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A statistical explanation of visual space


The subjective visual space perceived by humans does not reflect a simple transformation of objective physical space; rather, perceived space has an idiosyncratic relationship with the real world. To date, there is no consensus about either the genesis of perceived visual space or the implications of its peculiar characteristics for visually guided behavior. Here we used laser range scanning to measure the actual distances from the image plane of all unoccluded points in a series of natural scenes. We then asked whether the differences between real and apparent distances could be explained by the statistical relationship of scene geometry and the observer. We were able to predict perceived distances in a variety of circumstances from the probability distribution of physical distances. This finding lends support to the idea that the characteristics of human visual space are determined probabilistically.

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Figure 1: Anomalies in perceived distance.
Figure 2: A representative range image taken from one of the wide-field images acquired by laser range scanning.
Figure 3: Probability distributions of the physical distances from the image plane of points in the range image database of natural scenes.
Figure 4: The perceived distances predicted for objects located at eye level, and for objects on the ground.
Figure 5: Probability distribution of physical distances at different elevation angles.
Figure 6: Probability distributions of physical distances below eye level when the terrain has a local dip or a hump.
Figure 7: Statistical explanation of the effect of a dip in the ground-plane on perceived distance.
Figure 8: Statistical explanation of the effect of a hump in the ground plane on perceived distance.


  1. Hershenson, M. Visual Space Perception: a Primer (MIT Press, Cambridge, Massachusetts, 1999).

    Google Scholar 

  2. Gillam, B. The perception of spatial layout from static optical information. in Perception of Space and Motion (eds. Epstein, W. & Rogers, S.) 23–67 (Academic, New York, 1995).

    Chapter  Google Scholar 

  3. Sedgwick, H.A. Space perception. in Handbook of Perception and Human Performance Vol. 1 (eds. Boff, K.R., Kaufman, L. & Thomas, J. P.) 21.1–21.57 (Wiley, Toronto, 1986).

    Google Scholar 

  4. Loomis, J.M., Da Silva, J.A., Philbeck, J.W. & Fukusima, S.S. Visual perception of location and distance. Curr. Dir. Psych. Sci. 5, 72–77 (1996).

    Article  Google Scholar 

  5. Gogel, W.C. Equidistance tendency and its consequences. Psychol. Bull. 64, 153–163 (1965).

    Article  CAS  Google Scholar 

  6. Owens, D.A. & Leibowitz, H.W. Oculomotor adjustments in darkness and the specific distance tendency. Percept. Psychophys. 20, 2–9 (1976).

    Article  Google Scholar 

  7. Epstein, W. & Landauer, A.A. Size and distance judgments under reduced conditions of viewing. Percept. Psychophys. 6, 269–272 (1969).

    Article  Google Scholar 

  8. Gogel, W.C. & Tietz, J.D. A comparison of oculomotor and motion parallax cues of egocentric distance. Vis. Res. 19, 1161–1170 (1979).

    Article  CAS  Google Scholar 

  9. Morrison, J.D. & Whiteside, T.C.D. Binocular cues in the perception of distance to a point source of light. Perception 13, 555–566 (1984).

    Article  CAS  Google Scholar 

  10. Foley, J.M. Binocular distance perception: egocentric distance tasks. J. Exp. Psychol. Hum. Percept. Perform. 11, 133–149 (1985).

    Article  CAS  Google Scholar 

  11. Philbeck, J.W. & Loomis, J.M. Comparison of two indicators of perceived egocentric distance under full-cue and reduced-cue conditions. J. Exp. Psychol. Hum. Percept. Perform. 23, 72–85 (1997).

    Article  CAS  Google Scholar 

  12. Wallach, H. & O'Leary, A. Slope of regard as a distance cue. Percept. Psychophys. 31, 145–148 (1982).

    Article  CAS  Google Scholar 

  13. Ooi, T.L., Wu, B. & He, Z.J. Distance determined by the angular declination below the horizon. Nature 414, 197–200 (2001).

    Article  CAS  Google Scholar 

  14. Sinai, M.J., Ooi, T.L. & He, Z.J. Terrain influences the accurate judgment of distance. Nature 395, 497–500 (1998).

    Article  CAS  Google Scholar 

  15. Meng, J.C. & Sedgwick, H.A. Distance perception mediated through nested contact relations among surfaces. Percept. Psychophys. 63, 1–15 (2001).

    Article  CAS  Google Scholar 

  16. Knill, D.C. & Richards, W. Perception as Bayesian Inference (Cambridge Univ. Press, Cambridge, 1996).

    Book  Google Scholar 

  17. Purves, D. & Lotto, B. Why We See What We Do: an Empirical Theory of Vision (Sinauer, Sunderland, Massachusetts, 2003).

    Google Scholar 

  18. Kersten, D. High-level vision as statistical inference. in The New Cognitive Neurosciences. 2nd edn. (ed. Gazzaniga, M.S.) 353–363 (MIT Press, Cambridge, Massachusetts, 1999).

    Google Scholar 

  19. Geisler, W.S. & Kersten, D. Illusions, perception and Bayes. Nat. Neurosci. 5, 508–510 (2002).

    Article  CAS  Google Scholar 

  20. Belhumeur, P.N. A Bayesian approach to binocular stereopsis. Intl. J. Comp. Vision 19, 237–260 (1996).

    Article  Google Scholar 

  21. Bloj, M.G., Kersten, D. & Hurlbert, A.C. Perception of three-dimensional shape influences colour perception through mutual illumination. Nature 402, 877–879 (1999).

    Article  CAS  Google Scholar 

  22. Weiss, Y., Simoncelli, E. & Adelson, E.H. Motion illusions as optimal percepts. Nat. Neurosci. 5, 598–604 (2002).

    Article  CAS  Google Scholar 

  23. Geisler, W.S., Perry, J.S., Super, B.J. & Gallogly, D.P. Edge co-occurrence in natural images predicts contour grouping performance. Vis. Res. 41, 711–724 (2001).

    Article  CAS  Google Scholar 

  24. Mumford, D. & Gidas, B. Stochastic models for generic images. Q. J. Appl. Math. 59, 85–111 (2001).

    Article  Google Scholar 

  25. Lee, A.B., Mumford, D. & Huang, J. Occlusion models for natural images: a statistical study of a scale-invariant dead leaves model. Int. J. Comput. Vision 41, 35–59 (2001).

    Article  Google Scholar 

  26. Luneberg, R.K. Mathematical Analysis of Binocular Vision (Princeton Univ. Press, Princeton, New Jersey, 1947).

    Google Scholar 

  27. Indow, T. A critical review of Luneburg's model with regard to global structure of visual space. Psychol. Rev. 98, 430–453 (1991).

    Article  CAS  Google Scholar 

  28. Wagner, M. The metric of visual space. Percept. Psychophys. 38, 483–495 (1985).

    Article  CAS  Google Scholar 

  29. Todd, J.T., Oomes, A.H.J., Koenderink, J.J. & Kappers, A.M.L. On the affine structure of perceptual space. Psychol. Sci. 12, 191–196 (2001).

    Article  CAS  Google Scholar 

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

    Google Scholar 

  31. Simoncelli, E.P. & Olshausen, B.A. Natural image statistics and neural representation. Annu. Rev. Neurosci. 24, 1193–1216 (2001).

    Article  CAS  Google Scholar 

  32. Olshausen, B.A. & Field, D.J. Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature 381, 607–609 (1996).

    Article  CAS  Google Scholar 

  33. Vinje, W.E. & Gallant, J.L. Sparse coding and decorrelation in primary cortex during natural vision. Science 287, 1273–1276 (2000).

    Article  CAS  Google Scholar 

  34. Sigman, M., Cecchi, G.A., Gilbert, C.D. & Magnasco, M.O. On a common circle: natural scenes and Gestalt rules. Proc. Natl. Acad. Sci. USA 98, 1935–1940 (2001).

    Article  CAS  Google Scholar 

  35. Howe, C.Q. & Purves, D. The statistics of range images can explain the anomalous perception of length. Proc. Natl. Acad. Sci. USA 99, 13184–13188 (2002).

    Article  CAS  Google Scholar 

  36. Huang, J., Lee, A.B. & Mumford, D. Statistics of range images. Proc. IEEE Conf. CVPR 1, 324–331 (2000).

    Google Scholar 

  37. Brainard, D.H. & Freeman, W.T. Bayesian color constancy. J. Opt. Soc. Am. A 14, 1393–1411 (1997).

    Article  CAS  Google Scholar 

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We thank C. Howe, F. Long, S. Nundy, D. Schwartz and J. Voyvodic for useful comments, and M. Williams for help with the art. This project was supported by the National Institutes of Health and the Geller endowment.

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Correspondence to Zhiyong Yang.

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Yang, Z., Purves, D. A statistical explanation of visual space. Nat Neurosci 6, 632–640 (2003).

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