Shape constancy and depth-order violations in structure from motion: a look at non-frontoparallel axes of rotation

Abstract

Humans can recover the structure of a 3D object from motion cues alone. Recovery of structure from motion (SFM) from the projected 2D motion field of a rotating object has been studied almost exclusively in one particular condition, that in which the axis of rotation lies in the frontoparallel plane. Here, we assess the ability of humans to recover SFM in the general case, where the axis of rotation may be slanted out of the frontoparallel plane. Using elliptical cylinders whose cross section was constant along the axis of rotation, we find that, across a range of parameters, subjects accurately matched the simulated shape of the cylinder regardless of how much the axis of rotation is inclined away from the frontoparallel plane. Yet, we also find that subjects do not perceive the inclination of the axis of rotation veridically. This combination of results violates a relationship between perceived angle of inclination and perceived shape that must hold if SFM is to be recovered from the instantaneous velocity field. The contradiction can be resolved if the angular speed of rotation is not consistently estimated from the instantaneous velocity field. This, in turn, predicts that variation in object size along the axis of rotation can cause depth-order violations along the line of sight. This prediction was verified using rotating circular cones as stimuli. Thus, as the axis of rotation changes its inclination, shape constancy is maintained through a trade-off. Humans perceive the structure of the object relative to a changing axis of rotation as unchanging by introducing an inconsistency between the perceived speed of rotation and the first-order optic flow. The observed depth-order violations are the cost of the trade-off.

Figure 1

Experiments 1 and 2. (a) Subjects viewed a motion-defined elliptical cylinder textured with random dots. (b) Experiment 1: Subjects adjusted a cylindrical cross section to match the profile of the motion-defined cylinder (a). (c) Experiment 2: Subjects adjusted the inclination of a bar to match the inclination of the motion-defined cylinder (a).

Figure 2

Recovered depth. (a) The ratio C_obs/C_sim between perceived and simulated curvature, averaged across the four subjects and across the different simulated curvatures, does not change with the simulated angle of inclination θ_sim. (b) C_obs/C_sim, averaged across the four subjects and across simulated angles, does depend on the simulated curvature. (c) C_obs/C_sim as a function of viewing condition (dot lifetime and presence of an occluder). In all panels, error bars are individual subject’s standard error of the mean averaged across subjects and conditions.

Figure 3

Effect of inclination of the axis of rotation. (a) Perceived inclination as a function of the simulated inclination. The relationship is linear, with perceived angle substantially lower than the simulated angle. (b) Ratio between perceived and simulated depth along the line of sight (d), as a function of the simulated inclination of the axis of rotation. In both panels, error bars are standard error of the mean.

Figure 4

Nonaffine structure. (a) Schema showing the nomenclature utilized. (b) The effect of changing λ for a constant inclination of the axis of rotation. A change in λ results in a depth-order reversal of the bottom half of the object, as all the distances from the object to the axis of rotation (dotted line) double.

Figure 5

Nonaffine structure. (a) Variation of λ with the simulated angle of inclination. Error bars are standard error of the mean. (b) Same as Figure 4b, but this time showing the difference between the perceived and simulated object. The shape of the perceived object can be made independent of the inclination of the axis of rotation by using an adequate nonunity value of λ. It can lead to depth-order violations, as shown here. By contrast, a value of λ = 1, that is, an affine structure, would preserve depth order but result in changes in d₁, d₂, and α

Figure 6

Depth-order violation. (a) Experimental and control stimuli used in depth-order violation experiments. γ is shown as negative in the experimental example and positive in the control example. Drawings are not to scale. (b) λ recovered from depth-order experiments (black bars) and subjects’ intrinsic bias (gray bars). Error bars are standard error of the mean.

Figure 7

(a) Bird’s-eye view of the setup of Domini and Braunstein (1998) for testing internal consistency of perceived shape. Four probe dots (P₁ to P₄) were located in a surface (also defined by moving dots) consisting of two planar patches of different slants (σ₁ and σ₂). The two patches were united by a smooth curved surface (not shown). The circle with the cross inside represents the axis of rotation. The distance between dots P₁ and P₂ is the same as that between P₃ and P₄. (b) Perceived configuration predicted by our model of the simulated structure shown in Panel a.

Figure A1

Viewing geometry. See text for details.

Figure A2

Viewing geometry. See text for details.

Figure A3

Viewing geometry. See text for details.

Figure A4

Viewing geometry. See text for details.

Figure A5

(a) Bird’s-eye view of the setup of Domini et al. (1998). Probe dots P₁ and P₂ belong to planar patches that have different slants (σ₁ and σ₂). The distance between P₁ and P₃ is the same as that between P₂ and P₃. (b) Perceived configuration predicted by our model of the simulated structure shown in Panel a.