Perception, 1992, volume 21, pages 427-439
Perspective, orientation disparity, and anisotropy in stereoscopic slant perception Barbara Gillam, Colin Ryan Department of Psychology, University of New South Wales, PO Box 1, Kensington NSW 2033, Australia Received 26 April 1991, in revised form 22 August 1991
Abstract. Stereoscopic depth estimates are not predictable from the geometry of point disparities. The configural properties of surfaces (surface contours) may play an important role in determining, for example, slant responses to a disparity gradient, and the marked anisotropy in favour of slant around a horizontal axis. It has been argued that variation in slant magnitude are attributable to the degree of perspective conflict present and that anisotropy is attributable to orientation disparity, which varies with the axis of slant. Three experiments were conducted in which configural properties were varied to try and tease apart the respective roles of orientation disparity and conflicting perspective in determining stereoscopic slant perception and slant axis anisotropy. The results could not be accounted for by the magnitude of the orientation disparities present. Conflicting perspective cues appeared to play a role but only for slant around a vertical axis. It was concluded that there are important configural effects in stereopsis attributable neither to orientation disparity nor to perspective. 1 Introduction It has become obvious in recent years that stereoscopic depth estimates are not predictable from the geometry of point disparities. There is good evidence, for instance, that disparity discontinuities play a critical role in the perception of stereoscopic surface slant (Gillam et al 1984, 1988; Mitchison and Westheimer 1984) and, similarly, that configural properties (surface contours, markings, and details) are important in determining the slant response to a disparity gradient even in the absence of discontinuities (Gillam 1968; Westheimer and McKee 1979; Cagenello and Rogers 1988; Stevens and Brookes 1988). Furthermore, even for a given configuration, there is a marked anisotropy in slant perception, with slants around a horizontal axis demonstrating a lower detection threshold and faster resolution, as well as a greater magnitude of perceived slant, than slants around a vertical axis (Rogers and Graham 1983; Gillam et al 1988). Computational theories and physiological models of stereopsis will both need to take configural properties and anisotropics into account, and a thorough understanding of the role and basis of those effects is vital to any comprehensive account of stereoscopic vision. Two main factors have been proposed as important determinants of the nature and extent of configural effects in the stereoscopic slant perception of planar surfaces: (i) the presence or absence of orientation disparity; and (ii) the relative effectiveness of the perspective cues offered by such configurations. There has been an interesting divergence of views regarding orientation disparity; Cagenello and Rogers (1988) have argued that the degree of orientation disparity per se determines sensitivity to (and the perceived magnitude of) induced slant, regardless of the meridian along which it occurs, whereas others (Wallach and Bacon 1976; Gillam et al 1988) have suggested that the meridian in which orientation disparity occurs may be crucial, with orientation disparity about the vertical axis playing a special role. For example, Cagenello and Rogers (1988) have proposed that the now well-established slant anisotropy in favour of slant about a horizontal axis (Rogers and Graham 1983; Gillam et al 1988) can be accounted for by the fact that the magnitude of orientation
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disparity produced by rotation about a horizontal axis is greater than that produced by equivalent rotation about the vertical axis. It is important to note that when a planar surface is slanted, orientation disparities are created which vary with: (a) the axis of slant; (b) the absolute orientation of a contour on that surface, (eg vertical, horizontal, or diagonal); and (c) slant magnitude. Figure 1 illustrates how orientation disparity varies with contour orientation for two axes of slant, horizontal and vertical, for a slant magnitude of 30° relative to the frontal plane. It may be seen from figure 1 that neither axis of slant creates orientation disparity for horizontal surface contours. For vertical contours, no orientation disparity is generated by slanting the surface around a vertical axis, but maximal orientation disparity results from slant around a horizontal axis. For diagonal (45° and 135°) contour orientations, equal orientation disparity is generated by slants about the two axes under consideration. There are, however, differences in sign. Slants about a horizontal axis generate orientation disparities of the same sign for both diagonal orientations, whereas slants about the vertical axis generate orientation disparities opposite in sign for 45° and 135° contours. As figure 1 shows, the maximum orientation disparity for a given slant about a horizontal axis occurs for vertical contours and is twice the magnitude of the maximum orientation disparity for the same slant about a vertical axis, which occurs for contours oriented at 45° or 135°. Caganello and Rogers (1988) claim that this explains the better response to stereoscopic slant about a horizontal axis. Gillam et al (1988), however, equalized maximum orientation disparity for the two axes by halving the slant about the horizontal axis relative to the vertical axis, but this did not eliminate the much faster latencies found for the horizontal axis in a timed slantdetection task with random-dot stereograms. Cagenello and Rogers (1988) compared the slant thresholds for grids composed of horizontal and vertical lines with thresholds for grids composed of 45° and 135° diagonals. As figure 1 makes clear, orientation disparity should be unequal for the two axes of slant in the former case and equal in the latter. It was assumed by Cagenello and Rogers that the implicit diagonals in horizontal/vertical (H/V) grids, and the implicit horizontals and verticals in diagonal grids, played no significant role. This seems odd, as stereoscopic slant anisotropy was first demonstrated for randomdot stereograms (Rogers and Graham 1983; Gillam et al 1988), where orientation disparity must be carried by implicit lines. 2.5 r
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Cagenello and Rogers (1988) reported results in accordance with predictions based on the explicit orientation disparity present. For H / V grids, the horizontal-axis condition showed the usual lower slant threshold. For diagonal grids, on the other hand, thresholds for slants about horizontal and vertical axes were equal. These results have been questioned, however, by Mitchison and McKee (1990), who found lower thresholds and also greater magnitude for slants about a vertical axis both for H / V and for diagonal grids. One possible reason for this difference was the small size of Mitchison and McKee's targets: 1.6 to 3.3 deg, compared with the 20 deg fields of Cagenello and Rogers (1988). This possibility is explored below. As noted above, configural effects may also result from the failure of perspective to vary concomitantly with disparity (Gillam 1968; Stevens and Brookes 1988). It is usual, when investigating stereoscopic slant perception, to vary only disparity, keeping texture gradients and linear perspective consistent with a frontal plane. Thus, the perspective properties of a configuration may influence the magnitude of the stereoscopic slant response. Gillam (1968) showed that the very surface details which allow good judgments of slant monocularly, interfere most with a stereoscopic slant response. For example, subjects' slant responses to a field of horizontal dotted lines slanted stereoscopically about a vertical axis were greatly attenuated. Presumably, the lack of convergence of those lines provided good linear perspective information to suggest that the surface was not slanted. On the other hand, vertical lines (of irregular length) elicited an excellent stereoscopic slant response under the same conditions. This is consistent with findings that compression of texture does not compare with line convergence as an effective slant cue (Gillam 1968). One notable feature of these results (all of which were obtained with slant about a vertical axis), is that neither of the stimuli contained any orientation disparities in Cagenello and Rogers' terms, since neither had oblique lines. Hence, either perspective conflict was responsible for the differences between them, or else the patterns differed (in some presently unknown way) in their intrinsic effectiveness in driving the stereoscopic system. The purpose of the present experiments was to explore the effects of orientation disparity and perspective conflict on stereoscopic slant perception by making use of both vertical and horizontal axes of slant. In order to check the generality of their findings, and in order to provide a bridge between Mitchinson and McKee's (1990) findings and thosexof Cagenello and Rogers, we used stimuli some four to eight times larger (12 deg of visual angle) than those used by Mitchison and McKee. These authors measured slant by having subjects set cardboard rectangles, placed on a table beside them, to match perceived slant. This method did not equate conditions of measurement for the two meridians of slant under investigation. We sought to do this by measuring the magnitude of perceived slant by using a comparison technique designed to equate, for the two axes of slant, both the distance of the comparator from, and its geometrical relationship to, the surface to be measured (see section 2). 2 General method In all three experiments reported below, stereograms were^ generated with custom software routines on a PC driving a high-resolution Mitsubishi HJ6905 colour monitor via a Matrox PG-1281C graphics board. Separate images were generated for each eye and then photographed. Slides were made by using pin registration, and stereo pairs were centrally rear-projected onto a ground-glass screen as superimposed polarized images which subjects viewed through orthogonal Polaroid lenses from a distance of 1 m in a darkened room. The projected images were centred at eye level. Stereograms used were of two general classes: random-dot stereograms (Julesz 1971), and pattern stereograms in which surfaces were defined by vertical, horizontal, and/or diagonal lines forming fields or combined into grids. Across the three
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experiments reported below, six types of patterns were used: vertical-line fields, horizontal-line fields, diagonal line fields, H / V grids, diagonal grids, and grids formed by combining (overlaying) H / V and diagonal grids, which we term H / V / D grids. Randomly dotted lines were used in order t6 ensure that all stimuli contained ample disparities so that variations in response would be attributable to the way in which these were distributed or configured. In addition, the use of dotted lines reduced the quantization problem which produces discontinuities at each pixel shift. A customsmoothing algorithm was developed and applied which further reduced the quantization problem, corrected the x/y spacing of the dots for the aspect ratio of the Mitsubishi screen, and overcame the 'rounding' problem inherent in realizing continuous transformations via a digital graphics system. Interdot distances were determined by randomly sampling a uniform distribution. As projected, these interdot distances ranged from 0 mm (contiguous dots) to 3 mm. All the surface configurations used are illustrated in figure 2. It will be noted from this figure that an image-clipping algorithm made the edges of each stimulus irregular. However, all stimulus elements appeared in both fields; there were no monocular elements. In the case of random-dot stereograms, the field was divided into 1600 (40x40) cells, and a random position was assigned to one dot within each cell, subject to there being an arbitrary minimum Euclidean distance between any two dots. As projected, the minimum interdot distance was 2 mm, corresponding to a visual angle of 0.115 deg when viewed from a distance of 1 m. All stereograms used were geometrically consistent with whole-field planar surface slants around either a central vertical or a central horizontal axis. Slant about the vertical axis, was either front-right (positive slant), or front-left (negative slant). Slant about a central horizontal axis was either front-up (positive) or front-down (negative). The theoretical magnitude of those implicit planar slants was either 15° or 30°. Slants about the vertical axis were obtained by presenting the left (right) eye with an untransformed field and the right (left) eye with a field in which points were shifted away from the vertical centreline by an amount proportional to their distance from the centreline. Slants about the horizontal axis were achieved by presenting the left eye with a field in which the points were shifted away from the vertical centreline by a distance proportional (or inversely proportional) to their distance from the horizontal centreline, and the right eye with a field shifted in the opposite direction by the same distance.(1) Each surface configuration was presented eight times: at two slant magnitudes (15° and 30°) combined with two slant directions (positive and negative), for two axes of slant (horizontal and vertical). Within an experiment, each surface configuration was tested equally often. Presentation of all conditions was randomized independently for each subject. Projection was via twin Kodak Ektagraphic S-AV 2050 projectors with orthogonal Polaroid lens filters, positioned such that the visual angle subtended at the eye by each stereogram was 12 deg. The outline of the screen was dimly visible, subtending a visual angle of 26 deg in the horizontal and in the vertical meridians. All absolute disparities for each stereo pair were well above the conventional stereoscopic (1)
Slant is given geometrically by:
where M is the proportional magnification, y is the observation distance, and a is the interocular distance (Ogle 1950).
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threshold (20 s arc) and within conventional fusional areas (14min arc) across the whole pattern. Each pair of superimposed slides was exposed, in turn, by the experimenter, after which a computer-driven shutter was raised in the subject's line of sight. Subjects indicated their estimate of surface slant by adjusting the internally illuminated one of two circular-disk comparators adjacent to each projected stereogram. The front surface of each comparator was marked with concentric circular lines and was segmented by four narrow, black, diagonal bars. One comparator could be rotated about its vertical axis (thus enabling the subject to indicate slant around the vertical axis) by depressing either of two vertically opposed buttons on a desktop console conveniently located in front of the subject. The other comparator could be rotated about its horizontal axis by using either of two horizontally opposed buttons, similarly located. The former comparator was located 3 cm (visual angle, 1.72 deg) to the left of the projected image, midway between its top and bottom edges, and the latter was located 3 cm below the image, midway between its left and right edges. Placing the former beside and the latter below the surface to be matched ensured that the
Figure 2. Surface configurations used in experiments 1-3.
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comparator had the same geometric relationship to the surface in each case.*2* Each comparator was 75 mm (visual angle, 4.29 deg) in diameter and was positioned 75 mm in front of the plane of the screen. When each comparator was rotated, one edge became visible: each was 100 mm thick. The experimenter lighted one or other of the comparators prior to exposure of each stereo pair, indicating whether slant would be about the vertical or horizontal axis. Comparators were adjusted to the frontal plane prior to each trial. Subjects were instructed to use a bracketing technique when adjusting the comparator so as to be parallel to each slanted surface, ie first deliberately overestimating, then underestimating, then progressively finetuning the settings by using a homing-in strategy. Each comparator drove a potentiometer, calibrated prior to each testing session, to reflect veridically the magnitude (in degrees) of perceived slant. A computer automatically recorded and converted potentiometer output to estimates of slant. Subjects were screened for stereoscopic and visual acuity prior to testing and were accepted only if they had 20/20 (un)corrected vision in both eyes and stereoacuity of at least 20 s arc on the Randot test at a distance of 16 inches (40 cm). 3 Experiment 1 In this first experiment subjects' estimates of the slant of surfaces configured with H / V grids and diagonal grids, and stereoscopically rotated about either a horizontal or a vertical axis, were measured. These data were compared with the perceived slant of homologous random-dot stereograms which do not contain explicitly orientated lines (see figure 1 for an illustration of these surface configurations). Anisotropy in slant magnitude (as opposed to slant thresholds and latencies) had not previously been demonstrated for random-dot patterns. 3.1 Method Eleven volunteers from the departmental panel acted as subjects. A four-way, fullycrossed, within-subjects, factorial design was used to examine the effects of surface configuration (random dots, diagonal grids, and H / V grids), axis of slant (horizontal and vertical), theoretical slant (15° and 30°), and slant direction (positive and negative). Each subject was shown a total of twenty-four stereo pairs in a unique random sequence—one pair for each factorial combination: Surface Configuration^) x Axis of Rotation(2) x SlantMagnitude(2) x SlantDirection(2). 3.2 Results It is clear from figure 3 (and is confirmed by an appropriate four-way ANOVA) that estimates of perceived slant about a horizontal axis were markedly greater than those about the vertical axis (F 110 = 49.57, p < 0.001), and that the observed horizontalaxis superiority was characteristic of all three surface configurations. These data confirm Mitchison and McKee's principal finding but with larger stimuli. It is also clear from figure 3 that performance was much better for the diagonal and random-dot patterns than for the H / V grid patterns in the vertical-axis condition.
0.05). This is in accordance with Cagenello and Rogers' finding that diagonal grids improve performance relative to H / V grids when the configured surface is rotated about the vertical axis. However, it is unlikely that this superiority results from the presence of explicit diagonals and their orientation disparity, because it is shared by the random-dot pattern which, like the H / V grid pattern, has only implicit orientation disparities. It is more likely that the attenuated apparent slant for surfaces configured with H / V grids is attributable to the conflict produced by linear perspective indications that the patterns are really in the frontal plane. A diagonal grid also has linear perspective conflicting with stereopsis, of course, but in a much weaker form; slant for slant, the diagonal lines of the 45° grid converge only half as much as the horizontal lines of an H / V grid. It could be argued, therefore, that the absence of appropriate (congruent) line convergence should produce less extreme conflict in the case of diagonals. The same general argument with respect to linear perspective conflict applies, of course, when configured planar surfaces are rotated about a horizontal axis. Newman-Keuls decomposition confirmed that perceived slant in the H / V grids condition was significantly less than that for random-dot patterns (p < 0.05) and, again, dots and diagonal grids did not differ significantly from each other (p > 0.05). However, differences were rather less marked in this case, and the H / V griddiagonal grid contrast was not significant (p > 0.05). This could be taken as disconfirmation of the perspective-conflict hypothesis, or as an indication that there is some special feature of slants about the horizontal axis which makes them relatively immune to conflicting perspective. Gillam et al (1988) found that, even for slants about the vertical axis, the attenuating effects of perspective on perceived slant can be eliminated when powerful stereoscopic information is present in the form of a gradient of disparity discontinuities at the edge of a surface. As suggested by Gillam et al (1988), and earlier in the present paper, orientation disparity about the vertical characteristic of surfaces rotated about a horizontal axis may be the special factor which makes such surfaces resistant to the effects of perspective conflict.
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Axis of Slant • 1 horizontal liiliilj vertical
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H/V diagonal grids grids Surface Configuration Figure 3. Results of experiment 1. Perceived slant as a function of Surface Configuration with Axis of Slant as a parameter. Data are pooled across Slant Direction and Slant Magnitude. Slant Magnitude is shown as a parameter in figure 6. (3) Although the Surface Configuration x Axis of Rotation interaction was not significant in this case (p > 0.05) the Newman-Keuls decomposition is justified in that it was designed to check a priori hypotheses of interest (see Winer 1962). It is germane to note that the Surface Configuration x Axis of Rotation interaction was highly significant in experiments 2 and 3.
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Some of the slant judgments in the vertical-axis condition were reversed in direction. These were scored as zero in the analysis. An analysis and discussion of the reversals will be undertaken in a separate paper.(4) As might be expected, slant judgments were unaffected by the direction of surface recession (left or right, up or down); the effect of Slant Direction did not approach significance (F < 1). On the other hand, Slant Magnitude (15° and 30°) had a highly significant effect on performance, with a slant of 30° being reliably judged to be greater than a slant of 15° (F 1 1 0 = 37.53; p < 0.001). Under optimum conditions (the random-dot pattern and the horizontal-axis condition), the slants obtained in this experiment were of the geometrically predicted magnitude (see figure 6), and failed to show the general attenuation of slant found by Mitchison and McKee. 4 Experiment 2 Experiment 2 was designed to explore Gillam's (1968) finding that the perceived magnitude of stereoscopic slant about the vertical is even less for the horizontal components of an H / V grid than for the grid itself, whereas stereoscopic slant responses to the vertical components are markedly greater. It seemed important to replicate this finding since it demonstrates powerful configural effects on perceived slants about a vertical axis which cannot be attributed to explicit, or strong implicit, orientation disparity. By strong implicit orientation we mean alignments of line intersections. A second goal of experiment 2 was to see if equivalent effects of lines and their combination were to be found for surfaces stereoscopically slanted about a horizontal axis. Considering only the perspective implications of the configurations, the vertical axis findings should be reversed for horizontal axes, since, in the latter case, linear perspective dictates that it is the vertical lines which should converge in order to be consistent with the stereoscopic indicators and which may therefore show an attenuated stereoscopic response. The horizontal lines would be subject to compression, which is a much less powerful slant cue (Gillam 1968) than linear perspective. On the other hand, if orientation disparity at the vertical meridian (which occurs only in the horizontal axis condition) has a special stereoscopic status, this may offset the attenuation predicted for vertical line stimuli. 4.1 Method Fourteen volunteers drawn from the departmental panel acted as subjects. Again, a four-way, fully-crossed, within-subjects, factorial design was used. Experiment 2 differed from experiment 1 only in terms of the surface configurations used. Stimuli were either horizontal lines, vertical lines, or H / V grids obtained by overlaying the vertical and horizontal lines. 4.2 Results The key data are shown in figure 4. The effects of Surface Configuration (^2,26 = 4.62, p < 0.05) and Axis of Slant (Ful3 = 14.05, p < 0.01) were significant, as was their interaction (F226 = 21.48, p < 0.001). Consistent with the results of experiment 1, geometrically equivalent slants about a horizontal axis were judged to be greater than those about a vertical axis. Considering the vertical axis conditions alone, the data were consistent with Gillam's (1968) results. The mean slant response to horizontal-line fields was lower than that to surfaces configured with H / V grids. 0.05). In accordance with predictions from the type of perspective conflict present, the horizontal line stimulus in the horizontal axis condition did produce a similar slant response to the vertical line stimulus in the vertical axis condition. However, the vertical line stimulus in the horizontal axis condition appeared much more slanted than the horizontal line stimulus in the vertical axis condition. Perspective considerations cannot account for this difference. Again, as in experiment 1, direction of slant did not affect slant-magnitude estimates {F < 1), whereas the magnitude of slant (15° or 30°) had a highly significant effect on performance (Ful3 = 65.29, p < 0.001).
Axis of Slant horizontal liiiiilj vertical horizontal lines
vertical H/V lines grids Surface Configuration Figure 4. Results of experiment 2. Perceived slant as a function of Surface Configuration with Axis of Slant as a parameter. Data are pooled across Slant Direction and Slant Magnitude. Slant Magnitude is shown as a parameter in figure 6.
5 Experiment 3 To provide continuity and comparability, several conditions from the first two experiments were included in experiment 3: the H/V-grid and diagonal-grid conditions from experiment 1, and the vertical-line and horizontal-line conditions from experiment 2. Two new conditions were added: diagonal-line fields and H / V / D grids (see figure 2). The aims were (i) to confirm the configuration effects found in earlier experiments in a single experiment, thus enabling further comparisons to be made between the various effects; and (ii) to extend the range of configurations to enable additional hypotheses to be tested. The line density of the H / V and diagonal components of the H / V / D grids was reduced in comparison with their presentation in isolation so that the line density of the composite figure would not be so great as to constitute texture rather than pattern. Diagonal fields were included to see if the advantage for slants about the horizontal axis in the diagonal-grid condition would be preserved when (a) the strongly implicit H / V contours in a diagonal grid were eliminated, and (b) orientation disparities all had the same sign for both axes of slant, thus eliminating a major difference between them. Mitchison and McKee had found a horizontal-axis advantage for much smaller stimuli of this type.
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The purpose of including the composite H / V / D grid was to test the hypothesis for surfaces slanted around a vertical axis that the greater slant response to surfaces marked with diagonal grids compared with H/V grids resulted not from the presence of explicit orientation-disparity but from the absence of strong, conflicting, linear perspective. The H/V/D-grid stimuli contain explicit orientation-disparity information, but also provide strong, conflicting, linear perspective. If the presence of orientation disparity were responsible for the effectiveness of the diagonal grid, the H/V/D grid should be as effective as the diagonal grid. If the absence of strong linear perspective were responsible, then the H / V / D grids should be less effective than the diagonal grids and more like the H/V grids. 5.1 Method Again a four-way, fully-crossed, within-subjects, factorial design was used. The experiment differed from experiments 1 and 2 only in terms of the range of surface configurations used. Fourteen volunteers from the departmental panel acted as subjects. 5.2 Results The results for conditions used in previous experiments confirmed our earlier findings. The main effects of Surface Configuration (F 565 = 6.45, p < 0.001), Axis of Slant ( F U 3 = 68.81, p < 0.001), and Slant Magnitude (F1>13 = 126.14, p < 0.001) were all highly significant. The Surface Configuration x Axis of Slant interaction was significant (F5>65 = 10.33, p < 0.001). In all conditions (except vertical lines), the estimated magnitude of slants about a horizontal axis was significantly greater than that for equivalent surfaces rotated about the vertical (Newman-Keuls; p < 0.05 in all cases except vertical lines). Figure 5 shows the perceived slant for each Surface Configuration, with Axis of Slant as a parameter. As was the case in experiment 1, slant settings for diagonal grids were significantly greater than those for H / V grids for slants around a vertical axis (p < 0.05), whereas differences in perceived slant were considerably less (and not statistically significant, p > 0.05) for slants about the horizontal. Settings were substantially attenuated for horizontal-line fields, relative to all other configurations for slants about the vertical (Newman-Keuls, p< 0.05 for all contrasts) whereas, as in experiment 2, surfaces configured with vertical-line fields and rotated about the horizontal showed only a slight attenuation relative to other configurations, which failed to reach statistical significance except for the contrast with the diagonal line condition. Thus, equivalent perspective conflict had different effects for the two axes of slant.
Axis of Slant • H horizontal I vertical horizontal lines
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diagonal H/V lines grids Surface Configuration
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Figure 5. Results of experiment 3. Perceived slant as a function of Slant Configuration with Axis of Slant as a parameter. Data are pooled across Slant Direction and Slant Magnitude. Slant Magnitude is shown as a parameter in figure 6.
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In the vertical axis conditions the mean slant for the diagonal lines was lower than the mean for the diagonal grid, but this just failed to reach significance on a Newman-Keuls test (p < 0.05). Both diagonal lines and diagonal grids had significantly higher mean slants than H/V grids and horizontal lines (p < 0.05). This shows that the effectiveness of diagonals in eliciting a stereoscopic slant response does not depend on the presence of orientation disparity in two directions nor on implicit vertical lines. For slant around the horizontal axis the diagonal lines were as effective as any other condition (significantly more so than the vertical line condition) despite having only half the orientation disparity of the H / V grid and the vertical lines. The mean slant around a vertical axis of H/V/D grids was significantly greater than was obtained for H/V grids alone {p < 0.05) suggesting that the effectiveness of diagonals in this case is not simply due to weak perspective cues. In all three experiments reported here, the effect of Slant Magnitude was highly significant, with 30° slants being judged to be greater in magnitude than 15° slants. In figure 6, perceived slant is plotted as a function of Surface Configuration with Slant Magnitude and Axis of Slant as parameters. This figure shows data combined from all three experiments. Where conditions were common to two or more experiments, these data were pooled and the arithmetic mean calculated and plotted. All configurations (except the unique vertical-lines condition) showed the characteristic slantaxis anisotropy. All demonstrate a clear effect of Slant Magnitude.
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Figure 6. Combined data from experiments 1-3. Perceived slant as a function of Surface Configuration with Axis of Slant and Slant Magnitude as parameters. Data are pooled across Slant Direction. 6 General discussion Our data support the view that stereoscopic slant perception around a vertical axis is greater for diagonal grids than for H / V grids (Cagenello and Rogers 1988). However, it is clear that the presence or absence of explicit orientation disparity is unlikely to account for these differences. Slant estimates were greatest for surfaces slanted about a vertical axis when they were configured with vertical lines, despite the fact that these entirely lack explicit orientation disparity. We have proposed that diagonal grids are 'better' stimuli for slant about a vertical axis because they provide weaker perspective information than does a H / V grid. This conjecture awaits direct testing in monocular vision. Also consideration of the H/V/D grid results suggests that this is not the only reason.. Whereas for slants about the vertical, conflicting perspective provides a plausible account of the good response to vertical lines and of the poor response to horizontal
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lines and H / V grids, it is not sufficient to account for other aspects of our data. It does not account for the fact that horizontal lines are worse than H / V grids even though perspective information is (if anything) stronger in the latter case. Also, it sits uneasily with the data from the horizontal-axis conditions (which were much less affected by equivalent conflicting perspective information). In our view, one conclusion is inescapable: there are configural effects in stereopsis which have their origins neither in orientation disparity per se, nor in conflicting perspective. Vertical lines seem to be particularly powerful drivers of the stereoscopic response to surfaces rotated about a vertical axis. This is demonstrated by the fact that, when they are added to horizontal lines, subjects' slant estimates increase despite the fact that the verticals also increase conflicting linear perspective. This certainly requires further investigation, especially since the effects we obtained with irregular vertical lines do not seem to be obtained with regular vertical lines forming implicit rectangles (Mitchison and Westheimer 1984). Clearly, there is a property of configured surfaces rotated about a horizontal axis which makes estimates of surface slant relatively resistant to conflicting perspective information. This factor is unlikely to be orientation disparity. Consider the diagonal line condition of experiment 3: horizontal and vertical axis slants produce identical orientation disparity and identical conflicting perspective for this stimulus and yet the horizontal axis slant response was far greater than the vertical. One possibility is that a mechanism exists for detecting image shear per se. This is in line with the original observation by Wallach and Bacon (1976) on the importance of 'transverse disparity', and the findings of Rogers and Graham (1983) on the effectiveness of differential image shear relative to image expansion in both the stereo and motion parallax domains. The orientation disparity produced by rotating a surface configured with vertical lines about a horizontal axis is one manifestation of this shearing, but it is not essential to the effectiveness of shear and is, therefore, unlikely to be the mechanism by which shear is detected (Morgan 1989; Rogers and Cagenello 1989). In conclusion, the present data do not support the view that orientation disparity underlies stereoscopic surface slant perception, despite Ninio's (1985) finding that orientation disparity plays an important role in determining the depths of fine-grained 'needle' surface textures (in the absence of strong perspective conflict). The demonstration by Blakemore et al (1972) of a "second neural mechanism for depth perception" founded on orientation disparities has held out an enduring promise. We have, however, found nothing to suggest that orientation disparity plays a significant role in stereopsis. Acknowledgements. This research was supported by Grant No. B350307 from the Australian Research Council. The authors wish to thank Christopher Foster for assistance in running the experiments, and Hal Sedgwick for his comments on the manuscript in draft form. References Blakemore C, Fiorentini A, Maffei L, 1972 "A second neural mechanism of binocular depth discrimination" Journal of Physiology 226 725 - 749 Cagenello R, Rogers B J, 1988 "Local orientation differences affect the perceived slant of stereoscopic surfaces" Investigative Ophthalmology and Visual Science, Supplement 29 399 (abstract) GillamB, 1968 "Perception of slant when perspective and stereopsis conflict: experiments with aniseikonic lenses" Journal of Experimental Psychology 7 8 2 9 9 - 3 0 5 Gillam B, Chambers D, Russo T, 1988 "Postfusional latency in stereoscopic slant perception and the primitives of stereopsis" Journal of Experimental Psychology: Human Perception and Performance 14 163-175 GillamB, Flagg T, F inlay D, 1984 "Evidence for disparity change as the primary stimulus for stereoscopic processing" Perception &Psychophysics 36 559-564
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Julesz B, 1971 The Foundations of Cyclopean Perception (Chicago, IL: University of Chicago Press) MitchisonGJ, McKee S P, 1990 "Mechanisms underlying the anisotropy of stereoscopic tilt perception" Vision Research 301781-1791 Mitchison G J, Westheimer G, 1984 "The perception of depth in simple figures" Vision Research 24 1063-1073 Morgan M J, 1989 "Vision of solid objects" Nature (London) 339 101 - 1 0 3 NinioJ, 1985 "Orientational versus horizontal disparity in the stereoscopic appreciation of slant" Perception 1 4 3 0 5 - 3 1 4 Ogle K N, 1950 Researches in Binocular Vision (New York: Hafner) Rogers BM, Cagenello R, 1989 "Disparity curvature and the perception of three-dimensional surfaces" Nature (London) 339 135-137 Rogers B M, Graham M, 1983 "Anisotropics in the perception of three-dimensional surfaces" Science 111 1409-1411 Stevens K A, Brookes A, 1988 "Integrating stereopsis with monocular interpretations of planar surfaces" Vision Research 2 8 3 7 1 - 3 8 6 WallachH, Bacon J, 1976 "Two forms of retinal disparity" Perception & Psychophysics 19 375-382 Westheimer G, McKee S P, 1979 "What prior uniocular processing is necessary for stereopsis" Investigative Ophthalmology and Visual Science 18614-621 Winer B J, 1962 Statistical Principles in Experimental Design (New York: McGraw-Hill) p 85
Perspective, orientation disparity, and anisotropy in stereoscopic slant perception Barbara Gillam, Colin Ryan Department of Psychology, University of New South Wales, PO Box 1, Kensington NSW 2033, Australia Received 26 April 1991, in revised form 22 August 1991
Abstract. Stereoscopic depth estimates are not predictable from the geometry of point disparities. The configural properties of surfaces (surface contours) may play an important role in determining, for example, slant responses to a disparity gradient, and the marked anisotropy in favour of slant around a horizontal axis. It has been argued that variation in slant magnitude are attributable to the degree of perspective conflict present and that anisotropy is attributable to orientation disparity, which varies with the axis of slant. Three experiments were conducted in which configural properties were varied to try and tease apart the respective roles of orientation disparity and conflicting perspective in determining stereoscopic slant perception and slant axis anisotropy. The results could not be accounted for by the magnitude of the orientation disparities present. Conflicting perspective cues appeared to play a role but only for slant around a vertical axis. It was concluded that there are important configural effects in stereopsis attributable neither to orientation disparity nor to perspective. 1 Introduction It has become obvious in recent years that stereoscopic depth estimates are not predictable from the geometry of point disparities. There is good evidence, for instance, that disparity discontinuities play a critical role in the perception of stereoscopic surface slant (Gillam et al 1984, 1988; Mitchison and Westheimer 1984) and, similarly, that configural properties (surface contours, markings, and details) are important in determining the slant response to a disparity gradient even in the absence of discontinuities (Gillam 1968; Westheimer and McKee 1979; Cagenello and Rogers 1988; Stevens and Brookes 1988). Furthermore, even for a given configuration, there is a marked anisotropy in slant perception, with slants around a horizontal axis demonstrating a lower detection threshold and faster resolution, as well as a greater magnitude of perceived slant, than slants around a vertical axis (Rogers and Graham 1983; Gillam et al 1988). Computational theories and physiological models of stereopsis will both need to take configural properties and anisotropics into account, and a thorough understanding of the role and basis of those effects is vital to any comprehensive account of stereoscopic vision. Two main factors have been proposed as important determinants of the nature and extent of configural effects in the stereoscopic slant perception of planar surfaces: (i) the presence or absence of orientation disparity; and (ii) the relative effectiveness of the perspective cues offered by such configurations. There has been an interesting divergence of views regarding orientation disparity; Cagenello and Rogers (1988) have argued that the degree of orientation disparity per se determines sensitivity to (and the perceived magnitude of) induced slant, regardless of the meridian along which it occurs, whereas others (Wallach and Bacon 1976; Gillam et al 1988) have suggested that the meridian in which orientation disparity occurs may be crucial, with orientation disparity about the vertical axis playing a special role. For example, Cagenello and Rogers (1988) have proposed that the now well-established slant anisotropy in favour of slant about a horizontal axis (Rogers and Graham 1983; Gillam et al 1988) can be accounted for by the fact that the magnitude of orientation
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disparity produced by rotation about a horizontal axis is greater than that produced by equivalent rotation about the vertical axis. It is important to note that when a planar surface is slanted, orientation disparities are created which vary with: (a) the axis of slant; (b) the absolute orientation of a contour on that surface, (eg vertical, horizontal, or diagonal); and (c) slant magnitude. Figure 1 illustrates how orientation disparity varies with contour orientation for two axes of slant, horizontal and vertical, for a slant magnitude of 30° relative to the frontal plane. It may be seen from figure 1 that neither axis of slant creates orientation disparity for horizontal surface contours. For vertical contours, no orientation disparity is generated by slanting the surface around a vertical axis, but maximal orientation disparity results from slant around a horizontal axis. For diagonal (45° and 135°) contour orientations, equal orientation disparity is generated by slants about the two axes under consideration. There are, however, differences in sign. Slants about a horizontal axis generate orientation disparities of the same sign for both diagonal orientations, whereas slants about the vertical axis generate orientation disparities opposite in sign for 45° and 135° contours. As figure 1 shows, the maximum orientation disparity for a given slant about a horizontal axis occurs for vertical contours and is twice the magnitude of the maximum orientation disparity for the same slant about a vertical axis, which occurs for contours oriented at 45° or 135°. Caganello and Rogers (1988) claim that this explains the better response to stereoscopic slant about a horizontal axis. Gillam et al (1988), however, equalized maximum orientation disparity for the two axes by halving the slant about the horizontal axis relative to the vertical axis, but this did not eliminate the much faster latencies found for the horizontal axis in a timed slantdetection task with random-dot stereograms. Cagenello and Rogers (1988) compared the slant thresholds for grids composed of horizontal and vertical lines with thresholds for grids composed of 45° and 135° diagonals. As figure 1 makes clear, orientation disparity should be unequal for the two axes of slant in the former case and equal in the latter. It was assumed by Cagenello and Rogers that the implicit diagonals in horizontal/vertical (H/V) grids, and the implicit horizontals and verticals in diagonal grids, played no significant role. This seems odd, as stereoscopic slant anisotropy was first demonstrated for randomdot stereograms (Rogers and Graham 1983; Gillam et al 1988), where orientation disparity must be carried by implicit lines. 2.5 r
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Cagenello and Rogers (1988) reported results in accordance with predictions based on the explicit orientation disparity present. For H / V grids, the horizontal-axis condition showed the usual lower slant threshold. For diagonal grids, on the other hand, thresholds for slants about horizontal and vertical axes were equal. These results have been questioned, however, by Mitchison and McKee (1990), who found lower thresholds and also greater magnitude for slants about a vertical axis both for H / V and for diagonal grids. One possible reason for this difference was the small size of Mitchison and McKee's targets: 1.6 to 3.3 deg, compared with the 20 deg fields of Cagenello and Rogers (1988). This possibility is explored below. As noted above, configural effects may also result from the failure of perspective to vary concomitantly with disparity (Gillam 1968; Stevens and Brookes 1988). It is usual, when investigating stereoscopic slant perception, to vary only disparity, keeping texture gradients and linear perspective consistent with a frontal plane. Thus, the perspective properties of a configuration may influence the magnitude of the stereoscopic slant response. Gillam (1968) showed that the very surface details which allow good judgments of slant monocularly, interfere most with a stereoscopic slant response. For example, subjects' slant responses to a field of horizontal dotted lines slanted stereoscopically about a vertical axis were greatly attenuated. Presumably, the lack of convergence of those lines provided good linear perspective information to suggest that the surface was not slanted. On the other hand, vertical lines (of irregular length) elicited an excellent stereoscopic slant response under the same conditions. This is consistent with findings that compression of texture does not compare with line convergence as an effective slant cue (Gillam 1968). One notable feature of these results (all of which were obtained with slant about a vertical axis), is that neither of the stimuli contained any orientation disparities in Cagenello and Rogers' terms, since neither had oblique lines. Hence, either perspective conflict was responsible for the differences between them, or else the patterns differed (in some presently unknown way) in their intrinsic effectiveness in driving the stereoscopic system. The purpose of the present experiments was to explore the effects of orientation disparity and perspective conflict on stereoscopic slant perception by making use of both vertical and horizontal axes of slant. In order to check the generality of their findings, and in order to provide a bridge between Mitchinson and McKee's (1990) findings and thosexof Cagenello and Rogers, we used stimuli some four to eight times larger (12 deg of visual angle) than those used by Mitchison and McKee. These authors measured slant by having subjects set cardboard rectangles, placed on a table beside them, to match perceived slant. This method did not equate conditions of measurement for the two meridians of slant under investigation. We sought to do this by measuring the magnitude of perceived slant by using a comparison technique designed to equate, for the two axes of slant, both the distance of the comparator from, and its geometrical relationship to, the surface to be measured (see section 2). 2 General method In all three experiments reported below, stereograms were^ generated with custom software routines on a PC driving a high-resolution Mitsubishi HJ6905 colour monitor via a Matrox PG-1281C graphics board. Separate images were generated for each eye and then photographed. Slides were made by using pin registration, and stereo pairs were centrally rear-projected onto a ground-glass screen as superimposed polarized images which subjects viewed through orthogonal Polaroid lenses from a distance of 1 m in a darkened room. The projected images were centred at eye level. Stereograms used were of two general classes: random-dot stereograms (Julesz 1971), and pattern stereograms in which surfaces were defined by vertical, horizontal, and/or diagonal lines forming fields or combined into grids. Across the three
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experiments reported below, six types of patterns were used: vertical-line fields, horizontal-line fields, diagonal line fields, H / V grids, diagonal grids, and grids formed by combining (overlaying) H / V and diagonal grids, which we term H / V / D grids. Randomly dotted lines were used in order t6 ensure that all stimuli contained ample disparities so that variations in response would be attributable to the way in which these were distributed or configured. In addition, the use of dotted lines reduced the quantization problem which produces discontinuities at each pixel shift. A customsmoothing algorithm was developed and applied which further reduced the quantization problem, corrected the x/y spacing of the dots for the aspect ratio of the Mitsubishi screen, and overcame the 'rounding' problem inherent in realizing continuous transformations via a digital graphics system. Interdot distances were determined by randomly sampling a uniform distribution. As projected, these interdot distances ranged from 0 mm (contiguous dots) to 3 mm. All the surface configurations used are illustrated in figure 2. It will be noted from this figure that an image-clipping algorithm made the edges of each stimulus irregular. However, all stimulus elements appeared in both fields; there were no monocular elements. In the case of random-dot stereograms, the field was divided into 1600 (40x40) cells, and a random position was assigned to one dot within each cell, subject to there being an arbitrary minimum Euclidean distance between any two dots. As projected, the minimum interdot distance was 2 mm, corresponding to a visual angle of 0.115 deg when viewed from a distance of 1 m. All stereograms used were geometrically consistent with whole-field planar surface slants around either a central vertical or a central horizontal axis. Slant about the vertical axis, was either front-right (positive slant), or front-left (negative slant). Slant about a central horizontal axis was either front-up (positive) or front-down (negative). The theoretical magnitude of those implicit planar slants was either 15° or 30°. Slants about the vertical axis were obtained by presenting the left (right) eye with an untransformed field and the right (left) eye with a field in which points were shifted away from the vertical centreline by an amount proportional to their distance from the centreline. Slants about the horizontal axis were achieved by presenting the left eye with a field in which the points were shifted away from the vertical centreline by a distance proportional (or inversely proportional) to their distance from the horizontal centreline, and the right eye with a field shifted in the opposite direction by the same distance.(1) Each surface configuration was presented eight times: at two slant magnitudes (15° and 30°) combined with two slant directions (positive and negative), for two axes of slant (horizontal and vertical). Within an experiment, each surface configuration was tested equally often. Presentation of all conditions was randomized independently for each subject. Projection was via twin Kodak Ektagraphic S-AV 2050 projectors with orthogonal Polaroid lens filters, positioned such that the visual angle subtended at the eye by each stereogram was 12 deg. The outline of the screen was dimly visible, subtending a visual angle of 26 deg in the horizontal and in the vertical meridians. All absolute disparities for each stereo pair were well above the conventional stereoscopic (1)
Slant is given geometrically by:
where M is the proportional magnification, y is the observation distance, and a is the interocular distance (Ogle 1950).
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threshold (20 s arc) and within conventional fusional areas (14min arc) across the whole pattern. Each pair of superimposed slides was exposed, in turn, by the experimenter, after which a computer-driven shutter was raised in the subject's line of sight. Subjects indicated their estimate of surface slant by adjusting the internally illuminated one of two circular-disk comparators adjacent to each projected stereogram. The front surface of each comparator was marked with concentric circular lines and was segmented by four narrow, black, diagonal bars. One comparator could be rotated about its vertical axis (thus enabling the subject to indicate slant around the vertical axis) by depressing either of two vertically opposed buttons on a desktop console conveniently located in front of the subject. The other comparator could be rotated about its horizontal axis by using either of two horizontally opposed buttons, similarly located. The former comparator was located 3 cm (visual angle, 1.72 deg) to the left of the projected image, midway between its top and bottom edges, and the latter was located 3 cm below the image, midway between its left and right edges. Placing the former beside and the latter below the surface to be matched ensured that the
Figure 2. Surface configurations used in experiments 1-3.
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comparator had the same geometric relationship to the surface in each case.*2* Each comparator was 75 mm (visual angle, 4.29 deg) in diameter and was positioned 75 mm in front of the plane of the screen. When each comparator was rotated, one edge became visible: each was 100 mm thick. The experimenter lighted one or other of the comparators prior to exposure of each stereo pair, indicating whether slant would be about the vertical or horizontal axis. Comparators were adjusted to the frontal plane prior to each trial. Subjects were instructed to use a bracketing technique when adjusting the comparator so as to be parallel to each slanted surface, ie first deliberately overestimating, then underestimating, then progressively finetuning the settings by using a homing-in strategy. Each comparator drove a potentiometer, calibrated prior to each testing session, to reflect veridically the magnitude (in degrees) of perceived slant. A computer automatically recorded and converted potentiometer output to estimates of slant. Subjects were screened for stereoscopic and visual acuity prior to testing and were accepted only if they had 20/20 (un)corrected vision in both eyes and stereoacuity of at least 20 s arc on the Randot test at a distance of 16 inches (40 cm). 3 Experiment 1 In this first experiment subjects' estimates of the slant of surfaces configured with H / V grids and diagonal grids, and stereoscopically rotated about either a horizontal or a vertical axis, were measured. These data were compared with the perceived slant of homologous random-dot stereograms which do not contain explicitly orientated lines (see figure 1 for an illustration of these surface configurations). Anisotropy in slant magnitude (as opposed to slant thresholds and latencies) had not previously been demonstrated for random-dot patterns. 3.1 Method Eleven volunteers from the departmental panel acted as subjects. A four-way, fullycrossed, within-subjects, factorial design was used to examine the effects of surface configuration (random dots, diagonal grids, and H / V grids), axis of slant (horizontal and vertical), theoretical slant (15° and 30°), and slant direction (positive and negative). Each subject was shown a total of twenty-four stereo pairs in a unique random sequence—one pair for each factorial combination: Surface Configuration^) x Axis of Rotation(2) x SlantMagnitude(2) x SlantDirection(2). 3.2 Results It is clear from figure 3 (and is confirmed by an appropriate four-way ANOVA) that estimates of perceived slant about a horizontal axis were markedly greater than those about the vertical axis (F 110 = 49.57, p < 0.001), and that the observed horizontalaxis superiority was characteristic of all three surface configurations. These data confirm Mitchison and McKee's principal finding but with larger stimuli. It is also clear from figure 3 that performance was much better for the diagonal and random-dot patterns than for the H / V grid patterns in the vertical-axis condition.
0.05). This is in accordance with Cagenello and Rogers' finding that diagonal grids improve performance relative to H / V grids when the configured surface is rotated about the vertical axis. However, it is unlikely that this superiority results from the presence of explicit diagonals and their orientation disparity, because it is shared by the random-dot pattern which, like the H / V grid pattern, has only implicit orientation disparities. It is more likely that the attenuated apparent slant for surfaces configured with H / V grids is attributable to the conflict produced by linear perspective indications that the patterns are really in the frontal plane. A diagonal grid also has linear perspective conflicting with stereopsis, of course, but in a much weaker form; slant for slant, the diagonal lines of the 45° grid converge only half as much as the horizontal lines of an H / V grid. It could be argued, therefore, that the absence of appropriate (congruent) line convergence should produce less extreme conflict in the case of diagonals. The same general argument with respect to linear perspective conflict applies, of course, when configured planar surfaces are rotated about a horizontal axis. Newman-Keuls decomposition confirmed that perceived slant in the H / V grids condition was significantly less than that for random-dot patterns (p < 0.05) and, again, dots and diagonal grids did not differ significantly from each other (p > 0.05). However, differences were rather less marked in this case, and the H / V griddiagonal grid contrast was not significant (p > 0.05). This could be taken as disconfirmation of the perspective-conflict hypothesis, or as an indication that there is some special feature of slants about the horizontal axis which makes them relatively immune to conflicting perspective. Gillam et al (1988) found that, even for slants about the vertical axis, the attenuating effects of perspective on perceived slant can be eliminated when powerful stereoscopic information is present in the form of a gradient of disparity discontinuities at the edge of a surface. As suggested by Gillam et al (1988), and earlier in the present paper, orientation disparity about the vertical characteristic of surfaces rotated about a horizontal axis may be the special factor which makes such surfaces resistant to the effects of perspective conflict.
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H/V diagonal grids grids Surface Configuration Figure 3. Results of experiment 1. Perceived slant as a function of Surface Configuration with Axis of Slant as a parameter. Data are pooled across Slant Direction and Slant Magnitude. Slant Magnitude is shown as a parameter in figure 6. (3) Although the Surface Configuration x Axis of Rotation interaction was not significant in this case (p > 0.05) the Newman-Keuls decomposition is justified in that it was designed to check a priori hypotheses of interest (see Winer 1962). It is germane to note that the Surface Configuration x Axis of Rotation interaction was highly significant in experiments 2 and 3.
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Some of the slant judgments in the vertical-axis condition were reversed in direction. These were scored as zero in the analysis. An analysis and discussion of the reversals will be undertaken in a separate paper.(4) As might be expected, slant judgments were unaffected by the direction of surface recession (left or right, up or down); the effect of Slant Direction did not approach significance (F < 1). On the other hand, Slant Magnitude (15° and 30°) had a highly significant effect on performance, with a slant of 30° being reliably judged to be greater than a slant of 15° (F 1 1 0 = 37.53; p < 0.001). Under optimum conditions (the random-dot pattern and the horizontal-axis condition), the slants obtained in this experiment were of the geometrically predicted magnitude (see figure 6), and failed to show the general attenuation of slant found by Mitchison and McKee. 4 Experiment 2 Experiment 2 was designed to explore Gillam's (1968) finding that the perceived magnitude of stereoscopic slant about the vertical is even less for the horizontal components of an H / V grid than for the grid itself, whereas stereoscopic slant responses to the vertical components are markedly greater. It seemed important to replicate this finding since it demonstrates powerful configural effects on perceived slants about a vertical axis which cannot be attributed to explicit, or strong implicit, orientation disparity. By strong implicit orientation we mean alignments of line intersections. A second goal of experiment 2 was to see if equivalent effects of lines and their combination were to be found for surfaces stereoscopically slanted about a horizontal axis. Considering only the perspective implications of the configurations, the vertical axis findings should be reversed for horizontal axes, since, in the latter case, linear perspective dictates that it is the vertical lines which should converge in order to be consistent with the stereoscopic indicators and which may therefore show an attenuated stereoscopic response. The horizontal lines would be subject to compression, which is a much less powerful slant cue (Gillam 1968) than linear perspective. On the other hand, if orientation disparity at the vertical meridian (which occurs only in the horizontal axis condition) has a special stereoscopic status, this may offset the attenuation predicted for vertical line stimuli. 4.1 Method Fourteen volunteers drawn from the departmental panel acted as subjects. Again, a four-way, fully-crossed, within-subjects, factorial design was used. Experiment 2 differed from experiment 1 only in terms of the surface configurations used. Stimuli were either horizontal lines, vertical lines, or H / V grids obtained by overlaying the vertical and horizontal lines. 4.2 Results The key data are shown in figure 4. The effects of Surface Configuration (^2,26 = 4.62, p < 0.05) and Axis of Slant (Ful3 = 14.05, p < 0.01) were significant, as was their interaction (F226 = 21.48, p < 0.001). Consistent with the results of experiment 1, geometrically equivalent slants about a horizontal axis were judged to be greater than those about a vertical axis. Considering the vertical axis conditions alone, the data were consistent with Gillam's (1968) results. The mean slant response to horizontal-line fields was lower than that to surfaces configured with H / V grids. 0.05). In accordance with predictions from the type of perspective conflict present, the horizontal line stimulus in the horizontal axis condition did produce a similar slant response to the vertical line stimulus in the vertical axis condition. However, the vertical line stimulus in the horizontal axis condition appeared much more slanted than the horizontal line stimulus in the vertical axis condition. Perspective considerations cannot account for this difference. Again, as in experiment 1, direction of slant did not affect slant-magnitude estimates {F < 1), whereas the magnitude of slant (15° or 30°) had a highly significant effect on performance (Ful3 = 65.29, p < 0.001).
Axis of Slant horizontal liiiiilj vertical horizontal lines
vertical H/V lines grids Surface Configuration Figure 4. Results of experiment 2. Perceived slant as a function of Surface Configuration with Axis of Slant as a parameter. Data are pooled across Slant Direction and Slant Magnitude. Slant Magnitude is shown as a parameter in figure 6.
5 Experiment 3 To provide continuity and comparability, several conditions from the first two experiments were included in experiment 3: the H/V-grid and diagonal-grid conditions from experiment 1, and the vertical-line and horizontal-line conditions from experiment 2. Two new conditions were added: diagonal-line fields and H / V / D grids (see figure 2). The aims were (i) to confirm the configuration effects found in earlier experiments in a single experiment, thus enabling further comparisons to be made between the various effects; and (ii) to extend the range of configurations to enable additional hypotheses to be tested. The line density of the H / V and diagonal components of the H / V / D grids was reduced in comparison with their presentation in isolation so that the line density of the composite figure would not be so great as to constitute texture rather than pattern. Diagonal fields were included to see if the advantage for slants about the horizontal axis in the diagonal-grid condition would be preserved when (a) the strongly implicit H / V contours in a diagonal grid were eliminated, and (b) orientation disparities all had the same sign for both axes of slant, thus eliminating a major difference between them. Mitchison and McKee had found a horizontal-axis advantage for much smaller stimuli of this type.
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The purpose of including the composite H / V / D grid was to test the hypothesis for surfaces slanted around a vertical axis that the greater slant response to surfaces marked with diagonal grids compared with H/V grids resulted not from the presence of explicit orientation-disparity but from the absence of strong, conflicting, linear perspective. The H/V/D-grid stimuli contain explicit orientation-disparity information, but also provide strong, conflicting, linear perspective. If the presence of orientation disparity were responsible for the effectiveness of the diagonal grid, the H/V/D grid should be as effective as the diagonal grid. If the absence of strong linear perspective were responsible, then the H / V / D grids should be less effective than the diagonal grids and more like the H/V grids. 5.1 Method Again a four-way, fully-crossed, within-subjects, factorial design was used. The experiment differed from experiments 1 and 2 only in terms of the range of surface configurations used. Fourteen volunteers from the departmental panel acted as subjects. 5.2 Results The results for conditions used in previous experiments confirmed our earlier findings. The main effects of Surface Configuration (F 565 = 6.45, p < 0.001), Axis of Slant ( F U 3 = 68.81, p < 0.001), and Slant Magnitude (F1>13 = 126.14, p < 0.001) were all highly significant. The Surface Configuration x Axis of Slant interaction was significant (F5>65 = 10.33, p < 0.001). In all conditions (except vertical lines), the estimated magnitude of slants about a horizontal axis was significantly greater than that for equivalent surfaces rotated about the vertical (Newman-Keuls; p < 0.05 in all cases except vertical lines). Figure 5 shows the perceived slant for each Surface Configuration, with Axis of Slant as a parameter. As was the case in experiment 1, slant settings for diagonal grids were significantly greater than those for H / V grids for slants around a vertical axis (p < 0.05), whereas differences in perceived slant were considerably less (and not statistically significant, p > 0.05) for slants about the horizontal. Settings were substantially attenuated for horizontal-line fields, relative to all other configurations for slants about the vertical (Newman-Keuls, p< 0.05 for all contrasts) whereas, as in experiment 2, surfaces configured with vertical-line fields and rotated about the horizontal showed only a slight attenuation relative to other configurations, which failed to reach statistical significance except for the contrast with the diagonal line condition. Thus, equivalent perspective conflict had different effects for the two axes of slant.
Axis of Slant • H horizontal I vertical horizontal lines
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Figure 5. Results of experiment 3. Perceived slant as a function of Slant Configuration with Axis of Slant as a parameter. Data are pooled across Slant Direction and Slant Magnitude. Slant Magnitude is shown as a parameter in figure 6.
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In the vertical axis conditions the mean slant for the diagonal lines was lower than the mean for the diagonal grid, but this just failed to reach significance on a Newman-Keuls test (p < 0.05). Both diagonal lines and diagonal grids had significantly higher mean slants than H/V grids and horizontal lines (p < 0.05). This shows that the effectiveness of diagonals in eliciting a stereoscopic slant response does not depend on the presence of orientation disparity in two directions nor on implicit vertical lines. For slant around the horizontal axis the diagonal lines were as effective as any other condition (significantly more so than the vertical line condition) despite having only half the orientation disparity of the H / V grid and the vertical lines. The mean slant around a vertical axis of H/V/D grids was significantly greater than was obtained for H/V grids alone {p < 0.05) suggesting that the effectiveness of diagonals in this case is not simply due to weak perspective cues. In all three experiments reported here, the effect of Slant Magnitude was highly significant, with 30° slants being judged to be greater in magnitude than 15° slants. In figure 6, perceived slant is plotted as a function of Surface Configuration with Slant Magnitude and Axis of Slant as parameters. This figure shows data combined from all three experiments. Where conditions were common to two or more experiments, these data were pooled and the arithmetic mean calculated and plotted. All configurations (except the unique vertical-lines condition) showed the characteristic slantaxis anisotropy. All demonstrate a clear effect of Slant Magnitude.
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Figure 6. Combined data from experiments 1-3. Perceived slant as a function of Surface Configuration with Axis of Slant and Slant Magnitude as parameters. Data are pooled across Slant Direction. 6 General discussion Our data support the view that stereoscopic slant perception around a vertical axis is greater for diagonal grids than for H / V grids (Cagenello and Rogers 1988). However, it is clear that the presence or absence of explicit orientation disparity is unlikely to account for these differences. Slant estimates were greatest for surfaces slanted about a vertical axis when they were configured with vertical lines, despite the fact that these entirely lack explicit orientation disparity. We have proposed that diagonal grids are 'better' stimuli for slant about a vertical axis because they provide weaker perspective information than does a H / V grid. This conjecture awaits direct testing in monocular vision. Also consideration of the H/V/D grid results suggests that this is not the only reason.. Whereas for slants about the vertical, conflicting perspective provides a plausible account of the good response to vertical lines and of the poor response to horizontal
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lines and H / V grids, it is not sufficient to account for other aspects of our data. It does not account for the fact that horizontal lines are worse than H / V grids even though perspective information is (if anything) stronger in the latter case. Also, it sits uneasily with the data from the horizontal-axis conditions (which were much less affected by equivalent conflicting perspective information). In our view, one conclusion is inescapable: there are configural effects in stereopsis which have their origins neither in orientation disparity per se, nor in conflicting perspective. Vertical lines seem to be particularly powerful drivers of the stereoscopic response to surfaces rotated about a vertical axis. This is demonstrated by the fact that, when they are added to horizontal lines, subjects' slant estimates increase despite the fact that the verticals also increase conflicting linear perspective. This certainly requires further investigation, especially since the effects we obtained with irregular vertical lines do not seem to be obtained with regular vertical lines forming implicit rectangles (Mitchison and Westheimer 1984). Clearly, there is a property of configured surfaces rotated about a horizontal axis which makes estimates of surface slant relatively resistant to conflicting perspective information. This factor is unlikely to be orientation disparity. Consider the diagonal line condition of experiment 3: horizontal and vertical axis slants produce identical orientation disparity and identical conflicting perspective for this stimulus and yet the horizontal axis slant response was far greater than the vertical. One possibility is that a mechanism exists for detecting image shear per se. This is in line with the original observation by Wallach and Bacon (1976) on the importance of 'transverse disparity', and the findings of Rogers and Graham (1983) on the effectiveness of differential image shear relative to image expansion in both the stereo and motion parallax domains. The orientation disparity produced by rotating a surface configured with vertical lines about a horizontal axis is one manifestation of this shearing, but it is not essential to the effectiveness of shear and is, therefore, unlikely to be the mechanism by which shear is detected (Morgan 1989; Rogers and Cagenello 1989). In conclusion, the present data do not support the view that orientation disparity underlies stereoscopic surface slant perception, despite Ninio's (1985) finding that orientation disparity plays an important role in determining the depths of fine-grained 'needle' surface textures (in the absence of strong perspective conflict). The demonstration by Blakemore et al (1972) of a "second neural mechanism for depth perception" founded on orientation disparities has held out an enduring promise. We have, however, found nothing to suggest that orientation disparity plays a significant role in stereopsis. Acknowledgements. This research was supported by Grant No. B350307 from the Australian Research Council. The authors wish to thank Christopher Foster for assistance in running the experiments, and Hal Sedgwick for his comments on the manuscript in draft form. References Blakemore C, Fiorentini A, Maffei L, 1972 "A second neural mechanism of binocular depth discrimination" Journal of Physiology 226 725 - 749 Cagenello R, Rogers B J, 1988 "Local orientation differences affect the perceived slant of stereoscopic surfaces" Investigative Ophthalmology and Visual Science, Supplement 29 399 (abstract) GillamB, 1968 "Perception of slant when perspective and stereopsis conflict: experiments with aniseikonic lenses" Journal of Experimental Psychology 7 8 2 9 9 - 3 0 5 Gillam B, Chambers D, Russo T, 1988 "Postfusional latency in stereoscopic slant perception and the primitives of stereopsis" Journal of Experimental Psychology: Human Perception and Performance 14 163-175 GillamB, Flagg T, F inlay D, 1984 "Evidence for disparity change as the primary stimulus for stereoscopic processing" Perception &Psychophysics 36 559-564
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Julesz B, 1971 The Foundations of Cyclopean Perception (Chicago, IL: University of Chicago Press) MitchisonGJ, McKee S P, 1990 "Mechanisms underlying the anisotropy of stereoscopic tilt perception" Vision Research 301781-1791 Mitchison G J, Westheimer G, 1984 "The perception of depth in simple figures" Vision Research 24 1063-1073 Morgan M J, 1989 "Vision of solid objects" Nature (London) 339 101 - 1 0 3 NinioJ, 1985 "Orientational versus horizontal disparity in the stereoscopic appreciation of slant" Perception 1 4 3 0 5 - 3 1 4 Ogle K N, 1950 Researches in Binocular Vision (New York: Hafner) Rogers BM, Cagenello R, 1989 "Disparity curvature and the perception of three-dimensional surfaces" Nature (London) 339 135-137 Rogers B M, Graham M, 1983 "Anisotropics in the perception of three-dimensional surfaces" Science 111 1409-1411 Stevens K A, Brookes A, 1988 "Integrating stereopsis with monocular interpretations of planar surfaces" Vision Research 2 8 3 7 1 - 3 8 6 WallachH, Bacon J, 1976 "Two forms of retinal disparity" Perception & Psychophysics 19 375-382 Westheimer G, McKee S P, 1979 "What prior uniocular processing is necessary for stereopsis" Investigative Ophthalmology and Visual Science 18614-621 Winer B J, 1962 Statistical Principles in Experimental Design (New York: McGraw-Hill) p 85
