A perspective metric for visual slant and shape

Summary Linear retinal perspective as a cue for visual slant and shape is analyzed in terms of its geometrical and psychophysical properties. Geometrically, retinal perspective is defined as the visual-angle difference in the projective magnitudes of the near and far edges of slanted rectangles. The psychophysical effectiveness of retinal perspective is then determined in two experiments. In Experiment 1, the sensitivity of the eye to retinal perspective is shown to be inversely related to the visual-angle height of slanted rectangles which are rotated about a horizontal axis. In Experiment 2, the visual shape of monocularly observed slanted rectangles is shown to depend solely on a simple prediction derived from Weber's Law, in which it is assumed that perceived height is a linear function of visual-angle height.

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Zusammenfassung Die geometrischen und psychophysischen Eigenschaften der linearen retinalen Perspektive wurden analysiert, um festzustellen, welchen Einfluß die Perspektive auf gesehene Schräge und Form hat. Geometrisch wird die retinale Perspektive als die Sehwinkeldifferenz in der projektiven Größe der näheren und ferneren Kante schräger Rechtecke definiert. Die psychophysische Wirksamkeit der retinalen Perspektive wurde an Hand zweier Experimente gemessen. Im ersten wurde gezeigt, daß die Sensitivität des Auges für die retinale Perspektive eine negative Funktion der Sehwinkelhöhe schräger Rechtecke ist, wenn diese um eine waagerechte Achse gedreht sind. Im zweiten Experiment wurde festgestellt, daß die Sehform einäugig betrachteter schräger Rechtecke von einer einfachen Ableitung des Weberschen Gesetzes abhängig ist, in der angenommen wird, daß die wahrgenommene Körperhöhe des Rechteckes eine lineare Funktion der Sehwinkelhöhe des Retinabildes ist.

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These investigations were supported by research grants MH-08856 and MH-10,691 from the National Institute of Mental Health, United States Public Health Service. The advice of Robert Seibel in the calculations of the regression lines of Fig. 4, and the assistance of Robert Pasnak in the conduct of Experiment 2 are gratefully acknowledged. Technical assistance was rendered by J. L. Cohen, W. F. Garber, Martha Harsanyi, R. L. Kochman, S. Ludwig, and Judith Ann Muth.     

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.     

Monotone mapping of similarities into a general metric space

A method of “maximum variance nondimensional scaling” is described and tested that transforms similarity measures into distances that meet just three conditions: (C1) they exactly satisfy the metric axioms, (C2) they are, as nearly as possible, monotonically related to the similarity measures, (C3) they have maximum variance possible under the two preceding conditions. By achieving an appropriate balance between the last two conditions, one can determine the true underlying distances and the form of the unknown monotone function relating the similarity measures to those distances without assuming that the underlying space has any particular Euclidean, Minkowskian, or even dimensional strucutre. The method appears to have potential applications, e.g., to studies of stimulus generalization and the structure and processing of semantic information.     

Horizontal-vertical anisotropy in visual space

We investigated the structure of visual space with a 3D exocentric pointing task. Observers had to direct a pointer towards a ball. Positions of both objects were varied. We measured the deviations from veridical pointing-directions in the horizontal and vertical planes (slant and tilt resp.). The slant increased linearly with an increasing horizontal visual angle. We also examined the effect of relative distance, i.e., the ratio of the distances between the two objects and the observer. When the pointer was further away from the observer than the ball, the observer directed the pointer in between himself and the ball, whereas when the pointer was closer to the observer he directed the pointer too far away. Neither the horizontal visual angle nor the relative distance had an effect on the tilt. The vertical visual angle had no effect on the deviations of the slant, but had a linear effect on the tilt. These results quantify the anisotropy of visual space.     

Local autonomy in visual space

Abstract.— The adjacency principle is considered in the context of the two factor theory of perception which divides the sources of perceptual information into absolute and relative cues. The adjacency principle states that the effectiveness of relative cues between objects varies inversely with the perceived separation of the objects either in a frontoparallel plane or in depth. The evidence regarding this principle is discussed for paradigms in which a test object is displaced spatially with respect to either one induction object or two opposing induction objects. The mqjor cues examined for evidence regarding adjacency effects consists of binocular disparity, achromatic color induction. and relative motion.     

Splitting visual space with attention

If a right-left spatial compatibility effect is observed, it can be maintained that the space has been segmented into right- and left-side parts. The present study aimed at showing a spatial compatibility effect (and, by implication, a right-left subdivision of space) solely attributable to the orienting of attention. Five groups of 8 normal subjects were required to give right-left discriminative responses to stimuli presented within one of six empty boxes arranged in a horizontal row. Reaction times and errors were recorded. The first two experiments showed that a right-left grouping of the boxes occurred regardless of whether subjects' fixation was kept at the intermediate position (Experiment 1) or at one extremity (Experiment 2) of the row. In Experiments 3 and 4, subjects' attention was not aligned with a fixed position but was moved, through peripheral cues, from trial to trial and positioned between different pairs of adjacent boxes. The results showed that the display was again subdivided into two regions and that the reference point for the right-left subdivision was the focus of attention. In Experiment 5, eye position was instrumentally monitored, and subjects' attention was directed by central cues. The results confirmed that the focusing of attention leads to a right-left partitioning of space, with the boundary at the locus of focal attention. In conclusion, we demonstrated, by employing a right-left spatial compatibility paradigm, that directing attention to a position in space brings about a right-left perceptual organization that predominates over that provided by the other egocentric reference axes.     

Attentional distribution in visual space

Summary The horizontal extent of the visual attentive field was measured by the use of a two-choice-RT task and compatible and incompatible distractors. The target was a line that inclined either to the left or to the right. Whether or not the subject performed the choice RT was made contingent upon whether two other stimuli presented in the visual display matched or mismatched. The match-mismatch stimuli varied in locations so as to manipulate the relevant visual area (the attended area). The locations of the distractors were also varied. The increase in RT associated with incompatible distractors was found to vary inversely with their distance from the edge of the area attended to and independently of their distance from the target lines. The results were interpreted in terms of an inhibitory field that surrounds the area attended to.     

The role of space and time in object-based visual search

Recently we have provided evidence that observers more readily select a target from a visual search display if the motion trajectory of the display object suggests that the observer has dealt with it before. Here we test the prediction that this object-based memory effect on search breaks down if the spatiotemporal trajectory is disrupted. Observers searched a display for a target shape among multiple distractors. The entire search display then passed behind an occluder and reemerged in either the same or a different configuration. Experiment 1 shows that the same-object benefit for selection disappears whenever a spatial disruption is involved, but that it may survive a brief temporal disruption. Experiment 2 shows that with sufficiently long gaps, a temporal disruption also destroys the same-object benefit for selective attention. Experiment 3 demonstrates that the same-object effect is even stronger when there only is a very small probability that the target can be found at the same location as before. This also made the task sensitive to brief temporal disruptions. We conclude that a coherent spatiotemporal history of a display object supports the selection of its relevant subregions.     

Distribution of visual attention over space

Subjects viewed 3 X 3 grids in which different subsets of the nine squares were designated as "figure," either by physical shading of those squares or by a verbal instruction to imagine those squares as shaded. The time taken by participants to respond "on" or "off" the figure was measured for single or multiple probe dots, which all appeared on or off the figural subset together, and which had already been shown to be equally detectable against shaded or unshaded squares and in all nine locations within the grid. In contrast to the set-size effect generally found in experiments on memory scanning, reaction time did not necessarily increase with the number of squares in the figural subset. Instead, the critical variable, which in previous research may often have adventitiously covaried with set size, was the spatial compactness of the subsets (as indexed by square-root-area over perimeter): Probes of less compact figures required more time to classify correctly. Subjects were evidently more successful in confining their attention to sets of mutually proximal items. Reasons are given for believing that this principle may also apply in the more abstract representational or semantic spaces that determine reaction times and errors in various other cognitive tasks.     

The interacting effect of load and space on visual selective attention

In the study of visual attention, two major determinants of our ability to ignore distracting information have been isolated, namely, (1) the spatial separation from the focus of attention and (2) perceptual load. This study manipulated both factors using a dual-task adaptation of the flanker paradigm (Eriksen & Hoffman, 1973). It showed that (1) although attention followed a gradient profile under low perceptual load it followed a Mexican-hat profile under high perceptual load, consistent with the idea that increasing load focuses spatial attention; and (2) increasing perceptual load did not improve overall selectivity: Though selectivity improved at near separations, it was impaired at far ones. Load and spatial separation exert interacting effects.     

Integrating visual images and visual percepts across time and space

Recent studies suggest that visual information can be integrated over a relatively long delay (> 1500 ms) to form a more complete representation (image-percept integration). The current studies investigated whether this process can occur between stimuli that differ in their spatial properties. Participants viewed two dot arrays that filled all but one space in a square or rectangular grid when combined, and reported the missing space. The arrays differed either in size or orientation. Performance reached a comparable level as when spatial properties were matched. However, such performance depends on at least two processes. We suggest an early encoding process and a later image formation/spatial attention reallocation process are required. The flexibility of the image-percept integration process suggests a strong mechanism to form more complete or detailed representations over time, even when the retinal size and orientation of the scene may change between successive views.     

Sensory direction inhomogeneous binocular visual space

Shifts in subjective sensory direction (local signature) occur in all sense modalities, and the dynamics of these shifts are quite similar. This is also true for visual direction in homogeneous binocular space. Binocular vision may, therefore, be interpreted in accordance with principles of intersensory processing.     

Attention can retrospectively distort visual space

A brief visual cue that attracts attention repels the perceived location of a subsequent visual stimulus away from the focus of attention (attentional repulsion). In the first experiment reported here, we presented a visual cue after a visual target and found that the perceived location of the target stimulus shifted toward the location of the cue (attentional attraction). The subsequent experiments ruled out nonattentional hypotheses and indicated that the mislocalization effect is attributable to the attentional shift. The results of this study suggest that preceding and succeeding contexts differentially modulate the perceived location of a briefly presented stimulus. Our findings also underscore the importance of retrospective processes in visual attention.     

Distance perception within near visual space

We investigated the perception of distance of visual targets with constant size and luminance presented between 20 and 120 cm from subjects' eyes. When retinal disparity cues were present, the subjects could reproduce very accurately the distance of a seen reference in this area. When only extraretinal information was available, distance perception was still correct for distances of 40 cm or less. However, distances beyond 60 cm were underestimated. When forced to evaluate the distance between a reference and themselves, e.g. when evaluating the absolute distance or half the distance or twice the distance of a reference, subjects used an egocentric plane of reference located on average 10.4 cm in front of their eyes. Measurements of binocular eye movements indicated a clear relationship between vergence angle and target distance. The egocentric plane of reference at 10.4 cm also corresponds to the maximum achievable vergence. These results suggest that ocular convergence can be used as a reliable cue for distance within the arm's reaching space.     

The MBR metric

P. H. Schönemann's (1982, Bulletin of the Psychonomic Society, 19, 317–319; 1983, Journal of Mathematical Psychology, 27, 311–324) MBR metric is designed to account for distortion effects in distance estimates caused by upper bounds in the response scales. To test the MBR hypothesis, given data have to be transformed first into the half-open interval [0, 1). This is achieved by defining the greatest distance estimate as the upper bound, and then dividing all values by this bound plus some “small constant” e, which ensures that the interval is open. How e should be picked is left open. It is shown here that very small differences in the value chosen for e have massive effects on the fit of the MBR model. Moreover, if e → 0, then the fit will eventually get bad for all realistic data.     

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.