Limits of stereopsis explained by local cross-correlation.

Human stereopsis has two well-known constraints: the disparity-gradient limit, which is the inability to perceive depth when the change in disparity within a region is too large, and the limit of stereoresolution, which is the inability to perceive spatial variations in disparity that occur at too fine a spatial scale. We propose that both limitations can be understood as byproducts of estimating disparity by cross-correlating the two eyes' images, the fundamental computation underlying the disparity-energy model. To test this proposal, we constructed a local cross-correlation model with biologically motivated properties. We then compared model and human behaviors in the same psychophysical tasks. The model and humans behaved quite similarly: they both exhibited a disparity-gradient limit and had similar stereoresolution thresholds. Performance was affected similarly by changes in a variety of stimulus parameters. By modeling the effects of stimulus blur and of using different sizes of image patches, we found evidence that the smallest neural mechanism humans use to estimate disparity is 3-6 arcmin in diameter. We conclude that the disparity-gradient limit and stereoresolution are indeed byproducts of using local cross-correlation to estimate disparity.

Image-size differences worsen stereopsis independent of eye position.

With the eyes in forward gaze, stereo performance worsens when one eye's image is larger than the other's. Near, eccentric objects naturally create retinal images of different sizes. Does this mean that stereopsis exhibits deficits for such stimuli? Or does the visual system compensate for the predictable image-size differences? To answer this, we measured discrimination of a disparity-defined shape for different relative image sizes. We did so for different gaze directions, some compatible with the image-size difference and some not. Magnifications of 10-15% caused a clear worsening of stereo performance. The worsening was determined only by relative image size and not by eye position. This shows that no neural compensation for image-size differences accompanies eye-position changes, at least prior to disparity estimation. We also found that a local cross-correlation model for disparity estimation performs like humans in the same task, suggesting that the decrease in stereo performance due to image-size differences is a byproduct of the disparity-estimation method. Finally, we looked for compensation in an observer who has constantly different image sizes due to differing eye lengths. She performed best when the presented images were roughly the same size, indicating that she has compensated for the persistent image-size difference.

The surface of the empirical horopter.

The distribution of empirical corresponding points in the two retinas has been well studied along the horizontal and the vertical meridians, but not in other parts of the visual field. Using an apparent-motion paradigm, we measured the positions of those points across the central portion of the visual field. We found that the Hering-Hillebrand deviation (a deviation from the Vieth-Müller circle) and the Helmholtz shear of horizontal disparity (backward slant of the vertical horopter) exist throughout the visual field. We also found no evidence for non-zero vertical disparities in empirical corresponding points. We used the data to find the combination of points in space and binocular eye position that minimizes the disparity between stimulated points on the retinas and the empirical corresponding points. The optimum surface is a top-back slanted surface at medium to far distance depending on the observer. The line in the middle of the surface extending away from the observer comes very close to lying in the plane of the ground as the observer fixates various positions in the ground, a speculation Helmholtz made that has since been misunderstood.