Abstract

Many studies have demonstrated that the human visual system is sensitive to very small differences in relative binocular disparity. It is not known over what monocular regions information is spatially integrated to mediate performance in such tasks. In this study we present psychophysical observations that define the smallest spatial scale involved in disparity processing, and we indicate the nature of the computations performed by the units mediating that disparity discrimination. We show that human observers can identify the sign of disparity of a single target dot when it is embedded in a row of identical dots, with these noise dots presented either in the fixation plane or with a proportion binocularly uncorrelated. In conjunction with the psychophysical data, we explore how a class of simple correlator models of stereopsis must be constrained in order to account for human performance for the same fine-scale tasks. Such models can perform the task only when the correlation is carried out over a very small region of the image, for a very small range of disparities. Our results demonstrate that there is a fine-scale input to the stereo system, mediated by foveal mechanisms that spatially integrate visual signals over a region as small as 4–6 arcmin in diameter.

© 1997 Optical Society of America

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  1. R. N. Berry, “Quantitative relations among vernier, real depth and stereoscopic depth acuities,” J. Exp. Psychol. 38, 708–721 (1948).
    [Crossref] [PubMed]
  2. G. Westheimer, S. P. McKee, “Stereoscopic acuity for moving retinal images,” J. Opt. Soc. Am. 68, 450–455 (1978).
    [Crossref] [PubMed]
  3. C. A. Curcio, K. R. Sloan, O. Packer, A. E. Hendrickson, R. E. Kalina, “Distribution of cones in human and monkey retina: individual variability and radial symmetry,” Science 236, 579–582 (1987).
    [Crossref] [PubMed]
  4. C. A. Curcio, J. E. Sheedy, I. L. Bailey, M. Buri, E. Bass, “Binocular vs. monocular task performance,” Am. J. Optom. Physiol. Opt. 63, 839–846 (1986).
    [Crossref]
  5. R. Mestel, “Night of the strangest comet,” New Scientist 143 (No. 1933), 23–25 (1994).
  6. D. Marr, T. Poggio, “A computational theory of human stereo vision,” Proc. R. Soc. London, Ser. B 204, 301–328 (1979).
    [Crossref]
  7. H. K. Nishihara, “Practical real time imaging stereo matcher,” Opt. Eng. 23, 536–545 (1984).
    [Crossref]
  8. D. G. Jones, J. Malik, “Computational framework for determining stereo correspondence from a set of linear spatial filters,” Image Vision Comput. 10, 699–708 (1992).
    [Crossref]
  9. L. K. Cormack, S. B. Stevenson, C. M. Schor, “Interocular correlation, luminance contrast and cyclopean processing,” Vision Res. 31, 2195–2207 (1991).
    [Crossref] [PubMed]
  10. I. Ohzawa, G. C. DeAngelis, R. D. Freeman, “Stereoscopic depth discrimination in the visual cortex: neurons ideally suited as disparity detectors,” Science 249, 1037–1041 (1990).
    [Crossref] [PubMed]
  11. G. J. Mitchison, “New depth in stereopsis,” Nature 304, 123–124 (1983).
    [Crossref] [PubMed]
  12. D. Marr, T. Poggio, “The cooperative computation of stereo disparity,” Science 194, 283–287 (1976).
    [Crossref] [PubMed]
  13. D. Marr, Vision (Freeman, San Francisco, Calif., 1982).
  14. S. B. Pollard, J. E. W. Mayhew, J. P. Frisby, “PMF: a stereo correspondence algorithm using a disparity gradient limit,” Perception 14, 449–470 (1985).
  15. G. Sperling, “Binocular vision: a physical and neural theory,” J. Am. Psychol. 83, 461–534 (1970).
  16. J. I. Nelson, “Globality and stereoscopic fusion in binocular vision,” J. Theor. Biol. 49, 1–88 (1975).
    [Crossref] [PubMed]
  17. J. M. Harris, S. N. J. Watamaniuk, “Speed discrimination of motion-in-depth using binocular cues,” Vision Res. 35, 885–896 (1995).
    [Crossref] [PubMed]
  18. Note that the stimuli were presented on x–y CRT screens rather than on standard computer displays. The advantage of this is that points can be plotted with great precision. At the 1.5-m viewing distance, dots can be positioned 6.6 arcsec apart. However, because of the spreading of the electron beam on the screen and spreading caused by the eyes’ optics [F. W. Campbell, R. W. Gubisch, “Optical quality of the human eye,” J. Physiol. (London) 186, 558–578 (1966)], each dot appeared to have a width of approximately 1.5 s which we estimated by displaying a regular array of bright dots and reducing the interdot separation until the dots merged together.
  19. L. K. Cormack, S. B. Stevenson, C. M. Schor, “An upper limit to the binocular combination of stimuli,” Vision Res. 34, 2599–2608 (1994).
    [Crossref] [PubMed]
  20. B. M. Dow, A. Z. Snyder, R. G. Vautin, R. Bauer, “Magnification factor and receptive field size in foveal striate cortex of the monkey,” Exp. Brain Res. 44, 213–228 (1981).
    [Crossref] [PubMed]
  21. G. Westheimer, I. J. Tanzman, “Qualitative depth localization with diplopic images,” J. Opt. Soc. Am. 46, 116–117 (1956).
    [Crossref] [PubMed]
  22. W. Richards, “Stereopsis and stereoblindness,” Exp. Brain Res. 10, 380–388 (1970).
    [Crossref] [PubMed]
  23. A. L. Duwaer, “Patent stereopsis with diplopia in random-dot stereograms,” Percept. Psychophys. 33, 443–454 (1983).
    [Crossref] [PubMed]
  24. A. Glennerster, B. J. Rogers, “Stereo ‘dmax’ as a function of dot density,” Invest. Ophthalmol. Vis. Sci. (Suppl.) 33, 1369 (1992).
  25. In performing the simulations we had to make an arbitrary decision about how many arcminutes each pixel in the model array would represent. The question, in other words, is how does one choose the number of points in the memory array that is used to represent the image? For stimuli created on standard computer monitor screens, this is usually not considered a problem (although the problem of the optical transfer function of the eye is always present). One screen pixel is typically equated with one location in the array used for modeling purposes. However, here each dot was physically larger than the minimum separation between points. We expected model performance to vary as a function of dot density, and the model array dot density (in dots per total number of possible dot positions) would obviously increase as the allowed number of dot positions decreased. Over a wide range, varying both density and dot size in the model did not make much difference for the re- sults. In the simulations presented here, we chose each pixel to be equal to 0.5 arcmin.
  26. The results for a correlator exactly the same size as the image, with a disparity range including all possible disparities, are not shown. With no target present, such an ideal correlator will produce an exactly symmetrical correlation distribution. Adding a target dot will slightly distort the symmetry, but the existence of symmetry makes the performance very high. This is an unrealistic result: adding a small amount of noise or even a small amount of uncertainty to the correlation area size wipes out the symmetry and reduces performance to almost chance levels.
  27. H. S. Smallman, D. I. A. MacLeod, “Size–disparity correlation in stereopsis at contrast threshold,” J. Opt. Soc. Am. A 11, 2169–2183 (1994).
    [Crossref]
  28. In further simulations (not shown) we tested whether a larger correlator with a small disparity range would be able to mimic the human data. Although similar tuning was found, the overall performance level was much lower, suggesting that the data cannot be modeled at all with a large correlator.
  29. Had we used correlators in which the window and the range were not the same, then for this stimulus, which contains no noise, a small correlator with a large range would have been able to detect the whole range of disparities. It is possible that the human visual system contains such mechanisms, but we think it is unlikely. If we equate correlation mechanisms with single cortical units, such a mechanism would require the receptive fields in the right and left eyes to be laterally separated (they would have to be very far apart to represent large disparities). In our reading of the literature, no such units have been found.
  30. H. H. Baker, T. O. Binford, “Depth from edge- and intensity-based stereo,” Proceedings of the 7th International Joint Conference on Artificial Intelligence (Vancouver, B.C., 1981), pp. 631–636.
  31. P. Burt, B. Julesz, “A disparity gradient limit for binocular fusion,” Science 208, 615–617 (1980).
    [Crossref] [PubMed]
  32. K. Prazdny, “Detection of binocular disparities,” Biol. Cybern. 52, 93–99 (1985).
    [Crossref] [PubMed]
  33. Note that a correlator exactly the same size as the image would give different results for the smooth and the jagged stimuli. With the smooth stimulus, the correlation distribution would be almost symmetrical and hence would yield very high performance. With the jagged stimulus, the distribution would be random, leading to very poor performance.
  34. C. W. Tyler, “Stereoscopic vision: cortical limitations and a disparity scaling effect,” Science 181, 276–278 (1973).
    [Crossref] [PubMed]
  35. C. W. Tyler, “Depth perception in disparity gratings,” Nature 251, 140–142 (1974).
    [Crossref] [PubMed]
  36. C. W. Tyler, “Spatial organization of binocular disparity sensitivity,” Vision Res. 15, 583–590 (1975).
    [Crossref] [PubMed]
  37. C. W. Tyler, B. Julesz, “Binocular cross-correlation in time and space,” Vision Res. 18, 101–105 (1978).
    [Crossref] [PubMed]
  38. W. Richards, M. G. Kaye, “Local versus global stereopsis: two mechanisms?” Vision Res. 14, 1345–1347 (1974).
    [Crossref] [PubMed]
  39. C. M. Schor, I. Wood, “Disparity range for local stereopsis as a function of luminance spatial frequency,” Vision Res. 23, 1649–1654 (1983).
    [Crossref] [PubMed]
  40. F. W. Campbell, J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. (London) 197, 551–566 (1968).
  41. R. L. DeValois, K. K. DeValois, Spatial Vision (Oxford U. Press, New York, 1988).
  42. H. S. Smallman, D. I. A. MacLeod, S. He, R. W. Kentridge, “Fine grain of the neural representation of human spatial vision,” J. Neurosci. 16, 1852–1859 (1996).
    [PubMed]
  43. I. Ohzawa, G. C. DeAngelis, R. D. Freeman, “Encoding of binocular disparity by complex cells in the cat’s visual cortex,” J. Neurophysiol. 75, 1779–1805 (1996).
    [PubMed]
  44. A. J. Parker, M. J. Hawken, “Capabilities of monkey cortical cells in spatial-resolution tasks,” J. Opt. Soc. Am. A 2, 1101–1114 (1985).
    [Crossref] [PubMed]
  45. N. M. Grzywacz, S. N. J. Watamaniuk, S. P. McKee, “Temporal coherence theory for the detection and measurement of visual motion,” Vision Res. 35, 3183–3203 (1995).
    [Crossref] [PubMed]
  46. J. M. Harris, A. J. Parker, “Objective evaluation of human and computational stereoscopic visual systems,” Vision Res. 34, 2773–2785 (1994).
    [Crossref] [PubMed]
  47. J. M. Harris, A. J. Parker, “Constraints of human stereo dot matching,” Vision Res. 34, 2761–2772 (1994).
    [Crossref] [PubMed]
  48. S. B. Stevenson, L. K. Cormack, C. M. Schor, “Depth attraction and repulsion in random dot stereograms,” Vision Res. 31, 805–813 (1991).
    [Crossref] [PubMed]
  49. D. Weinshall, “Seeing ‘ghost’ planes in stereo vision,” Vision Res. 31, 1731–1748 (1991).
    [Crossref]
  50. G. J. Mitchison, S. P. McKee, “Mechanisms underlying the anisotropy of stereoscopic tilt perception,” Vision Res. 30, 1781–1791 (1990).
    [Crossref] [PubMed]
  51. R. Cagnello, B. R. Rogers, “Anisotropies in the perception of stereoscopic surfaces: the role of orientation disparity,” Vision Res. 33, 2189–2201 (1993).
    [Crossref]
  52. G. J. Mitchison, S. P. McKee, “Interpolation in stereoscopic matching,” Nature 315, 402–404 (1985).
    [Crossref] [PubMed]
  53. G. J. Mitchison, S. P. McKee, “Interpolation and the detection of fine structure in stereoscopic matching,” Vision Res. 27, 295–302 (1987).
    [Crossref] [PubMed]
  54. H. S. Smallman, “Fine-to-coarse scale disambiguation in stereopsis,” Vision Res. 35, 1047–1060 (1995).
    [Crossref] [PubMed]
  55. H. R. Wilson, R. Blake, D. L. Halpern, “Coarse spatial scales constrain the range of binocular fusion on fine scales,” J. Opt. Soc. Am. A 8, 229–236 (1991).
    [Crossref] [PubMed]
  56. A. M. Rohaly, H. R. Wilson, “The nature of coarse-to-fine constraints on binocular fusion,” J. Opt. Soc. Am. A 10, 2433–2441 (1993).
    [Crossref]
  57. A. M. Rohaly, H. R. Wilson, “Disparity averaging across spatial scales,” Vision Res. 34, 1315–1325 (1994).
    [Crossref] [PubMed]

1996 (2)

H. S. Smallman, D. I. A. MacLeod, S. He, R. W. Kentridge, “Fine grain of the neural representation of human spatial vision,” J. Neurosci. 16, 1852–1859 (1996).
[PubMed]

I. Ohzawa, G. C. DeAngelis, R. D. Freeman, “Encoding of binocular disparity by complex cells in the cat’s visual cortex,” J. Neurophysiol. 75, 1779–1805 (1996).
[PubMed]

1995 (3)

J. M. Harris, S. N. J. Watamaniuk, “Speed discrimination of motion-in-depth using binocular cues,” Vision Res. 35, 885–896 (1995).
[Crossref] [PubMed]

N. M. Grzywacz, S. N. J. Watamaniuk, S. P. McKee, “Temporal coherence theory for the detection and measurement of visual motion,” Vision Res. 35, 3183–3203 (1995).
[Crossref] [PubMed]

H. S. Smallman, “Fine-to-coarse scale disambiguation in stereopsis,” Vision Res. 35, 1047–1060 (1995).
[Crossref] [PubMed]

1994 (6)

A. M. Rohaly, H. R. Wilson, “Disparity averaging across spatial scales,” Vision Res. 34, 1315–1325 (1994).
[Crossref] [PubMed]

J. M. Harris, A. J. Parker, “Objective evaluation of human and computational stereoscopic visual systems,” Vision Res. 34, 2773–2785 (1994).
[Crossref] [PubMed]

J. M. Harris, A. J. Parker, “Constraints of human stereo dot matching,” Vision Res. 34, 2761–2772 (1994).
[Crossref] [PubMed]

R. Mestel, “Night of the strangest comet,” New Scientist 143 (No. 1933), 23–25 (1994).

L. K. Cormack, S. B. Stevenson, C. M. Schor, “An upper limit to the binocular combination of stimuli,” Vision Res. 34, 2599–2608 (1994).
[Crossref] [PubMed]

H. S. Smallman, D. I. A. MacLeod, “Size–disparity correlation in stereopsis at contrast threshold,” J. Opt. Soc. Am. A 11, 2169–2183 (1994).
[Crossref]

1993 (2)

R. Cagnello, B. R. Rogers, “Anisotropies in the perception of stereoscopic surfaces: the role of orientation disparity,” Vision Res. 33, 2189–2201 (1993).
[Crossref]

A. M. Rohaly, H. R. Wilson, “The nature of coarse-to-fine constraints on binocular fusion,” J. Opt. Soc. Am. A 10, 2433–2441 (1993).
[Crossref]

1992 (2)

A. Glennerster, B. J. Rogers, “Stereo ‘dmax’ as a function of dot density,” Invest. Ophthalmol. Vis. Sci. (Suppl.) 33, 1369 (1992).

D. G. Jones, J. Malik, “Computational framework for determining stereo correspondence from a set of linear spatial filters,” Image Vision Comput. 10, 699–708 (1992).
[Crossref]

1991 (4)

L. K. Cormack, S. B. Stevenson, C. M. Schor, “Interocular correlation, luminance contrast and cyclopean processing,” Vision Res. 31, 2195–2207 (1991).
[Crossref] [PubMed]

H. R. Wilson, R. Blake, D. L. Halpern, “Coarse spatial scales constrain the range of binocular fusion on fine scales,” J. Opt. Soc. Am. A 8, 229–236 (1991).
[Crossref] [PubMed]

S. B. Stevenson, L. K. Cormack, C. M. Schor, “Depth attraction and repulsion in random dot stereograms,” Vision Res. 31, 805–813 (1991).
[Crossref] [PubMed]

D. Weinshall, “Seeing ‘ghost’ planes in stereo vision,” Vision Res. 31, 1731–1748 (1991).
[Crossref]

1990 (2)

G. J. Mitchison, S. P. McKee, “Mechanisms underlying the anisotropy of stereoscopic tilt perception,” Vision Res. 30, 1781–1791 (1990).
[Crossref] [PubMed]

I. Ohzawa, G. C. DeAngelis, R. D. Freeman, “Stereoscopic depth discrimination in the visual cortex: neurons ideally suited as disparity detectors,” Science 249, 1037–1041 (1990).
[Crossref] [PubMed]

1987 (2)

C. A. Curcio, K. R. Sloan, O. Packer, A. E. Hendrickson, R. E. Kalina, “Distribution of cones in human and monkey retina: individual variability and radial symmetry,” Science 236, 579–582 (1987).
[Crossref] [PubMed]

G. J. Mitchison, S. P. McKee, “Interpolation and the detection of fine structure in stereoscopic matching,” Vision Res. 27, 295–302 (1987).
[Crossref] [PubMed]

1986 (1)

C. A. Curcio, J. E. Sheedy, I. L. Bailey, M. Buri, E. Bass, “Binocular vs. monocular task performance,” Am. J. Optom. Physiol. Opt. 63, 839–846 (1986).
[Crossref]

1985 (4)

S. B. Pollard, J. E. W. Mayhew, J. P. Frisby, “PMF: a stereo correspondence algorithm using a disparity gradient limit,” Perception 14, 449–470 (1985).

A. J. Parker, M. J. Hawken, “Capabilities of monkey cortical cells in spatial-resolution tasks,” J. Opt. Soc. Am. A 2, 1101–1114 (1985).
[Crossref] [PubMed]

K. Prazdny, “Detection of binocular disparities,” Biol. Cybern. 52, 93–99 (1985).
[Crossref] [PubMed]

G. J. Mitchison, S. P. McKee, “Interpolation in stereoscopic matching,” Nature 315, 402–404 (1985).
[Crossref] [PubMed]

1984 (1)

H. K. Nishihara, “Practical real time imaging stereo matcher,” Opt. Eng. 23, 536–545 (1984).
[Crossref]

1983 (3)

G. J. Mitchison, “New depth in stereopsis,” Nature 304, 123–124 (1983).
[Crossref] [PubMed]

A. L. Duwaer, “Patent stereopsis with diplopia in random-dot stereograms,” Percept. Psychophys. 33, 443–454 (1983).
[Crossref] [PubMed]

C. M. Schor, I. Wood, “Disparity range for local stereopsis as a function of luminance spatial frequency,” Vision Res. 23, 1649–1654 (1983).
[Crossref] [PubMed]

1981 (1)

B. M. Dow, A. Z. Snyder, R. G. Vautin, R. Bauer, “Magnification factor and receptive field size in foveal striate cortex of the monkey,” Exp. Brain Res. 44, 213–228 (1981).
[Crossref] [PubMed]

1980 (1)

P. Burt, B. Julesz, “A disparity gradient limit for binocular fusion,” Science 208, 615–617 (1980).
[Crossref] [PubMed]

1979 (1)

D. Marr, T. Poggio, “A computational theory of human stereo vision,” Proc. R. Soc. London, Ser. B 204, 301–328 (1979).
[Crossref]

1978 (2)

G. Westheimer, S. P. McKee, “Stereoscopic acuity for moving retinal images,” J. Opt. Soc. Am. 68, 450–455 (1978).
[Crossref] [PubMed]

C. W. Tyler, B. Julesz, “Binocular cross-correlation in time and space,” Vision Res. 18, 101–105 (1978).
[Crossref] [PubMed]

1976 (1)

D. Marr, T. Poggio, “The cooperative computation of stereo disparity,” Science 194, 283–287 (1976).
[Crossref] [PubMed]

1975 (2)

J. I. Nelson, “Globality and stereoscopic fusion in binocular vision,” J. Theor. Biol. 49, 1–88 (1975).
[Crossref] [PubMed]

C. W. Tyler, “Spatial organization of binocular disparity sensitivity,” Vision Res. 15, 583–590 (1975).
[Crossref] [PubMed]

1974 (2)

W. Richards, M. G. Kaye, “Local versus global stereopsis: two mechanisms?” Vision Res. 14, 1345–1347 (1974).
[Crossref] [PubMed]

C. W. Tyler, “Depth perception in disparity gratings,” Nature 251, 140–142 (1974).
[Crossref] [PubMed]

1973 (1)

C. W. Tyler, “Stereoscopic vision: cortical limitations and a disparity scaling effect,” Science 181, 276–278 (1973).
[Crossref] [PubMed]

1970 (2)

W. Richards, “Stereopsis and stereoblindness,” Exp. Brain Res. 10, 380–388 (1970).
[Crossref] [PubMed]

G. Sperling, “Binocular vision: a physical and neural theory,” J. Am. Psychol. 83, 461–534 (1970).

1968 (1)

F. W. Campbell, J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. (London) 197, 551–566 (1968).

1966 (1)

Note that the stimuli were presented on x–y CRT screens rather than on standard computer displays. The advantage of this is that points can be plotted with great precision. At the 1.5-m viewing distance, dots can be positioned 6.6 arcsec apart. However, because of the spreading of the electron beam on the screen and spreading caused by the eyes’ optics [F. W. Campbell, R. W. Gubisch, “Optical quality of the human eye,” J. Physiol. (London) 186, 558–578 (1966)], each dot appeared to have a width of approximately 1.5 s which we estimated by displaying a regular array of bright dots and reducing the interdot separation until the dots merged together.

1956 (1)

1948 (1)

R. N. Berry, “Quantitative relations among vernier, real depth and stereoscopic depth acuities,” J. Exp. Psychol. 38, 708–721 (1948).
[Crossref] [PubMed]

Bailey, I. L.

C. A. Curcio, J. E. Sheedy, I. L. Bailey, M. Buri, E. Bass, “Binocular vs. monocular task performance,” Am. J. Optom. Physiol. Opt. 63, 839–846 (1986).
[Crossref]

Baker, H. H.

H. H. Baker, T. O. Binford, “Depth from edge- and intensity-based stereo,” Proceedings of the 7th International Joint Conference on Artificial Intelligence (Vancouver, B.C., 1981), pp. 631–636.

Bass, E.

C. A. Curcio, J. E. Sheedy, I. L. Bailey, M. Buri, E. Bass, “Binocular vs. monocular task performance,” Am. J. Optom. Physiol. Opt. 63, 839–846 (1986).
[Crossref]

Bauer, R.

B. M. Dow, A. Z. Snyder, R. G. Vautin, R. Bauer, “Magnification factor and receptive field size in foveal striate cortex of the monkey,” Exp. Brain Res. 44, 213–228 (1981).
[Crossref] [PubMed]

Berry, R. N.

R. N. Berry, “Quantitative relations among vernier, real depth and stereoscopic depth acuities,” J. Exp. Psychol. 38, 708–721 (1948).
[Crossref] [PubMed]

Binford, T. O.

H. H. Baker, T. O. Binford, “Depth from edge- and intensity-based stereo,” Proceedings of the 7th International Joint Conference on Artificial Intelligence (Vancouver, B.C., 1981), pp. 631–636.

Blake, R.

Buri, M.

C. A. Curcio, J. E. Sheedy, I. L. Bailey, M. Buri, E. Bass, “Binocular vs. monocular task performance,” Am. J. Optom. Physiol. Opt. 63, 839–846 (1986).
[Crossref]

Burt, P.

P. Burt, B. Julesz, “A disparity gradient limit for binocular fusion,” Science 208, 615–617 (1980).
[Crossref] [PubMed]

Cagnello, R.

R. Cagnello, B. R. Rogers, “Anisotropies in the perception of stereoscopic surfaces: the role of orientation disparity,” Vision Res. 33, 2189–2201 (1993).
[Crossref]

Campbell, F. W.

F. W. Campbell, J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. (London) 197, 551–566 (1968).

Note that the stimuli were presented on x–y CRT screens rather than on standard computer displays. The advantage of this is that points can be plotted with great precision. At the 1.5-m viewing distance, dots can be positioned 6.6 arcsec apart. However, because of the spreading of the electron beam on the screen and spreading caused by the eyes’ optics [F. W. Campbell, R. W. Gubisch, “Optical quality of the human eye,” J. Physiol. (London) 186, 558–578 (1966)], each dot appeared to have a width of approximately 1.5 s which we estimated by displaying a regular array of bright dots and reducing the interdot separation until the dots merged together.

Cormack, L. K.

L. K. Cormack, S. B. Stevenson, C. M. Schor, “An upper limit to the binocular combination of stimuli,” Vision Res. 34, 2599–2608 (1994).
[Crossref] [PubMed]

L. K. Cormack, S. B. Stevenson, C. M. Schor, “Interocular correlation, luminance contrast and cyclopean processing,” Vision Res. 31, 2195–2207 (1991).
[Crossref] [PubMed]

S. B. Stevenson, L. K. Cormack, C. M. Schor, “Depth attraction and repulsion in random dot stereograms,” Vision Res. 31, 805–813 (1991).
[Crossref] [PubMed]

Curcio, C. A.

C. A. Curcio, K. R. Sloan, O. Packer, A. E. Hendrickson, R. E. Kalina, “Distribution of cones in human and monkey retina: individual variability and radial symmetry,” Science 236, 579–582 (1987).
[Crossref] [PubMed]

C. A. Curcio, J. E. Sheedy, I. L. Bailey, M. Buri, E. Bass, “Binocular vs. monocular task performance,” Am. J. Optom. Physiol. Opt. 63, 839–846 (1986).
[Crossref]

DeAngelis, G. C.

I. Ohzawa, G. C. DeAngelis, R. D. Freeman, “Encoding of binocular disparity by complex cells in the cat’s visual cortex,” J. Neurophysiol. 75, 1779–1805 (1996).
[PubMed]

I. Ohzawa, G. C. DeAngelis, R. D. Freeman, “Stereoscopic depth discrimination in the visual cortex: neurons ideally suited as disparity detectors,” Science 249, 1037–1041 (1990).
[Crossref] [PubMed]

DeValois, K. K.

R. L. DeValois, K. K. DeValois, Spatial Vision (Oxford U. Press, New York, 1988).

DeValois, R. L.

R. L. DeValois, K. K. DeValois, Spatial Vision (Oxford U. Press, New York, 1988).

Dow, B. M.

B. M. Dow, A. Z. Snyder, R. G. Vautin, R. Bauer, “Magnification factor and receptive field size in foveal striate cortex of the monkey,” Exp. Brain Res. 44, 213–228 (1981).
[Crossref] [PubMed]

Duwaer, A. L.

A. L. Duwaer, “Patent stereopsis with diplopia in random-dot stereograms,” Percept. Psychophys. 33, 443–454 (1983).
[Crossref] [PubMed]

Freeman, R. D.

I. Ohzawa, G. C. DeAngelis, R. D. Freeman, “Encoding of binocular disparity by complex cells in the cat’s visual cortex,” J. Neurophysiol. 75, 1779–1805 (1996).
[PubMed]

I. Ohzawa, G. C. DeAngelis, R. D. Freeman, “Stereoscopic depth discrimination in the visual cortex: neurons ideally suited as disparity detectors,” Science 249, 1037–1041 (1990).
[Crossref] [PubMed]

Frisby, J. P.

S. B. Pollard, J. E. W. Mayhew, J. P. Frisby, “PMF: a stereo correspondence algorithm using a disparity gradient limit,” Perception 14, 449–470 (1985).

Glennerster, A.

A. Glennerster, B. J. Rogers, “Stereo ‘dmax’ as a function of dot density,” Invest. Ophthalmol. Vis. Sci. (Suppl.) 33, 1369 (1992).

Grzywacz, N. M.

N. M. Grzywacz, S. N. J. Watamaniuk, S. P. McKee, “Temporal coherence theory for the detection and measurement of visual motion,” Vision Res. 35, 3183–3203 (1995).
[Crossref] [PubMed]

Gubisch, R. W.

Note that the stimuli were presented on x–y CRT screens rather than on standard computer displays. The advantage of this is that points can be plotted with great precision. At the 1.5-m viewing distance, dots can be positioned 6.6 arcsec apart. However, because of the spreading of the electron beam on the screen and spreading caused by the eyes’ optics [F. W. Campbell, R. W. Gubisch, “Optical quality of the human eye,” J. Physiol. (London) 186, 558–578 (1966)], each dot appeared to have a width of approximately 1.5 s which we estimated by displaying a regular array of bright dots and reducing the interdot separation until the dots merged together.

Halpern, D. L.

Harris, J. M.

J. M. Harris, S. N. J. Watamaniuk, “Speed discrimination of motion-in-depth using binocular cues,” Vision Res. 35, 885–896 (1995).
[Crossref] [PubMed]

J. M. Harris, A. J. Parker, “Objective evaluation of human and computational stereoscopic visual systems,” Vision Res. 34, 2773–2785 (1994).
[Crossref] [PubMed]

J. M. Harris, A. J. Parker, “Constraints of human stereo dot matching,” Vision Res. 34, 2761–2772 (1994).
[Crossref] [PubMed]

Hawken, M. J.

He, S.

H. S. Smallman, D. I. A. MacLeod, S. He, R. W. Kentridge, “Fine grain of the neural representation of human spatial vision,” J. Neurosci. 16, 1852–1859 (1996).
[PubMed]

Hendrickson, A. E.

C. A. Curcio, K. R. Sloan, O. Packer, A. E. Hendrickson, R. E. Kalina, “Distribution of cones in human and monkey retina: individual variability and radial symmetry,” Science 236, 579–582 (1987).
[Crossref] [PubMed]

Jones, D. G.

D. G. Jones, J. Malik, “Computational framework for determining stereo correspondence from a set of linear spatial filters,” Image Vision Comput. 10, 699–708 (1992).
[Crossref]

Julesz, B.

P. Burt, B. Julesz, “A disparity gradient limit for binocular fusion,” Science 208, 615–617 (1980).
[Crossref] [PubMed]

C. W. Tyler, B. Julesz, “Binocular cross-correlation in time and space,” Vision Res. 18, 101–105 (1978).
[Crossref] [PubMed]

Kalina, R. E.

C. A. Curcio, K. R. Sloan, O. Packer, A. E. Hendrickson, R. E. Kalina, “Distribution of cones in human and monkey retina: individual variability and radial symmetry,” Science 236, 579–582 (1987).
[Crossref] [PubMed]

Kaye, M. G.

W. Richards, M. G. Kaye, “Local versus global stereopsis: two mechanisms?” Vision Res. 14, 1345–1347 (1974).
[Crossref] [PubMed]

Kentridge, R. W.

H. S. Smallman, D. I. A. MacLeod, S. He, R. W. Kentridge, “Fine grain of the neural representation of human spatial vision,” J. Neurosci. 16, 1852–1859 (1996).
[PubMed]

MacLeod, D. I. A.

H. S. Smallman, D. I. A. MacLeod, S. He, R. W. Kentridge, “Fine grain of the neural representation of human spatial vision,” J. Neurosci. 16, 1852–1859 (1996).
[PubMed]

H. S. Smallman, D. I. A. MacLeod, “Size–disparity correlation in stereopsis at contrast threshold,” J. Opt. Soc. Am. A 11, 2169–2183 (1994).
[Crossref]

Malik, J.

D. G. Jones, J. Malik, “Computational framework for determining stereo correspondence from a set of linear spatial filters,” Image Vision Comput. 10, 699–708 (1992).
[Crossref]

Marr, D.

D. Marr, T. Poggio, “A computational theory of human stereo vision,” Proc. R. Soc. London, Ser. B 204, 301–328 (1979).
[Crossref]

D. Marr, T. Poggio, “The cooperative computation of stereo disparity,” Science 194, 283–287 (1976).
[Crossref] [PubMed]

D. Marr, Vision (Freeman, San Francisco, Calif., 1982).

Mayhew, J. E. W.

S. B. Pollard, J. E. W. Mayhew, J. P. Frisby, “PMF: a stereo correspondence algorithm using a disparity gradient limit,” Perception 14, 449–470 (1985).

McKee, S. P.

N. M. Grzywacz, S. N. J. Watamaniuk, S. P. McKee, “Temporal coherence theory for the detection and measurement of visual motion,” Vision Res. 35, 3183–3203 (1995).
[Crossref] [PubMed]

G. J. Mitchison, S. P. McKee, “Mechanisms underlying the anisotropy of stereoscopic tilt perception,” Vision Res. 30, 1781–1791 (1990).
[Crossref] [PubMed]

G. J. Mitchison, S. P. McKee, “Interpolation and the detection of fine structure in stereoscopic matching,” Vision Res. 27, 295–302 (1987).
[Crossref] [PubMed]

G. J. Mitchison, S. P. McKee, “Interpolation in stereoscopic matching,” Nature 315, 402–404 (1985).
[Crossref] [PubMed]

G. Westheimer, S. P. McKee, “Stereoscopic acuity for moving retinal images,” J. Opt. Soc. Am. 68, 450–455 (1978).
[Crossref] [PubMed]

Mestel, R.

R. Mestel, “Night of the strangest comet,” New Scientist 143 (No. 1933), 23–25 (1994).

Mitchison, G. J.

G. J. Mitchison, S. P. McKee, “Mechanisms underlying the anisotropy of stereoscopic tilt perception,” Vision Res. 30, 1781–1791 (1990).
[Crossref] [PubMed]

G. J. Mitchison, S. P. McKee, “Interpolation and the detection of fine structure in stereoscopic matching,” Vision Res. 27, 295–302 (1987).
[Crossref] [PubMed]

G. J. Mitchison, S. P. McKee, “Interpolation in stereoscopic matching,” Nature 315, 402–404 (1985).
[Crossref] [PubMed]

G. J. Mitchison, “New depth in stereopsis,” Nature 304, 123–124 (1983).
[Crossref] [PubMed]

Nelson, J. I.

J. I. Nelson, “Globality and stereoscopic fusion in binocular vision,” J. Theor. Biol. 49, 1–88 (1975).
[Crossref] [PubMed]

Nishihara, H. K.

H. K. Nishihara, “Practical real time imaging stereo matcher,” Opt. Eng. 23, 536–545 (1984).
[Crossref]

Ohzawa, I.

I. Ohzawa, G. C. DeAngelis, R. D. Freeman, “Encoding of binocular disparity by complex cells in the cat’s visual cortex,” J. Neurophysiol. 75, 1779–1805 (1996).
[PubMed]

I. Ohzawa, G. C. DeAngelis, R. D. Freeman, “Stereoscopic depth discrimination in the visual cortex: neurons ideally suited as disparity detectors,” Science 249, 1037–1041 (1990).
[Crossref] [PubMed]

Packer, O.

C. A. Curcio, K. R. Sloan, O. Packer, A. E. Hendrickson, R. E. Kalina, “Distribution of cones in human and monkey retina: individual variability and radial symmetry,” Science 236, 579–582 (1987).
[Crossref] [PubMed]

Parker, A. J.

J. M. Harris, A. J. Parker, “Constraints of human stereo dot matching,” Vision Res. 34, 2761–2772 (1994).
[Crossref] [PubMed]

J. M. Harris, A. J. Parker, “Objective evaluation of human and computational stereoscopic visual systems,” Vision Res. 34, 2773–2785 (1994).
[Crossref] [PubMed]

A. J. Parker, M. J. Hawken, “Capabilities of monkey cortical cells in spatial-resolution tasks,” J. Opt. Soc. Am. A 2, 1101–1114 (1985).
[Crossref] [PubMed]

Poggio, T.

D. Marr, T. Poggio, “A computational theory of human stereo vision,” Proc. R. Soc. London, Ser. B 204, 301–328 (1979).
[Crossref]

D. Marr, T. Poggio, “The cooperative computation of stereo disparity,” Science 194, 283–287 (1976).
[Crossref] [PubMed]

Pollard, S. B.

S. B. Pollard, J. E. W. Mayhew, J. P. Frisby, “PMF: a stereo correspondence algorithm using a disparity gradient limit,” Perception 14, 449–470 (1985).

Prazdny, K.

K. Prazdny, “Detection of binocular disparities,” Biol. Cybern. 52, 93–99 (1985).
[Crossref] [PubMed]

Richards, W.

W. Richards, M. G. Kaye, “Local versus global stereopsis: two mechanisms?” Vision Res. 14, 1345–1347 (1974).
[Crossref] [PubMed]

W. Richards, “Stereopsis and stereoblindness,” Exp. Brain Res. 10, 380–388 (1970).
[Crossref] [PubMed]

Robson, J. G.

F. W. Campbell, J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. (London) 197, 551–566 (1968).

Rogers, B. J.

A. Glennerster, B. J. Rogers, “Stereo ‘dmax’ as a function of dot density,” Invest. Ophthalmol. Vis. Sci. (Suppl.) 33, 1369 (1992).

Rogers, B. R.

R. Cagnello, B. R. Rogers, “Anisotropies in the perception of stereoscopic surfaces: the role of orientation disparity,” Vision Res. 33, 2189–2201 (1993).
[Crossref]

Rohaly, A. M.

A. M. Rohaly, H. R. Wilson, “Disparity averaging across spatial scales,” Vision Res. 34, 1315–1325 (1994).
[Crossref] [PubMed]

A. M. Rohaly, H. R. Wilson, “The nature of coarse-to-fine constraints on binocular fusion,” J. Opt. Soc. Am. A 10, 2433–2441 (1993).
[Crossref]

Schor, C. M.

L. K. Cormack, S. B. Stevenson, C. M. Schor, “An upper limit to the binocular combination of stimuli,” Vision Res. 34, 2599–2608 (1994).
[Crossref] [PubMed]

L. K. Cormack, S. B. Stevenson, C. M. Schor, “Interocular correlation, luminance contrast and cyclopean processing,” Vision Res. 31, 2195–2207 (1991).
[Crossref] [PubMed]

S. B. Stevenson, L. K. Cormack, C. M. Schor, “Depth attraction and repulsion in random dot stereograms,” Vision Res. 31, 805–813 (1991).
[Crossref] [PubMed]

C. M. Schor, I. Wood, “Disparity range for local stereopsis as a function of luminance spatial frequency,” Vision Res. 23, 1649–1654 (1983).
[Crossref] [PubMed]

Sheedy, J. E.

C. A. Curcio, J. E. Sheedy, I. L. Bailey, M. Buri, E. Bass, “Binocular vs. monocular task performance,” Am. J. Optom. Physiol. Opt. 63, 839–846 (1986).
[Crossref]

Sloan, K. R.

C. A. Curcio, K. R. Sloan, O. Packer, A. E. Hendrickson, R. E. Kalina, “Distribution of cones in human and monkey retina: individual variability and radial symmetry,” Science 236, 579–582 (1987).
[Crossref] [PubMed]

Smallman, H. S.

H. S. Smallman, D. I. A. MacLeod, S. He, R. W. Kentridge, “Fine grain of the neural representation of human spatial vision,” J. Neurosci. 16, 1852–1859 (1996).
[PubMed]

H. S. Smallman, “Fine-to-coarse scale disambiguation in stereopsis,” Vision Res. 35, 1047–1060 (1995).
[Crossref] [PubMed]

H. S. Smallman, D. I. A. MacLeod, “Size–disparity correlation in stereopsis at contrast threshold,” J. Opt. Soc. Am. A 11, 2169–2183 (1994).
[Crossref]

Snyder, A. Z.

B. M. Dow, A. Z. Snyder, R. G. Vautin, R. Bauer, “Magnification factor and receptive field size in foveal striate cortex of the monkey,” Exp. Brain Res. 44, 213–228 (1981).
[Crossref] [PubMed]

Sperling, G.

G. Sperling, “Binocular vision: a physical and neural theory,” J. Am. Psychol. 83, 461–534 (1970).

Stevenson, S. B.

L. K. Cormack, S. B. Stevenson, C. M. Schor, “An upper limit to the binocular combination of stimuli,” Vision Res. 34, 2599–2608 (1994).
[Crossref] [PubMed]

L. K. Cormack, S. B. Stevenson, C. M. Schor, “Interocular correlation, luminance contrast and cyclopean processing,” Vision Res. 31, 2195–2207 (1991).
[Crossref] [PubMed]

S. B. Stevenson, L. K. Cormack, C. M. Schor, “Depth attraction and repulsion in random dot stereograms,” Vision Res. 31, 805–813 (1991).
[Crossref] [PubMed]

Tanzman, I. J.

Tyler, C. W.

C. W. Tyler, B. Julesz, “Binocular cross-correlation in time and space,” Vision Res. 18, 101–105 (1978).
[Crossref] [PubMed]

C. W. Tyler, “Spatial organization of binocular disparity sensitivity,” Vision Res. 15, 583–590 (1975).
[Crossref] [PubMed]

C. W. Tyler, “Depth perception in disparity gratings,” Nature 251, 140–142 (1974).
[Crossref] [PubMed]

C. W. Tyler, “Stereoscopic vision: cortical limitations and a disparity scaling effect,” Science 181, 276–278 (1973).
[Crossref] [PubMed]

Vautin, R. G.

B. M. Dow, A. Z. Snyder, R. G. Vautin, R. Bauer, “Magnification factor and receptive field size in foveal striate cortex of the monkey,” Exp. Brain Res. 44, 213–228 (1981).
[Crossref] [PubMed]

Watamaniuk, S. N. J.

N. M. Grzywacz, S. N. J. Watamaniuk, S. P. McKee, “Temporal coherence theory for the detection and measurement of visual motion,” Vision Res. 35, 3183–3203 (1995).
[Crossref] [PubMed]

J. M. Harris, S. N. J. Watamaniuk, “Speed discrimination of motion-in-depth using binocular cues,” Vision Res. 35, 885–896 (1995).
[Crossref] [PubMed]

Weinshall, D.

D. Weinshall, “Seeing ‘ghost’ planes in stereo vision,” Vision Res. 31, 1731–1748 (1991).
[Crossref]

Westheimer, G.

Wilson, H. R.

Wood, I.

C. M. Schor, I. Wood, “Disparity range for local stereopsis as a function of luminance spatial frequency,” Vision Res. 23, 1649–1654 (1983).
[Crossref] [PubMed]

Am. J. Optom. Physiol. Opt. (1)

C. A. Curcio, J. E. Sheedy, I. L. Bailey, M. Buri, E. Bass, “Binocular vs. monocular task performance,” Am. J. Optom. Physiol. Opt. 63, 839–846 (1986).
[Crossref]

Biol. Cybern. (1)

K. Prazdny, “Detection of binocular disparities,” Biol. Cybern. 52, 93–99 (1985).
[Crossref] [PubMed]

Exp. Brain Res. (2)

W. Richards, “Stereopsis and stereoblindness,” Exp. Brain Res. 10, 380–388 (1970).
[Crossref] [PubMed]

B. M. Dow, A. Z. Snyder, R. G. Vautin, R. Bauer, “Magnification factor and receptive field size in foveal striate cortex of the monkey,” Exp. Brain Res. 44, 213–228 (1981).
[Crossref] [PubMed]

Image Vision Comput. (1)

D. G. Jones, J. Malik, “Computational framework for determining stereo correspondence from a set of linear spatial filters,” Image Vision Comput. 10, 699–708 (1992).
[Crossref]

Invest. Ophthalmol. Vis. Sci. (Suppl.) (1)

A. Glennerster, B. J. Rogers, “Stereo ‘dmax’ as a function of dot density,” Invest. Ophthalmol. Vis. Sci. (Suppl.) 33, 1369 (1992).

J. Am. Psychol. (1)

G. Sperling, “Binocular vision: a physical and neural theory,” J. Am. Psychol. 83, 461–534 (1970).

J. Exp. Psychol. (1)

R. N. Berry, “Quantitative relations among vernier, real depth and stereoscopic depth acuities,” J. Exp. Psychol. 38, 708–721 (1948).
[Crossref] [PubMed]

J. Neurophysiol. (1)

I. Ohzawa, G. C. DeAngelis, R. D. Freeman, “Encoding of binocular disparity by complex cells in the cat’s visual cortex,” J. Neurophysiol. 75, 1779–1805 (1996).
[PubMed]

J. Neurosci. (1)

H. S. Smallman, D. I. A. MacLeod, S. He, R. W. Kentridge, “Fine grain of the neural representation of human spatial vision,” J. Neurosci. 16, 1852–1859 (1996).
[PubMed]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

J. Physiol. (London) (2)

F. W. Campbell, J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. (London) 197, 551–566 (1968).

Note that the stimuli were presented on x–y CRT screens rather than on standard computer displays. The advantage of this is that points can be plotted with great precision. At the 1.5-m viewing distance, dots can be positioned 6.6 arcsec apart. However, because of the spreading of the electron beam on the screen and spreading caused by the eyes’ optics [F. W. Campbell, R. W. Gubisch, “Optical quality of the human eye,” J. Physiol. (London) 186, 558–578 (1966)], each dot appeared to have a width of approximately 1.5 s which we estimated by displaying a regular array of bright dots and reducing the interdot separation until the dots merged together.

J. Theor. Biol. (1)

J. I. Nelson, “Globality and stereoscopic fusion in binocular vision,” J. Theor. Biol. 49, 1–88 (1975).
[Crossref] [PubMed]

Nature (3)

G. J. Mitchison, “New depth in stereopsis,” Nature 304, 123–124 (1983).
[Crossref] [PubMed]

C. W. Tyler, “Depth perception in disparity gratings,” Nature 251, 140–142 (1974).
[Crossref] [PubMed]

G. J. Mitchison, S. P. McKee, “Interpolation in stereoscopic matching,” Nature 315, 402–404 (1985).
[Crossref] [PubMed]

New Scientist (1)

R. Mestel, “Night of the strangest comet,” New Scientist 143 (No. 1933), 23–25 (1994).

Opt. Eng. (1)

H. K. Nishihara, “Practical real time imaging stereo matcher,” Opt. Eng. 23, 536–545 (1984).
[Crossref]

Percept. Psychophys. (1)

A. L. Duwaer, “Patent stereopsis with diplopia in random-dot stereograms,” Percept. Psychophys. 33, 443–454 (1983).
[Crossref] [PubMed]

Perception (1)

S. B. Pollard, J. E. W. Mayhew, J. P. Frisby, “PMF: a stereo correspondence algorithm using a disparity gradient limit,” Perception 14, 449–470 (1985).

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

D. Marr, T. Poggio, “A computational theory of human stereo vision,” Proc. R. Soc. London, Ser. B 204, 301–328 (1979).
[Crossref]

Science (5)

C. A. Curcio, K. R. Sloan, O. Packer, A. E. Hendrickson, R. E. Kalina, “Distribution of cones in human and monkey retina: individual variability and radial symmetry,” Science 236, 579–582 (1987).
[Crossref] [PubMed]

I. Ohzawa, G. C. DeAngelis, R. D. Freeman, “Stereoscopic depth discrimination in the visual cortex: neurons ideally suited as disparity detectors,” Science 249, 1037–1041 (1990).
[Crossref] [PubMed]

D. Marr, T. Poggio, “The cooperative computation of stereo disparity,” Science 194, 283–287 (1976).
[Crossref] [PubMed]

P. Burt, B. Julesz, “A disparity gradient limit for binocular fusion,” Science 208, 615–617 (1980).
[Crossref] [PubMed]

C. W. Tyler, “Stereoscopic vision: cortical limitations and a disparity scaling effect,” Science 181, 276–278 (1973).
[Crossref] [PubMed]

Vision Res. (17)

G. J. Mitchison, S. P. McKee, “Interpolation and the detection of fine structure in stereoscopic matching,” Vision Res. 27, 295–302 (1987).
[Crossref] [PubMed]

H. S. Smallman, “Fine-to-coarse scale disambiguation in stereopsis,” Vision Res. 35, 1047–1060 (1995).
[Crossref] [PubMed]

A. M. Rohaly, H. R. Wilson, “Disparity averaging across spatial scales,” Vision Res. 34, 1315–1325 (1994).
[Crossref] [PubMed]

N. M. Grzywacz, S. N. J. Watamaniuk, S. P. McKee, “Temporal coherence theory for the detection and measurement of visual motion,” Vision Res. 35, 3183–3203 (1995).
[Crossref] [PubMed]

J. M. Harris, A. J. Parker, “Objective evaluation of human and computational stereoscopic visual systems,” Vision Res. 34, 2773–2785 (1994).
[Crossref] [PubMed]

J. M. Harris, A. J. Parker, “Constraints of human stereo dot matching,” Vision Res. 34, 2761–2772 (1994).
[Crossref] [PubMed]

S. B. Stevenson, L. K. Cormack, C. M. Schor, “Depth attraction and repulsion in random dot stereograms,” Vision Res. 31, 805–813 (1991).
[Crossref] [PubMed]

D. Weinshall, “Seeing ‘ghost’ planes in stereo vision,” Vision Res. 31, 1731–1748 (1991).
[Crossref]

G. J. Mitchison, S. P. McKee, “Mechanisms underlying the anisotropy of stereoscopic tilt perception,” Vision Res. 30, 1781–1791 (1990).
[Crossref] [PubMed]

R. Cagnello, B. R. Rogers, “Anisotropies in the perception of stereoscopic surfaces: the role of orientation disparity,” Vision Res. 33, 2189–2201 (1993).
[Crossref]

C. W. Tyler, “Spatial organization of binocular disparity sensitivity,” Vision Res. 15, 583–590 (1975).
[Crossref] [PubMed]

C. W. Tyler, B. Julesz, “Binocular cross-correlation in time and space,” Vision Res. 18, 101–105 (1978).
[Crossref] [PubMed]

W. Richards, M. G. Kaye, “Local versus global stereopsis: two mechanisms?” Vision Res. 14, 1345–1347 (1974).
[Crossref] [PubMed]

C. M. Schor, I. Wood, “Disparity range for local stereopsis as a function of luminance spatial frequency,” Vision Res. 23, 1649–1654 (1983).
[Crossref] [PubMed]

J. M. Harris, S. N. J. Watamaniuk, “Speed discrimination of motion-in-depth using binocular cues,” Vision Res. 35, 885–896 (1995).
[Crossref] [PubMed]

L. K. Cormack, S. B. Stevenson, C. M. Schor, “An upper limit to the binocular combination of stimuli,” Vision Res. 34, 2599–2608 (1994).
[Crossref] [PubMed]

L. K. Cormack, S. B. Stevenson, C. M. Schor, “Interocular correlation, luminance contrast and cyclopean processing,” Vision Res. 31, 2195–2207 (1991).
[Crossref] [PubMed]

Other (8)

D. Marr, Vision (Freeman, San Francisco, Calif., 1982).

Note that a correlator exactly the same size as the image would give different results for the smooth and the jagged stimuli. With the smooth stimulus, the correlation distribution would be almost symmetrical and hence would yield very high performance. With the jagged stimulus, the distribution would be random, leading to very poor performance.

In performing the simulations we had to make an arbitrary decision about how many arcminutes each pixel in the model array would represent. The question, in other words, is how does one choose the number of points in the memory array that is used to represent the image? For stimuli created on standard computer monitor screens, this is usually not considered a problem (although the problem of the optical transfer function of the eye is always present). One screen pixel is typically equated with one location in the array used for modeling purposes. However, here each dot was physically larger than the minimum separation between points. We expected model performance to vary as a function of dot density, and the model array dot density (in dots per total number of possible dot positions) would obviously increase as the allowed number of dot positions decreased. Over a wide range, varying both density and dot size in the model did not make much difference for the re- sults. In the simulations presented here, we chose each pixel to be equal to 0.5 arcmin.

The results for a correlator exactly the same size as the image, with a disparity range including all possible disparities, are not shown. With no target present, such an ideal correlator will produce an exactly symmetrical correlation distribution. Adding a target dot will slightly distort the symmetry, but the existence of symmetry makes the performance very high. This is an unrealistic result: adding a small amount of noise or even a small amount of uncertainty to the correlation area size wipes out the symmetry and reduces performance to almost chance levels.

In further simulations (not shown) we tested whether a larger correlator with a small disparity range would be able to mimic the human data. Although similar tuning was found, the overall performance level was much lower, suggesting that the data cannot be modeled at all with a large correlator.

Had we used correlators in which the window and the range were not the same, then for this stimulus, which contains no noise, a small correlator with a large range would have been able to detect the whole range of disparities. It is possible that the human visual system contains such mechanisms, but we think it is unlikely. If we equate correlation mechanisms with single cortical units, such a mechanism would require the receptive fields in the right and left eyes to be laterally separated (they would have to be very far apart to represent large disparities). In our reading of the literature, no such units have been found.

H. H. Baker, T. O. Binford, “Depth from edge- and intensity-based stereo,” Proceedings of the 7th International Joint Conference on Artificial Intelligence (Vancouver, B.C., 1981), pp. 631–636.

R. L. DeValois, K. K. DeValois, Spatial Vision (Oxford U. Press, New York, 1988).

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram illustrating the perceived depth profile of the stimuli. Only a small number of dots are shown; in the actual experiments, the stimuli subtended 3 deg at 1.5-m viewing distance, and there were 60 dots in total. (a) A target dot in front of the fixation plane (vertical dashed line) surrounded by a smooth row of noise dots in the fixation plane. (b) A target dot in front of the fixation plane surrounded by a small smooth region (marked by bracket) with a jagged surface on each side. The dots were binocularly uncorrelated but were perceived as having random depths and are thus shown here as though they had random depths.

Fig. 2
Fig. 2

(a) Interocular correlation distribution for a stimulus consisting of ten points, all at zero disparity. There are correlations at many disparities, because correlation obtains all possible matches, not just the correct ones. (b) Correlation distribution for a single point with a disparity of +2 pixels amid ten background points at a constant depth. This distribution is very similar to that for the flat surface, illustrating the difficulty of detecting a single disparate target dot embedded in a flat surface.

Fig. 3
Fig. 3

Kepler grids showing all possible matches between the left and right eyes’ images. The row of dots at the bottom left and bottom right of each figure show the images in each eye, with the target dot shaded gray. The possible matches between the target dot in the right eye and the dots in the left eye are black. (a) The gray area shows that when the correlation area is the same size as the image S and the disparity range is ±S, all dots contribute to the correlation. (b) The number of dots involved in the correlation (and hence the number of candidate matches) can be reduced by using both a smaller correlation area W and a narrower disparity range ±K.

Fig. 4
Fig. 4

Model performance for a stimulus containing a single disparate dot surrounded by a smooth row of noise dots, for a disparity of (a) 2 pixels and (b) 10 pixels. Each point on the graph shows proportion correct for a different combination of window size and disparity range (in pixels) and is the result of 500 simulated trials. The model performs the task best for a small correlation window and a small disparity range. Note that for the large disparity [(b)], performance was poor for all combinations of window size and correlation range.

Fig. 5
Fig. 5

Proportion correct as a function of disparity for two observers (a) and for the correlation model for four different correlation window sizes (b) when the stimulus contained a target dot embedded in 59 background dots in the fixation plane. To illustrate the effect of the noise dots, we also measured proportion correct for human observers (c) and for the model (d) for the same range of disparity but when there were no background dots present in the stimulus. (a) Human performance is good for a surprisingly small range of disparities. (b) The model data are shown for four different window sizes (with corresponding disparity range). Here we equated the model and human data by generating a stimulus that was 360 pixels wide and assuming that each pixel corresponded to 0.5 arcmin. Performance is good only for the smaller windows and only for a small range of disparities. (c) With no noise dots, human performance is good for a much wider range of disparity, suggesting that larger-scale correlators are accessed. (d) For the model, performance is high for all scales tested until the target falls outside the correlation window.

Fig. 6
Fig. 6

Proportion correct for the disparity-discrimination task for a disparate dot embedded in a 12-arcmin smooth region, with noise dots in the fixation plane. The rest of the stimulus was composed of binocularly uncorrelated dots. Both human (a) and model (b) data were very similar to the results found in the previous experiment, suggesting that human stereo performance uses a correlationlike procedure to detect depth.

Fig. 7
Fig. 7

Proportion correct for the disparity-discrimination task as a function of the size of the smooth region surrounding the target dot. The model (b) gives roughly constant performance for each window size. The human data (a) are like the model data for a large range of smooth-area sizes, but the results are different when the region is very small, peaking at 12 min and then falling rapidly.

Equations (4)

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correl(D)=i=-S/2S/2 I(xiL)I(xiR+D)
for-S<D<S,
correl(D)=i=-W/2W/2 I(xiL)I(xiR+D)
for-K<D<K.

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