In short, the percentile rank of image speeds in Figure 5 accurately predicts the anomalies of perceived speed that define the speed-distance effect. Discussion The ability to predict the psychophysical functions in Figures 6 and 7 supports the hypothesis that the basis for the speed-distance effect is accumulated experience with image speeds and their projected distances. When this evidence is combined with the success of an empirical framework in predicting other puzzling aspects of motion perception [2,3], as well as perceptions of brightness, color, and geometric form [9–11], a different concept of visual experience and its underlying mechanism emerges. Explaining motion perception empirically The rationale for this concept of perception stems from the fact that a moving stimulus cannot be directly linked to the speed and trajectory of a 3-D object (see Figure 1); thus, relying on retinal images alone to generate perceptions of motion could not elicit biologically useful visual behavior. Although counterintuitive, our results demonstrate that the motion we see is better understood in terms of the frequency of occurrence of a particular retinal stimulus relative to all other moving stimuli that have occurred in past experience. In this framework, perceived motion would be determined by linking retinal stimuli with moving objects according to the relative success of behavior over evolutionary and individual time. By accurately modeling the relationships between moving objects and the perspective projection of their corresponding images, the simulated environment served as a proxy for the relative success of visual behavior instantiated in visual circuitry. A corollary is that the motion percepts elicited from retinal stimuli correspond to neither the most likely physical speed and trajectory of an object, nor the properties of the stimulus itself, but to the locus (percentile rank) of a stimulus in accumulated past experience. In these terms, the discrepancies between the measured properties of a retinal stimulus sequence and perceived motion that define the speed-distance effect are simply a signature of an empirical strategy that evolved to contend with the inverse problem. The virtual environment Although there is at present no method for obtaining empirical information about the relationships between moving images and their possible sources in the real world, the simplicity of the virtual environment we used naturally raises questions about its adequacy as a proxy for human experience with moving objects (points in this case). For example, unlike natural objects, the objects in our simulated world are all the same, and key features such as object interactions, gravity, occlusion, and many other factors could not be incorporated (see Supporting Information S1). In addition, the model is noise-free, and does not include the stochastic variability that characterizes the biological generation of retinal images [12]. A further concern is the distribution of object directions and speeds employed in the simulation. Because there is at present no empirical information available about the distribution of object vectors in the real world, we randomly assigned vectors to 3-D objects from uniform distributions of direction and speed. Since objects in space can move in any direction, a uniform distribution of directions seemed a reasonable first approximation from which to obtain the cumulative probability distributions of the variety of natural motion stimuli that would have been encountered by observers (although gravity and other factors would of course affect the distribution in nature; see above). Similarly, because the Perceiving Motion Empirically distribution of speeds in the natural world is not known, we used a uniform distribution of 3-D speeds that generated projected speeds at approximately the speeds of retinal stimuli that elicit human motion percepts (0.1u/s–150u/s; see Materials and Methods). To test these assumptions about 3-D speed, we also generated data using an asymmetric normal speed distribution (mode = ,35 units/s), and a symmetric normal speed distribution (mode = ,75 units/s). Al- though it might seem that each distribution should result in markedly different image speed distributions (and therefore different predic- tions of the speed-distance effect), the projections arising from these different 3-D speed distributions were generally similar, all being skewed towards slower image speeds (see Figure S2). These additional data indicate that the distribution of 3-D speeds is not the basis for the data we collected. Given the ability to calculate image speed distributions from a priori assumptions about the motion of 3-D objects [13], the question arises whether the empirical data we extracted from the simulation could have been computed simply from geometrical principles alone. Because many different combinations of speeds and trajectories through space could produce the same distribution of projected speeds and distances, contending with the inverse problem (Figure 1) depends on associating 2-D images with 3-D objects over time. Since this information cannot be captured from a computed distribution of images, we adopted the more biologically relevant approach of simulating the experience of visual animals when linking 2-D projected stimuli with the perspective transformation of 3-D moving objects. This method therefore modeled more than simply the distribution of images that could be computed from 3-D objects; it also modeled the relationships between images and moving objects that the relative success of behavior would extract over time. Explaining the biases in the cumulative probability distributions Despite the limitations of the simulation, the predicted functions we derived are in good agreement with the psychophysical results. The reason is that the principles of perspective projection, which the simulation captures nearly perfectly, are the major determi- nant of the image-source statistics pertinent to perceived motion. The only potential concern in this regard is the design of the frustum, which necessarily affects the ability of objects to project images. The dimensions of frustum were therefore created to enable a range of projected speeds that mimic those normally experienced by humans (,0.1u/s–150u/s). Perspective projection also explains why the cumulative probability distributions of image-source relationships have the shape they do, and thus how they influence the percentile rank of motion stimuli (see Figures 5–7). The projected distances of moving stimuli in either the simulated or real world will always be equal to or shorter than the 3-D distances traveled by objects; in consequence a greater number of image sequences are experi- enced over smaller projected distances (see Figure 5). Furthermore, because perspective projection requires that the actual speeds of objects are always greater than or equal to their projected speeds, image speed distributions will always be weighted toward speeds slower than the range of physical speeds in 3-D space [12–15]; see also Figure S2). Together, these features of the projections of moving objects give rise to the non-linear biases apparent in the cumulative probabilities in Figure 5, which in turn determine the percentile rank of stimuli in past experience. Other models of motion processing The relationship between the physical movement of objects, their projected images, and perceived motion has long been a PLoS ONE | www.plosone.org 8 August 2009 | Volume 4 | Issue 8 | e6771