff, “Stereo integration, mean field theory and psychophysics,” Network 2, 423–442 (1991).
[CrossRef]

Gillam, B.

C. Ryan, B. Gillam, “Cue conflict and stereoscopic surface slant about horizontal and vertical axes,” Perception 23, 645–658 (1994).
[CrossRef] [PubMed]

Gillam, B. J.

B. J. Gillam, M. L. Cook, “Perspective based on stereopsis and occlusion,” Psychol. Sci. 12, 424–429 (2001).
[CrossRef] [PubMed]

B. J. Gillam, C. Ryan, “Perspective, orientation disparity, and anisotropy in stereoscopic slant perception,” Perception 21, 427–439 (1992).
[CrossRef] [PubMed]

B. J. Gillam, “Perception of slant when perspective and stereopsis conflict: experiments with aniseikonic lenses,” J. Exp. Psychol. 78, 299–305 (1968).
[CrossRef] [PubMed]

Gregory, R.

R. Gregory, The Intelligent Eye (Weidenfeld and Nicholson, London, 1970).

Grzywacz, N. M.

D. Ascher, N. M. Grzywacz, “A Bayesian model for the measurement of visual velocity,” Vision Res. 40, 3427–3434 (2000).
[CrossRef] [PubMed]

Hill, H.

H. Hill, V. Bruce, “Independent effects of lighting, orientation, and stereopsis on the hollow-face illusion,” Perception 22, 887–897 (1993).
[CrossRef] [PubMed]

Hogervorst, M. A.

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]

Hol, K.

R. van Ee, K. Hol, C. J. Erkelens, “Bistable stereoscopic percepts and depth cue combination,” Perception 30, S42 (2001).

Howard, I. P.

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]

R. S. Allison, I. P. Howard, “Stereopsis with persisting and dynamic textures,” Vision Res. 40, 3823–3827 (2000).
[CrossRef] [PubMed]

Hughes, P.

N. J. Wade, P. Hughes, “Fooling the eyes: trompe l’oeil and reverse perspective,” Perception 28, 1115–1119 (1999).
[CrossRef]

Jacobs, R. A.

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]

Johnston, E. B.

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]

Kersten, D.

D. Kersten, H. H. Bülthoff, B. L. Schwartz, K. J. Kurtz, “Interaction between transparency and SFM,” Neural Comput. 4, 573–589 (1992).
[CrossRef]

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).

Knill, D. C.

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).

D. C. Knill, W. Richards, Perception as Bayesian Inference (Cambridge U. Press, Cambridge, UK, 1996).

Kontsevich, L. L.

L. L. Kontsevich, C. W. Tyler, “Bayesian adaptive estimation of psychometric slope and threshold,” Vision Res. 39, 2729–2737 (1999).
[CrossRef] [PubMed]

Kurtz, K. J.

D. Kersten, H. H. Bülthoff, B. L. Schwartz, K. J. Kurtz, “Interaction between transparency and SFM,” Neural Comput. 4, 573–589 (1992).
[CrossRef]

Landy, M. S.

P. Mamassian, M. S. Landy, “Interaction of visual prior constraints,” Vision Res. 41, 2653–2668 (2001).
[CrossRef] [PubMed]

P. Mamassian, M. S. Landy, “Observer biases in the 3D interpretation of line drawings,” Vision Res. 38, 2817–2832 (1998).
[CrossRef] [PubMed]

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]

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]

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).

Laplace, P. S.

P. S. Laplace, Théorie Analytique des Probabilités (Courcier, Paris, 1812).

Lees, M.

K. A. Stevens, M. Lees, A. Brookes, “Combining binocular and monocular curvature features,” Perception 20, 425–440 (1991).
[CrossRef] [PubMed]

Little, J. J.

T. Poggio, E. B. Gamble, J. J. Little, “Parallel integration of vision modules,” Science 242, 436–440 (1988).
[CrossRef] [PubMed]

Mach, E.

E. Mach, “Über die physiologische Wirkung räumlich verteilter Lichtreize,” Sitzungsber. d. Wiener Akad. 54, 3 (1866).

Mallot, H. A.

H. H. Bülthoff, H. A. Mallot, “Integration of depth modules: stereo and shading,” J. Opt. Soc. Am. A 5, 1749–1758 (1988).
[CrossRef] [PubMed]

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).

Maloney, L. T.

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]

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]

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).

L. T. Maloney, “Statistical decision theory and biological vision,” in Perception and the Physical World, D. Heyer, R. Mausfeld, eds. (Wiley, Chichester, UK, 2002).

Mamassian, P.

P. Mamassian, M. S. Landy, “Interaction of visual prior constraints,” Vision Res. 41, 2653–2668 (2001).
[CrossRef] [PubMed]

P. Mamassian, M. S. Landy, “Observer biases in the 3D interpretation of line drawings,” Vision Res. 38, 2817–2832 (1998).
[CrossRef] [PubMed]

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).

Massaro, D. W.

D. W. Massaro, Speech Perception by Ear and Eye (Erlbaum, Hillsdale, N.J., 1987).

Papathomas, T. V.

T. V. Papathomas, “Experiments on the role of painted cues in Hughes’s reverspectives,” Perception 31, 521–530 (2002).
[CrossRef]

Poggio, T.

T. Poggio, E. B. Gamble, J. J. Little, “Parallel integration of vision modules,” Science 242, 436–440 (1988).
[CrossRef] [PubMed]

Porrill, J.

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]

Read, J. C. A.

J. C. A. Read, “A Bayesian model of stereopsis depth and motion direction discrimination,” Biol. Cybern. 86, 117–136 (2002).
[CrossRef] [PubMed]

Richards, W.

R. van Ee, W. Richards, “A planar and a volumetric test for stereoanomaly,” Perception 31, 51–64 (2002).
[CrossRef] [PubMed]

D. C. Knill, W. Richards, Perception as Bayesian Inference (Cambridge U. Press, Cambridge, UK, 1996).

Rogers, B. J.

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]

Rouse, M. W.

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]

Ryan, C.

C. Ryan, B. Gillam, “Cue conflict and stereoscopic surface slant about horizontal and vertical axes,” Perception 23, 645–658 (1994).
[CrossRef] [PubMed]

B. J. Gillam, C. Ryan, “Perspective, orientation disparity, and anisotropy in stereoscopic slant perception,” Perception 21, 427–439 (1992).
[CrossRef] [PubMed]

Schriever, W.

W. Schriever, “Experimentelle Studien über stereoskopisches Sehen,” Z. Psychol. Physiol. Sinnesorgane 96, 113–170 (1925).

Schwartz, B. L.

D. Kersten, H. H. Bülthoff, B. L. Schwartz, K. J. Kurtz, “Interaction between transparency and SFM,” Neural Comput. 4, 573–589 (1992).
[CrossRef]

Slyce, J.

J. Slyce, Patrick Hughes: Perverspective (Momentum, London, 1998).

Smith, A. H.

A. H. Smith, “Perceived slant as a function of stimulus contour and vertical dimension,” Percept. Mot. Skills 24, 167–173 (1967).
[CrossRef]

Sperling, G.

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]

Stevens, K. A.

K. A. Stevens, M. Lees, A. Brookes, “Combining binocular and monocular curvature features,” Perception 20, 425–440 (1991).
[CrossRef] [PubMed]

Tittle, J. S.

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]

Turner, J.

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]

Tyler, C. W.

L. L. Kontsevich, C. W. Tyler, “Bayesian adaptive estimation of psychometric slope and threshold,” Vision Res. 39, 2729–2737 (1999).
[CrossRef] [PubMed]

van Dam, L. C. J.

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]

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).

van der Meer, H. C.

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]

van Ee, R.

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]

R. van Ee, C. J. Erkelens, “Conscious selection of bi-stable 3D percepts described by neural population codes,” J. Vision 2, S549a (2002).

R. van Ee, W. Richards, “A planar and a volumetric test for stereoanomaly,” Perception 31, 51–64 (2002).
[CrossRef] [PubMed]

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]

R. van Ee, K. Hol, C. J. Erkelens, “Bistable stereoscopic percepts and depth cue combination,” Perception 30, S42 (2001).

R. van Ee, B. L. Anderson, “Motion direction, speed, and orientation in binocular matching,” Nature 410, 690–694 (2001).
[CrossRef] [PubMed]

R. van Ee, M. S. Banks, B. T. Backus, “An analysis of binocular slant contrast,” Perception 28, 1121–1145 (1999).
[CrossRef]

R. van Ee, C. J. Erkelens, “Temporal aspects of binocular slant perception,” Vision Res. 36, 43–51 (1996).
[CrossRef] [PubMed]

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).

von Helmholtz, H.

H. von Helmholtz, Handbuch der Physiologischen Optik (Voss, Hamburg, Germany, 1866), Vol. 3, Sec. 26.

Wade, N. J.

N. J. Wade, P. Hughes, “Fooling the eyes: trompe l’oeil and reverse perspective,” Perception 28, 1115–1119 (1999).
[CrossRef]

Wheatstone, C.

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]

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]

Wurst, S. A.

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]

Young, M.

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]

Youngs, W. M.

W. M. Youngs, “The influence of perspective and disparity cues on the perception of slant,” Vision Res. 16, 79–82 (1976).
[CrossRef] [PubMed]

Yuille, A. L.

A. L. Yuille, D. Geiger, H. H. Bülthoff, “Stereo integration, mean field theory and psychophysics,” Network 2, 423–442 (1991).
[CrossRef]

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).

J. J. Clark, A. L. Yuille, Data Fusion for Sensory Information Processing Systems (Kluwer Academic, Boston, 1990).

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]

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).

Arch. Hist. Exact Sci. (1)

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).

Biol. Cybern. (1)

J. C. A. Read, “A Bayesian model of stereopsis depth and motion direction discrimination,” Biol. Cybern. 86, 117–136 (2002).
[CrossRef] [PubMed]

J. Exp. Psychol. (1)

B. J. Gillam, “Perception of slant when perspective and stereopsis conflict: experiments with aniseikonic lenses,” J. Exp. Psychol. 78, 299–305 (1968).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (1)

J. Vision (2)

R. van Ee, C. J. Erkelens, “Conscious selection of bi-stable 3D percepts described by neural population codes,” J. Vision 2, S549a (2002).

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]

Nat. Neurosci. (1)

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]

Nature (3)

R. van Ee, B. L. Anderson, “Motion direction, speed, and orientation in binocular matching,” Nature 410, 690–694 (2001).
[CrossRef] [PubMed]

W. T. Freeman, “The generic viewpoint assumption in a framework for visual perception,” Nature 368, 542–545 (1994).
[CrossRef] [PubMed]

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]

Network (1)

A. L. Yuille, D. Geiger, H. H. Bülthoff, “Stereo integration, mean field theory and psychophysics,” Network 2, 423–442 (1991).
[CrossRef]

Neural Comput. (2)

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]

D. Kersten, H. H. Bülthoff, B. L. Schwartz, K. J. Kurtz, “Interaction between transparency and SFM,” Neural Comput. 4, 573–589 (1992).
[CrossRef]

Percept. Mot. Skills (1)

A. H. Smith, “Perceived slant as a function of stimulus contour and vertical dimension,” Percept. Mot. Skills 24, 167–173 (1967).
[CrossRef]

Percept. Psychophys. (3)

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]

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]

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]

Perception (9)

R. van Ee, M. S. Banks, B. T. Backus, “An analysis of binocular slant contrast,” Perception 28, 1121–1145 (1999).
[CrossRef]

K. A. Stevens, M. Lees, A. Brookes, “Combining binocular and monocular curvature features,” Perception 20, 425–440 (1991).
[CrossRef] [PubMed]

H. Hill, V. Bruce, “Independent effects of lighting, orientation, and stereopsis on the hollow-face illusion,” Perception 22, 887–897 (1993).
[CrossRef] [PubMed]

R. van Ee, K. Hol, C. J. Erkelens, “Bistable stereoscopic percepts and depth cue combination,” Perception 30, S42 (2001).

T. V. Papathomas, “Experiments on the role of painted cues in Hughes’s reverspectives,” Perception 31, 521–530 (2002).
[CrossRef]

B. J. Gillam, C. Ryan, “Perspective, orientation disparity, and anisotropy in stereoscopic slant perception,” Perception 21, 427–439 (1992).
[CrossRef] [PubMed]

C. Ryan, B. Gillam, “Cue conflict and stereoscopic surface slant about horizontal and vertical axes,” Perception 23, 645–658 (1994).
[CrossRef] [PubMed]

R. van Ee, W. Richards, “A planar and a volumetric test for stereoanomaly,” Perception 31, 51–64 (2002).
[CrossRef] [PubMed]

N. J. Wade, P. Hughes, “Fooling the eyes: trompe l’oeil and reverse perspective,” Perception 28, 1115–1119 (1999).
[CrossRef]

Philos. Trans. R. Soc. London (3)

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]

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]

T. Bayes, “An essay towards solving a problem in the doctrine of chances,” Philos. Trans. R. Soc. London 53, 370–418 (1763).
[CrossRef]

Proc. R. Soc. London Ser. B (1)

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]

Psychol. Sci. (1)

B. J. Gillam, M. L. Cook, “Perspective based on stereopsis and occlusion,” Psychol. Sci. 12, 424–429 (2001).
[CrossRef] [PubMed]

Psychonom. Monogr. (1)

R. B. Freeman, “Theory of cues and the psychophysics of visual space perception,” Psychonom. Monogr. 3, 171–181 (1970).

Q. J. Exp. Psychol. A (1)

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]

Science (1)

T. Poggio, E. B. Gamble, J. J. Little, “Parallel integration of vision modules,” Science 242, 436–440 (1988).
[CrossRef] [PubMed]

Sitzungsber. d. Wiener Akad. (1)

E. Mach, “Über die physiologische Wirkung räumlich verteilter Lichtreize,” Sitzungsber. d. Wiener Akad. 54, 3 (1866).

Vision Res. (11)

R. van Ee, C. J. Erkelens, “Temporal aspects of binocular slant perception,” Vision Res. 36, 43–51 (1996).
[CrossRef] [PubMed]

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]

R. S. Allison, I. P. Howard, “Stereopsis with persisting and dynamic textures,” Vision Res. 40, 3823–3827 (2000).
[CrossRef] [PubMed]

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]

W. M. Youngs, “The influence of perspective and disparity cues on the perception of slant,” Vision Res. 16, 79–82 (1976).
[CrossRef] [PubMed]

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]

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]

L. L. Kontsevich, C. W. Tyler, “Bayesian adaptive estimation of psychometric slope and threshold,” Vision Res. 39, 2729–2737 (1999).
[CrossRef] [PubMed]

P. Mamassian, M. S. Landy, “Observer biases in the 3D interpretation of line drawings,” Vision Res. 38, 2817–2832 (1998).
[CrossRef] [PubMed]

P. Mamassian, M. S. Landy, “Interaction of visual prior constraints,” Vision Res. 41, 2653–2668 (2001).
[CrossRef] [PubMed]

D. Ascher, N. M. Grzywacz, “A Bayesian model for the measurement of visual velocity,” Vision Res. 40, 3427–3434 (2000).
[CrossRef] [PubMed]

Z. Psychol. Physiol. Sinnesorgane (1)

W. Schriever, “Experimentelle Studien über stereoskopisches Sehen,” Z. Psychol. Physiol. Sinnesorgane 96, 113–170 (1925).

Other (27)

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

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

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).

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.

J. O. Berger, Statistical Decision Theory and Bayesian Analysis (Springer-Verlag, Berlin, 1985).

T. Ferguson, Mathematical Statistics: a Decision Theoretic Approach (Academic, New York, 1967).

D. W. Massaro, Speech Perception by Ear and Eye (Erlbaum, Hillsdale, N.J., 1987).

J. J. Clark, A. L. Yuille, Data Fusion for Sensory Information Processing Systems (Kluwer Academic, Boston, 1990).

L. T. Maloney, “Statistical decision theory and biological vision,” in Perception and the Physical World, D. Heyer, R. Mausfeld, eds. (Wiley, Chichester, UK, 2002).

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).

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).

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).

D. C. Knill, W. Richards, Perception as Bayesian Inference (Cambridge U. Press, Cambridge, UK, 1996).

See also other interesting contributions in Refs. 20-23.

R. Gregory, The Intelligent Eye (Weidenfeld and Nicholson, London, 1970).

J. Slyce, Patrick Hughes: Perverspective (Momentum, London, 1998).

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.

H. von Helmholtz, Handbuch der Physiologischen Optik (Voss, Hamburg, Germany, 1866), Vol. 3, Sec. 26.

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).

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).

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]

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).

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.

P. S. Laplace, Théorie Analytique des Probabilités (Courcier, Paris, 1812).

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.

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.

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]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

In this stereogram both perspective and binocular disparity specify surface slant about the vertical axis. In uncrossed fusion of the stereogram (the left eye views the left image, and the right eye views the right image), two stable percepts can be distinguished. In the first percept the grid recedes in depth with its right side farther away (it is perceived as a slanted rectangle). In the other percept the left side of the grid is farther away (it is perceived as a trapezoid with the near edge shorter than the far edge). In fact, the slants depend on the viewing distance; however, the slant signs are conflicting regardless of the viewing distance. Note that each of the two percepts can be selected and maintained at will in a relatively controlled fashion. In crossed fusion, perspective and disparity specify similar slants and the observer perceives a single stable slanted rectangular grid with its right side further away.

Fig. 2
Fig. 2

Schematic drawing of the slant-estimation method representing a top view of the viewing geometry. One of the lines was fixed, and the other two lines could be rotated about their centers. The fixed line represented zero slant (the image plane on the frontal screen); each of the other lines represented the perceived grid in either the perspective-dominated percept or in the disparity-dominated percept. Using this display, subjects matched the perceived slant(s) to the angle(s) between the fixed horizontal line and the rotatable intersecting line(s).

Fig. 3
Fig. 3

Perceived slant and the Bayesian fits as a function of disparity-specified slant for a range of different perspective-specified slants. Each row of panels represents the data of one subject. The top row depicts the best fit that accounted for 93% of the variance in the data. The bottom row depicts the worst fit that accounted for 79% of the variance in the data. The fits to the data produced by the Bayesian model are indicated by the gray and black curves. The gray curves indicate the strong rectangular assumption, and the black curves indicate the weak rectangular assumption. The subjects perceived either a slanted rectangular grid (square symbols) or a slanted trapezoid (triangles). The slants that were geometrically present in the stimulus are represented by the dashed prediction lines. Error bars represent ±1 standard deviation in the mean across the five trial repetitions.

Fig. 4
Fig. 4

Nonvertical stimulus lines change their orientation in the image plane when the stimulus is rotated about the vertical axis. (A) Frontal view of an unslanted trapezoidal object (w is the width, h is the central height, and d is the viewing distance). The orientation of the depicted stimulus line is θ. The 3D coordinates relative to the midpoint between the eyes are explicitly given. (B) A top view of the object after it has been rotated through an angle φ. (C) A projection of the slanted trapezoid on a frontal screen. The stimulus line whose orientation was originally θ is now projected with an orientation γ [see Eq. (A2) in Appendix A].

Fig. 5
Fig. 5

Normalized likelihood p(I|S) for perspective information computed from the geometry of perspective projection, assuming that the perceived orientation of each of the horizontal lines projected onto the image is subject to noise. The noise is assumed to be Gaussian centered on zero and with a standard deviation. Each curve shows the likelihood for one of the seven perspective-specified slants. For larger perspective-specified slants, the likelihood distribution is more peaked, reflecting an increased certainty in determining surface slant.

Fig. 6
Fig. 6

Same as Fig. 3 but for two observers who did not follow the data pattern shown by the other eight observers. Both AB and KM observed almost no bistability. The data of AB are dominated by disparity information. The percept of observer KM seems almost entirely dominated by perspective; all data lines are near horizontal, showing little effect of disparity. Note that in almost all panels the fits produced by the Bayesian model fall on top of each other for both the weak and the strong rectangularity assumptions. The fits for AB and KM accounted for 92% and 93% of the variance in the data, respectively.

Tables (1)

Tables Icon

Table 1 Model Parameters for the Individual Observers a

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

p(θ; r)=12πr2exp-θ22r2.
tan(γ)=d tan(θ)-h sin(φ)d cos(φ),
p(d|φ; σd)=12πσd2exp-(φ-φd)22σd2.
p(φ; σp)=12πσp2exp-φ22σp2.
p(S|I)p(I|S)p(S).
p(φ|γ, d; r, σd, σp)p(γ|φ; r)p(d|φ; σd)p(φ; σp).
G(φ; σg)=12πσg2exp-φ22σg2.
E(φ; r1, r2, σd, σp, σg)
=G(φ; σg)*[p(φ|γ, d; r1, σd, σp)
+p(φ|γ, d; r2, σd, σp)].

Metrics