Rotating a surface about a horizontal axis alters the retinal horizontal-shear disparities. Opposed torsional eye movements (cyclovergence) also change horizontal shear. If there were no compensation for the horizontal disparities created by cyclovergence, slant estimates would be erroneous. We asked whether compensation for cyclovergence occurs, and, if it does, whether it occurs by use of an extraretinal cyclovergence signal, by use of vertical-shear disparities, or by use of both signals. In four experiments, we found that compensation is nearly veridical when vertical-shear disparities are available and easily measured. When they are not available or easily measured, no compensation occurs. Thus, the visual system does not seem to use an extraretinal cyclovergence signal in stereoscopic slant estimation. We also looked for evidence of an extraretinal cyclovergence signal in a visual direction task and found none. We calculated the statistical reliabilities of slant-from-disparity and slant-from-texture estimates and found that the more reliable of the two means of estimation varies significantly with distance and slant. Finally, we examined how slant about a horizontal axis might be estimated when the eyes look eccentrically.

*HSR*), which is the ratio of horizontal angles a surface patch subtends at the left and right eyes (Rogers & Bradshaw, 1993). Changes in

*HSR*cause obvious changes in perceived slant, but

*HSR*by itself is an ambiguous slant indicator because it is also affected by the plane’s position relative to the head (Backus, Banks, van Ee, & Crowell, 1999; Gillam & Lawergren, 1983; Ogle, 1950). Thus, to estimate slant about a vertical axis, the visual system employs other signals to aid the interpretation of horizontal disparity. These signals include vertical disparity (which can be quantified by the vertical size ratio [

*VSR*]), eye-position signals (indicating the horizontal version and vergence), and other slant signals, such as the texture gradient (Backus et al, 1999). The horizontal disparity pattern associated with slant about a horizontal axis (right panel of Figure 1) can be represented locally as a horizontal-shear disparity. Ogle and Ellerbrock (1946) defined this disparity as follows. A line through the fixation point and perpendicular to the visual plane is a vertical line. There is a horizontal axis through the fixation point, in the visual plane, and parallel to the interocular axis. We rotate the vertical line about this axis and project the images of the line onto the two eyes. The horizontal-shear disparity (

*H*

_{R}) is the angle between the projections of the line in the two eyes. If the eyes are torsionally aligned (ie, the horizontal meridians of the eyes are coplanar) and fixating in the head’s median plane, slant about a horizontal axis is given by: where

*S*is the slant,

*i*is the interocular distance, and

*d*is the distance to the vertical line’s midpoint. When the distance to the surface is much greater than the interocular distance, slant is given to close approximation by: where

*μ*is the eyes’ horizontal vergence (right panel, Figure 1).1,2 Thus, estimating slant about a horizontal axis is straightforward when the eyes are aligned: the visual system must only measure the pattern of horizontal disparity (quantified by

*H*

_{R}) and the vergence distance (

*μ*), which could also be measured by use of the pattern of vertical disparities (Rogers & Bradshaw, 1995; Backus et al, 1999).

*H*

_{S}, a head-centric value, in order to distinguish it from the retinal shear disparity

*H*

_{R}. In the upper row, the eyes are torsionally aligned (τ = 0) and are fixating a frontoparallel plane.

*H*

_{S}is 0 near the fixation point. Slant can be recovered from Equations 1 and 2. In the middle row, the eyes are again torsionally aligned, but the plane is now slanted about a horizontal axis (

*S*< 0;

*H*

_{S}> 0; τ = 0); again slant can be recovered accurately from Equations 1 and 2. In the lower row, the plane is slanted by the same amount as in the middle row, but the eyes are extorted. The shear disparity at the retinas is

*H*

_{R}=

*H*

_{S}− τ. Thus, a particular combination of slant and extortion creates a pattern of horizontal-shear disparity identical to the pattern created by a frontoparallel plane when the eyes are aligned (upper row). If we do not know the torsional state of the eyes, the slant specified by

*H*

_{R}is ambiguous (Ogle & Ellerbrock, 1946; Howard & Kaneko, 1994).

*μ*). The horizontal-shear disparity observed at the retinas (

*H*

_{R}) is 0, −1, and −2 degrees in the upper, middle, and lower panels, respectively. Each panel shows five curves that correspond to the estimate from Equation 1 for cyclovergences of −4, −2, 0, 2, and 4 degrees. The correct surface slant is indicated by the thick curve in each panel (τ = 0). Estimates obtained from Equation 2 are indicated by the open circles; notice that the estimates are an excellent approximation to the correct estimate for all but very short distances (<10 cm). Clearly, failure to compensate for cyclovergence can have a profound effect on the estimated slant; for example, at a distance of 100 cm, the estimation error is −47.5, −28.6, 0, 28.6, and 47.5 degrees for cyclovergences of 4, 2, 0, −2, and −4 degrees, respectively. Likewise, failure to compensate for changes in horizontal vergence (correlate of distance) can have a large effect on the slant estimate; for example, when

*H*

_{R}= −2 degrees (lower panel) and the eyes are torsionally aligned (τ = 0), the correct slant varies from ∼0 degree at very near distances to 47.5 degrees at 200 cm. Here we ask whether the visual system compensates for changes in cyclovergence and horizontal vergence and, if it does compensate, the means by which the compensation is accomplished.

*V*

_{R}) can be defined as the angle between the projections of a horizontal line in the two eyes (lower panel, Figure 2). Slant about the horizontal axis is given to close approximation by:

*H*

_{R},

*V*

_{R}, and distance. This equation predicts that changes in perceived slant can be induced by altering

*H*

_{R}or

*V*

_{R}, and such an effect has been demonstrated by Ogle and Ellerbrock (1946), Howard and Kaneko (1994), and others.

- Perhaps compensation does not occur, so cyclovergence changes lead to errors in slant estimation such as those shown in Figure 3. We will refer to this as the
*no-compensation model*. It is represented quantitatively by Equations 1 and 2. - Perhaps compensation occurs via use of an extraretinal torsion signal. We will refer to this as the
*extraretinal-compensation model*. It is represented quantitatively by Equation 3. - Perhaps compensation occurs via use of vertical-shear disparity. We will refer to this as the
*vertical-disparity-compensation model*and it is represented by Equation 4.

*V*

_{S}, which is a head-centric disparity, to distinguish it from

*V*

_{R}, which is a retinal disparity. The solid gray curve is the predicted slant according to the vertical-disparity-compensation model (Equation 4). The data fell short of the prediction. For three reasons, we cannot determine from these data (nor from the data of the other reports listed above) precisely how the visual system compensates for cyclovergence. First, Howard and Kaneko,1994,(and the others listed above) did not measure the eyes’ cyclovergence during the experimental measurements. Vertical-shear disparity is known to stimulate cyclovergence (Rogers, 1992), so it is quite likely that cyclovergence covaried with vertical shear in this experiment. Thus, one cannot determine from these data how much of the observed compensation was due to vertical-disparity as opposed to extraretinal compensation. Second, the stimulus in the Howard and Kaneko experiment (and the others listed above) contained monocular slant signals (texture gradient and outline shape) and those signals always specified a slant of 0. Human observers take both monocular and stereoscopic estimates into account when judging surface slant (Buckley & Frisby, 1993; Banks & Backus, 1998), so Howard and Kaneko’s data are almost certainly contaminated by monocular slant signals. Third, Howard and Kaneko asked observers to adjust a paddle with the unseen hand until it was judged to have the same slant as the visual stimulus (others used a variety of estimation techniques). To do this, the observer has to convert the internal slant estimate into a manual response. The problem is that we do not know the function that maps internal estimate into response, so we cannot determine how much of the prediction shortfall was caused by this mapping function.

*both*settings, he indicated this with a key press and a new stimulus sequence was begun.

*H*

_{R}) of the observers’ settings is plotted as a function of the eyes’ cyclovergence. To determine the retinal disparity (

*H*

_{R}), we subtracted the measured cyclovergence for the appropriate condition from the head-centric horizontal shear at the CRTs (

*H*

_{S}). If observers failed to compensate for changes in cyclovergence, the data would be independent of the eyes’ torsion and would fall on a horizontal line at

*H*

_{R}= 0. On the other hand, if observers compensated for cyclovergence in the fashion suggested by Equation 3 (extraretinal compensation with an accurate torsion signal) or by Equation 4 (vertical-disparity compensation), the data would fall on the diagonal line (along which

*H*

_{R}= − τ).

*V*

_{S}). Specifically, vertical-shear disparities of −4, −2, 0, 2, or 4 degrees (head-centric coordinates) were added to the random-dot test stimulus. The vertical-shear disparity at the retinas was, therefore, the difference between the added vertical shear and the disparity created by the eyes’ cyclovergence:

*V*

_{R}=

*V*

_{S}− τ.

*H*

_{R}) of the observers’ settings is plotted as a function of the eyes’ cyclovergence. If observers failed to compensate for changes in cyclovergence, the data would fall on the horizontal line at

*H*

_{R}= 0. If observers compensated for cyclovergence by use of an extraretinal torsion signal (Equation 3), then only the horizontal-shear disparity and the eyes’ torsional state would matter. In this case, the data would fall on the middle diagonal line. Finally, if observers compensated for cyclovergence by use of the added vertical-shear disparity (Equation 4), then only the horizontal and vertical-shear disparities at the retinas would matter. Specifically, a slant of zero would be perceived whenever the vertical and horizontal shears were equal to one another. The data would fall on the series of diagonal lines, a different line for each added vertical-shear disparity (

*V*

_{S}).

*H*

_{R})of the observers’ settings is plotted as a function of the eyes’ cyclovergence. If observers failed to compensate for changes in cyclovergence, the data would fall on the horizontal line at

*H*

_{R}= 0. If they compensated for cyclovergence by use of an extraretinal torsion signal (Equation 3), the data would fall on the middle (green) diagonal line. We assume that the vertical-shear disparity cannot be measured reliably with the stimulus used in this experiment, so compensation by vertical disparity is unlikely.

*H*

_{R}= 0 is the prediction for no compensation, the five diagonal lines are the predictions for vertical-disparity compensation, and the middle (green) diagonal line is the prediction for extraretinal compensation. We conducted an analysis of variance on the data in Figure 12: there was no significant effect of vertical-shear disparity on any of the three observers’ data. Because there was no systematic effect of vertical-shear disparity, we conclude that observers were indeed unable to use this signal when it was made smaller. This specific finding is consistent with the results of Howard and Kaneko’s (1994) second experiment (see their Figure 4). The data also appear to be inconsistent with the prediction for extraretinal compensation. The analysis of variance revealed no significant effect of the eyes’ cyclovergence with the exception of observer B.T.B. who showed a small effect:

*P*< 0.01 (notice that the slope of B.T.B.’s data is much less than the slope of the extraretinal prediction, so his compensation was far short of veridical). Overall, the data are most consistent with a failure to compensate for changes in cyclovergence.

*μ*/2) (Somani et al, 1998; Tweed, 1997). To understand the consequence of Listing’s Extended Law, consider an observer looking at a gaze-normal surface consisting of a cross. The observer pitches the head upward or downward while maintaining fixation on the cross. According to Listing’s Extended Law, the eyes will move such that the vertical and horizontal limbs of the cross will continue to fall on the eyes’ vertical and horizontal meridians. Said another way, no horizontal- or vertical-shear disparity will be introduced as the observer pitches the head. Measurements of binocular eye movements reveal that not all observers follow Listing’s Extended Law (Somani et al, 1998); in upward gaze of near targets those observers tend to have the eyes extorted relative to prediction and in downward gaze they tend to have the eyes intorted relative to prediction. We used this to create changes in cyclovergence without presenting a vertical-shear stimulus.

*H*

_{R}= 0 is the prediction for no compensation and the diagonal line is the prediction for extraretinal compensation. For the purposes of determining whether compensation occurred, we are looking for an effect of head pitch on the disparity setting. The vertical position of the data points is unimportant because it is affected by the observer’s criterion for what constitutes a slant of 0. The figure shows that there was no systematic effect of the eyes’ cyclovergence. An analysis of variance was conducted on the data and revealed no significant effect of cyclovergence. Thus, the data are most consistent with the no-compensation model. When vertical-shear disparity is unmeasurable (in the test stimulus and in preceding stimuli), there appears to be no compensation for the eyes’ torsional state.

*H*

_{R}), vertical-shear disparity (

*V*

_{R}), and a distance estimate (

*μ*) that could be determined from an extraretinal signal of horizontal vergence or from the horizontal gradient of vertical disparity (Backus et al, 1999; Rogers & Bradshaw, 1995). We now consider how errors in the measurements of

*H*

_{R},

*V*

_{R}, and μ ought to affect slant estimates based on those signals alone and how those errors ought to affect slant estimates when the surface also provides useful perspective information.

*H*

_{R},

*V*

_{R}, and . We conducted a Monte Carlo simulation to determine the variance of the slant estimates from this equation (Backus & Banks, 1999). We assumed Gaussian noise (mean = 0) in the measurements of

*H*

_{R},

*V*

_{R}, and

*μ*. The assumed standard deviations of the noises for

*H*

_{R},

*V*

_{R}, and μ were 0.132, 0.132, and 0.5 degrees, respectively.9 With those assumed noises, the estimator would have a slant-discrimination threshold (71% correct) of ∼1.5 degrees at a viewing distance of 50 cm and base slant of 0 degree, a threshold value that is consistent with preliminary measurements (Banks, 2000).10

*H*

_{R}−

*V*

_{R}= 0). The reason for the high-reliability ridge can be seen by inspection of Equation 4. When surface slant is ∼0, the argument of the tangent is ∼0, and, therefore, variance due to error in the measurement of

*μ*is minimized. Third, ridges of high reliability occur at large slants. This occurs because smaller variations in slant cause increasingly large changes in horizontal-shear disparity as slant increases (the inverse tangent in Equation 4 asymptotes with increasing horizontal-shear disparity).

Dimensions | Distance | Texture Type | Estimate of w _{p} | |
---|---|---|---|---|

Gillam & Rogers (91) | 10 degree square | 90 cm | Random dot | 1.00 |

Howard & Kaneko (94) | ||||

Experiment 1 | 85 × 65 degrees | 61 cm | Random dot | 0.62 |

Experiment 2 | 60 degree circle | 94 cm | “ | 0.44 |

Experiment 2 | 30 degree circle | “ | “ | 0.50 |

Experiment 2 | 10 degree circle | “ | “ | 0.91 |

Kaneko & Howard (97) | ||||

Experiment 4 | 60 degree circle | 94 cm | Random dot | 0.52 |

Allison et al. (98) — 30 sec | 60 degree circle | 93 cm | Irregular | 0.52 |

Howard & Pierce (98) | ||||

Experiment 1 | 60 degree square | 89 cm | Various | 0.54 |

Experiment 2 | 60 degree square | “ | “ | 0.51 |

van Ee & Erkelens (98)— 25 sec | 70 degree square | 150 cm | Small circles | 0.55 |

*S*

_{d}; the “hat” indicates a visual estimate rather than the physically specified variable) and the perspective-specified slant (

*S*

_{p}). The observer then uses a weighted average to determine the best overall slant estimate: , where is the final slant estimate and the weights (

*w*

_{d}and

*w*

_{p}) add to 1 (Landy et al, 1995). For these experiments, , so .

*w*

_{p}to a small value.

*m*(

*m*/2 and −

*m*/2 in the left and right eyes, respectively) and a vertical-shear disparity of

*m*. The observer’s task was to adjust the slant of the stimulus (about a horizontal axis) until it looked gaze normal. Observers made settings for 10 magnifications (that added vertical-shear disparities of 0.67 to −0.84 degrees).

*V*

_{L}). The dashed line represents the predicted settings if there were no cyclovergence (or if cyclovergence occurred along with veridical extraretinal compensation). The solid curve represents the predictions if cyclovergence occurred and the visual system compensated for it veridically by using vertical-shear disparity (Equation 4) or if cyclovergence occurred and the visual system failed to compensate for it. The data points are from Figure 6 and Table 2 in Ogle and Ellerbrock (1946). Lower right panel: The horizontal-shear disparity (

*H*

_{L}) approaching the eyes at the slant setting. The dashed line represents the predicted settings if there were no cyclovergence (or extraretinal compensation). The solid diagonal line represents the predictions if cyclovergence occurs and the visual system compensates for it veridically by using vertical-shear disparity (Equation 4) or if cyclovergence occurs and the visual system fails to compensate for it. Again the data points are from Ogle and Ellerbrock.

*H*

_{S}. The corresponding vertical-shear disparity is always 0 because rotation of the stimulus about the horizontal axis has no effect on projections of a horizontal line. Second, there are the shear disparities as they leave the lenses and approach the eyes. These head-centric disparities are

*H*

_{L}and

*V*

_{L}. If the lenses add a horizontal shear of

*m*, then and . Finally, the shear disparities at the retina are

*H*

_{R}and

*V*

_{R}. These retino-centric values are affected by the eyes’ cyclovergence (τ), so and .

*g*which has a value between 0 and 1 (Howard & Zacher, 1991). Thus, .

*H*

_{R}is the horizontal-shear disparity at the retina and m is the vergence distance.

*H*

_{R}is affected by the lenses and the eyes’ torsion:

*V*

_{L}). If cyclovergence gain (

*g*) is 1, the predicted slant settings are the solid curve. If the gain is 0, the predictions are the dashed curve. The right-hand graph in Figure 17 plots the predicted horizontal-shear disparity approaching the eyes (

*H*

_{L}) as a function of the induced vertical-shear disparity (

*V*

_{L}). If no compensation for cyclovergence occurs,

*H*

_{L}=

*g V*

_{L}.

*H*

_{R}) is affected by the lenses and the eyes’ torsion, but now the visual system compensates for the part caused by the torsion. Thus, the observer’s task is to adjust the stimulus until . Because and , the task becomes: .

*H*

_{L}= 0). Ogle and Ellerbrock’s data are clearly inconsistent with these predictions, so we can reject this model. If we assume that the gain of the extraretinal signal (not the gain of the cyclovergence itself) is less than 1, the predictions move in the direction of the solid curve and line; of course, as the extraretinal gain goes to 0, the model becomes the no-compensation model.

*H*

_{R}is again affected by the lenses and the eyes’ torsion, but now the visual system compensates by using the vertical-shear disparity (

*V*

_{R}). The observer’s task in this model is to set the slant such that the horizontal and vertical-shear disparities at the retina are equal to one another; that is, . We can work out the model’s predictions by quantifying the lens and torsion effects on the horizontal and vertical-shear disparities separately. For the horizontal shear, For the vertical shear, Then subtracting

*V*

_{R}from

*H*

_{R}and rearranging, This prediction yields the solid curve in the left-hand part of Figure 17. Similarly, the prediction for the right-hand part of the figure is

*H*

_{L}=

*V*

_{L}, which is the solid diagonal line. Ogle and Ellerbrock’s data are quite consistent with the predictions of this model. Notice that the predictions do not depend on the gain of the cyclovergence response itself.

*V*

_{L}). More recent work on cyclovergence reveals that the gain is actually significantly less than 1. Indeed, cyclovergence gains are typically 0.4–0.6 for stimulus conditions like those of the Ogle and Ellerbrock (Howard & Zacher, 1991), so their data almost certainly cannot be explained by the no-compensation model. We conclude that their data manifest the operation of compensation based on vertical-shear disparity and that Ogle and Ellerbrock’s data demonstrated a direct effect of vertical-shear disparity on slant perception. They are not credited with the discovery because they did not understand that at the time.

*HSR*) is affected by slant about a vertical axis as well as the azimuth and distance of the surface from the head (Backus et al, 1999; Ogle, 1950). When the surface is straight ahead (in the head’s median plane),

*HSR*is 1 when the slant is 0. However, when the surface is 30 degrees to the left of the median plane,

*HSR*is 1 when the slant is 30 degrees. To recover the slant of surfaces at different azimuths, the visual system must “correct” the observed

*HSR*(Gåding, Porrill, Mayhew, & Frisby, 1995). It does so by using vertical-disparity and eye-muscle signals (Backus et al, 1999).

*V*) is positioned in the head’s median plane at distance d. The stimulus is a surface rotated about the horizontal axis through the fixation point (

*F*).

*P*lies on this surface and in the head’s median plane. The head is rotated about a vertical axis through the angle γ . The eyes’ vergence (

*μ*) is the angle subtended by the lines of sight at

*F*. We calculated the horizontal-shear disparity (

*H*

_{R}) by using standard cameras positioned at the left and right eyes and pointed at

*F*. We examined how two means of slant estimation—the ones expressed by Equations 1 and 4-are affected by head rotation:

*μ*) for the same purpose. The use of vergence is preferable because it compensates for the baseline change with eccentric viewing.

*HSR*) and for horizontal-axis slant, it measures horizontal shear disparity (

*H*

_{R}). Furthermore, in both cases, the system must “normalize” the disparities with a distance estimate. In some respects, however, vertical-axis slant estimation differs from horizontal-axis estimation. Changes in stimulus azimuth have a profound effect on the relationship between horizontal disparity and vertical-axis slant and, as we have shown here, none on the relationship between horizontal disparity and horizontal-axis slant. In contrast, changes in cyclovergence affect the relationship between horizontal disparity and horizontal-axis slant (Figure 3) and presumably do not affect the relationship between horizontal disparity and vertical-axis slant. The visual system must “correct” the disparities with an estimate of gaze azimuth in the vertical-axis case and with an estimate of cyclovergence in the horizontal case. Gaze azimuth varies substantially during natural viewing: human observers frequently adopt eye positions that are 20 degrees left or right of the median plane. Cyclovergence, on the other hand, does not vary substantially in natural viewing: as discussed earlier, a consequence of Listing’s Extended Law is that the eyes remain torsionally aligned (or nearly so) as an observer looks at a near target. The requirement for disparity correction is, therefore, generally greater in vertical- than in horizontal-axis estimation. As a consequence, the visual system may place greater reliance on stereoscopic slant estimation in the horizontal-axis case.

^{1}Equation 1 can be derived analytically. Impose a coordinate system on the head such that the x-axis is the interocular axis, y is straight up and down, and z is forward and back. Then the eyes are at (±

*i*/2, 0, 0) and a fixation point

*F*is at (0, 0,

*z*). A plane goes through

*F*and a line parallel to the x-axis. Line

*L*is within that plane and within the head’s median plane; the line also contains

*F*.

*L*has slant

*S*(the angle between

*L*and the y-axis).

*H*

_{R}can then be written in terms of

*S, i*, and

*d*by considering the projection of

*L*onto image planes at the eyes (ie, planes that are normal to lines of sight from each eye to

*F*, respectively):

*H*

_{R}is the difference in the deviation of these projected lines from the vertical. Ogle and Ellerbrock (1946) presented the following equation for estimating slant from horizontal-shear disparity: This same expression reappears in van Ee and Erkelens (1996). Equation 2 is slightly more accurate than Ogle and Ellerbrock’s equation when the surface is in the head’s median plane and, as we show in the discussion, it is much more accurate than Equation 1 and Ogle and Ellerbrock’s equation when the surface is to either side of this plane.

^{2}We use right-hand coordinates, so the sign conventions are the following. For positive

*H*

_{R}(and

*V*

_{R}), the right-eye’s image is rotated clockwise relative to the left eye’s. For positive τ, the right eye is rotated clockwise relative to the left eye. For positive

*S*, the surface is slanted top toward the observer.

^{3}Nakayama and Balliet (1977) have presented circumstantial evidence for an extraretinal signal for the cyclo

*version*state of the eyes, but to our knowledge, there is no evidence for an extraretinal signal for the cyclo

*vergence*state.

^{4}Because the lines of sight are perpendicular to both CRTs, the orientation difference at the eyes is given by the orientation difference in the calibration planes just in front of the CRTs (Backus et al, 1999).

^{5}Our procedure for eliminating intrusion due to monocular slant signals would not achieve the desired result if observers perceived a non-zero slant from the monocular signals alone. As a check against the possibility, Backus et al (1999) conducted a monocular control experiment. Stimuli like those in this paper were presented and observers made slant-nulling settings. The standard deviations of the settings were generally 10 times greater than when the task was performed binocularly. Thus, perceived slant from monocular signals was ill-defined and could have had little effect on slant settings in the binocular experiments. Admittedly, the task in Backus et al was different because it involved slant about a vertical axis, but there is no reason to believe that monocular signals are more informative for slant about a horizontal axis (eg, Buckley & Frisby, 1993). Thus, we are confident that monocular slant signals had no discernible effect on the data in the present report.

^{8}Kaneko and Howard (1997) did an experiment that was designed to reveal whether an extraretinal, cyclovergence signal affects perceived slant. In their fourth experiment, they adapted observers to displays with different amounts of cyclodisparity and then flashed planar stimuli in order to determine whether the adaptation affected subsequent percepts. They found no effect and concluded that the visual system does not use an extraretinal cyclovergence signal in slant estimation. This conclusion, however, is not warranted because they could not rule out the possibility that compensation due to vertical-shear disparity overrode an extraretinal-based compensation.

^{9}We assumed that the standard deviations for

*H*

_{R}and

*V*

_{R}were equal. We assumed that the standard deviation for

*μ*was 0.5 deg because that value seems reasonable and because we used it in a previous paper (Backus & Banks, 1999). We then found the values of

*H*

_{R}and

*V*

_{R}that yielded the desired slant-discrimination threshold of ∼1.5 deg at 50 cm.

^{10}We assumed fixed Gaussian noise with identical standard deviations for

*H*

_{R}and

*V*

_{R}. This assumption would be falsified by an observation that the measurement of

*H*

_{R}or

*V*

_{R}is more accurate for some values than others. The measurement of

*H*

_{R}and

*V*

_{R}would be less accurate if the disparities approached or exceeded the fusion range, but with small surfaces straddling the fixation point, the disparities remain small and fusable, so our assumption seems reasonable for the estimation of local slant near the fixation point. We also assumed that the standard deviations associated with

*H*

_{R}and

*V*

_{R}measurements are the same, but one might argue that the measurement of

*V*

_{R}is more accurate because the visual system seems to measure it over a large portion of the available stimulus (Stenton, Frisby, & Mayhew, 1984). In Equation 4, however, the argument of the tangent is

*H*

_{R}−

*V*

_{R}, so the effect on the slant estimate is the same for various combinations of errors in

*H*

_{R}and

*V*

_{R}as long as their variances add to a constant. The assumption that the vergence measurement is Gaussian distributed is presumably false because a sensible system would not accept vergence estimates less than 0. Fortunately, given the relatively short distances used in the simulation and the small vergence error, such estimates were extremely uncommon in the simulation, so again this assumption is reasonable for our purposes.

^{11}Variations in cyclovergence had no effect on the simulation. The reason is obvious from Equation 4. Changes in cyclovergence have the same effect on

*H*

_{R}and

*V*

_{R}, so it has no effect on the difference.

^{12}We explored how the choice of standard deviations for the

*H*

_{R},

*V*

_{R}, and

*μ*noises affected the outcome of the simulation. Changing the values by a factor of two had little effect. However by making larger changes, two discernible effects could be observed. When the

*μ*noise was large and

*H*

_{R}and

*V*

_{R}noises small, the high-reliability ridges at large slants diminished, leaving a single ridge in the middle. When the

*H*

_{R}and

*V*

_{R}noises were large and

*μ*noise was small, the reliability ridge in the middle of the panels diminished, leaving a wide valley with ridges on the sides.

^{13}The fact that the weight given a slant estimator ought to be a function of the slant it is trying to estimate seems odd. There is no logical problem, however, because the slant-from-disparity and slant-from-texture estimators could provide separate slant estimates before the weights are assigned. Then the estimator that combines the two could assign weights based on those estimates.

^{14}Howard and Kaneko (1994) and Kaneko and Howard (1997) circumvented this problem to some degree by using the results from a control experiment to normalize their data. In the control experiment, observers used the slant-estimation procedure while viewing a real, full-cue surface. Howard and Kaneko then used those data to scale the observers’ responses in their main experiments.

^{15}Porrill, Mayhew, & Frisby (1989) stated that they too observed a “vertical shear induced effect.”

^{16}The transformation can be quantified as a deformation as well (Gillam & Rogers, 1991; Koenderink & van Doorn, 1976).