The perceptual analysis of structure from motion for rotating objects undergoing affine stretching transformations

In two experiments, we evaluated the ability of human observers to make use of second-order temporal relations across three or more views of an apparent motion sequence for the perceptual analysis of three-dimensional form. Ratings of perceived rigidity were obtained in Experiment 1 for objects rotating in depth that were simultaneously subjected to sinusoidal affine stretching transformations along the line of sight or in a direction parallel to the image plane. Such transformations are theoretically interesting because they cannot be detected by analyses that are restricted to first-order temporal relations (i.e., two views), but they can be detected by more conventional analyses of structure from motion in which second-order temporal relations over three or more views are used. The current results show that human observers can perceive stretching transformations of a rotating 3-D object in a direction parallel to the image plane but that they fail to perceive stretching transformations along the line of sight. This result suggests that human observers can make use of some limited second-order temporal information. This finding was confirmed in Experiment 2, in which we investigated the effects of several specific optical consequences of sinusoidal stretching transformations applied in different directions. The results indicate that observers may be sensitive to the sign of acceleration, but that. they cannot make use of the precise magnitude of second-order relations necessary to recover euclidean metric structure.     

That’s the way the ball bounces: infants’ and adults’ perception of spatial and temporal contiguity in collisions involving bouncing balls

Three experiments investigated the perception of collisions involving bouncing balls by 7- and 10-month-old infants and adults. In previous research, 10-month-old infants perceived the causality of launching collisions (events in which one object moves along a smooth horizontal trajectory toward a second object, apparently launching it into motion) in relatively simple event contexts. In more complex event contexts, infants failed to discriminate among the events or respond to changes in individual features. Experiments 1 and 2 of the present investigation revealed that 7- and 10-month-old infants attended to spatial and temporal contiguity, but not causality, in collisions involving the movement of bouncing balls. In Experiment 3, both spatiotemporal contiguity and general knowledge about movement trajectories influenced adults’ judgments of causality for these collisions. The present results add to a growing understanding of infants’ event perception as constructive and a function, in part, of the complexity of the event context.     

The visual perception of smoothly curved surfaces from minimal apparent motion sequences

A series of four experiments was designed to investigate the minimal amounts of information required to perceive the structure of a smoothly curved surface from its pattern of projected motion. In Experiments 1 and 2, observers estimated the amplitudes of sinusoidally corrugated surfaces relative to their periods. Observers’ judgments varied linearly with the depicted surface amplitudes, but the amount of perceived relative depth was systematically overestimated by approximately 30%. The observers’ amplitude judgments were also influenced to a lesser extent by the amount of rotary displacement of a surface at each frame transition, and by increasing the length of the apparent motion sequences from two to eight frames. The latter effect of sequence length was quite small, however, accounting for less than 3% of the variance in the observers’ judgments. Experiments 3 and 4 examined observers’ discrimination thresholds for sinusoidally corrugated surfaces of variable amplitude and for ellipsoid surfaces of variable eccentricity. The results revealed that observers could reliably detect differences of surface structure as small as 5%. The length of the apparent motion sequences had no detectable effect on these tasks, although there were significant effects of angular displacement and surface orientation. These results are considered with respect to the analysis of affine structure from motion proposed by Todd and Bressan (1990).     

Effects of texture and shape on perceived time to passage: Knowing “what” influences judging “when”

In the present study, we examined whether it is easier to judge when an object will pass one’s head if the object’s surface is textured. There are three reasons to suspect that this might be so: First, the additional (local) optic flow may help one judge the rate of expansion and the angular velocity more reliably. Second, local deformations related to the change in angle between the object and the observer could help track the object’s position along its path. Third, more reliable judgments of the object’s shape could help separate global expansion caused by changes in distance from expansion due to changes in the angle between the object and the observer. We can distinguish among these three reasons by comparing performance for textured and uniform spheres and disks. Moving objects were displayed for 0.5–0.7 sec. Subjects had to decide whether the object would pass them before or after a beep that was presented 1 sec after the object started moving. Subjects were not more precise with textured objects. When the disk rotated in order to compensate for the orientation-related contraction that its image would otherwise undergo during its motion, it appeared to arrive later, despite the fact that this strategy increases the global rate of expansion. We argue that this is because the expected deformation of the object’s image during its motion is considered when time to passage is judged. Therefore, the most important role for texture in everyday judgments of time to passage is probably that it helps one judge the object’s shape and thereby estimate how its image will deform as it moves.     

Structure mapping in spatial reasoning

Four experiments examined whether spatial reasoning about non-spatial concepts is based on mapping concepts to space according to similarities in relational structure. Six- and 7-year-old children without any prior graphing experience were asked to reason with graph-like diagrams. In Experiments 1 and 2, children were taught to map time and quantity to vertical and horizontal lines, and then were asked to judge the relative value of a second-order variable (rate) or a first-order variable (quantity) represented in a function line. Children’s judgments indicated that they mapped concepts to space by aligning similar relational structures: quantity judgments corresponded to line height, and rate judgments corresponded to line slope. These correspondences entail a mapping of first-order concepts to first-order spatial dimensions and second-order concepts to second-order spatial dimensions. Experiments 3 and 4 investigated the role of context in establishing relational structure. Children were taught to map age and rate (Experiment 3) or age and size (Experiment 4) to vertical and horizontal lines, and were then asked to judge the rate or the size represented by a function line. In this context, both rate and size were first-order variables, and children’s judgments corresponded to line height, also a first-order variable. The results indicate that spatial reasoning involves a structure-sensitive mapping between concepts and space.     

Constructing naive theories of motion on the fly

People often make erroneous predictions about the trajectories of moving objects. McCloskey (1983a, 1983b) and others have suggested that many of these errors stem from well-developed, but naive, theories of motion. The studies presented here examine the role of naive impetus theory in people’s judgments of motion. Subjects with and without formal physics experience were asked to draw or select from alternatives the trajectories of moving objects that were presented in various manners. Results from two experiments indicate that both trajectory judgments and explanations were affected by specific response and display features of the problem. In addition, these data provide little evidence that naive impetus theory plays a significant role in subjects’ performance; instead, they suggest that motion judgments and explanations are constructed on the fly from contextual cues and knowledge that is not necessarily naive.     

The visual perception of smoothly curved surfaces from minimal apparent motion sequences.

A series of four experiments was designed to investigate the minimal amounts of information required to perceive the structure of a smoothly curved surface from its pattern of projected motion. In Experiments 1 and 2, observers estimated the amplitudes of sinusoidally corrugated surfaces relative to their periods. Observers' judgments varied linearly with the depicted surface amplitudes, but the amount of perceived relative depth was systematically overestimated by approximately 30%. The observers' amplitude judgments were also influenced to a lesser extent by the amount of rotary displacement of a surface at each frame transition, and by increasing the length of the apparent motion sequences from two to eight frames. The latter effect of sequence length was quite small, however, accounting for less than 3% of the variance in the observers' judgments. Experiments 3 and 4 examined observers' discrimination thresholds for sinusoidally corrugated surfaces of variable amplitude and for ellipsoid surfaces of variable eccentricity. The results revealed that observers could reliably detect differences of surface structure as small as 5%. The length of the apparent motion sequences had no detectable effect on these tasks, although there were significant effects of angular displacement and surface orientation. These results are considered with respect to the analysis of affine structure from motion proposed by Todd and Bressan (1990).     

Second-order reversed phi

In a first-order reversed-phi motion stimulus (Anstis, 1970), the black-white contrast of successive frames is reversed, and the direction of apparent motion may, under some conditions, appear to be reversed. It is demonstrated here that, for many classes of stimuli, this reversal is a mathematical property of the stimuli themselves, and the real problem is in perceiving forward motion, which involves the second- or third-order motion systems or both. Three classes of novel second-order reversed-phi stimuli (contrast, spatial frequency, and flicker modulation) that are invisible to first-order motion analysis were constructed. In these stimuli, the salient stimulus features move in theforward (feature displacement) direction, but the second-order motion energy model predicts motion in thereversed direction. In peripheral vision, for all stimulus types and all temporal frequencies, all the observers saw only the reversed-phi direction of motion. In central vision, the observers also perceived reversed motion at temporal frequencies above about 4 Hz, but they perceived movement in the forward direction at lower temporal frequencies. Since all of these stimuli are invisible to first-order motion, these results indicate that the second-order reversed-phi stimuli activate two subsequent competing motion mech-anisms, both of which involve an initial stage of texture grabbing (spatiotemporal filtering, followed by fullwave rectification). The second-order motion system then applies a Reichardt detector (or equiva-lently, motion energy analysis) directly to this signal and arrives at the reversed-phi direction. The third-order system marks the location of features that differ from the background (the figure) in a salience map and computes motion in the forward direction from the changes in the spatiotemporal location of these marks. The second-order system’s report of reversed movement dominates in peripheral vision and in central vision at higher temporal frequencies, because it has better spatial and temporal resolu-tion than the third-order system, which has a cutoff frequency of 3–4 Hz (Lu & Sperling, 1995b). In cen-tral vision, below 3–4 Hz, the third-order system’s report of resolvable forward movement of something salient (the figure) dominates the second-order system’s report of texture contrast movement.     

Discriminating rigid from nonrigid motion: minimum points and views

Theoretical investigations of structure from motion have demonstrated that an ideal observer can discriminate rigid from nonrigid motion from two views of as few as four points. We report three experiments that demonstrate similar abilities in human observers: In one experiment, 4 of 6 subjects made this discrimination from two views of four points; the remaining subjects required five points. Accuracy in discriminating rigid from nonrigid motion depended on the amount of nonrigidity (variance of the interpoint distances over views) in the nonrigid structure. The ability to detect a rigid group dropped sharply as noise points (points not part of the rigid group) were added to the display. We conclude that human observers do extremely well in discriminating between nonrigid and fully rigid motion, but that they do quite poorly at segregating points in a display on the basis of rigidity.     

How children know the relevant properties for generalizing object names

Young children’s novel word extensions indicate that their animal categories, like those of adults, are characterized by multiple similarities among instances; whereas their artifact categories, again like those of adults, are characterized more simply by commonalities among instances in shape. Three experiments shed light on the nature and development of a mechanism that enables children to organize novel lexical categories differently for different kinds of objects. Experiment 1 shows that, by adult judgments, animals and artifacts present different category organizations. Experiment 2 shows relations between both age and the number of nouns young children have acquired, and children’s kind‐specific generalizations of newly learned nouns. Experiment 3 is a training study in which even younger children show an ability to learn and then generalize highly abstract relations between different contextual cues and different category structures; and importantly, to learn more than one set of such relations at a time. Together, these three findings indicate one way in which children are able to rapidly and accurately form lexical categories that parallel those of adults in their language community.     

Confidence in judgment

Experiments have shown that, generally, people are overconfident about the correctness of their answers to questions. Cognitive psychologists have attributed this to biases in the way people generate and handle evidence for and against their views. The overconfidence phenomenon and cognitive psychologists' accounts of its origins have recently given rise to three debates. Firstly, ecological psychologists have proposed that overconfidence is an artefact that has arisen because experimenters have used question material not representative of the natural environment. However, it now appears that some overconfidence remains even after this problem has been remedied. Secondly, it has been proposed that overconfidence is an artefactual regression effect that arises because judgments contain an inherently random component. However, those claiming this appear to use the term overconfidence to refer to a phenomenon quite different from the one that the cognitive psychologists set out to explain. Finally, a debate has arisen about the status of perceptual judgments. Some claim that these evince only underconfidence and must, therefore, depend on mechanisms fundamentally different from those subserving other types of judgment. Others have obtained overconfidence with perceptual judgments and argue that a unitary theory is more appropriate. At present, however, no single theory provides an adequate account of the many diverse factors that influence confidence in judgment.     

Non-bayesian inference: causal structure trumps correlation

The study tests the hypothesis that conditional probability judgments can be influenced by causal links between the target event and the evidence even when the statistical relations among variables are held constant. Three experiments varied the causal structure relating three variables and found that (a) the target event was perceived as more probable when it was linked to evidence by a causal chain than when both variables shared a common cause; (b) predictive chains in which evidence is a cause of the hypothesis gave rise to higher judgments than diagnostic chains in which evidence is an effect of the hypothesis; and (c) direct chains gave rise to higher judgments than indirect chains. A Bayesian learning model was applied to our data but failed to explain them. An explanation-based hypothesis stating that statistical information will affect judgments only to the extent that it changes beliefs about causal structure is consistent with the results.     

Infants’ reasoning about ambiguous motion events: The role of spatiotemporal and dispositional status information

Two experiments investigate whether 7-month-olds reason about the origin of motion events by considering two sources of causally relevant information: spatiotemporal cues and dispositional status information derived from the identification of an object as either animate (with the enduring causal property of self-initiated motion) or inanimate (requiring an external cause of motion). Infants were shown a ball, a human hand, and an animal engaged in a motion event. While dispositional status information remained constant, spatiotemporal relations varied across conditions. Based on looking time data, we conclude that infants attend flexibly to both types of information. Without spatiotemporal cues, infants rely on dispositional status information. When two objects provide dispositional cues to motion origin, but only one also provides corresponding spatiotemporal information, infants attribute the motion to the object providing both types of information. Given an ambiguous motion event with two dispositional motion originators but no additional spatiotemporal cues, infants may prefer either of the two.     

Visible persistence is reduced by fixed-trajectory motion but not by random motion.

Despite the sluggish temporal response of the human visual system, moving objects appear clear and without blur, which suggests that visible persistence is reduced when objects move. It has been argued that spatiotemporal proximity alone can account for this modulation of visible persistence and that activation of a motion mechanism per se is not necessary. Experiments are reported which demonstrate that there is a motion-specific influence on visible persistence. Specifically, points moving in constant directions, or fixed trajectories, show less persistence than points moving with the same spatial and temporal displacements but taking random walks, randomly changing direction each frame. Subjects estimated the number of points present in the display for these two types of motion conditions. Under conditions chosen to produce 'good' apparent motion, ie small temporal and spatial increments, the apparent number of points for the fixed-trajectory condition was significantly lower than the apparent number in the random-walk condition. The traditional explanation of the suppression of persistence based on the spatiotemporal proximity of objects cannot account for these results. The enhanced suppression of persistence observed for a target moving in a consistent direction depends upon the activation of a directionally tuned motion mechanism extended over space and time.     

Effects of texture and shape on perceived time to passage: knowing "what" influences judging "when"

In the present study, we examined whether it is easier to judge when an object will pass one's head if the object's surface is textured. There are three reasons to suspect that this might be so: First, the additional (local) optic flow may help one judge the rate of expansion and the angular velocity more reliably. Second, local deformations related to the change in angle between the object and the observer could help track the object's position along its path. Third, more reliable judgments of the object's shape could help separate global expansioncaused by changes in distance from expansion due to changes in the angle between the object and the observer. We can distinguish among these three reasons by comparing performance for textured and uniform spheres and disks. Moving objects were displayed for 0.5-0.7 sec. Subjects had to decide whether the object would pass them before or after a beep that was presented 1 sec after the object started moving. Subjects were not more precise with textured objects. When the disk rotated in order to compensate for the orientation-related contraction that its image would otherwise undergo during its motion, it appeared to arrive later, despite the fact that this strategy increases the global rate of expansion. We argue that this is because the expected deformation of the object's image during its motion is considered when time to passage is judged. Therefore, the most important role for texture in everyday judgments of time to passage is probably that it helps one judge the object's shape and thereby estimate how its image will deform as it moves.     

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.     

Perceiving shape from profiles

Four experiments related human perception of shape from profiles to current theoretical predictions. In Experiment 1, judgments of structure and motion were obtained for single- and dualellipsoid displays rotating about various axes. Ratings were highest when the axis of rotation was in the image plane and were influenced by the number of ellipsoids and the orientation of a single ellipsoid. The subsequent experiments explored the effect of orientation on shape judgments of a single ellipsoid. The results of Experiments 2 and 3 suggested that the effect of orientation found in Experiment 1 was not due to either the inability of certain orientations to be perceived as three-dimensional objects or to two-dimensional artifacts. It was thus argued that this effect of orientation was due to points of correspondence in relative motion that arise when the major axis is not perpendicular to the axis of rotation. In Experiment 4, subjects provided judgments of both shape and angular velocity. The elevated ellipsoids that were judged as larger were also judged as rotating more slowly. The inverse relationship between size and angular velocity is consistent with current theories. The connection between theory and data was further demonstrated by applying a shape-recovery algorithm to the stimuli used in Experiment 4 and finding a similar tradeoff between angular velocity and shape.     

Distortions of depth-order relations and parallelism in structure from motion

Four experiments related human perception of depth-order relations in structure-from-motion dis-plays to current Euclidean and affine theories of depth recovery from motion. Discrimination between parallel and nonparallel lines and relative-depth judgments was observed for orthographic projections of rigidly oscillating random-dot surfaces. We found that (1) depth-order relations were perceived veridically for surfaces with the same slant magnitudes, but were systematically biased for surfaces with different slant magnitudes. (2) Parallel (virtual) lines defined by probe dots on surfaces with different slant magnitudes were judged to be nonparallel. (3) Relative-depth judgments were internally inconsistent for probe dots on surfaces with different slant magnitudes. It is argued that both veridical performance and systematic misperceptions may be accounted for by a heuristic analysis of the first-order optic flow.     

Integrating multiple cues to depth order at object boundaries

We examined the interaction between motion and stereo cues to depth order along object boundaries. Relative depth was conveyed by a change in the speed of image motion across a boundary (motion parallax), the disappearance of features on a surface moving behind an occluding object (motion occlusion), or a difference in the stereo disparity of adjacent surfaces. We compared the perceived depth orders for different combinations of cues, incorporating conditions with conflicting depth orders and conditions with varying reliability of the individual cues. We observed large differences in performance between subjects, ranging from those whose depth order judgments were driven largely by the stereo disparity cues to those whose judgments were dominated by motion occlusion. The relative strength of these cues influenced individual subjects' behavior in conditions of cue conflict and reduced reliability.     

The perception of 3-dimensional affine structure from minimal apparent motion sequences

The research described in the present article was designed to identify the minimal conditions for the visual perception of 3-dimensional structure from motion by comparing the theoretical limitations of ideal observers with the perceptual performance of actual human subjects on a variety of psychophysical tasks. The research began with a mathematical analysis, which showed that 2-frame apparent motion sequences are theoretically sufficient to distinguish between rigid and nonrigid motion and to identify structural properties of an object that remain invariant under affine transformations, but that 3 or more distinct frames are theoretically necessary to adequately specify properties of euclidean structure such as the relative 3-dimensional lengths or angles between nonparallel line segments. A series of four experiments was then performed to verify the psychological validity of this analysis. The results demonstrated that the determination of structure from motion in actual human observers may be restricted to the use of first order temporal relations, which are available within 2-frame apparent motion sequences. That is to say, the accuracy of observers' judgments did not improve in any of these experiments as the number of distinct frames in an apparent motion sequence was increased from 2 to 8, and performance on tasks involving affine structure was of an order of magnitude greater than performance on similar tasks involving euclidean structure.