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.     

Curvature discontinuities are cues for rapid shape analysis

Using the visual search method, we show that stimuli that contain curvature discontinuities (i.e., points where the second derivative along an image contour is not defined) are easily found among stimuli containing only smooth changes in curvature. Curved stimuli that lack curvature discontinuities, however, are difficult to find among distractors that have them. These results suggest that the visual system detects and analyzes abrupt changes in curvature in the image quickly to extract vital information about the 3-D structure of the visual environment.     

Curvature discontinuities are cues for rapid shape analysis.

Using the visual search method, we show that stimuli that contain curvature discontinuities (i.e., points where the second derivative along an image contour is not defined) are easily found among stimuli containing only smooth changes in curvature. Curved stimuli that lack curvature discontinuities, however, are difficult to find among distractors that have them. These results suggest that the visual system detects and analyzes abrupt changes in curvature in the image quickly to extract vital information about the 3-D structure of the visual environment.     

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.     

Illusory volumes from conformation

The purpose of this paper is to offer demonstrations of 'illusory volumes' in the spirit of the illusory flat surfaces described by Kanizsa. These demonstrations of illusory volumes exploit a new cue to the recovery of surface curvature from ambiguous images: conformation. In assuming conformation, the visual system assumes that the surface of a volume conforms to the curvature of its neighboring, underlying, or supporting surface, in the absence of image cues to the contrary. Demonstrations that exploit the assumption of conformation provide several insights into the nature of the inferential processing that underlies contour, surface, and volume formation. In particular, these demonstrations imply that the visual system does not calculate local surface curvature, illusory contours, or occlusion relationships before it analyzes global surface relationships.     

Using motor tasks to quantitatively judge 3-D surface curvatures.

The primary objective of this study was to quantitatively investigate the human perception of surface curvature by using virtual surfaces and motor tasks along with data analysis methods to estimate surface curvature from drawing movements. Three psychophysical experiments were conducted. In Experiment 1, we looked at subjects' sensitivity to the curvature of a curve lying on a surface and changes in the curvature as defined by Euler's formula, which relates maximum and minimum principal curvatures and their directions. Regardless of direction and surface shape (elliptic and hyperbolic), subjects could report the curvature of a curve lying on a surface through a drawing task. In addition, multiple curves drawn by subjects were used to reconstruct the surface. These reconstructed surfaces could be better accounted for by analysis that treated the drawing data as a set of curvatures rather than as a set of depths. A pointing task was utilized in Experiment 2, and subjects could report principal curvature directions of a surface rather precisely and consistently when the difference between principal curvatures was sufficiently large, but performance was poor for the direction of zero curvature (asymptotic direction) on a hyperbolic surface. In Experiment 3, it was discovered that sensitivity to the sign of curvature was different for perceptual judgments and motor responses, and there was also a difference for that of a curve itself and the same curve embedded in a surface. These findings suggest that humans are sensitive to relative changes in curvature and are able to comprehend quantitative surface curvature for some motor tasks.     

Identifying contours from occlusion events

Surface contours specified by occlusion events that varied in density, velocity, and type of motion (rotation or translation) were examined in four experiments. As a fourth experimental factor, there were both figure-motion trials (the occluding surface moved over a stationary background) and background-motion trials (the background moved behind a stationary surface) in each experiment. Displays contained line patterns and rotary motion (Experiment 1), line patterns and translatory motion (Experiment 2), textured surfaces and rotary motion (Experiment 3), and textured surfaces and translatory motion (Experiment 4). Results indicate that contour identifications are more accurate with translation than with rotation, and that background-motion trials are generally easier than figure-motion trials. Although density in all experiments affected identifications in both background- and figure-motion trials, velocity did so in Experiment 4 only. In Experiments 1, 2, and 3, velocity affected identifications in background-motion trials but not in figure-motion trials. In Experiments 3 and 4, the rate of accretion and deletion of texture was a poor predictor of identification accuracy. These results are not consistent with previous accounts of contour perception from occlusion events, and may reflect an involvement of ocular pursuit as a mechanism for registering contour information.     

Using motor tasks to quantitatively judge 3-D surface curvatures

The primary objective of this study was to quantitatively investigate the human perception of surface curvature by using virtual surfaces and motor tasks along with data analysis methods to estimate surface curvature from drawing movements. Three psychophysical experiments were conducted. In Experiment 1, we looked at subjects’ sensitivity to the curvature of a curve lying on a surface and changes in the curvature as defined byEuler’s formula, which relates maximum and minimum principal curvatures and their directions. Regardless of direction and surface shape (elliptic and hyperbolic), subjects could report the curvature of a curve lying on a surface through a drawing task. In addition, multiple curves drawn by subjects were used to reconstruct the surface. These reconstructed surfaces could be better accounted for by analysis that treated the drawing data as a set of curvatures rather than as a set of depths. A pointing task was utilized in Experiment 2, and subjects could report principal curvature directions of a surface rather precisely and consistently when the difference between principal curvatures was sufficiently large, but performance was poor for the direction of zero curvature (asymptotic direction) on a hyperbolic surface. In Experiment 3, it was discovered that sensitivity to the sign of curvature was different for perceptual judgments and motor responses, and there was also a difference for that of a curve itself and the same curve embedded in a surface. These findings suggest that humans are sensitive to relative changes in curvature and are able to comprehend quantitative surface curvature for some motor tasks.     

Effects of partial occlusion on perceived surface segregation

Tests are reported of the possibility that local information from contour junctions and from corners of intersecting surfaces is used for perceived surface segregation. Stimuli were two intersecting squares with small disks occluding different parts of the squares. The perceived segregation of the squares from one another decreased as the amount of occlusion of parts of the squares increased. This segregation was less when disks occluded parts of contours than when disks occluded parts entirely inside the squares. Occlusion of parts of contours reduced segregation independently of whether contour junctions or corners were visible or invisible, both where the intersecting surfaces were transparent squares and when they were outlined squares. The present findings show that local information from contour junctions or from corners is not used for surface segregation, and confirm that this segregation is determined by global processes of grouping of areas and of extrapolation of contours.     

17,000 years of depicting the junction of two smooth shapes

Competent realistic drawings preserve viewpoint-invariant shape characteristics of simple parts, such that a contour in the object that is straight or curved, for example, is depicted that way in the drawing. A more subtle invariant--a V-shaped singularity of the occluding boundary, containing a T-junction and a contour termination--is produced at the junction between articulated smooth surfaces, as with the leg joining the body of a horse. 45% of the drawings made in 2007 by individuals with only minimal art education correctly depicted such junctions, a proportion that is not reliably different from the incidence (42%) of correct depictions in a large sample of cave art made 17000 years ago. Whether a person did or did not include the invariant in their drawing, all agreed that it made for a better depiction.     

Short-term visual memory for contour curvature in a delayed discrimination task

Abstract: A two-interval forced-choice of constant stimuli was used to measure the point of subjective equality (PSE) and discrimination threshold for standard contour curvature (1.91, 3.24 deg⁻¹) held in short-term visual memory (STVM). At both standard curvatures, the PSE for remembered curvature was nearly constant for standard curvature from 2 s to 16 s retention intervals, while the discrimination threshold increased as a linear function of retention interval. These results show that the decay in STVM for contour curvature is due to the noisy representation of curvature, neither to fading of the represented curvature nor to converging to the constant curvature. Furthermore, the Weber fraction was nearly constant for both standard curvatures at any delay from 2 to 16 s.     

Perceptual saliency of points along the contour of everyday objects: a large-scale study

The aim of this large-scale study was to find out which points along the contour of a shape are most salient and why. Many subjects (N=161) were asked to mark salient points on contour stimuli, derived from a large set of line drawings of everyday objects (N=260). The database of more than 200,000 marked points was analyzed extensively to test the hypothesis, first formulated by Attneave (1954), that curvature extrema are most salient. This hypothesis was confirmed by the data: Highly salient points are usually very close to strong curvature extrema (positive maxima and negative minima). However, perceptual saliency of points along the contour is determined by more factors than just local absolute curvature. This was confirmed by an extensive correlational analysis of perceptual saliency in relation to ten different stimulus factors. A point is more salient when the two line segments connecting it with its two neighboring salient points make a sharp turning angle and when the 2-D part defined by the triplet of salient points is less compact and sticks out more.     

The role of junctions in surface completion and contour matching

It has been suggested that contour junctions may be used as cues for occlusion. Ecologically, T-junctions and L-junctions are concurrent with situations of occlusion: they arise when the bounding contour of the occluding surface intersects with that of the occluded surface. However, there are other image properties that can be used as cues for occlusion. Here the role of junctions is directly compared with other occlusion cues--specifically, relatability and surface-similarity--in the emergence of amodal completion and illusory contour perception. Stimuli have been constructed that differ only in the junction structure, with the other occlusion cues kept unchanged. L-junctions and T-junctions were eliminated from the image or manipulated so as to be locally inconsistent with the (still valid) global occlusion interpretation. Although the other occlusion cues of relatability and surface similarity still existed in the image, subjects reported not perceiving illusory contours or amodal completion in junction-manipulated images. Junction manipulation also affected the perceived stereoscopic depth and motion of image regions, depending on whether they were perceived to amodally complete with a disjoint region in the image. These results are interpreted in terms of the role of junctions in the processes of surface completion and contour matching. It is proposed that junctions, being a local cue for occlusion, are used to launch completion processes. Other, more global occlusion cues, such as relatability, play a part at a later stage, once completion processes have been launched.     

Neural dynamics of 3-D surface perception: figure-ground separation and lightness perception

This article develops the FACADE theory of three-dimensional (3-D) vision to simulate data concerning how two-dimensional pictures give rise to 3-D percepts of occluded and occluding surfaces. The theory suggests how geometrical and contrastive properties of an image can either cooperate or compete when forming the boundary and surface representations that subserve conscious visual percepts. Spatially long-range cooperation and short-range competition work together to separate boundaries of occluding figures from their occluded neighbors, thereby providing sensitivity to T-junctions without the need to assume that T-junction "detectors" exist. Both boundary and surface representations of occluded objects may be amodally completed, whereas the surface representations of unoccluded objects become visible through modal processes. Computer simulations include Bregman-Kanizsa figure-ground separation, Kanizsa stratification, and various lightness percepts, including the Münker-White, Benary cross, and checkerboard percepts.     

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 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.     

3-D vision and figure-ground separation by visual cortex

A neural network theory of three-dimensional (3-D) vision, called FACADE theory, is described. The theory proposes a solution of the classical figure-ground problem for biological vision. It does so by suggesting how boundary representations and surface representations are formed within a boundary contour system (BCS) and a feature contour system (FCS). The BCS and FCS interact reciprocally to form 3-D boundary and surface representations that are mutually consistent. Their interactions generate 3-D percepts wherein occluding and occluded object parts are separated, completed, and grouped. The theory clarifies how preattentive processes of 3-D perception and figure-ground separation interact reciprocally with attentive processes of spatial localization, object recognition, and visual search. A new theory of stereopsis is proposed that predicts how cells sensitive to multiple spatial frequencies, disparities, and orientations are combined by context-sensitive filtering, competition, and cooperation to form coherent BCS boundary segmentations. Several factors contribute to figure-ground pop-out, including: boundary contrast between spatially contiguous boundaries, whether due to scenic differences in luminance, color, spatial frequency, or disparity-partially ordered interactions from larger spatial scales and disparities to smaller scales and disparities; and surface filling-in restricted to regions surrounded by a connected boundary. Phenomena such as 3-D pop-out from a 2-D picture, Da Vinci stereopsis, 3-D neon color spreading, completion of partially occluded objects, and figure-ground reversals are analyzed. The BCS and FCS subsystems model aspects of how the two parvocellular cortical processing streams that join the lateral geniculate nucleus to prestriate cortical area V4 interact to generate a multiplexed representation of Form-And-Color-And-DEpth, orfacade, within area V4. Area V4 is suggested to support figure-ground separation and to interact with cortical mechanisms of spatial attention, attentive object learning, and visual search. Adaptive resonance theory (ART) mechanisms model aspects of how prestriate visual cortex interacts reciprocally with a visual object recognition system in inferotemporal (IT) cortex for purposes of attentive object learning and categorization. Object attention mechanisms of the What cortical processing stream through IT cortex are distinguished from spatial attention mechanisms of the Where cortical processing stream through parietal cortex. Parvocellular BCS and FCS signals interact with the model What stream. Parvocellular FCS and magnocellular motion BCS signals interact with the model Where stream. Reciprocal interactions between these visual, What, and Where mechanisms are used to discuss data about visual search and saccadic eye movements, including fast search of conjunctive targets, search of 3-D surfaces, selective search of like-colored targets, attentive tracking of multielement groupings, and recursive search of simultaneously presented targets.     

Visual search for direction of shading is influenced by apparent depth.

Recent reports of rapid visual search for some feature conjunctions suggested that preattentive vision might be sensitive to scene-based as well as to image-based features (Enns & Rensink, 1990a, 1990b). This study examined visual search for targets defined by the direction of a luminance gradient, a conjunction of luminance and relative location that often corresponds to object curvature and direction of lighting in naturalistic scenes. Experiment 1 showed that such search is influenced by several factors, including the type of gradient, the shape of the contour enclosing the gradient, and the background luminance. These factors were varied systematically in Experiment 2 in a three-dimensionality rating task and in a visual-search task. The factors combined interactively in the rating task, supporting the presence of an emergent property of three-dimensionality. In contrast, each factor contributed only additively to the speed of the visual-search task. This is inconsistent with the view that search is guided by specialized detectors for surface curvature or direction of lighting. Rather, it is in keeping with the view that search is governed by a number of "quick and dirty" processes that are implemented rapidly and in parallel across the visual field.     

Rapid Integration of Contour Fragments: From Simple Filling-in to Parts-based Shape Description

Line drawings are easy to recognize, although the only information to the visual system is the contour itself. Starting from information theory and a theory of decomposition in parts, we investigated whether certain regions of such a contour are perceptually more relevant than others, using a deletion detection paradigm. In this paradigm, high detectability means poor contour integration, and vice versa. Regions of interest were curvature singularities, namely positive maxima (M+), negative minima (m?) and inflection points (I), of smooth, closed contours. In Experiment 1, we performed a first exploration of the detectability of deletions around these three types of curvature singularities. M+ deletions were easier to detect than the deletions around the other two singularities, a result that is explained using a post hoc combination of both mentioned theoretical frameworks. In Experiment 2, we replicated these findings using figure-background reversal, so thatthe same physical deletions could either be M+ or m?. Again, the M+ deletions were easier to detect than m? deletions. Although both types of singularities involve regions of high curvature changes, they differ in that mdeletions create gaps that concur with spontaneous segmentation.