**Vertical line segments tend to be perceived as longer than horizontal ones of the same length, but this may in part be due to configuration effects. To minimize such effects, we used isolated line segments in a two-interval, forced choice paradigm, not limiting ourselves to horizontal and vertical. We fitted psychometric curves using a Bayesian method that assumes that, for a given subject, the lapse rate is the same across all conditions. The closer a line segment's orientation was to vertical, the longer it was perceived to be. Moreover, subjects tended to report the standard line (in the second interval) as longer. The data were well described by a model that contains both an orientation-dependent and an interval-dependent multiplicative bias. Using this model, we estimated that a vertical line was on average perceived as 9.2% ± 2.1% longer than a horizontal line, and a second-interval line was on average perceived as 2.4% ± 0.9% longer than a first-interval line. Moving from a descriptive to an explanatory model, we hypothesized that anisotropy in the polar angle of lines in three dimensions underlies the horizontal–vertical illusion, specifically, that line segments more often have a polar angle of 90° (corresponding to the ground plane) than any other polar angle. This model qualitatively accounts not only for the empirical relationship between projected length and projected orientation that predicts the horizontal–vertical illusion, but also for the empirical distribution of projected orientation in photographs of natural scenes and for paradoxical results reported earlier for slanted surfaces.**

^{2}. For brevity, we refer to line segments simply as lines. The midpoint of each line was at the center of the screen. Lines were 2 pixels (0.2 mm) wide. The standard line was always 3 cm long. The length of the comparison line could take values between 2 cm and 4 cm in 61 equal steps (equal apart from rounding to an integer number of pixels). Line orientations were 0° (horizontal), 30°, 45°, 60°, 90° (vertical), 120°, 135°, and 150°. The fixation dot was 7 pixels (0.7 mm) in diameter. All stimuli were created using Psychophysics Toolbox extensions (Brainard, 1997; Pelli, 1997) in MATLAB (The MathWorks, Natick, MA). For antialiasing, we set the smoothing option of “Screen(‘DrawLines')” to 2, which is “high-quality smoothing.” For Screen(‘BlendFunction'), we chose the most common alpha-blending factors ‘GL_SRC_ALPHA' (source) and ‘GL_ONE_MINUS_SRC_ALPHA' (destination).

- For the PSE, a discretized normal distribution with a mean of 3 cm and a standard deviation of 2 cm.
- For the logarithm of the noise parameter, a uniform distribution across the possible values; this corresponds to a Jeffreys' prior, which has the desirable property of being invariant under reparametrizations.
- For the lapse rate, a beta distribution with parameters 1 and 24. This prior is monotonically decreasing, so favors low lapse rates.

*i*th condition, we assume that the probability of reporting that the comparison line was longer takes the form (Wichmann & Hill, 2001) where

*s*is the length of the comparison line on that trial,

*μ*is the PSE in the

_{i}*i*th condition,

*σ*is the noise parameter in the

_{i}*i*th condition, and

*λ*is the lapse rate. We assume that the lapse rate is shared across all conditions. For each individual subject, we estimated the 64 values of

*μ*, the 64 values of

*σ*, and

*λ*using posterior mean estimation.

*; this function is defined as where*

_{i}*i*is the condition index,

*j*is the trial index within the condition, report

*is the subject's report (“comparison longer” or “standard longer”) on the*

_{j}*j*th trial,

*s*is the comparison length on the

_{j}*j*th trial, and

*p*(report

*|*

_{j}*s*,

_{j}*μ*,

_{i}*σ*,

_{i}*λ*) is given by Equation 1.

_{i}*μ*(between 2 and 4 cm in 51 steps),

_{i}*σ*(between 0.05 cm and 1.5 cm in 25 steps), and

_{i}*λ*(between 0 and 0.2 in 25 steps).

*F*(7, 56) = 18.43,

*p*< 10

^{−11}; a significant main effect of standard orientation,

*F*(7, 56) = 13.18,

*p*< 10

^{−9}; and no significant interaction,

*F*(49, 392) = 1.21,

*p*= 0.17.

*σ*of the psychometric curves, we did not find a significant effect of comparison orientation,

*F*(7, 56) = 1.30,

*p*= 0.27; a significant effect of standard orientation,

*F*(7, 56) = 1.36,

*p*= 0.24; or a significant interaction,

*F*(49, 392) = 1.28,

*p*= 0.11 (see Figure A2). This suggests that length was not encoded with substantially greater precision at some orientations than at others.

*θ*

_{comparison}=

*θ*

_{standard}, we obtained Figure 3B. This shows a clear bias toward reporting the standard line (the line in the second interval) as longer. For each subject, we defined the

*interval bias ratio*(IBR) as the average PSE across all conditions with

*θ*

_{comparison}=

*θ*

_{standard}, divided by the length of the standard. The mean IBR estimated in this way was 1.0224 ± 0.0088, or 2.24% ± 0.88% larger than 1.

*θ*equals the true length

*L*multiplied by an orientation-dependent bias factor OB(

*θ*) as well as by an interval-dependent bias factor IB(interval): The PSE is the true length

*L*of the comparison for which the average perceived length is equal to the average perceived length of the standard. Thus, where

*L*

_{standard}= 3 cm. It follows that By applying the definition of IBR from the previous subsection, we find IBR = [IB(second interval)] / [IB(first interval)]. Both

*θ*

_{comparison}and

*θ*

_{standard}take eight possible values, giving rise to 64 PSE values. To fit this descriptive model, we minimized the sum of the squares of the differences between the empirical PSEs and the ones described by Equation 6: Because this objective function is invariant to a common scaling of all OBs, we define the

*normalized orientation-dependent bias*as OB* (

*θ*) = [OB(

*θ*)] / [(OB(

*θ*= 0°)], so that the objective function becomes We then minimized this objective function over the eight parameters: IBR, OB* (30°), OB* (45°), …, OB* (150°), excluding OB* (0°), which by definition equals 1. We implemented the minimization, for each individual subject, using

*fmincon*in Matlab with 100 random initializations.

*θ*

_{comparison}=

*θ*

_{standard}conditions (reported in the previous subsection). Next, Figure 4 and Table 1 show the estimated normalized orientation-dependent bias. The orientation-dependent bias tends to be stronger for orientations closer to vertical. Signed-rank tests did not reveal significant differences between the normalized orientation-dependent biases at 30° and 150° (

*z =*1.24,

*p*= 0.21), between the ones at 45° and 135° (

*z*= 1.71,

*p*= 0.09), or between the ones at 60° and 120° (

*z*= 1.13,

*p*= 0.26). Thus, we did not find any evidence for asymmetry around vertical.

*orientation-dependent bias ratio*[OB(

*θ*

_{standard})] / [OB(

*θ*

_{comparison})], which in our data is equal to PSE / (IBR ×

*L*

_{standard}) , according to Equation 6. Comparison with these papers is complicated by two factors. First, we varied the orientations of both the standard and the comparison lines, and the earlier studies always fixed the orientation of one of them. Therefore, in the following analyses, we selected the corresponding subset of our trials. Second, the earlier studies used different metrics than we did to report the illusion; we map all metrics to ours, namely [OB(

*θ*

_{standard})] / [OB(

*θ*

_{comparison})]. We now review the details of the three studies.

_{standard}) + 1 based on the data given in table I of Pollock and Chapanis (1952) (this table only gives standard deviations, so we divided those values by

*x*and

*y*dimensions. Subjects reported whether the left or the right line appeared longer. The author reported mean and standard error of the mean of the apparent length of the nonhorizontal line with respect to the horizontal length; because the author used an additive model, we interpret this quantity as

*L*

_{standard}(10 cm).

*y*values using the piecewise cubic hermite interpolating polynomial algorithm (

*interp1*with “pchip” in Matlab). We did not allow for extrapolation, i.e., we limited the orientations to the narrower range between the two studies. We then evaluated goodness of fit by

*R*

^{2}on the interpolated means: The resulting values are shown in Table 2.

*shorter*than lines at orientations of 45°, 67.5°, 112.5°, and 135°. This is an effect that was emphasized by later authors (Craven, 1993; Howe & Purves, 2002). By contrast, in our data, signed-rank tests did not reveal significant differences between the normalized orientation-dependent biases at 60° and 90° (

*z =*0.18,

*p*= 0.86), between the ones at 90° and 120° (

*z*= 0.77,

*p*= 0.44), or between the ones at 90° and 135° (

*z*= 0.059,

*p*= 0.95); a signed-rank test

*did*reveal a significant difference between the normalized bias factors at 45° and 90° (

*z*= 2.43,

*p*= 0.015) but in the direction opposite to Cormack and Cormack. We have no good explanation for this discrepancy, but we suspect that it is due to a special property of the “+” configuration.

*anisotropy model*. We constrain this model by using not only the Howe and Purves (2002) data, but also the distribution of projected orientations in natural scenes.

*L*whose midpoint is the origin of a 3-D Cartesian coordinate system. The line segment has spherical coordinates

*θ*(azimuth, with

*θ*= 0 corresponding to the

*x*-axis) and

*φ*(polar angle, with

*φ*= 0 corresponding to a vertical line). This means that one of its end points has Cartesian coordinates

**v**=

*L*(cos

*θ*sin

*φ*, sin

*θ*sin

*φ*, cos

*φ*), and the other the negative of this. An observer's eye is located on the

*x*= 0 plane and views the origin at an angle

*α*; in other words, the eye's coordinates are

**n**= (0, −cos

*α*, sin

*α*). We define a projection plane

*P*through the origin, orthogonal to the observer's line of sight (i.e., a frontoparallel plane); in other words,

**n**is the unit normal vector of

*P*. The projection of the line segment onto

*P*will be a scaled version of the projection of the line segment onto the observer's retina, provided that we can reasonably approximate the retina as locally flat (this is reasonable when the observer is not too close to the line segment). Because we are interested in length bias

*ratios*, the scale factor is irrelevant.

**v**to properties of its projection onto the plane

*P*,

**v**

_{proj}: The projection of

**v**onto

*P*is

**v**− (

**v**·

**n**)

**n**, where · is the inner product. We work this out in our case: The length of this projected vector can be evaluated as We define the “foreshortening factor” (FF) as the ratio between the projected length and the physical length: The projected orientation

*θ*

_{proj}is defined within the projection plane

*P*as the angle between

**v**

_{proj}and the

*x*-axis (which lies in

*P*). The cosine of this angle is This can be simplified using the tangent: Two special cases provide useful sanity checks:

*θ*,

*φ*, and

*α*, projected length

*L*

_{proj}and projected orientation

*θ*

_{proj}will not be independent.

*φ*, the azimuth

*θ*, and the viewing angle

*α*will not be constant but will obey a probability distribution

*p*(

*θ*,

*φ*,

*α*). For a given true length, projected length and projected orientation will inherit their distribution from this distribution through the mapping in Equation 7. Formally,

*L*

_{proj}and

*θ*

_{proj}. First, to obtain a marginal distribution over projected orientation,

*p*(

*θ*

_{proj}), Girshick, Landy, and Simoncelli (2011) extracted orientations from photographs of natural scenes (for a predecessor of this work, see Coppola, Purves, McCoy, & Purves, 1998). The data show that a projected orientation is more often horizontal or vertical than oblique (Figure 7A). Second, regarding the conditional distribution

*p*(

*L*

_{proj}|

*θ*

_{proj},

*L*), which captures the relationship between projected length and projected orientation, we use the aforementioned data from Howe and Purves (2002) (Figure 7B).

*p*(

*L*

_{proj},

*θ*

_{proj}|

*L*) and, in particular, for the summary statistics in Figure 7A and B, by suitably choosing the distribution

*p*(

*θ*,

*φ*,

*α*). Without loss of generality, we set

*L*= 1. To further constrain the problem, we assume that

*θ*,

*φ*, and

*α*are independent so that their joint distribution factorizes:

*p*(

*θ*,

*φ*,

*α*) =

*p*(

*θ*)

*p*(

*φ*)

*p*(

*α*). We also assume that azimuth

*θ*is uniformly distributed between 0 and 2π; it seems strange to assume anything else. Then, anisotropy can only result from

*p*(

*φ*) or

*p*(

*α*) not being uniform. For simplicity, we choose

*p*(

*α*) to be uniform either on the interval [−45°, 45°] (and 0 outside it) or on [−90°, 90°].

*p*(

*φ*). In each simulation, we randomly drew

*θ*,

*φ*, and

*α*from their respective distributions and computed the histogram of projected orientations as well as the mean of the inverse of projected length for different projected orientation bins. The mean of the inverse of projected length corresponds to the mean of the ratio of physical to projected length that Howe and Purves (2002) use because our physical length equals one.

*p*(

*φ*) is uniform (Figure 7C, left), which means that the distribution of line orientation in 3-D space is

*isotropic*. Nevertheless, the distribution of projected orientation is not uniform at all but has a strong peak at vertical (Figure 7C, center). The projected length is

*greater*for vertical than for horizontal (Figure 7C, right), which contradicts the HVI. This suggests that, in the absence of true correlations between length and orientation in the world, anisotropy is needed to explain the HVI.

*z*= 0 or

*φ*= π/2. Then, the foreshortening factor is

*θ*

_{proj}= tan

*θ*sin

*α*. This predicts that projected orientation is much more often horizontal than any other orientation without vertical being special (Figure 7D, center). It also predicts that the projected length is smaller for projected orientations closer to vertical (Figure 7D, right) as already noted by Howe and Purves (2002). Both effects are qualitatively more consistent with the data, in particular with the HVI, than Simulation 1.

*θ*

_{proj}(Figure 7E, center) is qualitatively similar to the distribution of projected orientations reported by Girshick et al. (2011) (Figure 7A). As we might expect from Simulation 1, a polar angle of π/2 (horizontal) needs to be

*much more frequent*than a polar angle of 0 or π (vertical) for a

*projected*orientation of horizontal to be about as frequent as a

*projected*orientation of vertical. Furthermore, when

*p*(

*α*) is uniform on [−π/2, π/2], these choices produce a pattern of projected length as a function of orientation that is consistent with the HVI (Figure 7E, right). To compare with the data from Howe and Purves (2002) (Figure 7B), we normalized the mean inverse projected length by dividing by its value in the bin that includes horizontal (

*θ*

_{proj}= 0°); this yields a curve similar to the data (Figure 7B). The magnitude of the effect is off, but the empirical data are also equivocal on the magnitude because, for contours, the mean ratio might be much higher than the data in Figure 7B (Howe & Purves, 2002). We conclude that a high prevalence of ground plane lines with a broad distribution of other polar angles can qualitatively account for both the empirical distribution of orientation on the retina and for the empirical relationship between retinal length and retinal orientation.

*L*

_{proj}and projected orientation

*θ*

_{proj}whereas the length of the 3-D line segment

*L*, its angles

*θ*and

*φ*, and the viewing angle of the camera

*α*, are not observable. The generative model would be specified by the conditional distribution

*p*(

*L*

_{proj},

*θ*

_{proj}|

*L*). The observer would infer

*L*from

*L*

_{proj}and

*θ*

_{proj}while marginalizing over

*θ*,

*φ*, and

*α*. In this inference,

*p*(

*L*

_{proj},

*θ*

_{proj}|

*L*) would serve as the likelihood function over

*L*. The likelihood functions over

*L*from the first and second intervals would be used to make the decision. If, for a given projected orientation

*θ*

_{proj}, the FF

*L*

_{proj}/

*L*tends to be lower, the observer will be biased to judge the line at that projected orientation as longer to “compensate for” the foreshortening. Thus, the observer would infer that a vertical line (

*θ*

_{proj}= 90°) was longer than a horizontal line (

*θ*

_{proj}= 0) of the same retinal length. Thus, in a Bayesian view, the distribution

*p*(

*L*

_{proj}|

*θ*

_{proj},

*L*) would be the basis for orientation-dependent length biases. The Bayesian strategy would not be optimal with respect to the laboratory statistics; it would have been optimal if the laboratory had been the real world.

*L*/

*L*

_{proj}, not the distribution

*p*(

*L*

_{proj}|

*θ*

_{proj},

*L*). Given how many poorly justified assumptions we already had to make to match the data in Figure 7A and B, there is little point in trying to work out the inference process in detail. Instead, we view the anisotropy model as a proof of concept that anisotropy in polar angle might account for orientation-dependent length biases.

*θ*= ±90° (the plane formed by the observer's line of sight and the ground plane normal), this translates to the prior over slant depicted in Figure 8B (solid line), which has peaks at ±90° and a uniform distribution elsewhere. Multiplying this prior by the likelihood corresponding to the negative slant cue yields a posterior distribution that is nearly identical to the likelihood: This is because the region of high prior density corresponds to low likelihood, thereby removing the effect of the prior. Even if the peaks in the prior density are wider, the posterior shifts toward ±90° rather than toward 0°. In either case, the observer is predicted to set lower aspect ratios for surfaces slanted away from frontoparallel in either direction. In the extreme case that the slant cue provides perfect information about slant, we can simply apply Equation 8 with θ = ±90° to find that FF = |cos(

*φ α*)|, which has exactly the same cosine form as advocated by Hibbard et al. Thus, Hibbard et al.'s rejection of a Bayesian account might be premature.

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*i*. We assume we have a parameter log likelihood by condition—in our cases, LL

*(*

_{i}*μ*) as given by Equation 2. Here,

_{i}, σ_{i}, λ*λ*represents the parameter(s) that is (are) shared among conditions whereas

*μ*and

_{i}*σ*are both condition-specific parameters. We denote the set of all

_{i}*μ*s collectively by

_{i}'*σ*s by

_{i}'*λ*. We do this by marginalizing over the

*μ'*s and

*σ'*s and assuming uniform priors over all variables: where LL

** is the maximum log likelihood across parameter combinations in the*

_{k}*k*th condition. We separated this term off to prevent the exponentials inside the integrals from being numerically zero. Normalizing, we find for the posterior over

*λ,*

*λ*. We now compute the posterior over

*μ*conditioned on

_{i}*λ*. This is easy because

*λ*is the only variable that connects the conditions; therefore, when

*λ*is given, the posterior over

*μ*only depends on the data in the

_{i}*i*th condition: Normalizing, we find Similarly, the posterior of the noise parameter in the

*i*th condition,

*σ*, conditioned on

_{i}*λ*is

*μ*not conditioned on

_{i}*λ*by integrating over the conditioned posterior: where the first factor in the integrand is given by Equation 10 and the second factor by Equation 9. Similarly, the posterior of

*σ*, is given by

_{i}