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.     

Perception of,and adaptation to,a three-dimensional curvature distortion

The curvature distortion perceived under water was measured in three dimensions before and after 15 min of adaptation to the underwater environment. Subjects initially perceived distortions in all three dimensions. After 15 min under water, they exhibited significant adaptation to the curvature distortion in all dimensions. The type of task performed while in the water did not significantly affect the amount of adaptation. The results are compared with those of previous experiments which have reported evidence of counteradaptation when more than one kind of adaptation is measured simultaneously.     

The haptic perception of curvature

Ninety-two subjects, schoolchildren and undergraduate and postgraduate students, took part in a series of experiments on the haptic perception of curvature. A graded series of surfaces was produced using piano-convex lenses masked off to produce curved strips that could be explored without using arm movements. Thresholds were measured using the constant method and a staircase procedure. Experiments 1 and 2 yielded data on the absolute and difference thresholds for curvature. Experiment 3 demonstrated that the effective stimulus for curvature is represented by the overall gradient of a curved surface. Using this measure, it was shown that the present absolute thresholds for curvature are lower than those previously reported. In Experiment 4, absolute thresholds were compared using spherical and cylindrical curves: the results showed that, at least with the narrow strips used, the type of curvature does not exert a significant influence on performance. In Experiment 5, the subjective response to curvature was assessed using a rating procedure. Power functions are reported, although the relationship between stimuli and responses had a strong linear component. This suggests that haptically perceived curvature may be a metathetic rather than a prothetic continuum.     

Visual perception of surface curvature: psychophysics of curvature detection induced by motion parallax

The continuous approach to optic-flow processing shows that the curvature of a moving surface is related to a second spatial derivative of the velocity field, the spin variation (Droulez & Cornilleau-Pérès, 1989). With this approach as a theoretical framework, visual sensitivity to the curvature of a cylinder in motion was measured using a task of discrimination between cylindrical and planar patches. The results confirm the predictions suggested by the theory: (1) Sensitivity to curvature was always greater when the cylinder axis and the frontal translation were parallel than when they were orthogonal. The ratio of curvature detection thresholds in the two cases was between 1.3 and 2.5; the value predicted from the spin variation theory is about 2. (2) Sensitivity to curvature increased strongly with the velocity of the motion but was only weakly affected by its amplitude and the duration of viewing for the range of values used in our experiments.     

Perceived curvature of arcs and dot patterns as a function of curvature, arc length, and instructions

Arcs of circles, with six arc lengths and four radii of curvature, and an equivalent set of figures composed of three dots were used as stimuli. Subjects in Group I imagined the circle from which an arc or dot triplet was taken and indicated the centre of the circle. Group II subjects estimated the location of the point that was equidistant from the middle and ends of an arc, or equidistant from the three dots of a triplet. The results from arcs showed, in Group I, an underestimation of curvature that decreased as a function on the length of the arc. In Group II, however, overestimation of the curvature of most arcs occurred, indicating a strong influence of the difference in the perceptual task on the results. The effect of instructions was similar with the dot figures but, in general, more errors resembling overestimation of curvature occurred with these figures.     

Meta-analysis of raw mean differences

This article discusses the meta-analysis of raw mean differences. It presents a rationale for cumulating psychological effects in a raw metric and compares raw mean differences to standardized mean differences. Some limitations of standardization are noted, and statistical techniques for raw meta-analysis are described. These include a graphical device for decomposing effect sizes. Several illustrative data sets are analyzed.     

Overshoot of curvature in visual apparent motion

If a curved line and a straight line are presented briefly, one above the other, in sequence to the eye, then, under appropriate conditions, visual apparent motion is obtained. Subjects report that the illusory figure moving and changing from the curved line to the straight line appears to overshoot the latter, gaining a small curvature in the opposite sense. Three experiments are described. In the first, the magnitude of this apparent curvature was quantified as a function of the delay between the onsets of the curved line and straight line (the stimulus onset asynchrony, SOA). It is shown that overshoot in curvature cannot be attributed to inappropriate patterns of eye fixations. In the second experiment, the stimulus configuration was modified to reveal the contribution to apparent curvature of classical curvature-contrast effects. Curvature overshoot due to apparent motion alone was thus estimated as a function of SOA. In the third experiment, an analogous position overshoot was measured for apparent motion elicited by two brief sequentially presented parallel line segments. It is argued that a combination of such position overshoots cannot explain curvature overshoot. Two schemes of a more general kind that might be used to interpret curvature overshoot are then outlined. One scheme is based on a neural-net model of apparent motion, and the other on a functional model of apparent motion that operates by laws analogous to those governing real physical motion.     

Influence of shape on haptic curvature perception

Curvature discrimination of hand-sized doubly curved surfaces by means of static touch was investigated. Stimuli consisted of hyperbolical, cylindrical, elliptical and spherical surfaces of various curvatures. In the first experiment subjects had to discriminate the curvature along a specified orientation (the discrimination orientation) of a doubly curved surface from a flat surface. The curvature to be discriminated was oriented either along the middle finger or across the middle finger of the right hand. Independent of the shape of the surface, thresholds were found to be about 1.6 times smaller along the middle finger than across the middle finger. Discrimination biases were found to be strongly influenced by the shape of the surface; subjects judged a curvature to be more convex when the perpendicular curvature was convex than when this curvature was concave. With the results of the second experiment it could be ruled out that the influence of shape on curvature perception was simply due to a systematic error made by the subject regarding the discrimination orientation.     

Surface energy of free clusters of bubbles: An estimation

We propose an approximate equation for the surface energy of two-dimensional free bubble clusters which we compare with exact calculations of the surface energy of symmetrical clusters consisting of a central bubble surrounded by one or two shells of bubbles of two different areas. The accuracy of the equation is good for relatively narrow distributions of the areas and of the number of sides of the bubbles but underestimates the energy for large widths of those distributions. We propose a similar approximate equation for the surface energy of three-dimensional clusters.     

An abac for determining the mean deviation of a class from the general mean

The abac is presented with instructions for use.     

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.     

Adaptation to optically produced curvature of frontal planes

Attention, Perception, & Psychophysics - A new kind of adaptation was discovered, namely, to glasses that cause frontal planes to appear concave. The effect of this adaptation can be observed...     

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.     

The perception of surface curvature from optical motion

Observers viewed the optical flow field of a rotating quadric surface patch and were required to match its perceived structure by adjusting the shape of a stereoscopically presented surface. In Experiment 1, the flow fields included rigid object rotations and constant flow fields with patterns of image acceleration that had no possible rigid interpretation. In performing their matches, observers had independent control of two parameters that determined the surface shape. One of these, called the shape characteristic, is defined as the ratio of the two principle curvatures and is independent of object size. The other, called curvedness, is defined as the sum of the squared principle curvatures and depends on the size of the object. Adjustments of shape characteristic were almost perfectly accurate for both motion conditions. Adjustments of curvedness, on the other hand, were systematically overestimated and were not highly correlated with the simulated curvedness of the depicted surface patch. In Experiment 2, the same flow fields were masked with a global pattern of curl, divergence, or shear, which disrupted the first-order spatial derivatives of the image velocity field, while leaving the second-order spatial derivatives invariant. The addition of these masks had only negligible effects on observers’ performance. These findings suggest that observers’ judgments of three-dimensional surface shape from motion are primarily determined by the second-order spatial derivatives of the instantaneous field of image displacements.     

The effects of curvature on the grid illusions

Grid illusions, including the Hermann grid and scintillating grid (in which light disks are superimposed upon the grid intersections), are diminished by curving the alleys that limn the repeating pattern. Curvature might either disrupt the processes that induce the illusion, or simply make the illusory effects harder to see. To determine which mechanism might be invoked, we examined the effects of curving the alleys upon the vanishing-disk illusion, a phenomenon in which a single disk in a grid intersection is rendered less detectable. This illusion is of reduced visibility, rather than generating an illusory apparition as in the Hermann grid or scintillating grid. Thus, inhibition of illusory influence would enhance disk visibility, while a general reduction of visibility would render disks even harder to detect. We find that thresholds for both scintillation and the disk itself increase in a graded manner with increased curvature. Measuring the effect of curvature upon the vanishing disk with traditional forced-choice staircase methods demonstrates that the effect of curvature is upon detection, not subjective criterion. Furthermore, disks that are easy to detect within a rectilinear grid are more difficult to detect when the alleys are curved. Thus, curvature of the alleys induces a general tendency to inhibit the visibility of features, and is not specifically a repression of illusory effects.     

Quantification of three-dimensional structures

A mathematical model, based on additive subcomponents of grouping, subitizing and adding, was derived to account for quantification latencies of three-dimensional block arrangements. Subitizing is the process that people use to directly quantify a small number of objects without counting. It was found that most people consistently subitized up to four blocks. With more than four blocks, people resorted to grouping and adding, and the model was able to account for these data. The structural variables of compactness, symmetry, linearity, and planarity were shown to have small effects on quantification latencies relative to the large effect of number of blocks. Of these structural variables, compactness had the largest effect, and in terms of the model, it is suggested that visual structure had its effect on the perceptual grouping subcomponent.     

Haptic curvature discrimination at several regions of the hand

Static haptic discrimination of the curvature of convex, concave, or straight 20-cm-long strips was investigated for nine placements on the hand. In one condition, the strips were touched with the palmar side of the hand, and in the other condition, with the dorsal side. The influence of the lengths of the strips, and thus of contact lengths, was also investigated. For all placements, discrimination was poorer in the dorsal than in the palmar condition, owing to poorer cutaneous resolution on the dorsal side of the hand (the kinesthetic stimulation was the same in both conditions). Thus cutaneous stimulation is important. In both experiments, performance appeared to depend primarily on contact length. Moreover, the discrimination thresholds for all different placements and contact lengths followed the same trend. We conclude that in these experiments the effective stimulus for the discrimination of curved strips is the total difference of local surface attitude—that is, the slope difference over the far ends of the stimulus.     

Nucleation of He bubbles in amorphous FeBSi alloy irradiated with He ions

It is interesting to investigate the formation of He bubbles in amorphous alloys because point defects do not exist in amorphous materials. In the present study, the microstructural evolution of amorphous Fe₇₉B₁₆Si₅ alloy, either irradiated with 5?keV He⁺ ions or implanted with 150?eV He⁺ ions without causing displacement damage, and then annealed at a high temperature, was investigated using transmission electron microscopy (TEM). Vacancy-type defects were formed in the amorphous alloy after irradiation with 5?keV He⁺ ions, and He bubbles formed during annealing the irradiated samples at high temperature. On the other hand, for samples implanted with 150?eV He⁺ ions, although He atoms are also trapped in the free volume, no He bubbles were observed during annealing the samples even up to 873?K. In conclusion, the formation of He bubbles is related to the formation and migration of vacancy-type defects even in amorphous alloys.