Foreshortening and the perception of parallel projections

Does picture perception follow polar projective geometry? Parallel projection drawings, which are not produced by using rules of polar projection, are widely regarded as visually acceptable representations of three-dimensional (3-D) objects in free viewing. One explanation is that they are perceived by means of a system in which there is no foreshortening. If so, edges of a 3-D block in 1∶1 proportions should be denoted by lines in 1∶1 proportions on the picture surface. However, three experiments suggest that the perception of parallel projections of a block involves foreshortening. In Experiment 1, 90 subjects were shown a set of parallel projections of a cube, in which each drawing depicted three sides of the cube, drawn as a square with obliques—a frontal square with receding edges shown by parallel obliques of various lengths. The subjects preferred a drawing with a receding side length that was considerably foreshortened in relation to the front side. In Experiments 2 and 3, subjects viewed drawings of three blocks that differed in the ratios of the lengths of their receding edges to their frontal edges (1∶1, 1∶2, and 1∶0.65). In Experiment 2, the subjects were shown square-with-obliques drawings of the three blocks with receding edges shown by parallel obliques of various lengths. Again, the subjects preferred drawings with a receding side that was foreshortened. In Experiment 3, the drawings showed two sides of a block. The receding dimension was drawn with parallel or converging lines. The preferred foreshortening was not a fixed ratio of the dimensions of the 3-D blocks. We suggest that square-with-obliques parallel projections showing cubes are taken by vision to be approximations to projections using foreshortening. We suggest also that as the line showing the receding edge elongates, foreshortening becomes less of a factor.     

Face recognition in pictures is affected by perspective transformation but not by the centre of projection

Recognition of unfamiliar faces is susceptible to image differences caused by angular sizes subtended from the face to the camera. Research on perception of cubes suggests that apparent distortions of a shape due to large camera angle are correctable by placing the observer at the centre of projection, especially when visibility of the picture surface is low (Yang and Kubovy, 1999 Perception & Psychophysics 61 456-467). To explore the implication of this finding for face perception, observers performed recognition and matching tasks where face images with reduced visibility of picture surface were shown with observers either at the centre of projection or at other viewpoints. The results show that, unlike perception of cubes, the effect of perspective transformation on face recognition is largely unaffected by the centre of projection. Furthermore, the use of perspective cues is not affected by textured surfaces. The limitation of perspective in restoring 3-D information of faces suggests a stronger role for image-based, rather than model-based, processes in recognition of unfamiliar faces.     

Haptics and projection: drawings by Tracy, a blind adult

Outline drawings in a raised form were made by a blind woman, Tracy, who has been blind from very early in life. Highly practiced in drawing, she reports she is largely self-taught. To invoke matters of projection, she was asked to represent an object with faces slanting away from the observer, a fixed array from different vantage points, and sets of objects in depth. In particular, she drew a cube balanced on a vertex, three objects from different vantage points, receding rows of glasses, and a house. Her drawings included features of parallel and polar projection. Her use of these features may reflect an appreciation of direction from a vantage point, which observers deal with via haptics in everyday tasks. Tracy may have advanced drawing-development skills common to the blind and the sighted.     

The discrimination of heading from optic flow is not retinally invariant

Three experiments were conducted to determine whether the discrimination of heading from optic flow is retinally invariant and to determine the importance of acuity in accounting for heading eccentricity effects. In the first experiment, observers were presented with radial flow fields simulating forward translation through a three-dimensional volume of dots. The flow fields subtended 10° of visual angle and were presented at 0°, 10°, 20°, and 40° of retinal eccentricity. The observers were asked to indicate whether the simulated movement was to the right or the left of a target that appeared at the end of the display sequence. Eye movements were monitored with an electrooculogram apparatus. In a second experiment, static acuity thresholds were derived for each of the observers at the same retinal eccentricities. There was a significant increase in heading detection thresholds with retinal eccentricity (from 0.92° at 0° retinal eccentricity to 3.47° at 40°). An analysis of covariance indicated that the variation in sensitivity to radial flow, as a function of retinal eccentricity, is independent of acuity. Similar results were obtained when the Vernier acuity of observers was measured. These results suggest that the discrimination of heading from radial flow is not retinally invariant.     

The effect of the observer vantage point on perceived distortions in linear perspective images

Some features of linear perspective images may look distorted. Such distortions appear in two drawings by Jan Vredeman de Vries involving perceived elliptical, instead of circular, pillars and tilted, instead of upright, columns. Distortions may be due to factors intrinsic to the images, such as violations of the so-called Perkins's laws, or factors extrinsic to them, such as observing the images from positions different from their center of projection. When the correct projection centers for the two drawings were reconstructed, it was found that they were very close to the images and, therefore, practically unattainable in normal observation. In two experiments, enlarged versions of images were used as stimuli, making the positions of the projection centers attainable for observers. When observed from the correct positions, the perceived distortions disappeared or were greatly diminished. Distortions perceived from other positions were smaller than would be predicted by geometrical analyses, possibly due to flatness cues in the images. The results are relevant for the practical purposes of creating faithful impressions of 3-D spaces using 2-D images.     

Discriminating the volume of motion-defined solids

We investigated the ability of human observers to discriminate an important global 3-D structural property, namely volume, of motion-defined objects. We used convex transparent wire-frame objects consisting of about 12 planar triangular facets. Two objects, vertically separated by 7°, were shown simultaneously on a computer display. Both revolved at 67°/sec around a common vertical axis through their centers of mass. Observers watched the objects monocularly for an average of three full rotations before they responded. We measured volume discrimination as a function of absolute volume (3–48 cm³; 1 m viewing distance) and shape (cubes, rods, and slabs of different regularity). We found that (1) volume discrimination performance can be described by Weber’s law, (2) Weber fractions depend strongly on the particular combination of shapes used (regular shapes, especially cubes, are easiest to compare, and similar shapes are easier to compare than different shapes), and (3) humans use a representation of volume that is more veridical and stable in the sense of repeatability than a strategy based on the average visible (2-D) area would yield.     

Perceived direction of rotation of simulated three-dimensional patterns

Direction of rotation judgments were obtained from 72 subjects for computer-generated dot patterns simulating points randomly distributed in a sphere rotating about a vertical axis. The displays were produced either with normal polar projections or with perspective effects limited to the horizontal or to the vertical dimension of the projection. The simulated viewing distance used in the projections and the visual angle subtended by the projected displays were also varied. Accuracy of direction judgments was about the same with perspective effects limited to the vertical dimension as with normal polar projections but did not exceed chance expectations with perspective effects limited to the horizontal dimension. Accuracy was lower at the greater simulated viewing distance and at the greater visual angle.     

Aging and the perception of slant from optical texture, motion parallax, and binocular disparity

The ability of younger and older observers to perceive surface slant was investigated in four experiments. The surfaces possessed slants of 20°, 35°, 50°, and 65°, relative to the frontoparallel plane. The observers judged the slants using either a palm board (Experiments 1, 3, and 4) or magnitude estimation (Experiment 2). In Experiments 1–3, physically slanted surfaces were used (the surfaces possessed marble, granite, pebble, and circle textures), whereas computer-generated 3-D surfaces (defined by motion parallax and binocular disparity) were utilized in Experiment 4. The results showed that the younger and older observers' performance was essentially identical with regard to accuracy. The younger and older age groups, however, differed in terms of precision in Experiments 1 and 2: The judgments of the older observers were more variable across repeated trials. When taken as a whole, the results demonstrate that older observers (at least through the age of 83 years) can effectively extract information about slant in depth from optical patterns containing texture, motion parallax, or binocular disparity.     

Terrestrial passage theory of the moon illusion

Theories of the celestial, or moon, illusion have neglected geometric characteristics of movement along and above the surface of the earth. The illusion occurs because the characteristics of terrestrial passage are attributed to celestial passage. In terrestrial passage, the visual angle subtended by an object changes discriminably as an essentially invariant function of elevation above the horizon. In celestial passage, by contrast, change in visual angle is indiscriminable at all elevations. If a terrestrial object gains altitude, its angular subtense fails to follow the expansion projected for an orbital course: Angular diminution or constancy is equivalent to distancing. On the basis of terrestrial projections, a similar failure of celestial objects in successive elevations is also equivalent to distancing. The illusion occurs because of retinal image constancy, not--as traditionally stated--despite it.     

Brightness as a function of retinal locus

Brightness functions were determined for the dark-adapted fovea and periphery. In one series of experiments, observers matched numbers to the brightness of a 1° white target at various intensities, presented half the time to the fovea, the other half to one of five peripheral loci: 5°, 12°, 20°, 35°, and 60°. In a second series, observers matched the brightness of a 1° white target in the fovea of one eye to the brightness of an identical target in the periphery of the other eye at various intensities. Thresholds were also determined for the fovea and for the five peripheral loci by a staircase procedure. The magnitude estimations and the interocular matches concur in showing that a stimulus of fixed luminance appears brighter in the periphery than in the fovea. The brightness was found to be maximal at 20°. Brightness grows as a similar power function of luminance at all six retinal positions.     

Brightness as a function of retinal locus

Brightness functions were determined for the dark-adapted fovea and periphery. In one series of experiments, observers matched numbers to the brightness of a 1° white target at various intensities, presented half the time to the fovea, the other half to one of five peripheral loci: 5°, 12°, 20°, 35°, and 60°. In a second series, observers matched the brightness of a 1° white target in the fovea of one eye to the brightness of an identical target in the periphery of the other eye at various intensities. Thresholds were also determined for the fovea and for the five peripheral loci by a staircase procedure. The magnitude estimations and the interocular matches concur in showing that a stimulus of fixed luminance appears brighter in the periphery than in the fovea. The brightness was found to be maximal at 20°. Brightness grows as a similar power function of luminance at all six retinal positions.     

The effect of polar projection on the perception of euclidean structure from motion

The aim of the present study was to examine whether euclidean structure could be recovered from apparent motion sequences under polar projection. In Experiment 1, length judgments of two sides of a simulated triangle rotating in depth did not reveal effects of type of projection, polar or parallel, on the perception of euclidean structure. However, there was a significant correlation between simulated and produced slants. The results also indicated that absolute depth judgments could not be accounted for by a random mechanism suggested by Todd and Bressan (1990). Experiments 2 and 3, in which a continuous dot surface was substituted for the triangle, showed that polar projection information from a relatively large visual angle, 17.40°, as compared with a small visual angle, 4.35°, facilitated discrimination of depth. Produced height:width ratios were consistently related to simulated shape, although the depth dimension was underestimated. Finally, Experiment 4 showed significant correlations between simulated and produced height:width ratios that could be accounted for only by an analysis in whichX andY velocities were treated independently. As in previous experiments, the variation in the depth dimension was underestimated. It was concluded that the visual system utilizes the additional information that is available in polar projection when recovering structure from motion, but that for different reasons the perceived structure does not become euclidean. These reasons are discussed briefly.     

Discriminating direction of motion trajectories from angular speed and background information

The effects of a background scene on the perception of the trajectory of an approaching object and its relation to changes in angular speed and angular size were examined in five experiments. Observers judged the direction (upward or downward) of two sequentially presented motion trajectories simulating a sphere traveling toward the observer at a constant 3-D speed from a fixed distance. In Experiments 1–4, we examined the effects of changes in angular speed and the presence of a scene background, with changes in angular size based either on the trajectories being discriminated or on an intermediate trajectory. In Experiment 5, we examined the effects of changes in angular speed and scene background, with angular size either constant or consistent with an intermediate 3-D trajectory. Overall, we found that (1) observers were able to judge the direction of object motion trajectories from angular speed changes; (2) observers were more accurate with a 3-D scene background, as compared with a uniform background, suggesting that scene information is important for recovering object motion trajectories; and (3) observers were more accurate in judging motion trajectories based on angular speed when the angular size function was consistent with motion in depth than when the angular size was constant.     

The use of occlusion to resolve ambiguity in parallel projections

Previous research has shown that the three-dimensional structure of an object usually can be perceived when viewing a parallel projection of the object rotating in depth. Accurate judgments of direction of rotation, however, have been found only with polar projections. The present study demonstrated that accurate direction judgments can occur with parallel projections if occlusion is included in the displays. The stimuli were parallel projections of pentagonal texture elements on the surface of a rotating sphere. In one condition, the elements were occluded as they rounded the edge of an opaque sphere. In another condition, elements on the far surface of a transparent sphere were occluded by elements on the near surface. Accuracy of direction judgments was consistently high in the first condition and increased monotonically with element size in the second condition, from chance to over 80% correct. The relationship of these results to the general issue of perceptually combining structure in depth information from one source with relative distance information from another source is discussed.     

Eye height scaling of absolute size in immersive and nonimmersive displays

Eye-height (EH) scaling of absolute height was investigated in three experiments. In Experiment 1, standing observers viewed cubes in an immersive virtual environment. Observers' center of projection was placed at actual EH and at 0.7 times actual EH. Observers' size judgments revealed that the EH manipulation was 76.8% effective. In Experiment 2, seated observers viewed the same cubes on an interactive desktop display; however, no effect of EH was found in response to the simulated EH manipulation. Experiment 3 tested standing observers in the immersive environment with the field of view reduced to match that of the desktop. Comparable to Experiment 1, the effect of EH was 77%. These results suggest that EH scaling is not generally used when people view an interactive desktop display because the altitude of the center of projection is indeterminate. EH scaling is spontaneously evoked, however, in immersive environments.     

Visual rays are parallel

We show that human observers using monocular viewing treat the pencil of 'visual rays' that diverges from the vantage point as experientally parallel. This oddity becomes very noticeable in the case of wide-angle presentations, where the angle subtended by a pair of visual rays may be as large as the angular size of the display. In our presentations such angles subtended over 100 deg. There are various ways to demonstrate the effect; in this study we measure the attitudes of pictorial objects that appear to be situated in mutually parallel attitudes in pictorial space. Our finding is that such objects appear parallel if they are similarly oriented with respect to the local visual rays. This leads to 'errors' in the judgment of mutual orientations of up to 100 deg. Although this appears to be the first quantitative study of the effect, we trace it to qualitative reports by Helmholtz (late 19th century) and Kepler (early 17th century) as well as speculation by early authors (AD 500). The effect has apparently been noticed by visual artists from the late middle ages to the present day.     

Drawings from Gaia,a blind girl

Gaia, a totally blind girl, was asked to make raised-line drawings. Gaia's vision at best was peripheral. She draws out of interest, and has drawn since preschool with encouragement from her mother. She was asked to draw objects and scenes involving depth from a vantage point, eg a table from below, two cars (one behind the other), and two parallel rows of apples (receding from her, on a table top). Gaia represented space in her drawings using T-junctions for overlap, height in the picture plane, parallel projection, and inverse projection. That is, Gaia uses features of systems common in sighted children's drawings. The development of drawing in blind and sighted children may be similar in good measure because haptics provides access to many of the same spatial principles as vision.     

Stereo disparity facilitates view generalization during shape recognition for solid multipart objects

Current theories of object recognition in human vision make different predictions about whether the recognition of complex, multipart objects should be influenced by shape information about surface depth orientation and curvature derived from stereo disparity. We examined this issue in five experiments using a recognition memory paradigm in which observers (N = 134) memorized and then discriminated sets of 3D novel objects at trained and untrained viewpoints under either mono or stereo viewing conditions. In order to explore the conditions under which stereo-defined shape information contributes to object recognition we systematically varied the difficulty of view generalization by increasing the angular disparity between trained and untrained views. In one series of experiments, objects were presented from either previously trained views or untrained views rotated (15°, 30°, or 60°) along the same plane. In separate experiments we examined whether view generalization effects interacted with the vertical or horizontal plane of object rotation across 40° viewpoint changes. The results showed robust viewpoint-dependent performance costs: Observers were more efficient in recognizing learned objects from trained than from untrained views, and recognition was worse for extrapolated than for interpolated untrained views. We also found that performance was enhanced by stereo viewing but only at larger angular disparities between trained and untrained views. These findings show that object recognition is not based solely on 2D image information but that it can be facilitated by shape information derived from stereo disparity.     

The box alignment illusion: An orientation illusion induced by pictorial depth

In four experiments, observers attempted to align two sets of oblique edges to parallel. The contexts for these alignments included lines in isolation (2-D control), lines embedded in orthogonal drawings of same-oriented and different-oriented boxes (3-D objects), and each of these viewed against backgrounds depicting strong linear perspective (3-D backgrounds). A consistent distortion was observed in the alignments of different-oriented boxes relative to control lines, indicating that the parallel lines in these stimuli appeared to diverge toward the top of the picture. Furthermore, thisbox alignment illusion decreased with interstimulus distance, whereas alignment distortions in control lines and same-oriented boxes increased with distance. Viewing the stimuli against 3-D backgrounds produced a dramatic reversal of the illusion, with control lines now appearing to converge more than the boxes. These results suggest that the illusion reflects basic processes involved in pictorial depth perception.     

Quality of Drawing as a Factor in the Interpretation of Figure Drawings

Examined the drawings of 32 poliomyelitis patients and their matched controls to see whether figure drawings primarily reflect the subject's projection of psychological state, ability to draw, or some combination of these two factors. An overview of the literature is also given. Drawings from disabled and nondisabled subjects were reliably rated for quality, with no significant quality difference found between groups. Analyses of variance were then used to compare the drawings on several different measures of drawing size, completion and movement that might be assumed on the basis of the literature, to reflect the subjects' projection of disability status. Results showed that quality of drawing was a significant factor in 13 of the 17 comparisons while disability status proved to be a significant factor in only one of the 17 comparisons. There were no significant interactions. Therefore, the overall findings are consistent with the hypothesis that quality of drawing—rather than projective mechanisms—may at times be the overwhelming determinant of clinical and research findings with figure drawings.