**Abstract**:

**Abstract**
**Three studies, involving a total of 145 observers examined quantitative theories of the overestimation of perceived optical slant. The first two studies investigated the depth/width anisotropies on positive and negative slant in both pitch and yaw at 2 and 8 m using calibrated immersive virtual environments. Observers made judgments of the relative lengths of extents that were frontal with those that were in depth. The physical aspect ratio that was perceived as 1:1 was determined for each slant. The observed anisotropies can be modeled by assuming overestimation in perceived slant. Three one-parameter slant perception models (angular expansion, affine depth compression caused by mis-scaling of binocular disparity, and intrinsic bias) were compared. The angular expansion and the affine depth compression models provided significantly better fits to the aspect ratio data than the intrinsic bias model did. The affine model required depth compression at the 2 m distance; however, that was much more than the depth compression measured directly in the third study using the same apparatus. The present results suggest that depth compression based on mis-scaling of binocular disparity may contribute to slant overestimation, especially as viewing distance increases, but also suggest that a functional rather than mechanistic account may be more appropriate for explaining biases in perceived slant in near space.**

*β*′ is perceived slant,

*β*is actual slant,

*D*is viewing distance, and

*k*is a constant.

*affine*depth compression, Wagner, 1985), and the intrinsic bias hypothesis (Ooi, Wu, & He, 2006).

*k*ln(

*D*) becomes negligible (for example, when

*D*= 1 m, ln(

*D*) = 0). For shallow slants, sin(

*β*) approximates

*β*(in radians), while 90° is 1.57 (in radians). Thus, both models would approximately degenerate to

*β*′ = 1.5

*β*for shallow slants at short viewing distance. It is possible to unify the two models by keeping the sine function in Equation 2 (for the sake of the 90° saturation) while also including the log-distance term in Equation 1 (to simulate the distance effect). Equation 3 describes a candidate for such unified model. The cos(

*β*) added to the log-distance term is to force the perceived optical slant

*β*′ to 90° when actual optical slant

*β*is 90°. We will refer to Equation 3 as the new optical slant model. where

*β*′ is perceived slant,

*β*is actual slant,

*D*is viewing distance, and

*k*is a constant.

^{−8}) and the other started with a big physical aspect ratio (i.e., x

^{15}). The base x decreased with the slant (i.e., x was 1.15, 1.14, 1.13, 1.12, 1.11, 1.10, 1.09, 1.08, 1.07, and 1.06 for the 10 slants, respectively). The stimuli were shown in blocks. In each block, 1 of the 20 staircases was shown once in a random order. There were 12 blocks for each participant. On each trial, a two-alternative forced-choice (2AFC) response was collected by means of key presses to indicate whether the in-depth extent appeared longer or shorter than the frontal extent. The physical aspect ratio of the next trial in that staircase was adjusted up or down by a variable multiplicative step size (achieved by adding or subtracting from the exponent), depending on the response given and the block number in that staircase. Initial step size was by increasing or decreasing the exponent by eight; this step size declined to four after the first block and to two after the fourth block, where it remained thereafter.

*axis of slant*(yaw or pitch),

*direction of slant*(positive or negative),

*amount of slant*(degrees), and

*viewing distance*(meters), and all two-way interactions between these four factors revealed no interaction between axis of slant and the other factors, and no effects of direction of slant, but it did show that the slants in yaw produced larger aspect ratios than did slants in pitch,

*t*(943) = 2.049,

*p*= 0.041. Because this effect is consistent with a horizontal-vertical illusion (HVI), and Hibbard et al. (2012) have shown that the HVI is not caused by slant perception, we normalized each aspect ratio for each axis of slant by the matched aspect ratio (PSE) for each observer in the case of the frontal (90°) slants. When the same mixed-effects linear model was conducted on such normalized data, there was no longer any main effect of axis of slant,

*t*(943) = 0.67,

*p*= 0.503, nor were there any reliable interactions with axis of slant. Thus, the first observation to be made is that, once the horizontal-vertical illusion is taken into account, the implicitly estimated slant did not differ across the four directions we tested.

*t*(959) = 4.54,

*p*< 0.0001; higher aspect ratios at farther distances,

*t*(959) = 4.32,

*p*< 0.0001; and it also indicated that the distance effect was more pronounced for lower slants,

*t*(959) = 4.02,

*p*< 0.0001. Figure 4 depicts the relationship between distance and perceived aspect ratio (with HVI correction) as a function of slant, collapsed across slant direction.

*F*(1, 88) = 27.3,

*p*< 0.0001. (There was no difference between uphill and downhill slants.) This difference between the yaw and pitch conditions is interesting because the frontal stimuli in the two slant conditions were essentially the same, so that the absence of a prominent HVI effect in the 90° yaw slant is most likely related to the presence of yaw slants on other trials. An HVI magnitude of about 5%–6% is quite typical, and the difference between these two conditions corresponds to the sum of two 6% HVI effects. Hibbard et al. (2012) argued that the HVI is not due to a bias in perceived slant in depth. We thought it worth noting that the HVI measured on frontal surfaces in our study differed as a function of the possibility of surface tilt would be in yaw or in pitch. This may show that, independent of perceived slant, the visual system seeks to take into account likely slant when evaluating exocentric extents that might or might not be in depth.

*k*, for the new optical slant model is a free parameter. The parameter

*c*, for the affine model, refers to the amount (ratio) of depth compression along the line of sight at the fixation distance. The parameter

*η*, for the intrinsic bias model, refers to the additive slant exaggeration error supposed by that model. The best fit for each of the three models is shown in Figure 5 along with the observed aspect ratio data. Note, the best-fitting model parameters were computed separately for the two viewing distances in all three models to make the fits comparable, although

*k*is expected to be largely independent of viewing distance. The sum of squared error (SSE) was computed for each model fit at each viewing distance. The SSEs for the new optical slant model (0.012 at 2 m and 0.005 at 8 m) are the smallest among the three. The SSEs for the intrinsic bias model (0.074 at 2 m and 0.103 at 8 m) are the largest. The SSEs for the affine model (0.034 at 2 m and 0.062 at 8 m) are in the middle. Thus, the new optical slant model provided the best fit to the averaged aspect ratio data at both viewing distances.

*t*test of the SSEs between each pair of the models at the two viewing distances revealed no significant difference between the new optical slant model and the affine model at both 2 m,

*t*(48) = 1.33,

*p*= 0.19, and 8 m,

*t*(47) = 0.52,

*p*= 0.60; but the SSEs of the intrinsic bias model were significantly greater than that of the new optical slant model at 2 m,

*t*(48) = 3.00,

*p*< 0.01, and 8 m,

*t*(47) = 4.63,

*p*< 0.001; and they were also significantly larger than that of the affine model at 2 m,

*t*(48) = 4.54,

*p*< 0.001, and at 8 m,

*t*(47) = 3.73,

*p*< 0.001 (Figure 6).

*c*in the affine model were reliably smaller at 8 m (

*M*= 0.35) than at 2 m (

*M*= 0.52),

*t*(96) = 4.04,

*p*= 0.00016, consistent with greater depth compression at farther distance. In contrast, and as expected, for the new optical slant model there was no reliable difference between the individual estimates of

*k*in the 2 m (

*M*= 8.2) and the 8 m (

*M*= 7.8) conditions,

*t*(96) = 0.13,

*p*= 0.897, suggesting that

*k*in the new optical slant model is independent of viewing distance.

*k*in the new optical slant model remains independent of viewing distance, which suggests that the intercept of the slant function increases linearly with log distance for shallow slants (i.e., when cos(

*β*) is close to one), as proposed by Li and Durgin (2010).

*t*test was used to test whether the aspect ratio at the 18° slant differed between the two experiments for each of the four (Direction of Slant × Viewing Distance) combinations. The

*t*test showed no difference for the pitch slant at 2 m,

*t*(36) = 0.39,

*p*= 0.70; no difference for the pitch slant at 8 m,

*t*(36) = 0.29,

*p*= 0.77; no difference for the yaw slant at 2 m,

*t*(37) = 1.44,

*p*= 0.16; and no difference for the yaw slant at 8 m,

*t*(36) = 0.19,

*p*= 0.85. This result supports the conclusion that there was no range effect. Figure 7B shows the aspect ratio data of the texture-scaling group. Compared to the aspect ratio data of the no-texture-scaling group, the aspect ratio data of the scaling group show a consistent elevation. A mixed effects regression of the complete data set found a main effect of texture scaling,

*t*(441) = 2.13,

*p*= 0.034. To compare the aspect ratio data from the texture-scaling group to that of Li and Durgin (2010), the aspect ratio data of Li and Durgin are also plotted (Figure 7B, colored lines). The data from the two studies are quite similar. Note that this suggests that the use of a fixed IPD in the present paper did not alter the results substantially.

*t*(441) = 10.97,

*p*< 0.0001. Moreover, aspect ratios were higher for pitch slant than for yaw slant,

*t*(441) = 8.09,

*p*< 0.0001, which replicates the results in Experiment 1 in the absence of HVI correction. Because there was no frontal slant in Experiment 2, we could not correct the HVI effect for individual participant's data in Experiment 2 as we did in Experiment 1.

*c*, would be smaller than its true value (i.e., the affine model alone would require stronger depth compression than was actually present). To test whether this is the case, in Experiment 3, we sought to directly measure the depth compression ratio in our virtual environment. One technique to assess perceived viewing distance for stereoscopic scaling is the apparently circular cylinder (ACC) task (Johnston, 1991). Glennerster, Rogers, and Bradshaw (1996) pointed out some problems with this task (including the self-occlusion of the edges of a convex cylinder), and suggested using an apparent right dihedral angle task. So as to deal with some of the shortcomings of the ACC task without using a slope task, we instead implemented a version of the ACC task with a concave hemicylinder so that the entire surface would be visible. We also developed a simple ditch version of the experiment in which participants had to compare the height and depth of a horizontal cyclopean ditch specified by binocular disparity between the near and far surfaces.

^{−10}) and the other started with a big depth to half-height ratio (1.1

^{9}). Each staircase was sampled in random order in each of 15 mini blocks of two trials per block (30 trials total). On each trial, a two-alternative forced-choice (2AFC) response was collected by means of key presses (on a radio mouse) to indicate whether the depth extent appeared longer or shorter than the half height. The simulated depth to half-height ratio of the next trial in that staircase was adjusted up or down by a variable multiplicative step size (achieved by adding or subtracting from the exponent), depending on the response given and the block number in that staircase. Initial step size was by increasing or decreasing the exponent by eight; this step size declined to four after the first block and to two after the sixth block, where it remained thereafter. In the ditch task, the simulated viewing distance to the near plane of the ditch was fixed at 2 or 8 m. The vertical height of the ditch subtended a fixed visual angle of 12°. The depth of the ditch also varied from trial to trial. A similar logarithmic up-down staircase procedure as that used in the hemicylinder task was used, with the only exception that the PSE between the depth and whole-height of the ditch was measured. No feedback of the participants' performance was given in any of the three tasks.

Task | Distance | PSE | JND | WF |

Hemicylinder | 2 m | 1.18 | 0.11 | 9.3% |

Hemicylinder | 8 m | 2.62 | 0.24 | 9.2% |

Ditch | 2 m | 1.34 | 0.23 | 17% |

Ditch | 8 m | 6.73 | 1.16 | 17% |

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*The geometries of visual space**R*is the actual aspect ratio,

*R′*is the perceived aspect ratio,

*β*is the actual optical slant, and

*β*′ is the perceived optical slant.

*R′*= 1. Thus, to predict the aspect ratios at the PSEs, we only need to know the perceived slant

*β*′. The three models (i.e., the new optical slant model, the affine model based on mis-scaling of binocular disparity, and the intrinsic bias model) each assumes a different perceived slant function. Replacing

*β*′ with the corresponding slant function in Equation A1 would give us the formula to predict the aspect ratio data for the corresponding model. Equation A2 and A3 represent the assumed slant function for the optical slant model and for the intrinsic bias model respectively. The slant function for the affine model based on mis-scaling of binocular disparity is more complicated and is deduced in next section.

*α*. The center of the observer's head is at the origin and the fixation point is

*F. P*is any given point on the slant and also on the sagittal plane of the observer.

*P*and fixation point

*F*can be expressed by

*θ*−

_{p}*θ*. Since the interpupillary distance (ipd) for each observer is fixed,

_{f}*θ*and

_{p}*θ*only depend on the viewing distance to the two points (i.e.,

_{f}*D*and

_{p}*D*in Figure A1, right panel). That is, the disparity,

_{0}*δ*, between

*P*and

*F*can be expressed as a function of

*D*,

_{p}*D*, and ipd: As shown in Figure A1 right panel, let's assume that the viewing distance to the fixation point is foreshortened. That is,

_{0}*F*is misperceived as

*F*′. Similarly, the position of point

*P*is also misperceived as

*P*′ (note, this is based on the assumption that the viewing direction to

*P*is perceived accurately). Although the positions of

*P*and

*F*are both foreshortened, the relative disparity,

*δ*, between them remained unchanged: Replacing

*δ*in Equation A4 with Equation A5, and simplifying all the tan

^{−1}(

*x*) with

*x*(because

*θ*and

_{p}*θ*should always be small), we obtain the formula to calculate the perceived viewing distance to point

_{f}*P*: Because the origin, point

*P*and point

*P*′ are collinear, the coordinate of

*P′*(

*x*′,

*y*′) can be expressed by the coordinate of

*P*(

*x*,

*y*) with the following equations, Thus, if we know the perceived viewing distance to the fixation point

*F*(equivalent to multiplying actual distance by the depth compression ratio), we can calculate the perceived position of any given point

*P*on the slant and also on the sagittal plane of the observer using Equations A6, A7, and A8. That is how the affine model fits in Figure 2 and Figure 5 in the main text were obtained.