Perceiving Motion Empirically 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 2u, 4u, or 6u of visual arc; the distances traversed by the test stimulus were 1u, 3u, 5u, or 7u. 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). doi:10.1371/journal.pone.0006771.g003 probabilities of their possible sources have different percentile ranks, the perception of their speeds should be different. We therefore assessed whether the cumulative probability distributions of image speeds in Figure 5 accurately predicted the psychophysical functions derived from observers viewing similar motion stimuli (see Figure 3). As described in Materials and Methods, observers compared the apparent speed generated by a reference stimulus with the apparent speed generated by a test stimulus traversing the different projected distances assessed in Figure 5; their task was to indicate when the speed elicited by the test stimulus appeared to match the speed elicited by the reference stimulus in a randomized double staircase procedure. We first examined the accuracy of the percepts predicted on this basis when the reference and test stimuli had different image speeds but the same percentile rank. Consider, for example, a reference stimulus traversing 2u on the image plane at a speed of 6.5u/s compared with test stimuli traversing distances 1u, 3u, 5u, or 7u (Figure 6A). When the frequency of occurrence of image speeds of test stimuli has the same percentile rank as the reference stimulus, the frequency of occurrence of physical sources generating these image speeds is, by definition, the same. Thus, as indicated by the vertical dashed lines in Figure 6A, when an object with a projected image speed of 6.5u/s traverses 2u on the retina, test stimuli that traverse projected distances of 3u, 5u, and 7u at the same percentile rank have always had image speeds that are progressively greater than 6.5u/s; conversely, a test stimulus with a projected distance of 1u at the same rank has always had an image speed less than 6.5u/s. If the hypothesis concerning the speed-distance effect is correct, then these empirical functions should predict the relative speeds seen by the observers. As shown in Figure 6B-C, the predictions made on this basis are in close agreement with the observed results. Thus, a reference stimulus traversing 2u at an image speed of 6.5u/s is correctly predicted to elicit the same apparent speed as stimuli with greater image speeds translating over distances of 3u, 5u, and 7u. Conversely, the same reference stimulus is correctly predicted to elicit the same apparent speed as a slower stimulus translating over 1u. Based on the sum of squared errors, the empirical functions explain.92% of the variance in the psychophysical data (1u test: 95.5%; 3u test: 95.1%; 5u test: 95.6%; 7u test: 92.5%). Similarly accurate predictions were made for reference stimuli of 4u and 6u (see Figure S1). The cumulative probability distributions in Figure 5 also predict the phenomenology elicited by stimuli having the same image speed but traversing different projected distances (Figure 7). Consider, for instance, the different percentile ranks generated by PLoS ONE | www.plosone.org 4 August 2009 | Volume 4 | Issue 8 | e6771