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.

---

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.

---

---

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.     

A necessary and sufficient formula for matric factoring

For the purpose of extracting factors from matrices, it is proved that a certain formula is both necessary and sufficient. In factor analysis, the formula may be applied either to the correlation matrix, or directly to the score matrix (assuming the communality problem is solved). As many factors as desired can be extracted in one operation. Having such a compact formulation is useful for teaching as well as computing purposes, since it includes all techniques of factor extraction as special cases.On leave from the Israel Institute for Applied Social Research. This research was facilitated in part by a grant from the Lucius N. Littauer Foundation to the American Committee for Social Research in Israel, Inc.     

Note on two necessary and sufficient axioms for a well-graded knowledge space

We introduce two pedagogically sound axioms for a knowledge structure modeling a student’s learning: outer fringe coherence and learning smoothness. We show that the knowledge structures satisfying these two axioms are equivalent to well-graded knowledge spaces.     

Identifying the necessary and sufficient number of risk factors for predicting academic failure

Identifying the point at which individuals become at risk for academic failure (grade point average [GPA] < 2.0) involves an understanding of which and how many factors contribute to poor outcomes. School-related factors appear to be among the many factors that significantly impact academic success or failure. This study focused on 12 school-related factors. Using a thorough 5-step process, we identified which unique risk factors place one at risk for academic failure. Academic engagement, academic expectations, academic self-efficacy, homework completion, school relevance, school safety, teacher relationships (positive relationship), grade retention, school mobility, and school misbehaviors (negative relationship) were uniquely related to GPA even after controlling for all relevant covariates. Next, a receiver operating characteristic curve was used to determine a cutoff point for determining how many risk factors predict academic failure (GPA < 2.0). Results yielded a cutoff point of 2 risk factors for predicting academic failure, which provides a way for early identification of individuals who are at risk. Further implications of these findings are discussed.     

Grid analysis: Continuing the search for a metric of shape

A model of form perception is proposed, based on the concept that shape properties can be assessed by the measurement of the dimensions of a form, taken at regular intervals over several directions in a grid of lines. These measurements are then collected into a frequency distribution, which serves as a translationally invariant summary of shape properties for purposes of perceptual judgment. The present article illustrates the basic approach using simple convex forms, with primary emphasis on grid properties and the plausibility of the model with respect to biological substrates.     

Spatial memory over long retention intervals: nonmemorial factors are not necessary for accurate performance on the radial-arm maze by rats

A. Markowska, O. Buresová, and J. Bures (1983, Behavioral and Neural Biology, 38, 97-112) argued that the apparent persistence of accurate spatial working memory over delays of several hours arises from the formation of response strategies and the use of olfactory stimuli that develop with extended training at long delays. To test this explanation rats with extensive prior training at long delays were forced to enter the first four arms in a random order. On test days, the maze was rotated 180 degrees during the 2-h retention interval to determine whether the rats were using intramaze or extramaze (i.e., spatial) cues to guide their choices. On both rotation and control days, postdelay choices were spatially guided, averaging over 90% correct. Accurate spatial working memory at long delays is a reproducible phenomenon and does not appear to result from nonmemorial artifacts.     

Distortions of distance and shape are not produced by a single continuous transformation of reach space

We investigated whether distortions of perceived distance and shape could be captured by a single continuous one-to-one transformation of the underlying space. In Experiment 1, the participants reached to touch points around the perimeter of spherical targets viewed at five different distances, to yield simultaneous measures of perceived distance and shape. Different participants reached while using dynamic monocular, static binocular, or dynamic binocular vision. Thin plate spline (TPS) analysis was applied so as to transform a Cartesian grid in such a way as to carry the original target points to the mean reach locations. In all cases, discontinuities appeared in the transformed grid from folding of the space. In Experiment 2, the participants reached to points that lay at the same locus in reach space, but on different portions of the visible target spheres (e.g., front vs. side). The participants reached to different locations when the points were different with respect to shape (e.g., front vs. side) but reached to the same locations when the points were the same with respect to shape (left vs. right side). TPS analysis revealed discontinuities from holes torn in the underlying space. The results show that perceived distance and perceived shape entail different distortions and cannot be captured by a single continuous transformation of reach space.     

Distortions of distance and shape are not produced by a single continuous transformation of reach space

We investigated whether distortions of perceived distance and shape could be captured by a single continuous one-to-one transformation of the underlying space. In Experiment 1, the participants reached to touch points around the perimeter of spherical targets viewed at five different distances, to yield simultaneous measures of perceived distance and shape. Different participants reached while using dynamic monocular, static binocular, or dynamic binocular vision. Thin plate spline (TPS) analysis was applied so as to transform a Cartesian grid in such a way as to carry the original target points to the mean reach locations. In all cases, discontinuities appeared in the transformed grid from folding of the space. In Experiment 2, the participants reached to points that lay at the same locus in reach space, but on different portions of the visible target spheres (e.g., front vs. side). The participants reached to different locations when the points were different with respect to shape (e.g., front vs. side) but reached to the same locations when the points were the same with respect to shape (left vs. right side). TPS analysis revealed discontinuities from holes torn in the underlying space. The results show that perceived distance and perceived shape entail different distortions and cannot be captured by a single continuous transformation of reach space.     

Intrinsic array structure is neither necessary nor sufficient for nonegocentric coding of spatial layouts

Mou and McNamara (2002) have recently theorized that nonegocentric reference frames (viz., intrinsic reference frames, based on the spatial structure of a configuration of objects) are used to organize spatial relationships in memory. The theory has not made claims about whether the intrinsic structure of a stimulus array is necessary or sufficient for such nonegocentric coding. We demonstrate that salient intrinsic axes in a layout of objects are neither necessary nor sufficient for people to use a nonegocentric reference frame in organizing spatial memory. In Experiment 1, participants were successfully instructed to adopt a nonegocentric preferred direction in memory for an array of objects with no salient intrinsic reference axes. In Experiment 2, with no instructions, participants adopted an egocentric preferred direction for an array with a salient intrinsic axis. These results suggest that physically salient array structure exerts a minimal influence in the coding of spatial memory through nonegocentric reference frames.     

Domain general mechanisms account for imagined transformations of whole body perspective

Three experiments investigated the contribution of domain general processes to imagined transformations of whole body perspective. In a task where participants make left–right judgements about a schematic human figure, reaction times made from the point of view of the figure are longer when the figure does not share the same spatial orientation as the participant and are substantially longer than those made from the participant's own point of view. These phenomena have been attributed to a specialised mechanism for imagined perspective transformations. In the present experiments, the effect of orientation on performance was influenced by factors that affect spatial stimulus–response (S–R) compatibility; it was attenuated with reduced dimensional overlap and reversed by crossed S–R mappings. Performance in a control experiment in which no figure was present suggests that processes moderating S–R mappings may account for the effect of adopting another's point of view. Our findings demonstrate a role for domain general processes in imagined transformations of whole body perspective, imply that egocentric codes for spatial relationships within visually presented bodies are automatically generated, and undermine evidence for a specialised mechanism for imagined perspective transformations.     

The visual and haptic perception of natural object shape

In this study, we evaluated observers' ability to compare naturally shaped three-dimensional (3-D) objects, using their senses of vision and touch. In one experiment, the observers haptically manipulated 1 object and then indicated which of 12 visible objects possessed the same shape. In the second experiment, pairs of objects were presented, and the observers indicated whether their 3-D shape was the same or different. The 2 objects were presented either unimodally (vision-vision or haptic-haptic) or cross-modally (vision-haptic or haptic-vision). In both experiments, the observers were able to compare 3-D shape across modalities with reasonably high levels of accuracy. In Experiment 1, for example, the observers' matching performance rose to 72% correct (chance performance was 8.3%) after five experimental sessions. In Experiment 2, small (but significant) differences in performance were obtained between the unimodal vision-vision condition and the two cross-modal conditions. Taken together, the results suggest that vision and touch have functionally overlapping, but not necessarily equivalent, representations of 3-D shape.     

Children’s multiplicative transformations of discrete and continuous quantities

Recent studies have documented an evolutionarily primitive, early emerging cognitive system for the mental representation of numerical quantity (the analog magnitude system). Studies with nonhuman primates, human infants, and preschoolers have shown this system to support computations of numerical ordering, addition, and subtraction involving whole number concepts prior to arithmetic training. Here we report evidence that this system supports children’s predictions about the outcomes of halving and perhaps also doubling transformations. A total of 138 kindergartners and first graders were asked to reason about the quantity resulting from the doubling or halving of an initial numerosity (of a set of dots) or an initial length (of a bar). Controls for dot size, total dot area, and dot density ensured that children were responding to the number of dots in the arrays. Prior to formal instruction in symbolic multiplication, division, or rational number, halving (and perhaps doubling) computations appear to be deployed over discrete and possibly continuous quantities. The ability to apply simple multiplicative transformations to analog magnitude representations of quantity may form a part of the toolkit that children use to construct later concepts of rational number.     

Development of the perception of invariants: substance and shape

Three experiments investigated the perception of substance and shape as invariant properties of objects by three-month-old infants. In experiment 1, infants were habituated to two differently shaped objects undergoing a rigid motion. After habituation of the infants, the objects were presented undergoing a different rigid motion, or undergoing a deforming motion, or undergoing the same rigid motion. Habituation was maintained to the new rigid motion, indicating that the two rigid motions were perceived as sharing an invariant property. Dishabituation, on the other hand, occurred when a deforming motion followed a rigid one. In experiment 2, infants were habituated to one shape undergoing two different rigid motions. After habituation, the shape was changed but the same two motions continued. Dishabituation occurred, compared to a group with no shape change, indicating that shape is distinguished as an invariant property over two rigid motions. In experiment 3, habituation to a shape undergoing two rigid motions was followed by a new shape presented motionless, or the same shape presented motionless. Cessation of motion did not prevent recognition of shape as invariant. Two properties of an object, substance and shape, thus appear to be detectable as invariant in an event sequence, an instance of "phenomenal doubling" at an early age.     

Aging and the haptic perception of 3D surface shape

Two experiments evaluated the ability of older and younger adults to perceive the three-dimensional (3D) shape of object surfaces from active touch (haptics). The ages of the older adults ranged from 64 to 84 years, while those of the younger adults ranged from 18 to 27 years. In Experiment 1, the participants haptically judged the shape of large (20 cm diameter) surfaces with an entire hand. In contrast, in Experiment 2, the participants explored the shape of small (5 cm diameter) surfaces with a single finger. The haptic surfaces varied in shape index (Koenderink, Solid shape, 1990; Koenderink, Image and Vision Computing, 10, 557-564, 1992) from -1.0 to +1.0 in steps of 0.25. For both types of surfaces (large and small), the participants were able to judge surface shape reliably. The older participants' judgments of surface shape were just as accurate and precise as those of the younger participants. The results of the current study demonstrate that while older adults do possess reductions in tactile sensitivity and acuity, they nevertheless can effectively perceive 3D surface shape from haptic exploration.