What does the occluding contour tell us about solid shape?

A new theorem is discussed that relates the apparent curvature of the occluding contour of a visual shape to the intrinsic curvature of the surface and the radial curvature. This theorem allows the formulation of general laws for the apparent curvature, independent of viewing distance and regardless of the fact that the rim (the boundary between the visible and invisible parts of the object) is a general, thus twisted, space curve. Consequently convexities, concavities, or inflextions of contours in the retinal image allow the observer to draw inferences about local surface geometry with certainty. These results appear to be counterintuitive, witness to the treatment of the problem by recent authors. It is demonstrated how well-known examples, used to show how concavities and convexities of the contour have no obvious relation to solid shape, are actually good illustrations of the fact that convexities are due to local ovoid shapes, concavities to local saddle shapes.     

The role of constant curvature in 2-D contour shape representations

In early cortex, visual information is encoded by retinotopic orientation-selective units. Higher-level representations of abstract properties, such as shape, require encodings that are invariant to changes in size, position, and orientation. Within the domain of open, 2-D contours, we consider how an economical representation that supports viewpoint-invariant shape comparisons can be derived from early encodings. We explore the idea that 2-D contour shapes are encoded as joined segments of constant curvature. We report three experiments in which participants compared sequentially presented 2-D contour shapes comprised of constant curvature (CC) or non-constant curvature (NCC) segments. We show that, when shapes are compared across viewpoint or for a retention interval of 1000 ms, performance is better for CC shapes. Similar recognition performance is observed for both shape types, however, if they are compared at the same viewpoint and the retention interval is reduced to 500 ms. These findings are consistent with a symbolic encoding of 2-D contour shapes into CC parts when the retention intervals over which shapes must be stored exceed the duration of initial, transient, visual representations.     

Recovery of 3-D shape from deforming contours

Three experiments were conducted to examine the accuracy of 3-D shape recovery from deforming-contour displays. The displays simulated silhouettes of ellipsoids rotating about a vertical axis. Subjects judged the horizontal cross-section of the ellipsoids. The shape of the ellipsoid, the position of the axis of rotation, and the type of projection were manipulated in Experiment 1. The results indicated relatively accurate shape recovery when the major axis of the ellipsoid was small. In Experiment 2, the shape of the ellipsoid and the velocity and curvature of the contour were manipulated. When the rate of deformation of curvature was decreased, more eccentric shapes were reported. In Experiment 3, the shape of the object and the amount of simulated rotation were manipulated. Subjects made both shape and extent of rotation judgments. The results showed that eccentricity of shape responses could be accurately predicted from rotation responses, suggesting that the recovery of 3-D shape from smooth, deforming contours is dependent on the perceived extent of rotation.     

Haptic curvature comparison of convex and concave shapes

A sculpture and the mould in which it was formed are typical examples of objects with an identical, but opponent, surface shape: each convex (ie outward pointing) surface part of a sculpture has a concave counterpart in the mould. The question arises whether the object features of opponent shapes can be compared by touch. Therefore, we investigated whether human observers were able to discriminate the curvatures of convex and concave shapes, irrespective of whether the shape was convex or concave. Using a 2AFC procedure, subjects had to compare the curvature of a convex shape to the curvature of a concave shape. In addition, results were also obtained for congruent shapes, when the curvature of either only convex shapes or only concave shapes had to be compared. Psychometric curves were fitted to the data to obtain threshold and bias results. When subjects explored the stimuli with a single index finger, significantly higher thresholds were obtained for the opponent shapes than for the congruent shapes. However, when the stimuli were touched by two index fingers, one finger per surface, we found similar thresholds. Systematic biases were found when the curvature of opponent shapes was compared: the curvature of a more curved convex surface was judged equal to the curvature of a less curved concave surface. We conclude that human observers had the ability to compare the curvature of shapes with an opposite direction, but that their performance decreased when they sensed the opponent surfaces with the same finger. Moreover, they systematically underestimated the curvature of convex shapes compared to the curvature of concave shapes.     

Information along contours and object boundaries

F. Attneave (1954) famously suggested that information along visual contours is concentrated in regions of high magnitude of curvature, rather than being distributed uniformly along the contour. Here the authors give a formal derivation of this claim, yielding an exact expression for information, in C. Shannon's (1948) sense, as a function of contour curvature. Moreover, they extend Attneave's claim to incorporate the role of sign of curvature, not just magnitude of curvature. In particular, the authors show that for closed contours, such as object boundaries, segments of negative curvature (i.e., concave segments) literally carry greater information than do corresponding regions of positive curvature (i.e., convex segments). The psychological validity of this informational analysis is supported by a host of empirical findings demonstrating the asymmetric way in which the visual system treats regions of positive and negative curvature.     

Curvature and the visual perception of shape: theory on information along object boundaries and the minima rule revisited

Previous empirical studies have shown that information along visual contours is known to be concentrated in regions of high magnitude of curvature, and, for closed contours, segments of negative curvature (i.e., concave segments) carry greater perceptual relevance than corresponding regions of positive curvature (i.e., convex segments). Lately, Feldman and Singh (2005, Psychological Review, 112, 243-252) proposed a mathematical derivation to yield information content as a function of curvature along a contour. Here, we highlight several fundamental errors in their derivation and in its associated implementation, which are problematic in both mathematical and psychological senses. Instead, we propose an alternative mathematical formulation for information measure of contour curvature that addresses these issues. Additionally, unlike in previous work, we extend this approach to 3-dimensional (3D) shape by providing a formal measure of information content for surface curvature and outline a modified version of the minima rule relating to part segmentation using curvature in 3D shape.     

Detection of convexity and concavity in context

Sensitivity to shape changes was measured, in particular detection of convexity and concavity changes. The available data are contradictory. The author used a change detection task and simple polygons to systematically manipulate convexity/concavity. Performance was high for detecting a change of sign (a new concave vertex along a convex contour or a new convex vertex along a concave contour). Other things being equal, there was no evidence of an advantage for detecting a new concavity compared with a new convexity, for detecting a change of angle to a concave vertex compared with a convex vertex, for detecting a change within a concave region compared with a change within a convex region, or for an interaction between convexity and concavity and changes affecting or not affecting a vertex. The author concludes that change detection is affected by changes of sign of curvature (leading to changes in part structure). However, contrary to previous proposals, there is no special role for negative curvature or minima of curvature in guiding attention.     

The structure of three-dimensional object representations in human vision: evidence from whole-part matching

This article examines how the human visual system represents the shapes of 3-dimensional (3D) objects. One long-standing hypothesis is that object shapes are represented in terms of volumetric component parts and their spatial configuration. This hypothesis is examined in 3 experiments using a whole-part matching paradigm in which participants match object parts to whole novel 3D object shapes. Experiments 1 and 2, consistent with volumetric image segmentation, show that whole-part matching is faster for volumetric component parts than for either open or closed nonvolumetric regions of edge contour. However, the results of Experiment 3 show that an equivalent advantage is found for bounded regions of edge contour that correspond to object surfaces. The results are interpreted in terms of a surface-based model of 3D shape representation, which proposes edge-bounded 2-dimensional polygons as basic primitives of surface shape.     

Surface contours, Glass patterns, and a slant illusion

A surface contour pattern constructed from continuous sine waves is subject to several visual interpretations, whereby the separate regions containing the maxima and the minima of the sine waves may be seen as representing either convex or concave areas of a three-dimensional surface. In a pattern of segments of contours comprising only the regions containing the maxima and minima of the sine waves, a set of surfaces is perceived, each of which tends to be seen as convex, and which possesses an illusory slant which is different for columns of contour segments containing maxima as compared with columns containing minima. It is conjectured that the slant illusion is a manifestation of the processes by which depth is derived from surface contour information. It is demonstrated that corresponding figures constructed from sinusoidal Glass patterns produce similar effects. From this it is concluded that the structure of Glass patterns provides a sufficient input representation for the processes by which surface shape is recovered from surface contours.     

Early computation of contour curvature and part structure: evidence from holes

We used holes to study unilateral border ownership and in particular the information carried by the sign of the curvature along the contour (ie the difference between convex and concave regions). When people perceive a hole, its shape has a reversed curvature polarity (ie a changed sign of curvature) compared to the same region perceived as an object. Bertamini (2001 Perception 30 1295-1310), and Bertamini and Croucher (2003 Cognition 87 33-54) suggested and found evidence to support the hypothesis that, because convex regions are perceived as parts, positional information is more readily available for convex regions. Therefore a change is predicted when a given region is perceived as either a hole or a figure. We confirm that finding in this study, using holes defined by binocular disparity. We conclude that a change from figure to hole always reverses the encoding of curvature polarity. In turn, polarity obligatorily affects perceived part structure and the processing of position.     

The role of convexity in perception of symmetry and in visual short-term memory

Visual perception of shape is affected by coding of local convexities and concavities. For instance, a recent study reported that deviations from symmetry carried by convexities were easier to detect than deviations carried by concavities. We removed some confounds and extended this work from a detection of reflection of a contour (i.e., bilateral symmetry), to a detection of repetition of a contour (i.e., translational symmetry). We tested whether any convexity advantage is specific to bilateral symmetry in a two-interval (Experiment 1) and a single-interval (Experiment 2) detection task. In both, we found a convexity advantage only for repetition. When we removed the need to choose which region of the contour to monitor (Experiment 3) the effect disappeared. In a second series of studies, we again used shapes with multiple convex or concave features. Participants performed a change detection task in which only one of the features could change. We did not find any evidence that convexities are special in visual short-term memory, when the to-be-remembered features only changed shape (Experiment 4), when they changed shape and changed from concave to convex and vice versa (Experiment 5), or when these conditions were mixed (Experiment 6). We did find a small advantage for coding convexity as well as concavity over an isolated (and thus ambiguous) contour. The latter is consistent with the known effect of closure on processing of shape. We conclude that convexity plays a role in many perceptual tasks but that it does not have a basic encoding advantage over concavity.     

Modal and amodal completion generate different shapes

Mechanisms of contour completion are critical for computing visual surface structure in the face of occlusion. Theories of visual completion posit that mechanisms of contour interpolation operate independently of whether the completion is modal or amodal--thereby generating identical shapes in the two cases. This identity hypothesis was tested in two experiments using a configuration of two overlapping objects and a modified Kanizsa configuration. Participants adjusted the shape of a comparison display in order to match the shape of perceived interpolated contours in a standard completion display. Results revealed large and systematic shape differences between modal and amodal contours in both configurations. Participants perceived amodal (i.e., partly occluded) contours to be systematically more angular--that is, closer to a corner--than corresponding modal (i.e., illusory) contours. The results falsify the identity hypothesis in its current form: Corresponding modal and amodal contours can have different shapes, and, therefore, mechanisms of contour interpolation cannot be independent of completion type.     

Infants' perception of information along object boundaries: concavities versus convexities

Object parts are signaled by concave discontinuities in shape contours. In seven experiments, we examined whether 5- and 6 1/2-month-olds are sensitive to concavities as special aspects of contours. Infants of both ages detected discrepant concave elements amid convex distractors but failed to discriminate convex elements among concave distractors. This discrimination asymmetry is analogous to the finding that concave targets among convex distractors pop out for adults, whereas convex targets among concave distractors do not. Thus, during infancy, as during adulthood, concavities appear to be salient regions of shape contours. The current study also found that infants' detection of concavity is impaired if the contours that define concavity and convexity are not part of closed shapes. Thus, for infants, as for adults, concavities and convexities are defined more readily in the contours of closed shapes. Taken together, the results suggest that some basic aspects of part perception from shape contours are available by at least 5 months of age.     

Intrinsic orientation and study viewpoint in recognizing spatial structure of a shape

Two experiments dissociated the roles of intrinsic orientation of a shape and participants’ study viewpoint in shape recognition. In Experiment 1, participants learned shapes with a rectangular background that was oriented differently from their viewpoint, and then recognized target shapes, which were created by splitting study shapes along different intrinsic axes, at different views. Results showed that recognition was quicker when the study shapes were split along the axis parallel to the orientation of the rectangular background than when they were split along the axis parallel to participants’ viewpoint. In Experiment 2, participants learned shapes without the rectangular background. The results showed that recognition was quicker when the study shape was split along the axis parallel to participants’ viewpoint. In both experiments, recognition was quicker at the study view than at a novel view. An intrinsic model of object representation and recognition was proposed to explain these findings.     

The absence of depth constancy in contour stereograms

Stereoscopic surfaces constructed from Kanizsa-type illusory contours or explicit luminance contours were tested for three-dimensional (3-D) shape constancy. The curvature of the contours and the apparent viewing distance between the surface and the observer were manipulated. Observers judged which of two surfaces appeared more curved. Experiment 1 allowed eye movements and revealed a bias in 3-D shape judgment with changes in apparent viewing distance, such that surfaces presented far from the observer appeared less curved than surfaces presented close to the observer. The lack of depth constancy was approximately the same for illusory-contour surfaces and for explicit-contour surfaces. Experiment 2 showed that depth constancy for explicit-contour surfaces improved slightly when fixation was required and eye movements were restricted. These experiments suggest that curvature in depth is misperceived, and that illusory-contour surfaces are particularly sensitive to this distortion.     

Investigating local and global effects of surface colours and contours in amodal completion

ABSTRACT

We studied interpretations of partly occluded shapes. Models that account for amodal completion mostly deal with local and global contour characteristics. In the current study, we were interested in the effects of colour on local and global contour completions. In our stimuli, local contour completions comprised simple linear extensions of the partly occluded contours, whereas global contour completions accounted for global shape regularities. Our stimuli were designed such that the visible surface colour could also be completed in a local or global fashion, being consistent or inconsistent with contour completions. We tested the preferred interpretations of the partly occluded shapes by using a sequential matching task. Participants had to judge whether a test shape could be a previously shown partly occluded shape. We found that interpretations of partly occluded shapes depend on both colour and contour characteristics. Additional time bin analyses revealed that for fast responses colour and contour completions already depend on the visible context of the partly occluded shapes, while for slow responses the congruency between colour and contour completions play a role as well.     

The effect of contour closure on shape recognition

Recent research on the Gestalt principle of closure has focused on how the presence of closure affects the ability to detect contours hidden in cluttered visual arrays. Some of the earliest research on closure, however, dealt with encoding and recognizing closed and open shapes, rather than detection. This research re-addresses the relation between closure and shape memory, focusing on how contour closure affects the ability to learn to recognize novel contour shapes. Of particular interest is whether closed contour shapes are easier to learn to recognize and, if so, whether this benefit is due to better encoding of closed contour shapes or easier comparison of closed contour shapes to already learned shapes. The results show that closed contours are indeed easier to recognize and, further, that this advantage appears to be related to better encoding.