Figure 1.
The inverse optics problem and the speed of moving objects.
Due to perspective transformation, an infinite number of objects (black dots) at various distances and moving in different trajectories with different speeds (arrows) in 3-D space all generate the same 2-D image speed. Therefore, a moving image cannot specify the speeds of real-world sources (After Wojtach et al., 2008).
Figure 2.
The virtual environment and sampling templates.
(A) The frustum (red outline) embedded in a larger spherical space; moving 3-D objects in the frustum projected different speeds onto the image plane (blue outline), as indicated. (B) Example of the templates used to sample the image plane. For distances from 1° to 7° on the projection surface (2° in this example), image speeds from 0.1°/s to 150°/s were sampled by systematically moving the template (filled circles) to tile the entire image plane. This procedure was repeated for different orientations of the template at 30° increments, as indicated by the unfilled circles.
Figure 3.
Psychophysical determination of perceived motion.
Presentations of reference and test stimuli were separated by 400 ms. The distances traversed by the reference stimulus were 2°, 4°, or 6° of visual arc; the distances traversed by the test stimulus were 1°, 3°, 5°, or 7°. Observers adjusted the speed of the test stimulus, indicating when its speed appeared to be equal to the speed elicited by the reference stimulus in a random double staircase procedure (see Materials and Methods).
Figure 4.
Image-source relationships derived from the virtual environment.
(A) The physical speeds of 3-D objects generate a range of projected speeds over a given distance on the image plane (2° in the examples shown). As a consequence of perspective projection, relatively slow image speeds (e.g., 1°/s; solid line) are generated by objects moving at relatively slow physical speeds in 3-D space; somewhat faster image speeds (e.g., 5°/s; dashed line) arise from a wider distribution of objects in 3-D space with a larger range of physical speeds; relatively fast image speeds (e.g., 20°/s; dotted line) tend to be generated by an even wider distribution of objects moving at still greater physical speeds. Note that any given image speed can only be produced by objects moving with speeds equal to or greater than the stimulus, explaining the biased frequency distributions in (B). (B) The overall frequency distribution of image speeds generated by empirical sampling. The diamonds on the 2° projected distance function indicate the summed data from each of the three specific distributions in (A).
Figure 5.
Cumulative probability distributions derived from the analyses of objects moving in the simulated environment.
By transforming the frequency distribution of projected images obtained in the virtual environment (see Figure 4), the cumulative distributions order how often objects in 3-D space produced images of different speeds over different projected distances. These functions provide the basis for predicting the motion observed in psychophysical testing (see Figures 6 and 7).
Figure 6.
Predicting the psychophysical results elicited by image sequences traversing different distances on the image plane at different speeds but having the same percentile rank.
(A) A 2° reference stimulus with an image speed of 6.5°/s (dark blue arrowhead on the abscissa) has a percentile rank at the 76th percentile (black arrowhead on the ordinate). If our hypothesis of motion perception is correct, then test stimuli of 1°, 3°, 5°, and 7° with the same rank should generate perceptions of the same speed, despite their different actual speeds on the image plane (indicated by the other colored arrowheads along the abscissa). (B) The cumulative distribution data from (A) are re-plotted to indicate the predicted motion percepts elicited by the various test stimuli matched to a 2° reference stimulus as a function of the reference image speed. (C) Psychophysical functions produced by the 6 observers for test stimuli relative to the speed of the 2° reference stimulus (dashed blue line). The perceived speed reported for each test stimulus is plotted as a function of the image speed of the reference stimulus, as in (B). Bars indicate±1 standard error.
Figure 7.
Predicting the psychophysical results elicited by image sequences traversing different distances on the image plane at the same speed but having different percentile ranks.
(A) Stimuli moving across different distances on the image plane at a particular speed have different percentile ranks. For example, test stimuli traversing the image plane at a speed of 6.5°/s (black arrowhead on the abscissa) have ranks that range from the 65th to the 82nd percentile (colored arrowheads on the ordinate). If motion percepts are generated empirically, then the same image speed should be perceived as slower when traversing distances of 3°, 5°, or 7° in comparison with a 2° reference, but faster when traversing a test distance of 1°. (B) The blue curve indicates the projected speeds at which test stimuli traversing different projected distances (1°, 3°, 5°, and 7°) appeared the same to observers as an image speed traversing 2° at 6.5°/s (data re-plotted from Figure 6C). The area below the curve represents image speed-distance combinations perceived as slower than the reference stimulus, whereas the area above the curve represents combinations perceived as faster than the reference stimulus. Thus, test stimuli presented at 6.5°/s (dashed horizontal line) traversing distances of 3°, 5° or 7° (dashed vertical lines) are seen as moving more slowly than a 2° reference stimulus traveling at the same speed, whereas test stimuli traversing 1° at 6.5°/s are seen as moving faster.
Figure 8.
Bayesian posterior probability distributions for stimuli traversing projected distances of 1°–7° at 6.5°/s.
For each projected distance from 1°–7°, the probability of 3-D speeds that can give rise to an image speed of 6.5°/s are shown. Calculating the mean, median, or mode of each distribution results in the predicted percept for the specific projected distance. Similar distributions for the other image speeds tested (2.6°/s, 3.9°/s, 7.8°/s, and 10.4°/s) were generated, but are not shown.
Figure 9.
Bayesian predictions of the psychophysical results elicited by image sequences traversing different projected distances at the same speed.
To predict the relevant percepts, the mean of each posterior in Figure 8 was calculated (indicated by the corresponding colored points). In contrast to the observed psychophysical results, a Bayesian model predicts little or no change of perceived motion in response to a given image speed (2.6°/s, 3.9°/s, 6.5°/s, 7.8°/s, or 10.4°/s) traversing different projected distances (1°–7°) on the image plane (cf. Figure 7B; see also Figure 6C).