THREE-DIMENSIONAL BILATERAL SYMMETRY BIAS IN JUDGMENTS OF FIGURAL IDENTITY AND ORIENTATION

Abstract— The two experiments reported explored a bias toward symmetry in judging identity and orientation of indeterminate two-dimensional shapes. Subjects viewed symmetric and asymmetric filled, random polygons and described, "what each figure looks like" and its orientation. Viewers almost universally interpreted the shapes as silhouettes of bilaterally symmetric three-dimensional (3-D) objects. This assumption of 3-D symmetry tended to constrain perceived vantage of the identified objects such that symmetric shapes were interpreted as straight-on views, and asymmetric shapes as profile or oblique views. Because most salient objects in the world are bilaterally symmetric, these findings are consistent with the view that assuming 3-D symmetry can be a robust heuristic for constraining orientation when identifying objects from indeterminate patterns.     

Center of mass perception: Perturbation of symmetry

Bingham and Muchisky (1993) found that observers were very accurate in determining the location of the center of mass in planar objects. Systematic errors were affected primarily by object orientation, while random errors varied with the amount of symmetry. Radial and axial reflective symmetry affected errors in different ways. In the current study, we investigated the different effects of axial reflective versus rotational symmetry. All random errors decreased with increasing rotational symmetry. Axial reflective symmetry further reduced errors in the direction perpendicular to the axis. We replicated the effect on systematic error of orientation. However, we also found an effect of the perturbation of symmetry that suggested that observers used an approximation to symmetry. To investigate this possibility, we constructed a series of objects in which axial reflective symmetry was established and then perturbed by varying amounts. We found that systematic errors were structured by the underlying approximate symmetries, and we discuss the problem of quantifying symmetry.     

Illusory contours and spatial judgment

We investigated whether, in the human visual system, the mechanisms responsible for relative location judgments are the same when those judgments are made in the context of illusory contours and in the context of mentally joining two points. We asked subjects to align a dot with the oblique contour of an illusory surface or to align a dot with two markers at an oblique orientation. The systematic errors differed in direction for these two conditions. All the systematic errors were orientation dependent. The errors in aligning a dot with an illusory contour seem to be related to the asymmetrical shape of the single objects, which are able to induce an illusory contour, as well as figure-ground segregation.     

The role of axes of elongation and symmetry in rotated object naming

Many theorists have postulated that axes of elongation and/or symmetry play an important role in the recognition of objects. In this paper, evidence is presented that mitigates this claim from independent assessments of the effects of axes of elongation or symmetry on the time to name rotated line drawings of common objects. This conclusion was further supported in a stronger test in which both of these variables were orthogonally controlled, the aspect ratio of elongation was manipulated,and only objects that were completely geometrically symmetrical or asymmetrical were used. In all the experiments, objects were named for several blocks to determine the influence of these variables on effects of orientation with practice. Symmetry was found to diminish the effects of orientation after practice in naming the object set, and the effects of the most extreme orientation tested (120 degrees from upright) were diminished when both axes defined the same orientation, relative to when they defined different orientations. Contrary to many theories, these findings relegate the axes of symmetry and elongation to relatively minor roles during object identification.     

Haptic discrimination of bilateral symmetry in 2-dimensional and 3-dimensional unfamiliar displays

In five experiments, we tested the accuracy and sensitivity of the haptic system in detecting bilateral symmetry of raised-line shapes (Experiments 1 and 2) and unfamiliar 3-D objects (Experiments 3–5) under different time constraints and different modes of exploration. Touch was moderately accurate for detecting this property in raised displays. Experiment 1 showed that asymmetric judgments were systematically more accurate than were symmetric judgments with scanning by one finger. Experiment 2 confirmed the results of Experiment 1 but also showed that bimanual exploration facilitated processing of symmetric shapes without improving asymmetric detections. Bimanual exploration of 3-D objects was very accurate and significantly facilitated processing of symmetric objects under different time constraints (Experiment 3). Unimanual exploration did not differ from bimanual exploration (Experiment 4), but restricting hand movements to one enclosure reduced performance significantly (Experiment 5). Spatial reference information, signal detection measures, and hand movements in processing bilateral symmetry by touch are discussed.     

Visual and haptic discrimination of symmetry in unfamiliar displays extended in the z-axis

We investigated, in two experiments, the discrimination of bilateral symmetry in vision and touch using four sets of unfamiliar displays. They varied in complexity from 3 to 30 turns. Two sets were 2-D flat forms (raised-line shapes and raised surfaces) while the other two were 3-D objects constructed by extending the 2-D shapes in height (short and tall objects). Experiment 1 showed that visual accuracy was excellent but latencies increased for raised-line shapes compared with 3-D objects. Experiment 2 showed that unimanual exploration was more accurate for asymmetric than for symmetric judgments, but only for 2-D shapes and short objects. Bimanual exploration at the body midline facilitated the discrimination of symmetric shapes without changing performance with asymmetric ones. Accuracy for haptically explored symmetric stimuli improved as the stimuli were extended in the third dimension, while no such a trend appeared for asymmetric stimuli. Unlike vision, haptic response latency decreased for 2-D shapes compared with 3-D objects. The present results are relevant to the understanding of symmetry discrimination in vision and touch.     

Determining the orientation of depth-rotated familiar objects

How does the human visual system determine the depth-orientation of familiar objects? We examined reaction times and errors in the detection of 15° differences in the depth orientations of two simultaneously presented familiar objects, which were the same objects (Experiment 1) or different objects (Experiment 2). Detection of orientation differences was best for 0° (front) and 180° (back), while 45° and 135° yielded poorer results, and 90° (side) showed intermediate results, suggesting that the visual system is tuned for front, side and back orientations. We further found that those advantages are due to orientation-specific features such as horizontal linear contours and symmetry, since the 90° advantage was absent for objects with curvilinear contours, and asymmetric object diminished the 0° and 180° advantages. We conclude that the efficiency of visually determining object orientation is highly orientation-dependent, and object orientation may be perceived in favor of front-back axes.     

Perceiving shape from profiles

Four experiments related human perception of shape from profiles to current theoretical predictions. In Experiment 1, judgments of structure and motion were obtained for single- and dualellipsoid displays rotating about various axes. Ratings were highest when the axis of rotation was in the image plane and were influenced by the number of ellipsoids and the orientation of a single ellipsoid. The subsequent experiments explored the effect of orientation on shape judgments of a single ellipsoid. The results of Experiments 2 and 3 suggested that the effect of orientation found in Experiment 1 was not due to either the inability of certain orientations to be perceived as three-dimensional objects or to two-dimensional artifacts. It was thus argued that this effect of orientation was due to points of correspondence in relative motion that arise when the major axis is not perpendicular to the axis of rotation. In Experiment 4, subjects provided judgments of both shape and angular velocity. The elevated ellipsoids that were judged as larger were also judged as rotating more slowly. The inverse relationship between size and angular velocity is consistent with current theories. The connection between theory and data was further demonstrated by applying a shape-recovery algorithm to the stimuli used in Experiment 4 and finding a similar tradeoff between angular velocity and shape.     

Perceptual plane geometry: collinearity judgments probe the perceived orientation of an angle's sides

The task was to position a dot to lie one line length beyond the vertex of an angle by mentally extending one line segment forming the angle (and later the other line segment). Eight angles, variously oriented, provided judgmental errors attributable to the size of the subtended angle and line orientation. Collinearity errors are consistent with the hypothesis that the sizes of all subtended angles are underestimated. Horizontal-vertical assimilation describes the line-orientation effect. The largest error component is associated with the interaction of the two independent variables. Judgmental errors were modeled by a set of theoretically meaningful additive terms. Modeling was facilitated by symmetry assumptions (associated with odd and even mathematical functions) about the absolute magnitude and sign of judgmental errors that lead to the comparison of judgments derived from "interesting pairs" of stimuli.     

Can shape be perceived by dynamic touch?

The possibility that some aspects of the shapes of solid objects can be perceived through dynamic touch, even when the objects are not touched, but simply wielded with a handle, was investigated in four experiments. Wooden solids were constructed of three sizes and five shapes: hemisphere, cylinder, parallelepiped, cone, and pyramid. Experiments 1 and 2 involved comparisons (judgments of same or different) between and among wielded objects of the same mass. In Experiments 3 and 4, subjects were required to wield an object and to select a match from a visible arrangement of objects of the five shapes; the wielded objects were of two sizes, each different from that of the visible objects. The success of subjects at these tasks, and the patternings of errors, are seen to involve the characteristic moment of inertia profiles of each shape, and a ratio of the object's resistances to rotation around orthogonal axes is shown to be a strong predictor of performance in the identification experiments. The results are discussed with reference to dynamic touch and to the notion of shape invariants that do not reduce to aspects of object surface.     

Symmetry and elongation of objects influence perceived direction of translational motion

Five experiments were conducted to examine how perceived direction of motion is influenced by aspects of shape of a moving object such as symmetry and elongation. Random polygons moving obliquely were presented on a computer screen and perceived direction of motion was measured. Experiments 1 and 2 showed that a symmetric object moving off the axis of symmetry caused motion to be perceived as more aligned with the axis than it actually was. However, Experiment 3 showed that motion did not influence perceived orientation of symmetry axis. Experiment 4 revealed that symmetric shapes resulted in faster judgments on direction of motion than asymmetric shapes only when the motion is along the axis. Experiment 5 showed that elongation causes a bias in perceived direction of motion similar to effects of symmetry. Existence of such biases is consistent with the hypothesis that in the course of evolution, the visual system has been adapted to regularities of motion in the animate world.     

A polarity effect in misoriented object recognition: The role of polar features in the computation of orientation-invariant shape representations

This study investigated the contribution of polar features and internal shape axes to misoriented object recognition. A recognition memory paradigm was used to examine the effects of stimulus orientation on the recognition of previously memorized 2-D novel objects. In contrast to some recent reports, Experiment 1 showed that orientation-invariant performance can be found from the outset of testing with objects containing a salient axis of symmetry. In Experiments 2 and 3 it was found that the removal of a single salient polar feature, while preserving the axis of elongation, was sufficient to increase stimulus orientation time costs. This polarity effect suggests that polar features, and shape axes, play a role in the computation of orientation-invariant shape representations. It is proposed that shape axes facilitate the localization of polar features, which, in turn, are used to resolve the polarity of shape representations during recognition.     

Overestimation of the projected size of objects on the surface of mirrors and windows

Four experiments investigated judgments of the size of projections of objects on the glass surface of mirrors and windows. The authors tested different ways of explaining the task to overcome the difficulty that people had in understanding what the projection was, and they varied the distance of the observer and the object to the mirror or window and varied the size of the mirror. The authors compared estimations of projected size with estimations of the physical size of the object that produced the projection. For both mirrors and windows, observers accurately judged the physical size of objects but greatly overestimated the projected size of the same objects. Indeed, judgments of projected size were more similar to physical than to projected size. People were also questioned verbally about their knowledge of projected size relative to physical size. The errors produced for these conceptual questions were similar to those found in the perceptual estimation tasks. Together, these results suggest that projections of objects on mirrors and windows are treated in the same way and that observers cannot perceive such projections as distal objects.     

Coordinate frame for symmetry detection and object recognition

Can subjects voluntarily set an internal coordinate frame in such a way as to facilitate the detection of symmetry about an arbitrary axis? If so, is this internal coordinate frame the same as that involved in determining perceived top and bottom in object recognition and shape perception? Subjects were required to determine whether dot patterns were symmetric. Cuing the subjects in advance about the orientation of the axis of symmetry produced a substantial speedup in performance (Experiments 1 and 3) and an increase in accuracy with brief displays (Experiment 2). The effects appeared roughly additive, with an overall advantage for vertical symmetry; thus, the vertical axis effect is not due to a tendency to prepare for the vertical axis. The cuing advantage was found to depend upon the subject's knowing in advance the spatial location as well as orientation of the frame of reference (Experiment 4). The fifth experiment provided evidence that the frame of reference responsible for these effects is the same as the one that determines shape perception: Subjects viewed displays containing a letter (at an unpredictable orientation) and a dot pattern, rapidly naming the letter and then determining whether the dots were symmetric about a prespecified axis. When the top-bottom axis of the letter was oriented the same way as the axis of symmetry for the dots, symmetry judgments were significantly more accurate. Thus, the results suggest a single frame of reference for both types of judgment. The General Discussion proposes a theory of how visual symmetry may be computed, which might account for these phenomena and also characterize their relation to "mental rotation" effects.     

Utilization of static and kinetic information for depth by young Malaŵians

The utilization of static and kinetic information for depth by Mala?ian children and young adults in making monocular relative size judgments was investigated. Subjects viewed pairs of objects or photographic slides of the same pairs and judged which was the larger of each pair. The sizes and positions of the objects were manipulated such that the more distant object subtended a visual angle equal to, 80% of, or 70% of the nearer object. Motor parallax information was manipulated by allowing or preventing head movement. All subjects displayed sensitivity to static information for depth when the two objects subtended equal visual angles. When the more distant object was larger but subtended a smaller visual angle than the nearer object, subjects tended to base their judgments on retinal size. Motion parallax information increased accuracy of judgments of three-dimensional displays but reduced accuracy of judgments of pictorial displays. Comparisons are made between these results and those for American subjects.     

Regularity of symmetry verticality guides perceptual judgments of objects

Previous research indicated that most salient, real-world objects possess natural regularities that observers commonly assume in perceptual judgments of figural orientation and interpretation. Regularities include 3-dimensionality, bilateral symmetry, and the tendency for object tops to possess more salient information than bottoms. Thus, when observers interpret randomly shaped figures, they reliably impose volume, bilateral symmetry, and top and front orientation directions, even when figures are 2-dimensional and asymmetric. We confirmed generalizability for observers to assume these regularities with stimuli that vary in complexity, and we found evidence supporting another regularity, that of symmetry verticality (symmetry about a vertical axis). Findings support use of a family of perceptual heuristics corresponding to natural regularities that constrain stimulus indeterminacy and help guide judgment of object orientation and interpretation.     

Object pose: perceiving 3-d shape as sticks and slabs

Estimating the pose (three-dimensional orientation) of objects is an important aspect of 3-D shape perception. We studied the ability of observers to match the pose of the principal axes of an object with the pose of a cross consisting of three perpendicular axes. For objects, we used a long and a flat spheroid and eight symmetric objects with aspect ratios of dimensions of approximately 4:2:1. Stimulus cues were the contour and stereo for the spheroids, and contour, stereo, and shading for the symmetric objects. In addition, the spheroids were shown with or without surface texture and with or without active motion. Results show that observers can perform the task with standard deviations of a few degrees, though biases could be as large as 30 degrees. The results can be naturally decomposed in viewer-centered coordinates, and it turns out that the estimation of orientation in the frontoparallel plane (tilt) is more precise than estimation of orientation in depth (slant, roll). A comparison of long and flat spheroids shows that sticks lead to better performance than do slabs. This can even be the case within the same object; the pose of the stick-like aspect is seen with more precision than is the pose of the slab-like aspect. The largest biases occurred when the spheroids were displayed with the binocular contour as the only cue. We can explain these biases by assuming that subjects' settings are influenced by the orientation of the rim.     

The influence of visual symmetry on the encoding of objects

Two experiments examined how symmetry affects the visual encoding of simple flat objects (lamellae). Experiment 1 showed that subjects encode a lamella's shape as both a facsimile and its enantiomorph. In Experiment 2 the errors made when responding to symmetrical and asymmetrical stimuli showed that there were three factors affecting the reproduction of a symmetrical stimulus. These are, in order of decreasing influence, (i) the orientation of the axis of symmetry, (ii) the presence of overall symmetry (irrespective of orientation), and (iii) identity of the orientation of the elements of the response in relation to those of the stimulus. They also show that symmetry is likely to be reproduced even where the reproduction is not of the same shape as the original stimulus. The relevance of these results to representation of objects is discussed.     

Typicality in logically defined categories: Exemplar-similarity versus rule instantiation

A rule-instantiation model and a similarity-to-exemplars model were contrasted in terms of their predictions of typicality judgments and speeded classifications for members of logically defined categories. In Experiment 1, subjects learned a unidimensional rule based on the size of objects. It was assumed that items that maximally instantiated the rule were those farthest from the category boundary that separated small and large stimuli. In Experiment 2, subjects learned a disjunctive rule of the form "x or y or both". It was assumed that items that maximally instantiated the rule were those with both positive values (x and y). In both experiments, the frequency with which different exemplars were presented during classification learning was manipulated across conditions. These frequency manipulations exerted a major impact on subjects' postacquisition goodness-of-example judgments, and they also influenced reaction times in a speeded classification task. The results could not be predicted solely on the basis of the degree to which the rules were instantiated. The goodness judgments were predicted fairly well by a mixed exemplar model involving both relative-similarity and absolute-similarity components. It was concluded that even for logically defined concepts, stored exemplars may form a major component of the category representation.     

Layout geometry in the selection of intrinsic frames of reference from multiple viewpoints

Four experiments investigated the roles of layout geometry in the selection of intrinsic frames of reference in spatial memory. Participants learned the locations of objects in a room from 2 or 3 viewing perspectives. One view corresponded to the axis of bilateral symmetry of the layout, and the other view(s) was (were) nonorthogonal to the axis of bilateral symmetry. Judgments of relative direction using spatial memory were quicker for imagined headings parallel to the symmetric axis than for those parallel to the other viewing perspectives. This advantage disappeared when the symmetric axis was eliminated. Moreover, there was more consistency across participants in the selection of intrinsic axes when the layout contained an axis of bilateral symmetry than when it did not. These results indicate that the layout geometry affects the selection of intrinsic frames of reference supporting the intrinsic model of spatial memory proposed by W. Mou and T. P. McNamara (2002) and by A. L. Shelton and T. P. McNamara (2001).