Depth estimates from disparity are most precise when the visual input stimulates corresponding retinal points or points close to them. Corresponding points have uncrossed disparities in the upper visual field and crossed disparities in the lower visual field. Due to these disparities, the vertical part of the horopter—the positions in space that stimulate corresponding points—is pitched top-back. Many have suggested that this pitch is advantageous for discriminating depth in the natural environment, particularly relative to the ground. We asked whether the vertical horopter is adaptive (suited for perception of the ground) and adaptable (changeable by experience). Experiment 1 measured the disparities between corresponding points in 28 observers. We confirmed that the horopter is pitched. However, it is also typically convex making it ill-suited for depth perception relative to the ground. Experiment 2 tracked locations of corresponding points while observers wore lenses for 7 days that distorted binocular disparities. We observed no change in the horopter, suggesting that it is not adaptable. We also showed that the horopter is not adaptive for long viewing distances because at such distances uncrossed disparities between corresponding points cannot be stimulated. The vertical horopter seems to be adaptive for perceiving convex, slanted surfaces at short distances.

*empirical corresponding points*(Blakemore, 1970). It is useful to describe those points with respect to geometric points.

*Geometric corresponding points*are pairs of points with the same coordinates in the two retinas: By definition, they have zero disparity. The two anatomical vertical meridians of the eyes (great circles with zero azimuth) are an example of a set of geometric corresponding points. The locations in the world that stimulate geometric corresponding points define the

*geometric horopter*. When fixation is in the head's mid‐sagittal plane, the geometric vertical horopter is a vertical line through fixation (Figure 1a).

*Empirical corresponding points*are generally defined by determining positions in the two retinas that, when stimulated, yield the same perceived‐direction. Empirical and geometric points differ in that empirical points have uncrossed disparities in the upper visual field and crossed disparities in the lower field (i.e., in the upper field, points are offset leftward in the left eye relative to their corresponding points in the right eye; in the lower field, they are offset rightward). This pattern of offsets is often described as a horizontal shear between the empirical corresponding meridians, and this causes the

*empirical vertical horopter*—the locus of points in the world that stimulate empirical corresponding points near the vertical meridians—to be pitched top-back (Figure 1b). The qualitative similarity between the disparities of empirical corresponding points and the disparities cast on the retinas by natural scenes has led to the hypothesis that corresponding points are adaptive for precisely perceiving the 3D structure of the natural environment (Breitmeyer, Battaglia, & Bridge, 1977; Helmholtz 1925; Nakayama, 1977).

*θ*

_{v}) shows the measured angle between corresponding points near the vertical meridians in all published experiments that used the criterion of equal perceived‐direction. The angle is positive in every case, consistent with corresponding points having crossed disparities in the lower visual field and uncrossed disparities in the upper field. However, the measured angle could be a consequence of cyclovergence, the disconjugate rotation of the eyes around the visual axes (Amigo, 1974). Cyclovergence causes equal rotations between the vertical and horizontal meridians, whereas the hypothesized shear of corresponding points should affect only the horizontal offsets between corresponding points near the vertical meridians. Therefore, to quantify the retinal shear angle, the cyclovergence angle must be subtracted from the measured angle between corresponding points near the vertical meridians. Specifically,

*θ*

_{r}is the true retinal horizontal shear angle between corresponding points near the vertical meridians,

*θ*

_{v}is the measured angle, and

*θ*

_{h}is cyclovergence. (Obviously, if cyclovergence is zero, the true retinal shear and measured shear are equal.) The last column of Table 1 shows this adjustment for the studies in which both measurements were made. After adjustment, the shear angles are still all positive.

Citation | Subject | θ _{v} (deg) | θ _{h} (deg) | θ _{r} (θ _{v} − θ _{h}) (deg) |
---|---|---|---|---|

Helmholtz (1925) | HH^{a} | 2.66 | 0.3 | 2.36 |

WV^{b} | 2.13 | – | – | |

WV^{a} | 2.15 | – | – | |

FS^{b} | 1.32 | – | – | |

FS^{a} | 1.44 | – | – | |

Nakayama (1977) | AC | 3.4 | 0.0 | 3.4 |

CWT | 4.8 | 0.0 | 4.8 | |

Ledgeway and Rogers (1999) | TL | 3.9 | 0.6 | 3.3 |

BJR | 5.8 | 1.3 | 4.5 | |

MLG | 2.9 | 0.5 | 2.4 | |

Siderov, Harwerth, and Bedell (1999) | AK^{1} | 0.56 | – | – |

AK^{2} | 0.48 | – | – | |

HB^{1} | 0.49 | – | – | |

HB^{2} | 0.40 | – | – | |

LB^{1} | 0.62 | – | – | |

LB^{2} | 0.41 | – | – | |

MG^{1} | 0.32 | – | – | |

MG^{2} | 0.41 | – | – | |

Grove, Kaneko, and Ono (2001) | PG | 1.9 | – | – |

HK | 1.7 | – | – | |

NU | 1.6 | – | – | |

Schreiber et al. (2008) | PRM | 2.8 | 0.0 | 2.8 |

KMS | 6.1 | 0.0 | 6.1 | |

HRF | 3.6 | 0.0 | 3.6 |

*θ* _{v} is the angle between the vertical meridians. *θ* _{h} is the angle between the horizontal meridians (cyclovergence). *θ* _{r} is the difference between the two, indicating the amount of retinal shear between corresponding points. All of these studies used apparent-motion except: ^{a}binocular apparent vertical and horizontal; ^{b}monocular apparent vertical. For Siderov et al. (1999): ^{1}viewing distance = 200 cm; ^{2}viewing distance = 50 cm.

*I*is the observer's inter-ocular distance and

*h*is the observer's eye height. We will call

*θ*

_{o}the

*optimal shear angle*. With

*I*= 6.5 cm and

*h*= 160 cm,

*θ*

_{o}= 2.3° (Schreiber, Hillis, Fillipini, Schor, & Banks, 2008). The experimental measurements of the shear angle are reasonably consistent with this optimal value (Table 1). The similarity between observed and optimal shear angles suggests that the vertical horopter may be adaptive for making depth discriminations in the natural environment. Furthermore, for fixations on the ground in the head's sagittal plane, Listing's Law dictates that the horopter will remain coincident with the ground for an observer with an optimal shear value (Helmholtz, 1925; Schreiber et al., 2008). This is also illustrated in Figure 2.

*adaptability hypothesis*and the

*hard-coded hypothesis*. The adaptability hypothesis is that an individual's shear angle is determined by his/her experience with the natural environment. According to this hypothesis, corresponding points adapt to optimize precision in depth estimation based on each individual's experience: If experience changes, the shear should change. The hard-coded hypothesis claims that the shear is hard-coded into the visual system because it confers an evolutionary advantage; that is, the shear is adaptive but not adaptable.

*adaptability hypothesis*makes the additional prediction that the shear should change with individual experience. We tested this prediction by determining whether observers with different inter-ocular distances and eye heights have different shear angles (Equation 2) and by determining whether an observer's shear angle changes when the experienced patterns of disparities are systematically altered by distorting lenses.

*θ*

_{v}) as the angle between the best-fit regression lines for the left and right eyes. Azimuth and elevation are plotted in Hess coordinates, a spherical coordinate system in which azimuth and elevation are both measured along major circles (i.e., longitudes). Lines in Cartesian coordinates project to major circles in spherical coordinates, and major circles are plotted as lines in Hess coordinates. Therefore, lines in the world map to lines in Hess coordinates. Using this coordinate system enabled us to readily assess whether the empirical horopter could lie in a plane.

*θ*

_{h}) to correct the measurements near the vertical meridians (

*θ*

_{v}) and thereby obtain an estimate of the retinal shear angle (

*θ*

_{r}; Equation 2). This is shown in Figure 5c.

*θ*

_{r}) and their optimal shear (

*θ*

_{o}from Equation 2). The hard-coded hypothesis predicts no correlation.

*x*-intercept of a regression line fit to the data (we used quadratic regression lines because much of the data was poorly fit by lines). Rotations due to cyclovergence (

*θ*

_{h}) were also subtracted as shown in Figure 5 (Equation 1). In agreement with the previous literature, all but two observers had corresponding points with uncrossed disparity above fixation and crossed disparity below fixation. (Observer KKD had uncrossed disparity above but no clear pattern of disparity below fixation; LAT had no clear pattern at all.) This means that the vertical horopters of nearly all observers are pitched top-back.

*θ*

_{r}). The mean shear angle was 1.6° and the standard deviation was 0.8°. Figure 8b plots each observer's measured shear value against their optimal shear value. The two values were not significantly correlated (

*r*(26) = 0.07,

*p*= 0.72). The non-significant correlation between measured and optimal shear suggests that corresponding points are not adjusted to keep the horopter in the ground plane for individuals. This is counter to a prediction of the adaptability hypothesis. However, the average measured value was similar to the average optimal value (2.1°), which is consistent with the hypothesis that the shear is hard-coded to be adaptive for the population in general.

*t*-test also revealed that the coefficients on the quadratic terms for the fits near the horizontal meridians were not significantly different from zero (mean = 0.001,

*df*= 26,

*p*= 0.496), which means, as we expected, that there is no curvature in the pattern of correspondence near the horizontal meridians. In contrast, the coefficients for the fits near the vertical meridians were significantly less than zero (mean = −0.009,

*df*= 26,

*p*= 0.004), consistent with convex horopters.

*θ*

_{r}) changed in response to the distorting lenses, we had to take into account the effects of the optical shear caused by the lenses (because observers wore them during the experimental measurements) and of cyclovergence. To take the optical shear into account, we subtracted the horizontal shear due to the lenses (i.e., +3° or −3°) from the empirical measurements. To take the cyclovergence into account, we subtracted the measured cyclovergence values as in Experiment 1. We found that cyclovergence changed slightly during lens wear: The average increase was 0.5° for observers wearing lenses with

*ω*= +3° and −0.1° for those wearing lenses with

*ω*= −3°. Vertical shear disparity along the horizontal meridians induces cyclovergence (Crone & Everhard-Halm, 1975), so the change in cyclovergence we observed was surely due to the vertical disparities of the lenses near the horizontal meridians.

*ω,*frontoparallel surfaces appeared slanted top-back or top-forward, and perceived height was increased or decreased, respectively). However, after 5 days of lens wear, everyone reported that the world appeared undistorted. Four of them also reported perceptual distortions in the opposite direction when the lenses were removed. Closing an eye eliminated the perceptual aftereffect, which suggests that the perceptual effects were due to changes in the interpretation of binocular information and not due to changes in monocular shape representation. However, additional tests would be necessary to confirm this.

*t*-test;

*df*= 8,

*p*= 0.01). They were significantly greater than 0° in the first condition (blank) but not in the second condition (hallway). Assuming that the second condition is more representative of natural viewing than the first condition, these results illustrate the need to measure and correct cyclovergence to zero in Experiments 1 and 2. Thus, our correction procedure was justified.

*linear*because the offsets can be fit with lines. As noted earlier, we observed systematic deviations from linearity in most observers (Figure 7).

*PG, PRM, KMS*) and perhaps present in others (

*CWT, AC, NU*). Importantly, whenever a deviation from linearity occurs (in our data and theirs), it is always convex (i.e., centers bent toward zero azimuth). Convex patterns of correspondence are evidently common.

*Vieth–Müller Circle*: the circle containing the fixation point and the nodal points of the eyes. Figure 14a shows a plan view of the geometric horizontal horopter. As shown in Figure 14b, the empirical horizontal horopter is less concave than the geometric horopter, and the

*Hering–Hillebrand deviation*(

*H*) quantifies the difference:

*α*

_{L}and

*α*

_{R}are the angular locations of corresponding points along the horizontal meridians in the left and right eyes, respectively. Note that the empirical and geometric horizontal horopters are the same when

*H*= 0. Table 2 shows the

*H*values reported from several previous studies that used the perceived‐direction criterion.

*H*is always greater than zero except for observer

*HRF*in Schreiber et al. (2008) and she has intermittent strabismus.

Paper | Observer | Distance (cm) | H |
---|---|---|---|

von Liebermann (1910)* | 97 | 0.04 | |

Lau (1921)* | 150 | 0.10 | |

Helmholtz (1925)* | 71 | 0.07 | |

Ogle (1950) | KNO | 76 | 0.08 |

AA | 76 | 0.05 | |

FDC | 76 | 0.05 | |

WH | 60 | 0.04 | |

Amigo (1967) | PDL | 67 | 0.13 |

GA | 67 | 0.07 | |

Hillis and Banks (2001) | MSB | 172 | 0.24 |

JMH | 172 | 0.22 | |

ND | 172 | 0.43 | |

Schreiber et al. (2008) | PRM | 40 | 0.25 |

KMS | 40 | 0.36 | |

HRF | 40 | −0.11 |

*Values obtained from Ogle (1950). When values were given for various fixation distances, the farthest distance was used.

^{2}

*Z*/∂

*E*

^{2}). Figure 15b plots the second derivative as a function of fixation distance for the vertical and horizontal horopters. Positive and negative values indicate convex and concave shapes, respectively. Greater magnitudes indicate greater curvature. As expected, the vertical horopter is convex at all distances and becomes increasingly so with increasing distance. The horizontal horopter is concave at near distance (see inset where the second derivative is less than zero), becomes planar at approximately 0.46 m (the abathic distance), and becomes increasingly convex at greater distances. For comparison, we also plot the results of the same analysis for a basketball (men's size 7, radius = 11.8 cm). We used a cross-section of the basketball to determine the osculating circle for a parabola and calculated the second derivative. The basketball is more convex than the horizontal horopter at all plotted distances (at sufficiently great distance, the horopter becomes more convex than the ball). The basketball is more convex than the vertical horopter for distances less than 2 m and less convex than the horopter at greater distances.

*Z*

_{0}while a point at distance

*Z*

_{1}stimulates the retina at locations

*α*

_{L}and

*α*

_{R}relative to the foveae. The horizontal disparity due to

*Z*

_{1}is the difference in those locations. The horizontal disparity in radians is given by

*I*is the inter-ocular separation (Held, Cooper, O'Brien, & Banks, 2010). Rearranging, we obtain

*Z*

_{0}) and object distances (

*Z*

_{1}) for positive (uncrossed) disparities and negative (crossed) disparities, respectively; the disparities have been converted to degrees. For each positive disparity, there is a greatest fixation distance

*Z*

_{0}at which it is possible for that disparity to arise from the natural environment. That greatest distance is

*I*/

*δ*(Equation 5 for

*Z*

_{1}= ∞). For disparities of +0.1° and +1.0°, the greatest fixation distances are 34.4 and 3.44 m, respectively (indicated by arrows in the figure). Greater distances could not possibly give rise to the observed disparity.

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