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.     

Haptic perception of spatial relations

There are some indications that haptic space like visual space is not Euclidean (e.g. Blumenfeld, 1937 Acta Psychologica 2 125-174). In a series of experiments, we investigated the haptic perception of spatial relations in a systematic way. We restricted ourselves to a horizontal plane at waist height. Blindfolded subjects were asked to perform three tasks with their right hand: (i) a reference bar was presented under four different orientations and subjects were asked to rotate a test bar such that it felt to be parallel to the reference bar; (ii) subjects had to rotate two test bars in such a way that they felt collinear; (iii) subjects had to point a test bar in the direction of a marker. Bars and marker could appear at nine different locations. In all experiments large systematic deviations (up to 40 degrees) were made. The deviations strongly correlated with horizontal (right-left) but not with vertical (forward-backward) distance. Subjects showed qualitatively identical trends but the size of the deviations was strongly subject-dependent. In addition, a significant haptic oblique effect was found. These results provide strong evidence that haptic space in non-Euclidean.     

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.     

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.     

Orientation tuning of shape from shading

Four experiments were performed to assess the effect of different orientations and direction of lighting on the visual processing of shaded or bipartite disks. In the first two experiments, observers were presented with nine different shading orientations from 0 degree to 180 degrees. Targets were detected in a rapid and parallel fashion for shaded disks when the orientation of the shading gradient was not horizontal (90 degrees) or oriented at 67.5 degrees. Search asymmetries favoring the detection of "pock" targets over "ball" targets were found for all orientations. The search rates for bipartite disks were similar to the shaded disks at 0 degree, 22.5 degrees, and 90 degrees but not for intermediate orientations, and no search asymmetries were found. These differences suggest that shaded displays and bipartite displays are processed by different underlying mechanisms. The third experiment showed that the direction of the light source (left or right) had no influence on search asymmetries around the 90 degrees point. Shading gradient orientation affected magnitude estimates of depth in the fourth experiment. These experiments show that the visual system's "assumption" of overhead lighting is broadly tuned.     

The Thatcher illusion: rotating the viewer instead of the picture

Faces are difficult to recognise when presented upside down. This effect of face inversion was effectively demonstrated with the 'Thatcher illusion' by Thompson (1980 Perception 9 483-484). It has been tacitly assumed that this effect is due to inversion relative to retinal coordinates. Here we tested whether it is due to egocentric (i.e. retinal) inversion or whether the orientation of the body with respect to gravity also influences the face-inversion effect. A 3-D human turntable was used to test subjects in 5 different body-tilt (roll) orientations: 0 degree, 45 degrees, 90 degrees, 135 degrees, and 180 degrees. The stimuli consisted of 4 'normal' and 4 'thatcherised' faces and were presented in 8 different orientations in the picture plane. The subjects had to decide in a yes-no task whether the faces were 'normal' or 'thatcherised'. Analysis of the d' values revealed a significant effect of stimulus orientation and body tilt. The significant effect of body tilt was due to a drop in d' values in the 135 degrees orientation. This result is compared to findings of studies on the subjective visual vertical, where larger errors occurred in body-tilt orientations between 90 degrees and 180 degrees. The present findings suggest that the face-inversion effect relies mainly on retinal coordinates, but that in head-down body-tilt orientations around 135 degrees the gravitational reference frame has a major influence on the perception of faces.     

Direct measurement of the curvature of visual space

We consider the horizontal plane at eye height, that is all objects seen at the horizon. Although this plane visually degenerates into a line in the visual field, the 'depth' dimension nevertheless gives it a two-dimensional structure. We address the problem of intrinsic curvature of this plane. The classical geometric method is based on Gauss's original definition: The angular excess in a triangle equals the integral curvature over the area of the triangle. Angles were directly measured by a novel method of exocentric pointing. Experiments were performed outside, in the natural environment, under natural viewing conditions. The observers were instructed not to move from a set location and to maintain eye height, but were otherwise free to perform eye, head, and body movements. We measured the angular excess for equilateral triangles with sides of 2-20 m, the vantage position at the barycenter. We found angular excesses and deficits of up to 30 degrees. From these data we constructed the metric. The curvature changes from elliptic in near space to hyperbolic in far space. At very large distances the plane becomes parabolic.     

Two reference frames for visual perception in two gravity conditions

The processing and storage of visual information concerning the orientation of objects in space is carried out in anisotropic reference frames in which all orientations are not treated equally. The perceptual anisotropies, and the implicit reference frames that they define, are evidenced by the observation of 'oblique effects' in which performance on a given perceptual task is better for horizontally and vertically oriented stimuli. The question remains how the preferred horizontal and vertical reference frames are defined. In these experiments cosmonaut subjects reproduced the remembered orientation of a visual stimulus in 1g (on the ground) and in 0g, both attached to a chair and while free-floating within the International Space Station. Results show that while the remembered orientation of a visual stimulus may be stored in a multimodal reference frame that includes gravity, an egocentric reference is sufficient to elicit the oblique effect when all gravitational and haptic cues are absent.     

The visual perception coordinate system uses axes defined by the earth, trunk, and vision

Eight young adults adjusted a line located on one side of a computer display parallel to internally specified Earth-fixed vertical (display in frontal plane), to the horizontal trunk-fixed anterior-posterior axis (display in horizontal plane), and to an oblique line (display in horizontal and vertical planes). All tasks were completed in a dark room with the head and trunk in both a standard erect posture and varied postures. Errors were lowest when setting the line to internally specified vertical in the frontal plane and to an oblique line in the horizontal plane when head and trunk orientations were varied. Constant errors for setting one line parallel to a second line were in opposite directions when the second line was located on the left versus right side of the display, but did not differ in direction when setting the line parallel to internally specified axes. Also, the oblique effect was preserved when the head and trunk were tilted to various orientations, suggesting that it results from integration of an internally specified gravitational reference with visual input. We conclude that the visual perceptual coordinate system uses internally specified vertical and, when available, a visually specified horizontal reference axis to define object orientation.     

The structure of frontoparallel haptic space is task dependent

In three experiments, we investigated the structure of frontoparallel haptic space. In the first experiment, we asked blindfolded participants to rotate a matching bar so that it felt parallel to the reference bar, the bars could be at various positions in the frontoparallel plane. Large systematic errors were observed, in which orientations that were perceived to be parallel were not physically parallel. In two subsequent experiments, we investigated the origin of these errors. In Experiment 2, we asked participants to verbally report the orientation of haptically presented bars. In this task, participants made errors that were considerably smaller than those made in Experiment 1. In Experiment 3, we asked participants to set bars in a verbally instructed orientation, and they also made errors significantly smaller than those observed in Experiment 1. The data suggest that the errors in the matching task originate from the transfer of the reference orientation to the matching-bar position.     

Multidimensional scaling in riemannian space

A metric which is a function of position is proposed for the analysis of the intrinsic geometry involved in preference or similarity judgments. Variation in the distance function or metric is characteristic of the Riemannian spaces and may be interpreted as curvature, stress or distortion in distance estimates and thus in the subjective perceptual space. It is possible to find the coefficients in the distance function at selected points by fitting a least-squares Riemannian surface to the Euclidean plane. The functional form of the distance can then be obtained by an application of the Laplace equation. Several examples are worked out for the two-dimensional solution but extension to higher spaces appears to be quite feasible.     

A Riemannian geometry theory of human movement: The geodesic synergy hypothesis

Mass-inertia loads on muscles change with posture and with changing mechanical interactions between the body and the environment. The nervous system must anticipate changing mass-inertia loads, especially during fast multi-joint coordinated movements. Riemannian geometry provides a mathematical framework for movement planning that takes these inertial interactions into account. To demonstrate this we introduce the controlled (vs. biomechanical) degrees of freedom of the body as the coordinate system for a configuration space with movements represented as trajectories. This space is not Euclidean. It is endowed at each point with a metric equal to the mass-inertia matrix of the body in that configuration. This warps the space to become Riemannian with curvature at each point determined by the differentials of the mass-inertia at that point. This curvature takes nonlinear mass-inertia interactions into account with lengths, velocities, accelerations and directions of movement trajectories all differing from those in Euclidean space. For newcomers to Riemannian geometry we develop the intuitive groundwork for a Riemannian field theory of human movement encompassing the entire body moving in gravity and in mechanical interaction with the environment. In particular we present a geodesic synergy hypothesis concerning planning of multi-joint coordinated movements to achieve goals with minimal muscular effort.     

Catching and matching bars with different orientations

The hypothesis that perception enslaves action is examined by assessing whether systematic distortions in perceptual judgments are reflected by inaccuracies in catching. In the first experiment, participants had to align manually the orientation of a reference bar placed at different distances in the frontoparallel plane. In the second experiment participants had to catch differently orientated moving bars, which became invisible at different distances from the interception point. In the matching experiment, systematic errors in the alignment of orientation were found in particular for oblique orientations, the magnitude of which increased with increasing distance of the reference bar. The inaccuracies in the final hand orientation during the catching task, however, did not mirror this pattern of deviations. The findings are interpreted to be more consistent with recent views that vision for perception (i.e., matching) and vision for action (i.e., catching) are dissociated than with the view that perception enslaves action.     

Effects of hand orientation and delay on the verbal judgment of haptically perceived orientation

We examined the haptic perception of orientations of a single bar throughout the horizontal plane using a verbal response: participants were to assign a number of minutes to the orientation of a bar defined with respect to the stimulus table. Performance was found to be systematically biased. Deviations were consistent with, yet much smaller than, those resulting from haptic motor matching tasks. The size and direction of the deviations were found to correlate with hand orientation, and not to depend on spatial location per se, suggesting a role for hand-centred reference frames in biasing performance. Delaying the response by 10 s led to a small improvement only of right-hand perceptions, indicating different hemispheric involvement in processes involved in retaining and/or recoding of haptic orientation information. Also the haptic oblique effect was found with the current verbal response. Importantly, it was affected neither by hand orientation nor by delay, suggesting that the oblique effect is independent of the aforementioned deviations in orientation perception.     

Experimental disorientation in the horizontal plane

Twenty-eight subjects were examined on a visual matching task for their ability to maintain an orientation with respect to a particular direction in the horizontal plane following a voluntary rotary body movement through 180 degrees. Each subject was examined with respect to eight different directions.

Numerous gross errors occurred when visual information was reduced to the display of an arrow indicating a direction and a second arrow manipulated by the subject. The magnitude and distribution of the errors suggest that, under the conditions of this experiment, visual information as to direction in the horizontal plane is analysed according to the two horizontal dimensions defined by the sagittal and coronal planes of the head. In correcting for the rotary body movement, failure may occur with respect to either or both of these two dimensions. The frequency of a failure to make any correction at all (i.e. 180-degree errors) is consistent with independent failure in each of the two horizontal dimensions.

Failure is markedly more frequent in the fore-aft dimension than in the left-right dimension. It is suggested that this may be explained in terms of the ambiguous spatial significance of vertical disposition on the retina and the possibility of contamination between the two systems of conceptual analysis which identify the vertical and the fore-aft dimensions of visual space.

It is demonstrated that when minimal “landmarks” are provided they tend to be utilized as reference points in attempts to maintain orientation, even when the subject is aware that the “landmarks” are misleading. Such a use of “landmarks” does not suppress the previously mentioned mechanism of dimensional orientation.

The relevance of these results to normal human orientation is discussed.     

The reproduction of vertical and oblique orientations in the visual, haptic, and somatovestibular systems

This study investigates whether the vertical orientation may be predominantly used as an amodal reference norm by the visual, haptic, and somato-vestibular perceptual systems to define oblique orientations. We examined this question by asking the same sighted adult subjects to reproduce, in the frontal (roll) plane, the vertical (0°) and six oblique orientations in three tasks involving different perceptual systems. In the visual task, the subjects adjusted a moveable rod so that it reproduced the orientation of a visual rod seen previously in a dark room. In the haptic task, the blindfolded sighted subjects scanned an oriented rod with one hand and reproduced its orientation, with the same hand, on a moveable response rod. In the somato-vestibular task, the blindfolded sighted subjects, sitting in a rotating chair, adjusted this chair in order to reproduce the tested orientation of their own body. The results showed that similar oblique effects (unsigned angular error difference between six oblique orientations and vertical orientation) were observed across the three tasks. However, there were no positive correlations between the visual, haptic,     

Perceived size and perceived direction: the interplay of the two descriptors of visual space

The stimulus in the outdoor psychophysical experiment was formed by two rods placed on the ground plane over a range of possible distances and orientations. Observers estimated its size and direction by positioning the third rod in the neighbouring space to form an evenly spaced collinear triple of rods. The data revealed interesting similarities between the profiles of the deviations in both judgments: for size judgments, the variability of the responses was least when the targets were at a frontal orientation and gradually increased as the orientation approached the medial plane. For direction judgments, on the other hand, the variability of the responses was least when the stimuli were aligned with the observer's line of sight and gradually increased as the orientation approached the frontoparallel plane. The finding of inverse relationship between the precision of size and direction judgments is interpreted as a consequence of the unequal precision in localisation between the frontal and in-depth dimensions of visual space. The question of the best parameterisation of the observers' responses is discussed.