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). doi:10.1371/journal.pone.0006771.g001 analysis was also in accord with the psychophysical stimuli we used, which did not include effects of distance on size (see below). When an object reached the boundary of the virtual environment a new object was randomly generated to take its place. The distribution of object speed ranged from 0.1 to 200 units per second, giving rise to image speeds within the boundaries of human motion perception (,0.1u to ,150u/s) [4]. In this way, we generated more than 2.4 million objects in the environment, approximately 624,000 of which entered the frustum volume and projected onto the image plane as proxies for motion stimuli; the other 1.7 million objects did not enter the frustum, and therefore did not create projections. Determining the frequency of occurrence of image speeds and distances The probability distributions of image speeds and projected distances were determined by systematically analyzing the entire Perceiving Motion Empirically image plane of the frustum at 30u increments over 360u with templates configured to correspond with the distances traversed by stimuli in the psychophysical testing (1u–7u; see Figure 2B). The projections were sampled at a resolution of 0.1 unit, or ,6 minutes of visual arc. The slight discrepancy in angular measurement on the image plane that occurred with eccentricity from the negative z-axis was accounted for by appropriate scaling of the templates (never more than 0.18 units). Because linearly constant motion in 3-D space does not produce linearly constant 2-D projections, average image speed between the sampling points of the template was calculated. Objects whose projections satisfied distances of 1u–7u and speeds of 0.1u/s–150u/s on the image plane were then compiled to represent the accumulated relationship between moving images and their sources. These procedures yielded approximately 598,000 valid samples that were then used to compute the relevant probability distributions reported in the Results. Although a combination of factors determined a valid sample, the parameters of the sampling templates were the primary constraint on the data collected from the image plane; other features of the simulation—e.g., the speed and trajectory of objects, the length of time an object took to travel through the frustum—affected the data only indirectly. Psychophysical testing Six adults (2 female; ages 18–69 years) with normal or corrected- to-normal vision participated in the psychophysical study (the authors and three participants na¨ıve to the purposes of the experiment). Ten sessions lasting ,30 min each were needed to complete the full range of testing. The perceptual functions were derived using a matching paradigm in which the speeds of two moving objects (the reference and test stimuli) were compared (see Figure 3). The stimuli were digitized using MATLAB 7.1 Psychophysics Toolbox [5] on a Dell Dimension E510 computer, and were presented as white disks (170 cd/m2) subtending 0.3u of visual arc on a black background. The moving objects appeared centrally on a Sony FD Trinitron 210 CRT monitor at a frame rate of 100 Hz in a darkened testing room; a 4-pixel central fixation point was present throughout each trial. Participants viewed the stimuli binocularly with their heads stabilized by a chin rest from a distance of 140 cm. The output file for each block of trials was analyzed using JMP statistical software (v.6.0, SAS Institute), and graphed using Microsoft Excel (v.11.1.1 for Macintosh, Microsoft Corp.). The psychophysical tests were given as a three-alternative, forced choice task, arranged in randomized double staircase format [6]. As illustrated in Figure 3, observers were first presented with a reference stimulus moving left-to-right, followed 400 ms later by a test stimulus, also moving left-to-right and centered on the fixation point. All pairs of reference and test stimulus distances were randomly presented. In any given trial, the reference stimulus traversed 2u, 4u, or 6u of visual arc at a constant speed of 2.6u/s, 3.9u/s, 6.5u/s, 7.8u/s, or 10.4u/s; the test stimulus traversed 1u, 3u, 5u, or 7u of visual arc at a constant speed that was initially either faster than or slower than the speed of the reference stimulus, depending on the staircase. Observers indicated by a keystroke whether the test stimulus generated a percept that appeared to move faster than, slower than, or the same as the speed elicited from the reference stimulus. Depending on the response, the speed of the test stimulus either decreased (for the ‘‘faster’’ response) or increased (for the ‘‘slower’’ response) in steps of 0.65u/s in the next presentation of that particular staircase. When two consecutive ‘‘same speed’’ responses were reported for each staircase, the image speed of the test stimulus was recorded. These values were averaged to derive a psychophys- ical function for each participant. The similarity of responses across PLoS ONE | www.plosone.org 2 August 2009 | Volume 4 | Issue 8 | e6771