Effects of changes in size, speed, and distance on the perception of curved 3-D trajectories

Previous research on the perception of 3-D object motion has considered time to collision, time to passage, collision detection, and judgments of speed and direction of motion but has not directly studied the perception of the overall shape of the motion path. We examined the perception of the magnitude of curvature and sign of curvature of the motion path for objects moving at eye level in a horizontal plane parallel to the line of sight. We considered two sources of information for the perception of motion trajectories: changes in angular size and changes in angular speed. Three experiments examined judgments of relative curvature for objects moving at different distances. At the closest distance studied, accuracy was high with size information alone but near chance with speed information alone. At the greatest distance, accuracy with size information alone decreased sharply, but accuracy for displays with both size and speed information remained high. We found similar results in two experiments with judgments of sign of curvature. Accuracy was higher for displays with both size and speed information than with size information alone, even when the speed information was based on parallel projections and was not informative about sign of curvature. For both magnitude of curvature and sign of curvature judgments, information indicating that the trajectory was curved increased accuracy, even when this information was not directly relevant to the required judgment.     

Exocentric pointing to opposite targets

We use an exocentric pointing task to study exocentric visual directions to targets that are opposite to a pointer relative to the observer. (The apparent distance between the target and the pointer always exceeded 90 degrees of visual angle.) All pointing takes place in the horizontal plane at eye height. Observers could not see both target and pointer at a single glance. They had to look back and forth between them, using combinations of eye movements, head turns, twists at the waist and turning on the feet. In the limit of diametrically opposite targets we find that the observers pick either one of two distinct orientations of the pointer as equally "visually correct". Which one results depends on the stance assumed by the observer. The difference between the two equally acceptable pointings is between 5 degrees and 10 degrees. Such a result is predicted from earlier measurements in the context of a model that describes the geometry of the horizon as a Riemannian space with varying intrinsic curvature. The present results thus fit--perhaps surprisingly--very well in such a picture.     

Visual perception of collinearity

In a metrical space, there exists an intimate relation between collinearity and parallelity. In particular, in a Riemannian space collinearity is just a special case of parallelity. Is this true for visual space as well? We investigated the visual perception of collinearity by having subjects align two bars in the horizontal plane at eye height. The distances of the bars from the subject and the angles at which they were placed were varied. We found deviations of up to 22 degrees. The deviations of the left and right bars could be split into two independent components: namely, the sum and the difference of the deviations of the left and right bars. We found that the former depended only on the ratio between the distances of each bar from the subject, whereas the latter was largely independent of the positions of the bars. The difference in deviations corresponded to the deviation from parallelity. Compared with the results in the parallelity task (Cuijpers, Kappers, & Koenderink, 2000b), the deviations from parallel were much smaller. As a consequence, the results of the two experiments cannot be described by the same Riemannian geometry. This indicates that the intrinsic geometry of visual space differs across tasks. This is conceivable if the intrinsic geometry of visual space is operationally defined.     

Large-scale visual frontoparallels under full-cue conditions

We determined the curvature of apparent frontoparallels in a natural scene (a large lawn in broad daylight). Data on frontoparallels in these conditions are very sparse and reveal idiosyncratic curvatures of frontoparallels and irregular variation with distance. We used a method of bisection of linear segments indicated through pairs of stakes at angular separations (from the vantage point) of up to 120 deg. Distances of 2 m and 10 m (in the forward direction) were used. The bisection was carried out by the observer through maneuvering a radio-controlled vehicle carrying a third stake. Four observers participated in the experiment; they had no problems with the task and yielded mutually consistent results. We found that the frontoparallels are significantly curved and are concave towards the observer. Surprisingly, the sign of the curvature is opposite to that found when the frontoparallels are defined through an exocentric pointing task. Available theory (Luneburg's) does not predict this, but the theory is hardly applicable to the case of vision in natural scenes. This interesting discrepancy has not been reported before.     

Perceptual space in the dark affected by the intrinsic bias of the visual system

Correct judgment of egocentric/absolute distance in the intermediate distance range requires both the angular declination below the horizon and ground-surface information being represented accurately. This requirement can be met in the light environment but not in the dark, where the ground surface is invisible and hence cannot be represented accurately. We previously showed that a target in the dark is judged at the intersection of the projection line from the eye to the target that defines the angular declination below the horizon and an implicit surface. The implicit surface can be approximated as a slant surface with its far end slanted toward the frontoparallel plane. We hypothesize that the implicit slant surface reflects the intrinsic bias of the visual system and helps to define the perceptual space. Accordingly, we conducted two experiments in the dark to further elucidate the characteristics of the implicit slant surface. In the first experiment we measured the egocentric location of a dimly lit target on, or above, the ground, using the blind-walking-gesturing paradigm. Our results reveal that the judged target locations could be fitted by a line (surface), which indicates an intrinsic bias with a geographical slant of about 12.4 degrees. In the second experiment, with an exocentric/relative-distance task, we measured the judged ratio of aspect ratio of a fluorescent L-shaped target. Using trigonometric analysis, we found that the judged ratio of aspect ratio can be accounted for by assuming that the L-shaped target was perceived on an implicit slant surface with an average geographical slant of 14.4 degrees. That the data from the two experiments with different tasks can be fitted by implicit slant surfaces suggests that the intrinsic bias has a role in determining perceived space in the dark. The possible contribution of the intrinsic bias to representing the ground surface and its impact on space perception in the light environment are also discussed.     

An approach to geometry of visual space with no a priori mapping functions: Multidimensional mapping according to Riemannian metrics

A method of multidimensional mapping is described which constructs a configuration of points {P_i} in a Euclidean map of Riemannian space of constant curvature (hyperbolic, Euclidean, and elliptic) from the dissimilarity matrix (d_ij). The method was applied to the distance matrix in visual space where stimulus points Q_i were either small light points in the dark or small black points in the illuminated field surrounded by white curtains and d_ij represent scaled values of perceptual distances. Configuration of points {Q_i} were at intersections of parallel or distance alleys and horopters for the subject in the horizontal plane of the eye level. In contrast to the theoretical equations for {Q_i} by Luneburg and Blank, no a priori assumption on mapping functions between {Q_i} and {P_i} is necessary in this procedure to fit theoretical curves to {P_i} in the Euclidean map. The data were accounted for better by equations in the hyperbolic plane than by ones in the Euclidean plane. Discussions are made on robustness of Euclidean representation and on how to approach geometry of visual space as a dynamic entity under more natural conditions than the traditional frameless condition for alley and horopter experiments.     

Alleys in visual space

Geometry of frameless visual space is dealt with. First, parallel and equidistant alleys, horopters in the horizontal plane of eyes' level are discussed within the framework of the Luneburg's model that the frameless visual space is a Riemannian space of constant curvature. That basic postulate and the specific mapping functions assumed by Luneburg between the euclidean map of visual space and the physical space are kept separate, and efforts are directed to make the model applicable to more natural conditions of our visual space. A possibility is pointed out to remove the constraint “frameless” in the sense that perceptual geometrical properties are primarily determined by the convergence of optic axes. So far, only alleys in the horizontal plane extending from us toward infinity have been studied, but more often we perceive parallel lines, horizontal or vertical, in front of us like shelves of a bookcase. Hence, equations are derived for horopter plane appearing fronto-parallel in the three-dimensional visual space and alleys running horizontally or vertically on the horopter plane. It is shown that parallel and equidistant alleys are not the same in the horopter plane as in the horizontal plane, if the visual space is not euclidean. A method to evaluate the discrepancy between the two alleys without using any mapping functions is stated with some numerical examples.     

Distance estimation in vista space

Although distance estimation has been extensively studied in the laboratory, our ability to judge large distances in the field is not well researched. We challenge the notion that large distances are uniformly underestimated. We presented different targets to observers at distances ranging from 25 to 500 m to obtain egocentric distance judgments in natural environments. Three experiments showed that observers tend to underestimate distances below 75 m in a large open field, whereas they overestimate farther distances. Both the eye height of the observer and the size of the target also influenced distance estimation. We conclude that the notion of a uniform vista space has to be reconceived.     

Stereo disparity facilitates view generalization during shape recognition for solid multipart objects

Current theories of object recognition in human vision make different predictions about whether the recognition of complex, multipart objects should be influenced by shape information about surface depth orientation and curvature derived from stereo disparity. We examined this issue in five experiments using a recognition memory paradigm in which observers (N = 134) memorized and then discriminated sets of 3D novel objects at trained and untrained viewpoints under either mono or stereo viewing conditions. In order to explore the conditions under which stereo-defined shape information contributes to object recognition we systematically varied the difficulty of view generalization by increasing the angular disparity between trained and untrained views. In one series of experiments, objects were presented from either previously trained views or untrained views rotated (15°, 30°, or 60°) along the same plane. In separate experiments we examined whether view generalization effects interacted with the vertical or horizontal plane of object rotation across 40° viewpoint changes. The results showed robust viewpoint-dependent performance costs: Observers were more efficient in recognizing learned objects from trained than from untrained views, and recognition was worse for extrapolated than for interpolated untrained views. We also found that performance was enhanced by stereo viewing but only at larger angular disparities between trained and untrained views. These findings show that object recognition is not based solely on 2D image information but that it can be facilitated by shape information derived from stereo disparity.     

Saccades to targets in three-dimensional space: Dependence of saccadic latency on target location

The latency of saccadic movements to targets appearing at various positions in three-dimensional visual space was measured in four experiments. The first experiment confirmed that latencies of saccades to visual targets are greater in the lower visual field and showed that the increase is not influenced by the vertical starting position of the eye in the orbit, nor by a time gap between the fixation offset and the target onset. A hypothesis that this visual field difference was caused by a link between downward saccades and convergence movements was tested by recording saccade latencies when the targets were in a different depth plane from that of the original fixation. We did not find any direct support for the vergence involvement hypothesis, although the lower/upper visual field effect was shown to decrease consistently in monocular viewing. It was also shown that saccades to targets positioned in a different depth plane have longer latencies. In a final experiment, the visual field effect was shown to depend on the egocentric rather than the gravitational vertical.     

Do observers like curvature or do they dislike angularity?

Humans have a preference for curved over angular shapes, an effect noted by artists as well as scientists. It may be that people like smooth curves or that people dislike angles, or both. We investigated this phenomenon in four experiments. Using abstract shapes differing in type of contour (angular vs. curved) and complexity, Experiment 1 confirmed a preference for curvature not linked to perceived complexity. Experiment 2 tested whether the effect was modulated by distance. If angular shapes are associated with a threat, the effect may be stronger when they are presented within peripersonal space. This hypothesis was not supported. Experiment 3 tested whether preference for curves occurs when curved lines are compared to straight lines without angles. Sets of coloured lines (angular vs. curved vs. straight) were seen through a circular or square aperture. Curved lines were liked more than either angular or straight lines. Therefore, angles are not necessary to generate a preference for curved shapes. Finally, Experiment 4 used an implicit measure of preference, the manikin task, to measure approach/avoidance behaviour. Results did not confirm a pattern of avoidance for angularity but only a pattern of approach for curvature. Our experiments suggest that the threat association hypothesis cannot fully explain the curvature effect and that curved shapes are, per se, visually pleasant.     

Saccades to targets in three-dimensional space: dependence of saccadic latency on target location.

The latency of saccadic movements to targets appearing at various positions in three-dimensional visual space was measured in four experiments. The first experiment confirmed that latencies of saccades to visual targets are greater in the lower visual field and showed that the increase is not influenced by the vertical starting position of the eye in the orbit, nor by a time gap between the fixation offset and the target onset. A hypothesis that this visual field difference was caused by a link between downward saccades and convergence movements was tested by recording saccade latencies when the targets were in a different depth plane from that of the original fixation. We did not find any direct support for the vergence involvement hypothesis, although the lower/upper visual field effect was shown to decrease consistently in monocular viewing. It was also shown that saccades to targets positioned in a different depth plane have longer latencies. In a final experiment, the visual field effect was shown to depend on the egocentric rather than the gravitational vertical.     

The effects of stimulus size on bisection judgements in near and far space

Primate data suggest that near (peripersonal) and far (extrapersonal) space are coded within distinct representations. Support for this claim has been gained from human studies of line bisection, many of which have focused on neuropsychological, rather than normative, samples. One important aspect of these bisection studies has been to control for the changes in angular extent of stimuli that normally accompany changes in viewing distance. The control of angular information, however, requires alterations in the linear dimensions (actual stimulus size) of stimuli. We report two experiments in which normal subjects made manual bisection judgements on stimuli positioned in near or far space, and which were oriented in either the left-right (Experiment 1) or radial plane (Experiment 2). Both experiments were designed to enable the separable effects of linear and angular extent to be disentangled. Viewing distance effects were obtained when angular information was controlled, but many of these were dependent on changes in linear extent, and were only apparent at the individual subject level. Our data confirm that genuine near/far effects may be observed in normative bisection, but that many previous studies which appeared to support a near/far distinction in both normal and brain-damaged bisection behaviour may reflect a failure to control for changes in stimulus size.     

Discovery of the Pendulum and Spring Dynamics in the Early Stages of Walking

The authors investigated the self-selected, overground walking patterns of 7 children (aged 11 months to 1 year, 5 months) at the initiation of walking (brand-new walkers [BNWs]) and for the next 6 months at 1-month intervals. Walking speed, stride length, and stride frequency increased significantly between the first 2 visits without significant changes in height and weight. The authors calculated sagittal plane angular accelerations of the center of mass over the foot for each step as an indicator of the escapement pulse. Results for the acceleration profiles changed after the 1st visit to positive, single-peaked accelerations that occurred < 0.20 s after initial foot contact. Increases in sagittal plane hip angular displacement and decreases in frontal plane pelvic angular displacement were observed. The pattern changes suggest that children quickly discover appropriately timed and directed escapements that initiate and support the conservative sagittal plane pendulum and spring dynamics observed in older children.     

Discovery of the pendulum and spring dynamics in the early stages of walking

The authors investigated the self-selected, overground walking patterns of 7 children (aged 11 months to 1 year, 5 months) at the initiation of walking (brand-new walkers [BNWs]) and for the next 6 months at 1-month intervals. Walking speed, stride length, and stride frequency increased significantly between the first 2 visits without significant changes in height and weight. The authors calculated sagittal plane angular accelerations of the center of mass over the foot for each step as an indicator of the escapement pulse. Results for the acceleration profiles changed after the 1st visit to positive, single-peaked accelerations that occurred < 0.20 s after initial foot contact. Increases in sagittal plane hip angular displacement and decreases in frontal plane pelvic angular displacement were observed. The pattern changes suggest that children quickly discover appropriately timed and directed escapements that initiate and support the conservative sagittal plane pendulum and spring dynamics observed in older children.     

Infants' localization of sounds in the median vertical plane: estimates of minimum audible angle

Infants 6, 9, 12, 15, and 18 months of age were seated in a dark room directly facing an array of nine loudspeakers positioned along the median vertical plane. One loudspeaker was positioned at ear level, 0 degree, and four others each were positioned above and below 0 degree. To examine infants' resolution of auditory space in the median vertical plane we sought to determine the smallest angular shift in the vertical location of a sound that infants could reliably detect (i.e., minimum audible angle). A two-alternative forced-choice procedure was used in which a sequence of white noise bursts was presented initially at 0 degree, and then shifted vertically (i.e., above or below 0 degree) and continued to be presented until the infant made a directional response; correct responses were visually reinforced. The smallest angular shift in vertical location that was reliably detected systematically decreased with increasing age between 6 months (15 degrees) and 18 months (4 degrees), suggesting a finer partitioning of auditory space along the vertical axis over this age range. By 18 months infants' performance matched that of a group of adults tested under the same circumstances.     

The effects of attenuation of frequency segments on binaural localization of sound

Perceived location of tonal stimuli d narrow noise bands presented in two-dimensional space varies in an orderly manner with changes in stimulus frequency. Hence, frequency has a referent in space that is most apparent during monaural listening. The assumption underlying the present study is that maximum sound pressure level measured at the ear canal entrance for the various frequencies serves as a prominent spectral cue for their spatial referents. Even in binaural localization, location judgments in the vertical plane are strongly influenced by spatial referents. We measured sound pressure levels at the left ear canal entrance for 1.0-kHz-wide noise bands, centered from 4.0 kHz through 10.0 kHz, presented at locations from 60° through ?45° in the vertical plane; the horizontal plane coordinate was fixed at ?90°. On the basis of these measurements, we fabricated three different band-stop stimuli in which differently centered 2.0-kHz-wide frequency segments were filtered from a broadband noise. Unfiltered broadband noise served as the remaining stimulus. Localization accuracy differed significantly among stimulus conditions (p<.01). Where in the vertical plane most errors were made depended on which frequency segment was filtered from the broadband noise.     

Systematic angular biases in the representation of visual space

Representing spatial information is one of our most foundational abilities. Yet in the present work we find that even the simplest possible spatial tasks reveal surprising, systematic misrepresentations of space—such as biases wherein objects are perceived and remembered as being nearer to the centers of their surrounding quadrants. We employed both a placement task (in which observers see two differently sized shapes, one of which has a dot in it, and then must place a second dot in the other shape so that their relative locations are equated) and a matching task (in which observers see two dots, each inside a separate shape, and must simply report whether their relative locations are matched). Some of the resulting biases were shape specific. For example, when dots appeared in a triangle during the placement task, the dots placed by observers were biased away from certain parts of the symmetry axes. But other systematic biases were not shape specific, and seemed instead to reflect differences in the grain of resolution for different regions of space. For example, with both a circle and even a shapeless configuration (with only a central landmark) in the matching task, observers were better at discriminating angular differences (when a dot changed positions around the circle, as opposed to inward/outward changes) in cardinal versus oblique sectors. These data reveal a powerful angular spatial bias, and highlight how the resolution of spatial representation differs for different regions and dimensions of space itself.

     

A distance judgment function based on space perception mechanisms: revisiting Gilinsky's (1951) equation

In her seminal article in Psychological Review, A. S. Gilinsky (1951) successfully described the relationship between physical distance (D) and perceived distance (d) with the equation d = DA/(A + D), where A = constant. To understand its theoretical underpinning, the authors of the current article capitalized on space perception mechanisms based on the ground surface to derive the distance equation d = Hcosalpha/sin(alpha + eta), where H is the observer's eye height, alpha is the angular declination below the horizon, and eta is the slant error in representing the ground surface. Their equation predicts that (a) perceived distance is affected by the slant error in representing the ground surface; (b) when the slant error is small, the ground-based equation takes the same form as Gilinsky's equation; and (c) the parameter A in Gilinsky's equation represents the ratio of the observer's eye height to the sine of the slant error. These predictions were empirically confirmed, thus bestowing a theoretical foundation on Gilinsky's equation.