Binocular disparity is the input to stereopsis, which is a very strong depth cue in humans. However, the distribution of binocular disparities in natural environments has not been quantitatively measured. In this study, we converted distances from accurate range maps of forest scenes and indoor scenes into the disparities that an observer would encounter, given an eye model and fixation distances (which we measured for the forest environment, and simulated for the indoor environment). We found that the distributions of natural disparities in these two kinds of scenes are centered at zero, have high peaks, and span about 5 deg, which closely matches the macaque MT cells' disparity tuning range. These ranges are fully within the operational range of human stereopsis determined psychophysically. Suprathreshold disparities (>10 arcsec) are common rather than exceptional. There is a prevailing notion that stereopsis only operates within a few meters, but our finding suggests that we should rethink the role of stereopsis at far viewing distances because of the abundance of suprathreshold disparities.

*x*axis goes from left to right, the

*y*axis goes from down to up, and the

*z*axis points toward the viewer. In each range map, the origin of the coordinate system is the location of the laser scanner.

*z*axis.

*F*(

*x*

_{ f},

*y*

_{ f},

*z*

_{ f}). We define the fixation distance to be the distance between the fixation point

*F*and

*O*

_{ c}, which is also the midpoint between the two eyes. To simplify the computation, we assume the two eyes have a midsagittal fixation, which means

*x*

_{ f}= 0 and

*y*

_{ f}= 0. By doing so, we can use the fixation distance

*z*

_{ f}to represent a fixation. The disparity of point P can be specified by:

*O*

_{ c}. If we think of the laser scanner as the cyclopean eye, then its retina is hit by laser beams at all directions with the same angular sampling step size. Of course, this geometrical description of the world may not be the same when the laser scanner is located at the left eye's position

*O*

_{ l}, or the right eye's position

*O*

_{ r}. But since the interpupillary distance is much smaller than the distances in the range maps, it is reasonable to assume that the range maps are valid for two eyes at

*O*

_{ l}and

*O*

_{ r}.

*z*

_{ f},

*x*

_{ p},

*y*

_{ p}, and

*z*

_{ p}. The range map gives

*x*

_{ p},

*y*

_{ p}, and

*z*

_{ p}. We must decide the position of the fixation point

*F*(0, 0,

*z*

_{ f}) beforehand to calculate the disparity. In order to simulate the human fixation in the forest scenes, we conducted the following experiment to collect human fixation distances in an environment similar to the one in which the range data were collected.

*d*

_{1},

*d*

_{2}, …

*d*

_{255}. These fixation distances range from 2.7 m to 123.8 m.

*d*

_{ i}, we went through the 23 forest range maps to find a matched distance

*d*on the horizontal plane, with a criterion of the error (

*d*

_{ i}−

*d*) /

*d*

_{ i}< 5%. This matching process gave us (hopefully) a good intuitive association between the range maps and our fixation data, the idea being “Given that observers generally fixate on objects, and given that this observer was fixating at a distance

*d,*where could he or she have been plausibly looking in the range data?” After we found the matched distance

*d,*we placed a virtual observer with the midpoint between two eyes located at the origin

*O*

_{ c}and directly facing the direction of the matched distance

*d*.

*f*(

*x*∣

*μ, b*) = exp (−∣

*x*−

*μ*∣/

*b*)/2

*b,*to the disparity data using the MLE (maximum likelihood estimation) method. The fit is clearly better. Moreover, a plot of log-density vs. disparity is shown in Figure 4B, and the roughly linear flanks are consistent with the Laplace distribution (though the kurtosis Laplace distribution is 3, which is still substantially lower than our obtained value of close to 6). Crucially, however, 98% of the distribution comprises disparity values greater than ±10 arcsec (a large but not unreasonable stereothreshold).

^{1}It is unclear how this could be the case in the current study given that, even if there were some systematic orientation bias in the original range data, it is difficult to imagine how this could propagate to the disparity data given the randomness inherent in our simulations. Nevertheless, we verified the stationarity of the range data (as a whole) with respect to orientation by computing the range distributions (and the means and standard deviations) at many orientations in the original data and, indeed, the mean range, the standard deviation, and (as far as we could tell) the shape of the range distribution were independent of orientation. The only thing remaining that could have produced the orientation dependence was the nature of the disparity calculation, specifically, the VM circle. Referring to Figure 1, assume that the fixation

*F*is such that the mean disparity along

*OcF*is zero. Given the independence of the range data, this condition will hold (at least approximately) for any other fixation point on the iso-distance circle. Thus, if fixation were to remain at

*F,*then the mean disparity at any eccentricity will be shifted by the disparity between the VM circle and the iso-distance circle at that eccentricity. This disparity shift is plotted as the red dashed line in Figure 7A. The agreement is good given the standard deviations associated with the data, and the remaining discrepancy is probably due to the particular sample of ranges we acquired. Note also that the subjective horopter measured by psychophysical methods is a little different from the VM circle (Howard & Rogers, 1995) and is generally between the VM circle and the frontal parallel plane located at the fixation point. Thus, the effective dependence of mean disparity on eccentricity should be less in practice than estimated here.

^{2}

Eccentricity | Standard deviation of spread of receptive field disparities | Standard deviation of the natural binocular disparities in the 23 forest scenes | Standard deviation of the natural binocular disparities in the 27 indoor scenes |
---|---|---|---|

0–4 deg | 0.50 deg (90 cells) | 0.23 deg | 0.29 deg |

4–8 deg | 0.76 deg (74 cells) | 0.26 deg | 0.32 deg |

8–12 deg | 0.79 deg (39 cells) | 0.35 deg | 0.40 deg |

12–16 deg | 0.90 deg (10 cells) | 0.41 deg | 0.46 deg |

- 1. strong depth perception in the Panum's fusional area (±5 arcmin at the fovea), where objects are fused,
- strong depth perception with diplopia, and
- vague depth perception with diplopia.

*patent stereopsis,*and the third

*qualitative stereopsis*.

*patent stereopsis*extends to about ±10 arcmin at the fovea. There is a clear disparity–depth relationship in this area, which makes fine depth judgment possible. Beyond the ±10 arcmin limit, the performance of judging the stimulus to be nearer or further than the fixation is still above chance, but the fine disparity–depth relationship is lost. The range of

*qualitative stereopsis*at fovea is about ±15 arcmin. There is no reliable depth perception beyond the

*qualitative stereopsis*range. At the periphery, the range of reliable depth perception is much larger. For example,

*patent stereopsis*extends to about 70 arcmin and

*qualitative stereopsis*about 2 deg at a 6-deg eccentricity.

^{2}Of course people deploy their fixations differently (in some sense) for different tasks in different environments, and this may influence the shape of the distribution of fixation distances relative to the distribution of environmental distances, but until we can simultaneously measure ranges and fixations in the same environment and co-register the data, we will proceed under the above assumption.