Probability distribution of the directions of moving lines on the image plane in the absence of occlusion. (A) Projected lines in a given orientation (measured from the horizontal axis) arising from objects of different lengths and orientations, moving at different speeds in all possible 3D directions in the simulated environment. (B) The probability distribution of the directions of movement on the image plane determined by sampling all of the image sequences generated by objects indicated in A. Angles around the perimeter of the polar coordinate graph indicate the projected directions of movement; 0° in this and subsequent figures indicates the direction of motion perpendicular to the orientation of the projected line. The distance from the center to any point in the coordinate system is the probability of image sequences moving in that direction.

Empirical predictions compared with psychophysical results for moving lines in a vertical slit aperture. (A) The aperture. (B) Probability distributions of the directions of linear image sequences in different orientations constrained by a vertical slit. The red arrows are the empirical predictions; the green arrows are results of psychophysical testing. As the angle of orientation increases, an increasing bias away from vertical (arrows under each graph) is apparent in both the perceived and predicted directions [F(6, 36) = 15.41; mean square error = 17.00; P < 0.001; η_p² = 0.996].

Geometrical explanations of the biased directions generated by projection onto an image plane. (A) Because longer lines necessarily include shorter lines, the occurrence of projections that fill the aperture generated by the red line moving in the direction indicated by the black arrow on the left will be always be greater than the occurrence of projections generated by the blue line moving in another direction indicated by black arrow on the right. Thus, the most frequently occurring projected direction when linear objects move behind a circular aperture will always be the direction orthogonal to the orientation of the projected line (gray arrows). For other apertures, however, the direction of the shortest line will be different (see Fig. S5 for example). (B) A further bias arising from perspective. In this example, 2 physically different lines, x and y, can generate projected lines of the same (red) or different (red vs. blue) lengths on the image plane. Because the longer line (y) can, in a different 3D orientation, generate the same projected length (red) as the shorter line (x), projected lines of different lengths have different numbers of possible real-world sources. Moreover, the same linear source, z, can generate different projected lengths (red vs. blue) depending on the distance of line z from the image plane. Because of the shape of the frustum volume (or the visual field), more distant sources are available, leading to a bias in favor of shorter line projections. Both of these geometrical considerations lead to more short than long line projections, further influencing the distributions of the directions projected through apertures.

The inverse optics problem as it pertains to moving objects. Because of perspective transformation, linear objects of different sizes, at different distances, in different orientations, and moving at different speeds in 3D space (the gray, blue, and orange examples illustrated here) all generate the same sequence of projections on the image plane. Therefore, an image sequence cannot uniquely specify its source in 3D space.

Empirical predictions compared with psychophysical results for moving lines in a circular aperture. (A) The aperture. Line orientations (θ) from 30° to 60° were sampled in 5° increments; directions of movement in the polar coordinate graphs in B are indicated as positive or negative deviations from the direction normal to the line orientation (0°). (B) Probability distributions of the directions of linear image sequences in different orientations constrained by a circular aperture. The red arrows indicate the directions predicted by the empirical analysis; the green arrows indicate the directions reported in psychophysical testing. Repeated measures ANOVA on the observed directions showed no significant difference as a function of the line orientation [F(6, 36) = 1.88; mean square error = 6.36].

Empirical predictions compared with psychophysical results for moving lines in a triangular aperture. (A) Example of a right triangular aperture; note that to have the moving lines sequences fully fill the aperture in this instance the lengths of the 2 arms must vary as a function of orientation. (B) Probability distributions of 2D directions of line sequences constrained by a right triangular aperture. As before, the red arrows are the empirical predictions, and the green arrows are the results of psychophysical testing. Both the predicted and perceived directions differ systematically from the direction normal to the line orientation for orientations that are not 45°, increasing either positively or negatively for line orientations that deviate from this value. Repeated measures ANOVA again showed a significant difference in the observed directions as a function of the line orientation [F(6, 36) = 179.63; mean square error = 15.35; P < 0.001, η_p² = 0.997].