An Empirical Explanation of the Speed-Distance Effect William T. Wojtach*, Kyongje Sung, Dale Purves Center for Cognitive Neuroscience, Department of Neurobiology, Duke University, Durham, North Carolina, United States of America Abstract Understanding motion perception continues to be the subject of much debate, a central challenge being to account for why the speeds and directions seen accord with neither the physical movements of objects nor their projected movements on the retina. Here we investigate the varied perceptions of speed that occur when stimuli moving across the retina traverse different projected distances (the speed-distance effect). By analyzing a database of moving objects projected onto an image plane we show that this phenomenology can be quantitatively accounted for by the frequency of occurrence of image speeds generated by perspective transformation. These results indicate that speed-distance effects are determined empirically from accumulated past experience with the relationship between image speeds and moving objects. Citation: Wojtach WT, Sung K, Purves D (2009) An Empirical Explanation of the Speed-Distance Effect. PLoS ONE 4(8): e6771. doi:10.1371/journal.pone.0006771 Editor: Eric Warrant, Lund University, Sweden Received April 29, 2009; Accepted July 11, 2009; Published August 26, 2009 Copyright: 2009 Wojtach et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Funding: Duke University. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing Interests: The authors have declared that no competing interests exist. * E-mail: wtw3@duke.edu Introduction To succeed in their environments, humans and other visual animals must generate perceptual responses appropriate to the objects that give rise to sensory stimuli. As has long been recognized, however, the transformation of three-dimensional (3- D) space into two-dimensional (2-D) projections on the retina means that an image cannot uniquely specify its source, and therefore that visual perception cannot be determined simply by encoding the physical characteristics of retinal stimuli. This quandary is referred to as the inverse optics problem (Figure 1). Consequently, it has been appreciated for at least a century that the visual system must in some way use past experience to promote successful visual behavior [1]. Recent work on the flash-lag effect [2] and the aperture effect [3] has suggested that to contend with the inverse optics problem in the context of motion, the visual system has evolved to generate percepts empirically—i.e., according to the relative frequency of retinal images accumulated in past experience. In light of this evidence, we investigated the variable perception of speed that occurs when stimuli traverse different projected distances on the retina (the speed-distance effect). Thus, when two objects generate images with the same speed traversing different projected distances, the perception of speed elicited by such stimuli is different; conversely, when objects produce images with different speeds traversing different projected distances, the perception of speed can be the same. In keeping with earlier evidence [2,3], we hypothesized that this effect is determined by the frequency of occurrence of image speeds traversing different projected distances in accumulated experience. Perceiving motion in this way would allow observers to produce generally successful visual responses toward objects whose actual motions cannot be derived in any direct way from retinal stimuli alone (see Figure 1). To test this hypothesis, we used a computer-simulated environment that accurately represented the perspective transfor- mations between moving 3-D objects and their image speeds and directions, these data serving as a proxy for the link between images and their sources that would be extracted behaviorally over time [2]. In this way, we could determine the frequency with which 3-D objects generated different image speeds traversing a range of projected distances. If the speed-distance effect indeed arises from an empirical strategy of vision, then the relative occurrence of image speeds traveling over different projected distances on the image plane should accurately predict the responses of observers in psychophysical testing. Materials and Methods Ethics statement Informed written consent for the human psychophysical testing described below was obtained as required by the Duke University Institutional Review Board. The virtual environment To obtain data about the frequency of occurrence of images needed to predict the speed-distance effect, we situated a frustum in the center of a spherical virtual environment to approximate the process of retinal image formation (Figure 2A; see [2] for details). Space within the environment was defined in uniform arbitrary units. The image plane measured 50 units in both azimuth and elevation, and was positioned at a distance from the apex of the frustum such that one square degree of visual angle corresponded to one square unit on the projection surface, resulting in a 50u650u visual field. Since introducing objects directly into the frustum would bias their distribution, object movement was initiated outside the frustum within the uniform space of the spherical environment. Point objects were set in motion in this space, each with a direction vector and speed assigned randomly from uniform distributions (the real-world distribution of object speeds and directions are not known; see Discussion). While the use of point objects precluded an analysis of size-distance relationships, it preserved the association between speed and distance inherent in perspective projection. Omitting size-distance effects from the statistical PLoS ONE | www.plosone.org 1 August 2009 | Volume 4 | Issue 8 | e6771