We investigated the perception of three-dimensional plane orientation—focusing on the perception of tilt—from optic flow generated by the observer’s active movement around a simulated stationary object, and compared the performance to that of an immobile observer receiving a replay of the same optic flow. We found that perception of plane orientation is more precise in the active than in the immobile case. In particular, in the case of the immobile observer, the presence of shear in optic flow drastically diminishes the precision of tilt perception, whereas in the active observer, this decrease in performance is greatly reduced. The difference between active and immobile observers appears to be due to random rather than systematic errors. Furthermore, perceived slant is better correlated with simulated slant in the active observer. We conclude with a discussion of various theoretical explanations for our results.

*xy*-plane is co-planar with the monitor screen, with the

*x*-axis pointing to the subject’s right, the

*y*-axis upward, the

*z*-axis toward the subject, and the origin at the center of the monitor. Lengths will be expressed in centimeters.

*xy*-plane. The stimulus was centered at the point directly opposite the subject’s eye at the beginning of each trial [i.e., if the subject’s eye was at point (

*x,y,z*), the center of the stimulus was at (

*x,y*,0)].

*σ*cos

*τ*, sin

*σ*sin

*τ*, cos

*σ*), where

*σ*is the slant and

*τ*is the tilt. Tilt varied from 0° to 345° in increments of 15°. Slant was 30°, 45°, or 60°. A red fixation mark (a circle of size 0.05 cm) was visible in the center of the stimulus during the entire duration of the trial. Other than the stimulus, nothing was visible, including the borders of the display monitor.

*x*- or

*y*-axis in horiz and vert trials, respectively) exceeded 6 cm from the central point. To make sure that motion was primarily in the required direction, at the end of each trial, we calculated the RMS of the subject’s motion in that direction, normalized by the RMS of the motion in the two perpendicular directions; if this ratio exceeded 0.5, the trial was restarted. Furthermore, a trial was restarted when the duration of the visible stimulus was less than 2 s or greater than 5 s.

_{p}) versus simulated (τ

_{s}) tilt. We define the quantity with angular differences always taken in the shortest way around the circle, and therefore ranging from –180° to 180°. The histograms in Figure 4 show the distributions of Δ

_{τ}. The figure shows that, especially in the immobile (immob) condition, in many trials Δ

_{τ}was close to ±180°. This corresponds to the perception of a reversal (see “Introduction” and Figure 1). Because the optic flow is almost ambiguous — there are really two simulated tilts differing by 180° — we also define a second tilt error, with respect to the reversed tilt: (with angular differences as above). Using these two quantities, we introduce an absolute-value tilt error measure, E

_{τ}, which is the absolute value of the angular difference between the response and either the regular or the reversed simulated tilt, whichever is closer: . As defined, E

_{τ}ranges from 0° to 90°. In the remainder of this article, we will refer to E

_{τ}as the tilt error.

*p*< .01). A higher tilt error in immob than in act was observed in all subjects. However, the average error is not fully informative, as there is a large effect of shear angle.

*p*< 10

^{−4}). Further analysis showed that in both immob and act, tilt error increased significantly with increasing shear (both

*p*< 10

^{−3}). However, the tilt errors increased differently in the two conditions. The ANOVA showed a significant selfmotion × shear angle interaction (

*p*< 10

^{−4}): the tilt error rose faster in the immob than in the act condition. The magnitude of this effect can be demonstrated by a linear regression: the mean slope of the tilt error versus shear is 0.224 in immob but only 0.067 in act. Moreover, this slope is lower in act than in immob in all subjects. As far as the effect of slant was concerned, the ANOVA revealed that tilt errors decreased with increasing slant (

*p*< 10

^{−3}). Finally, the ANOVA showed a significant influence of direction, which we will return to in section “Anisotropy With Respect to Movement Direction” below.

*p*< 10

^{−4}), although this dependence is rather weak (the slope of the linear regression is 0.21); however, all subjects showed a significant positive correlation between simulated and response slant in both selfmotion conditions (all

*p*< 10

^{−2}). However, slant response was better correlated with simulated slant in act than immob (mean slope 0.25 in act, 0.16 in immob). Indeed, there was a significant selfmotion-slant interaction (

*p*< .05).

*p*< .05).

*p*< 10

^{−4},

*z*test for independent proportions). The rate of reversals in immob is significantly lower than 50% (

*p*< 10

^{−4}) — a 50% reversal rate would have been expected if subjects had ignored second-order information in optic flow (Wexler et al., 2001a)

*p*< .05), but not in immob. Nevertheless, even when errors were greater in reversal trials, the responses were not random: tilt responses in reversal trials were centered around reversed tilt and almost absent in the region which could be interpreted as large deviations in the percept of the simulated tilt—see, for example, the histograms in Figure 4. Absolute slant errors in immob did not increase when reversals occurred, but a significant increase was seen in act (

*p*< .05).

_{τ}, which confounds random with systematic errors in tilt. Here, we wish to examine systematic errors in tilt corrected for reversals, and therefore we define a new, signed error measure: with Δ

_{τ}and Δ

_{τ}′ defined in Equations 2 and 3. S

_{τ}is a signed tilt error that corrects for possible tilt reversals (i.e., the error is with respect to either the regular or the reversed simulated tilt, whichever is closer); it therefore ranges from −90° (clockwise errors) to +90° (counterclockwise errors). Averaging S

_{τ}for a given value of simulated tilt permits us to study any systematic bias that is present at that point, independently of reversals.

*p*< 10

^{−4}, Bonferroni corrected, for both), with mean bias tilt of 85° in act and 88° in immob. In individual subject data, mean tilt biases were 91°, 84°, 85°, 82°, and 176° in act, and 99°, 85°, 80°, 100°, and 178° in immob (keeping the order of the subjects the same). All 10 tests were significant at

*p*< 10

^{−4}and remained so when Bonferroni corrected.

*t*test for the mean values of S

_{τ}on individual subject data at the 24 values of simulated tilt. None of the tests reached the Bonferroni-corrected threshold for significance at

*p*= .05.

_{τ}was significantly different in the two direction conditions, being greater in the vert (26.8°) than in the horiz condition (22.5°,

*p*< .05). Further analysis showed that the two curves in immob were not significantly different, but the ones in act were. Quantitatively, however, the difference between the two curves of the immob conditions and the difference between the two act curves was very similar.

*z*-axis than horiz trials. The displacement along the

*x*-axis during up/down movement was also greater than the displacement along the

*y*-axis during left/right movement. We homogenized the trajectories post hoc by only considering trials whose movement amplitudes fell within a certain range. After this homogenization, the anisotropy with respect to movement was, however, still present.

*α*, a reasonable assumption for experiment where the average maximum angle was about 4°), the change in the surface normal ^n = (sin

*σ*cos

*τ*, sin

*σ*sin

*τ*, cos

*σ*is given by (^A × ^n)

*α*plus terms higher order in

*α*, where ^A = (−sin(

*τ*−

*η*), cos(

*τ*−

*η*),0) is the axis of rotation and

*η*the shear angle. The corresponding changes in slant and tilt are therefore .

*η*=0° and 15° and subtracting from the average for

*η*=75° and 90°, we found tilt error differences of 14.6°, 18.7°, and 15.6° for slant 30°, 45°, and 60°, respectively. Third, the sin

*η*term in Equation 7 would predict that the slope of the tilt error versus shear curve approach zero as shear approaches 90°, which is not observed in our data. Fourth, Equation 6 would predict that slant errors decrease with shear, but they actually increase significantly.

_{0}(ρ,ϑ) and R(ρ,ϑ) be the initial and final positions in 3D space of a point on the circle with 2D polar coordinates (ρ,ϑ). When we average the square length of 3D displacements generated by this rotation, we find the following expression Equation 8 shows that nonstationarity rises with the shear, η, which would seem to be in agreement with our tilt error results in immob (see Figure 5). However, our virtual objects were not circles in space but in the image plane, and were then projected onto the simulated surface; therefore, in space, these objects were ellipses. When we perform the above calculation for these elliptical objects (in parallel projection), we find the following mean square displacement: which is independent of shear. (In perspective projection, the first correction to Equation 9 is in second order, which can be safely ignored for our small stimuli.)

**R**and

**T**the rotation and translation of the plane, (

*x,y*) a point on the retina, and

*Z*the

*z*-coordinate of the plane in that direction, we have the following optic flow: If the flow u

_{x,y}is known and the goal is to solve for 3D structure (

*Z*) and motion (

**R**,

**T**), Equations 10 and 11 are a complex nonlinear system, due to the

**T**/

*Z*terms.

**R**and

**T**is integrated into the process — the reduced SfM problem of solving Equations 10 and 11 for

*Z*becomes linear and therefore simple. In our experiments, any self-motion information would have to be extra-retinal, as optic flow was the same in the act and immob conditions. We hypothesize that quantitative, extra-retinal information about self-motion is integrated into the SfM process.

^{2}By “shear” (a term that is used in somewhat different ways in the literature), we mean the extent to which optic flow is perpendicular to its gradient. Cornilleau-Pérès et al. (2002) used the “winding angle” for what we call “shear angle.”

^{3}We also searched for but found no evidence of any oblique effect such as that found in Oomes & Dijkstra (2002).

^{4}The fact that we found systematic errors in tilt perception, whereas other studies have not (Domini & Caudek, 1999; Norman et al., 1995; Stevens, 1983; Todd & Perotti, 1999), may be due to the way we analyzed the data. Most of the above-mentioned studies calculate regression coefficients between perceived and simulated tilts over the entire range of tilt values (i.e., 360°). Consequently, it is hardly surprising that the slope of the regression line one finds is always near unity (remember that tilt is a circularly periodic variable, in contrast to the way slant is normally defined). Such an analysis passes over the possibility of systematic errors with a period of less than 360°, nor is it a very strong indicator of random errors (errors in precision) because these errors are not necessarily uniform over the entire range of tilt. This is precisely what we have found.

^{5}This is only true, of course, if the subject estimates relative motion to the object, and self-motion in an allocentric frame, accurately. There is converging evidence that self-motion is underestimated, but only by about 30%–40% in actively moving subjects (Wexler, in press). Therefore, the above argument still holds.

**p**

_{0}= (0,0,

*z*

_{0}) and fixates the origin and then moves to point

**p**=

*ρ*(sin

*θ*cos

*φ*,sin

*θ*sin

*φ*,cos

*θ*) while still fixating the origin, it rotates about an axis parallel to

**p**

_{0}×

**p**, although any other rotation would have done equally well (see below). In this law, rotations about the line of sight are neglected. The corresponding rotation matrix is Because

*L*is orthogonal,

*L*

^{−1}=

*L*

^{T}.

**p**, having rotated according to Listing’s law from point

**p**

_{0}. If

**r**is a point on the virtual stimulus (in the world frame), where is it in the eye’s frame (

**r**

_{e})? Because the eye’s frame is parallel to the world frame when the eye is on the

*z*-axis (i.e., the rotation matrix between them is the identity), we have

**r**

_{e}=

*L*(

**p**)(

**r**−

**p**).

**P**and fixates the origin. Define point

**R**so that it is at the same position in the eye frame in the immobile trial as point

**r**was in the active trial. In other words, the changes in the active condition must be the same as in the immob condition, in the eye’s frame: Equation 13 guarantees the same optic flow in immobile and active trials. Ideally, the observer in the immobile condition should not move, in which case

*L*(

**P**) would be the identity, but as we wanted to correct for any spurious motion in the immobile condition, we need a rotation matrix here, too. Equation 13 can be easily solved for

**R**, giving .

*a*

_{i,j}depend on 3D structure and motion (see Equations 10 and 11). One way of solving the SfM problem, suggested by Longuet-Higgins (1984), is by forming the matrix and solving its eigenvalue problem,

*Hv*

_{i}=

*λ*

_{i}

*v*

_{i}. If we order the eigenvectors so that λ

_{1}≤

*λ*

_{2}≤

*λ*

_{3}and normalize them so that |

*v*

_{1}|

^{2}= (

*λ*

_{3}−

*λ*

_{1})(

*λ*

_{2}−

*λ*

_{1}) and |

*v*

_{3}|

^{2}= (

*λ*

_{3}−

*λ*

_{1})(

*λ*

_{3}−

*λ*

_{2}), we have a simple expression for the plane’s normal:

**^n**=

*v*

_{1}+

*v*

_{3}(Longuet-Higgins, 1984).

*a*

_{2,i}will be much greater than for first derivatives, we can perturb the noise-free matrix (Equation 17) By and expand to first order in

*ɛ*. Using standard perturbation-theory techniques (e.g., see Courant & Hilbert, 1953), we find the estimated normal

**^n**

_{e}from the perturbed eigenvectors, and from it the estimated tilt

*τ*

_{e}= arctan

*n*

_{y}/

*n*

_{x}, obtaining where τ is the exact tilt, obtained from Equation 17, and η is the shear angle. Terms proportional to

*a*

_{2,1}disappear in the first-order correction to τ

_{e}. Thus, the first-order tilt error “gain” (i.e., the sensitivity of the tilt estimate to noise) is the term in parentheses in Equation 19. The first-order tilt error gain as a function of shear is shown in Figure 8 for slant σ = 30°, 45°, and 60°.

*a*

_{2,i}were drawn randomly from a Gaussian distribution with a SD of 0.03 centered around zero. The eigenvalues and eigenvectors of the resulting (perturbed) matrix were calculated, and the tilt error gain was calculated as above. Ten thousand iterations were performed for each slant-shear combination. The average tilt error gain, plotted as points in Figure 8, closely agrees with the analytic result (Equation 19).