Perceiving Motion Empirically 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 1u to 7u on the projection surface (2u in this example), image speeds from 0.1u/s to 150u/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 30u increments, as indicated by the unfilled circles. doi:10.1371/journal.pone.0006771.g002 participants allowed the data to be merged. It was apparent from these results that observers were responding to the speed elicited by stimuli, and not the duration of presentation. If the responses had been based on the duration of presentation, then two stimuli with the same duration (e.g., a slower moving reference stimulus traversing 2u of arc and a faster moving test stimulus traversing 5u of arc) would have generated equivalent percepts. In agreement with findings reported by McKee [7] and Orban et al. [8], such responses were not observed. Results Deriving probability distributions from the virtual environment To assess the hypothesis that the speed-distance effect is determined by the frequency of occurrence of image speeds over various projected distances, we first established how images traversing 1u–7u on the image plane were related to moving objects in the 3-D virtual environment. Since the environment accurately modeled perspective projection (see Figure 2A), objects traveling at different distances in depth but at the same speed generated projections that traversed different distances with different speeds on the image plane; similarly, objects traveling at different depths and speeds could project the same distance and speed on the image plane (see Figure 1). As an example of how we determined the relationship between ambiguous image sequences and their moving 3-D sources, consider the frequency of occurrence of image speeds traversing 2u on the image plane. By repeatedly sampling the full extent of the image plane with a 2u template (see Figure 2B), the distribution of object speeds and trajectories in 3-D space that could have produced image speeds traversing this distance was determined (Figure 4A). By compiling these data, we were able to determine how often 3-D sources generated projected speeds that traversed a distance of 2u on the image plane. Frequency distributions for projected distances from 1u–7u were acquired using this method (Figure 4B). Generating cumulative probability distributions The frequency distributions for projected distances of 1u–7u (Figure 4B) were then normalized and re-plotted as cumulative probability distributions (Figure 5). Ordering the data in this way indicated how often moving objects with different velocities in 3-D space underlie projected image speeds traversing different projected distances. These cumulative probability distributions thus provide a normalized empirical scale of image speeds, each point along a given distribution in Figure 5 showing the percentage of possible physical sources that generated projections equal in speed or slower than the stimulus in question. Predicting the speed-distance effect using percentile rank If the speed-distance effect is determined by accumulated experience, then the corresponding percepts should be predicted by the percentile rank (cumulative probability6100%) of any given image speed when presented as a visual stimulus. In this conceptual framework, the higher an image speed ranks on the scale of accumulated image-source relationships, the faster the motion perceived. This approach therefore predicts that when two moving objects generate images that translate different projected distances at different speeds but nonetheless have the same percentile rank, their perceived speeds should be the same. Conversely, when two objects generate the same rate of image translation over different projected distances, but the cumulative PLoS ONE | www.plosone.org 3 August 2009 | Volume 4 | Issue 8 | e6771