Perception, 1999, volume 28, pages 375 ^ 386

DOI:10.1068/p2711

Constancy scaling and conflict when the Zo«llner illusion is seen in three dimensions David Phillips 5 Rowsley Avenue, West Didsbury, Manchester M20 2XD, UK Received 19 November 1997, in revised form 15 February 1999

Abstract. If a standard Zo«llner illusion is seen as a staircase in depth, pairs of long lines flanking convex stair edges appear to diverge as usual, but divergence in pairs flanking concave edges can appear reduced. If the stair is reversed perceptually in the manner of the Schro«der staircase, convex and concave shapes exchange and the extent of apparent divergence in the long line pairs exchanges with them. The effect is enhanced if explicit stair edges are added, and reduced if the standard Zo«llner pattern is replaced by one in which segments of the long lines are offset in the direction of the usual illusory effect. The observations suggest that the three-dimensional potential of the pattern cannot be excluded from explanations of the illusion, and are compatible with the view of Gregory and Harris that inappropriate constancy scaling is its primary cause, triggered `bottom ^ up' by pattern properties or `top ^ down' by cognitive inference. However, these two mechanisms would have to be acting in conflict to generate suppression of divergence in the concave steps. Pattern processing for properties, such as orientation, that are not associated with the potential of the Zo«llner illusion as a three-dimensional configuration, but that have been suggested as sources of the illusion in recent studies, could also be acting in opposition to hypothesis scaling in the concave steps.

1 Introduction If the standard Zo«llner illusion is seen as a staircase in three dimensions, pairs of long lines that flank convex step edges appear to diverge as usual in the illusion, but the divergence can appear to some observers to be suppressed for pairs that flank concave step edges. If, therefore, the staircase is made to `flip' perceptually in depth in the manner of the Schro«der staircase (Schro«der 1858), in which case convexities and concavities of the staircase reverse, the divergence of each pair of long lines can appear to vary with the apparent three-dimensional orientation of the stair. The effect may to an extent be learned or voluntary. I can obtain it with any Zo«llner array, and find suppression in the concaves enhanced if explicit step-edge lines are added, but reduced if the standard Zo«llner pattern is replaced with one or more of the characteristics of a twisted cord pattern. However, other observers vary substantially in response. Some do not obtain the effect I see at all, and others only in part. Two experiments were therefore set up to see how robust the effect is, and to explore some of the variables that affect it. 2 Experiment 1 In figure 1, standard Zo«llner arrays are shown as staircases seen from above (bottom left) and from below (top right) within a pictorial context intended to be assertively threedimensional. The columns of obliques are quite short, since preliminary, informal studies suggested that the effects described are hard to obtain for most observers if the columns are longer. Two further versions of the picture were also prepared, identical to the first except that in one, as in figure 2a, edges of the steps were added as explicit lines within the Zo«llner array, and in the other, as in figure 2b, the standard Zo«llner illusion was replaced by a zigzag stimulus in which segments of the long lines were offset in the direction of the usual tilt effect. The three pictures were prepared as laminated A3 sheets, and shown to thirty observers, all naive to the subject, with normal or corrected-tonormal vision, of both sexes and aged from 12 years to their mid-fifties.

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Figure 1. If a standard Zo«llner illusion is seen steadily as a staircase in depth in a consistent three-dimensional context, pairs of long lines flanking convex stair edges appear to diverge as usual, but divergence in pairs flanking concave edges can tend to appear reduced. If the stair is reversed perceptually, as seen upper right and lower left in this picture, convex and concave shapes exchange and the extent of apparent divergence in the long line pairs exchanges with them. (With apologies to Rubens for the angels and to the artist known as the Supposed Juste de Juste for the acrobats.)

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(c) Figure 2. Three versions of figure 1 were presented to observers as A3 sheets, one as in figure 1, and the other two identical except that the stairs were as in (a) and (b), respectively. The observers were asked to match the apparent divergence of pairs of long Zo«llner lines in the pictures, with line pairs on the scale shown in (c).

Because the effect depends upon seeing the array as three-dimensional, some time was spent in discussing the picture showing the standard Zo«llner array with each observer. The picture was intended to be taken as a view into a stage-set-like room space, with the rectangle behind the angels as a back wall to the room and the trapezoid below them as a floor. The figures to the top right were intended to enforce this orientation. However, in spite of these figures, some observers saw the space more readily as if seen from above, as if looking down from the ceiling in an out-of-body experience, with the rectangle behind the angels as the floor. This was discussed, and observers were encouraged to see the space as intended instead. Also, in spite of the context, both stairs can spontaneously reverse for some observers, in the manner of the Schro«der staircase, in particular the upper right one. This was intended to be seen from below, so that the upper, shaded end is nearer to the viewer, and the lower end further away. Many observers remarked that it tends to appear with the orientation reversed, as if seen from above, again in spite of the figures standing on the stair, and some had great difficulty in obtaining the intended orientation. To help with perceiving the stair as intended, concave and convex stair edges were discussed, but of course carefully avoiding in this connection any reference to the other long lines on the stairs. Once observers' remarks suggested that they could hold the perception as intended at least briefly, they were told that the study was of the degree and direction of apparent divergence, if any, of pairs of the long lines labelled a to e in each stair, whilst the scene was held perceptually in the intended orientation. For comparison they were given a laminated strip of line pairs of varying divergence, as in figure 2c, and of a size similar to that of the line pairs on the stairs in the picture. They were then asked to look at one of the line pairs on one of the staircases in the picture at a time, for example lines b and c on the lower left stair, to judge the apparent direction and degree of divergence if any of that line pair, and to find the best match between its appearance and one of the line pairs on the strip. Each observer made one such judgment of every line pair, in both staircases in the case of the picture with the standard Zo«llner (figure 1), but just in the lower left-hand stair, seen from above, for the other two pictures (in which the stairs were as in figures 2a and 2b). The sequence in which linepair judgments were requested in each presentation was varied informally. Each stair presents two convexities and two concavities, so for each of four `states' of the illusion

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(first, standard Zo«llner from above, as in figure 1 lower left; second, standard Zo«llner from below, as in figure 1 upper right; third, standard Zo«llner just from above, but with edges as in figure 2a; and fourth zigzag Zo«llner just from above, as in figure 2b), each observer made two matches for line pairs flanking convex stair edges, and two for line pairs flanking concave edges. These were converted into a single numerical value for each two matches, after substituting a numerical scale for the alphabetical scale labelling the line pairs on the comparison strip, and taking the average of each two matches. For ease of visual comparison of the results in figure 3, apparent variation from parallelism in the line pairs, whether divergent or convergent, is always shown increasing to the right of the graphs. This was done by making the substitution of an ordinal scale for the R-to-Z scale, identifying the line pairs on the comparison strip, run from 0 to 8 in the case of matchings with picture line pairs ab and cd, and from 8 to 0 for matchings with picture line pairs bc and de. V, therefore, always converted to 4, represents an average assessment that the lines appeared parallel. Line pairs bc and de As concave, standard Z, from above

Line pairs ab and cd As convex, standard Z, from above

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Figure 3. Distributions of average observations from experiment 1. Conversion from the alphabetical scale in figure 2c to a numerical scale was organised so that in these distributions apparent variation from parallelism in the line pairs, whether divergent or convergent, is always shown increasing to the right of the graphs. The average was taken of two observations by each observer of line pairs flanking convex step edges, and of two observations of line pairs flanking concave edges, for four comparisons of effect on different stairs, as specified in superscripts to the distributions.

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2.1 Results The resulting distributions appear in figure 3, and were compared with two-sample t-tests as follows. Distributions i and ii (observations of line pairs ab and cd, flanking convex edges, and of line pairs bc and de flanking concaves, in the standard Zo«llner seen from above): t29 ˆ 7:58, p 5 0:05; distributions i and iii (observations of line pairs ab and cd, first seen as flanking convex edges in the standard Zo«llner when seen from above, and then as flanking concave edges seen from below): t29 ˆ 6:13, p 5 0:05; distributions iii and iv (observations of line pairs ab and cd, flanking concave edges, and of line pairs bc and de flanking convex ones, when all are seen in the standard Zo«llner, but from below): t29 ˆ 0:8, p 4 0:05; distributions vii and viii (observations of line pairs ab and cd, flanking convex edges, and of pairs bc and de flanking concaves, with the zigzag Zo«llner seen from above): t29 ˆ 3:09; p 5 0:05. Thus the differences in these comparisons of distributions were significant, with the exception of the judgments of the standard Zo«llner seen from below, in the upper right in figure 1, where there was no significant difference between observations of concave and convex steps. That may have been due to the difficulty remarked by a number of observers in holding the representation of the stair in that orientation. If these distributions are otherwise evidence of an acceptable level of robustness in the observations reported at the start of the paper, informal visual comparisons of the other distributions also seem to indicate that when explicit edge lines are added (as in figure 2a and distributions v and vi in figure 3) apparent divergence is even more strongly suppressed in the line pairs flanking concave steps, and reduced even for line pairs flanking convex ones. By contrast when the standard Zo«llner array is replaced with the zigzag alternative (as in figure 2b and distributions vii and viii in figure 3) apparent divergence seems much less inhibited in the line pairs flanking concave steps, though it is not eliminated. The final t-test was intended to test for any remaining inhibition with the zigzag Zo«llner, and (perhaps surprisingly) still indicates a significant, though smaller, difference in perceptions of divergence of the long Zo«llner lines in convex and concave parts of the stair. 2.2 Discussion Explanations of the Zo«llner illusion can be divided into those which focus upon the three-dimensional potential of the array, and those that concentrate on the interaction of two-dimensional pattern properties in the stimulus in varieties of proposed neural channels at various stages in pattern processing. The effects presented in the present study suggest that, whereas recent studies have concentrated on two-dimensional pattern properties (for example, Tyler and Nakayama 1984; Morgan and Casco 1990; Earle and Maskell 1995; Parlangeli and Roncato 1995; Ninio and O'Regan 1996), the three-dimensional potential of the array cannot be ignored. It might be objected that, if threedimensional potential played more than an occasional role in exceptional circumstances, the suppression of divergence in the concaves reported here would have been much more often remarked upon than it has been. However, it arises only if perception of the entire array is held in a consistent three-dimensional orientation, whereas a three-dimensional effect can appear in much less constant ways in informal presentations of the illusion, with major shifts in fixation. Sinusoidal Zo«llner arrays are particularly unstable. If one flute is viewed as convex, with a fixation near to the top of the array, the one right next to it, which should be concave, readily appears convex too if the next fixation jumps down to near the bottom. In general, whatever point in a Zo«llner array is fixated can appear convex in this way unless an effort is made to see it as flat or to envisage the whole array in a consistent spatial orientation. This could be why the columns of obliques in figures 1 and 2 have to be short for most observers to obtain the suppression of divergence in the concaves at all. If they are longer, it may be the scope for large fixation shifts that makes a consistent spatial configuration of the array harder to hold.

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A more difficult question is whether these observations suggest that the three-dimensional potential of the Zo«llner illusion is not merely a contributor, but the dominant factor in inducing the illusion. The idea is an old one. Wundt associated with such potential all illusions of the kind, which he called variable, but discovered in the Zo«llner array a quite different spatial arrangement from the stair configuration (Wundt 1902, page 554). His white line plate is reproduced as my figure 4a, and my diagrammatic representation of the spatial arrangement he discovered in it appears as figure 4b. With a roving fixation, he noted, individual lines can appear in spatial orientations, but inconsistently. With a monocular fixation a global array in space appears. Titchener (1901, page 316) called this a very strong perspective effect with white line figures, but I find it hard to obtain at all, and do not share Wundt's surprise, in a slightly reproachful footnote, that what he called such a striking effect had escaped Helmholtz. With the abandonment of theories such as Wundt's, that such illusions arise when spatial perceptions are compromised by eye movements, there has been less discussion of spatial readings of the geometric illusions in general. A study by Ward (Ward et al 1977) explored the extent to which observers could discover three-dimensional readings at all in various illusions, including the Zo«llner illusion, but Julesz (1971, page 234) noted that, at least for broad-line versions of the Zo«llner illusion, the effect is strikingly reduced in a Cyclopean presentation, suggesting that it arises primarily in low-level processing.

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(b) Figure 4. Wundt obtained a completely different spatial configuration in the Zo«llner illusion. His plate (1902, page 553 and figure 262) is 4a, and the configuration he discovered in it is suggested in the diagram in 4b. ``As long as one scans the figure with a restless glance, it is true that occasionally the inclination of individual lines can appear to be in depth, but not within a clear perspective. It is quite different with a monocular fixation; then all the lines appear arrayed in depth, in such a way that ... going from left to right, first the upper end of the top left hand line and then the lower end of the second line lies nearer to the observer, thereafter the third again with its upper end, etc.''

A much more important exception to the trend is Gregory and Harris's argument for a decisive role for three-dimensional potential of the array in generating the effect, which they propose is available in the stimulus through two quite distinct routes: a `top ^ down', cognitive route, when the array is inferred as three dimensional; and a `bottom ^ up', signal route, which the authors propose involves two-dimensional pattern properties in the array that are automatically associated in the brain with three-dimensional configuration, even when none is consciously perceived (Gregory and Harris 1975; Gregory 1998, pages 217 ^ 243). In either case, the illusion is then proposed to arise from inappropriate constancy scaling, distinguished by the authors as hypothesis scaling and cue scaling (Gregory and Harris 1975, page 204), as the brain attempts to reconcile the three-dimensional suggestion of the array with the absence of perspective and stereo disparity in the stimulus. Let us start by accepting this point of view, and see how results of experiment 1 can be accommodated within it. The first point to address is that if inappropriateconstancy-scaling effects are to play a key role, we need to establish whether we would

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expect them to be of equal strength in the convex and concave steps, and as seen from above and below. Informal experience suggested that they might not be, and an experiment was devised to explore that. 3 Experiment 2 Gregory (1973, page 92) pointed out that an extended, oblique but parallel-edged step shape, with striations across each surface of the kind that would appear if the step was made of Lego 2 bricks, offers a striking effect of size-constancy divergence with distance. An A3 version of figure 5(i) ^ (iv) was therefore prepared, presenting solid and hollow step shapes, seen from above and below. F B Hofmann observed at the turn of the century (cited in Robinson 1972, page 68) that striations in similar, herringbone-type configurations can cause apparent divergence of the long edge lines (as they do in my figure 6), so to standardise any such effect striation was limited to just one horizontal surface in each view, and to minimise it the steps were orientated so that the striations do not form very acute angles with the edges. The step surfaces were impaled with long rods to make the intended spatial configurations easier to obtain. Near

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Figure 5. To test the strength of size-constancy effects, in experiment 2 observers were asked to choose between divergence, parallelism, or convergence with distance as characteristic of the long lines H and O in each of these four configurations of a step shape: (i) solid seen from above and (ii) hollow seen from above (near ends lower down page); (iii) solid seen from below and (iv) hollow seen from below (near ends higher on page). Results reflected the difficulty for some observers of the hollow views, but suggest size-constancy scaling is stronger for solid shapes.

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(a)

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Figure 6. Two reversing `Mach books', but with added striations running in different directions, are shown horizontally, and with un-book-like contextual detail to aid three-dimensional reversal. Pattern and distance size-constancy effects seem to combine to produce divergence with distance in (a) and (d), but to compete to suppress it in (b) and (c). Distortion of superimposed circles follows the direction of the striations, but appears unaffected by reversal of spatial direction of the surfaces below.

The laminated A3 sheet was shown to twenty-four of the observers from experiment 1. Once again each of the four spatial configurations and the viewpoint from which it should be seen was first discussed with each observer, stressing which end in each case should be seen as near and which as far. Several observers took some time to obtain figure 5(iv) in the intended configuration (a hollow shape seen from below) and one or two reported that they could hold it in that configuration only momentarily before it flipped to a solid view, even though the impaling rods would have had to be bent in that case. These observers were encouraged to try to make the judgments that would be requested on the basis of what they saw in even a momentary view of the image in the intended configuration. Observers were then asked for each viewpoint whether the long lines (labelled H and O) appeared to diverge with distance, to remain parallel, or to converge with distance. Those observers who reported that the lines appeared to diverge in both the solid and the hollow views were then additionally asked whether the divergence in the hollow view was greater than, the same as, or less than that in the solid view. 3.1 Results The observations are included in figure 5, and indicate that divergence does tend to be less in the hollow shapes than the solid ones. The indication is not quite as straightforward as the presentation suggests, since five observers reported more divergence with distance in the hollow than in the solid forms, either seeing the edges of solids as parallel and those of hollows as divergent, or seeing divergence in each case, but to a greater extent in the hollows. However, these observations were outnumbered by reports that, where both solid and hollow steps offered divergence, it was greater in the solids. Also, in the observations of experiment 2, as in those of experiment 1, there was more spread in the data from the hollow shape seen from below, perhaps because these configurations seem to be so hard for some observers to resolve, so that confusion enters into the judgments. 3.2 Discussion In seeing how well Gregory and Harris's scheme fits with the results of experiment 1, therefore, we have to bear in mind that we can expect the size-constancy contribution in the concaves to be less than in the convex steps. But that of course does not in any simple way help us with the reduction of divergence seen in the concave steps in experiment 1, because the reduced divergence reported is still in the direction opposite to the divergence with distance that size-constancy scaling would induce in these concave steps. What might be happening is competition between the two hypothesised

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mechanisms, inference and signal. When the Zo«llner illusion is seen as a stair in depth, for long line pairs flanking convex edges, both inferred and signalled effects would act in the same direction, to swing the apparent orientation of the long lines so as to increase the apparent angle at which they meet the obliques, as usual with the Zo«llner illusion. However, whenever line pairs flank a concave step, any inferred size-constancy effect should produce a shift in the direction opposite to that of the usual Zo«llner tilt. For example, in figure 1, in the lower left stair, note lines b and c flanking a concave step edge. If the whole stair is held as in a coherent configuration seen from above, hypothesis size-constancy scaling should induce in line b an anticlockwise rotation and in line c a clockwise one, counter to the usual divergence seen in the Zo«llner effect. However, observers in the great majority of comparisons reported some divergence in the other direction. What may be happening in the concaves is therefore that any weak hypothesis size-constancy scaling is opposing and reducing rotations that are in the usual Zo«llner direction, signalled (if we follow Gregory and Harris), by the spatial implications of the patterns of the columns of obliques, which can surely only signal the pair of columns of obliques involved as if flanking a convex step. If that interpretation is right, we can predict that, if a reversing figure can be given striations that can run in different directions, it should be possible to associate cooperation of cues with concavities, and competition with convexities, counter to the effect in experiment 1. To my eye that happens in figure 6, in which Mach-book-type stimuli are shown, but horizontal and striated, each in a convex (figures 6a and 6c) and concave (figures 6b and 6d) configuration. To aid three-dimensional reversal they are represented as houses when seen as convex and as wedge-shaped excavations in blocks when concave. These reversals are equivalent to those of Schro«der stairs, since in each case the apparent axis of recession into depth of the figures swings through 908 with reversal. In figures 6a and 6b divergence is associated with convexity and inhibition with concavity as in experiment 1, but with the striations turned around in figures 6c and 6d, distance and pattern now work together to enhance divergence in the hollow view (figure 6d), and compete to inhibit divergence in the convex view (figure 6c). At the same time the superimposed circles appear to show the effect just of two-dimensional pattern cues. They seem to float, their distortions following the direction of the obliques but quite unchanged as the surfaces below reverse perceptually in depth, as if on sheets of glass that are balanced horizontally on the roof ridges in the convex views, and rest on the long edges of the concavities. The figures therefore seem to separate out signal and hypothesis cues, and to be consistent with the idea that they can act in concert or in opposition on the long edges. So long as the idea that the two kinds of size-constancy cues can on occasion act in opposition is acceptable, in other respects the results of experiment 1 are quite compatible with Gregory and Harris's account. It has no problem in accommodating the relative resistance of the zigzag Zo«llner in figure 2b to suppression of divergence in the concaves. Resistance in this case may arise because here neural channeling of pattern properties that have nothing to do with signalling spatial effect is coming into play. This would then be just one of many examples of the way that the characteristics of the long lines in Zo«llner arrays can affect the strength of the illusion, but only as a minor variable. As was remarked by Zo«llner himself (1860, page 501), and recently confirmed by Tyler and Nakayama (1984), and then by Ninio and O'Regan (1996), the Zo«llner effect is often obtained even in columns of obliques with no explicit long line at all. What seems to matter is just the long axis of the column of obliques, whether explicitly marked or not. However, the effect is stronger if the axes are explicitly marked with long lines, though these can take many forms. In the standard versions of the illusion they are explicit black or white lines, but alternatively, for example, they may be subjective white ones, or even blurry white bars underlying sharp black obliques on a grey ground, as demonstrated

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by Earle and Maskell (1995, page 1401). The Zo«llner effect on the axis may be stronger still if the lines marking it also present two-dimensional pattern properties such that processing through channels optimised for different spatial frequencies causes segments of the lines to be signalled as rotated in the direction of the usual overall tilt illusion. This accounts for the strength of tilt in twisted cord type patterns (Woodhouse and Taylor 1987), though it remains unclear whether it can arise from line crossings in standard Zo«llner arrays. However, the illusion can certainly appear enhanced if segments of the lines are explicitly offset, as in the zigzag Zo«llner, figure 2b. Just why subjective or actual segment offsets should enhance apparent axial tilt is of course not yet understood, but there is no problem in accommodating these effects as special cases within Gregory and Harris's scheme, and Gregory (1998, page 209) notes effects in illusions of retinal edge processing. The second observation from experiment 1 that is consistent with Gregory and Harris's account is the increase in suppression of the usual Zo«llner tilt, particularly in line pairs flanking concave steps, when explicit step edges are outlined. That seems compatible with their demonstration that illusion vanished when stimuli with the two-dimensional properties of a number of illusions were presented to observers as three-dimensional visual experiences, but with perspective and binocular depth-disparity properties exactly as they should have been in real-space binocular views of the arrays (Gregory and Harris 1975, with an anaglyph of a Zo«llner type array on page 219). These demonstrations seem extreme instances of what may be a general rule, that the more explicitly that Zo«llner illusions (and some others) are seen to correspond with surfaces in a coherent spatial array, the less the strength of the illusion. Wundt (1902) based his (now outdated) theory of the complementarity of perspective and plane illusory views of variable illusions on numerous observations of this kind. Though at times it seems not quite clear to what extent he understood illusory distortions to vanish in perspective views, and to what extent he simply saw them as stabilised and appropriate when seen in depth, it is clear that in many cases he did understand illusion to vanish in the depth views. In my figure 7, divergence in an Orbison square appears reduced when the inducing lines present surfaces in a clear spatial array. For Zo«llner illusions, as Ninio and O'Regan point out (1996, page 92), Judd had also observed in 1905 that even a line at the edge of the obliques suppresses the usual Zo«llner effect. In their own study of orientation effects of isolated columns of obliques, without even a long transverse line, they note a similar effect of any line at the edge of a stack. They also found that the Zo«llner effect was larger for isolated columns of obliques than for arrays assembled in the classic Zo«llner configuration, and that even in the classic configuration the effect increased, at first dramatically, as the columns were placed wider apart. With the columns close together, the white spaces between them can present subjective step edges, so this could be another instance in which strength of the illusion reduces as edges are more explicit. Some of the reduction in illusion reported in these cases might be due to two-dimensional framing effects, but generally the extra suppression of Zo«llner divergence in the concaves of figure 2a, when edge lines are added,

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Figure 7. Divergence in an Orbison square appears reduced when the inducing lines form edged surfaces in a clear spatial array.

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is certainly not in conflict with Gregory and Harris's observations of the reduction in illusion as appropriate spatial cues are added. However, though nothing in the results reported here seems inconsistent with their account, one earlier report in particular does seem to require pattern effects at work other than as size-constancy signal cues in the Zo«llner illusion. One of Ninio and O'Regan's experiments (1996, page 82, test configuration C) on `half-Zo«llners', short columns of obliques not paired with columns of counter oblique orientation as in the usual Zo«llner array (and with no explicit axial lines), effectively measured errors in aligning the axis of an isolated column of obliques. The variations in degrees of error observed as the subjects attempted the alignment of sixteen different orientations of the column (minimum when the axis of the column was vertical or horizontal, maximum when it was at 458) were in line with those in more conventional Zo«llner configurations. Yet in the case of an isolated column of obliques, even if imagined as a surface in space, it is hard to see how inappropriate constancy scaling could be causing the shift in the observed orientation. Two-dimensional pattern properties that may have no role in space perception could therefore be at work in inducing the perceptual shifts, both in the single column of obliques and in normal Zo«llners. In my experiment 1, such pattern properties would be at work instead of (or as well as) cue scaling, enhanced and opposed by hypothesis scaling. But then why should all illusory effect have vanished when Gregory and Harris presented illusion arrays to observers, but in appropriate perspective and stereo-disparity contexts? Might it be that inappropriate constancy scaling is not the only process triggered when a stimulus proves anomalous in terms of signalling or inference of expected spatial configuration? It could be that as well as that process, the relative sparseness of data in these patterns (absence of either edge or texture information) and the anomalies of the stimuli (apparent depth without perspective adjustment or stereo-disparity data) can also leave the brain attributing too much value to, for example, the output of mechanisms in pattern processing, whose usual role is only to distinguish areas of pattern of different properties, rather than specify pattern properties precisely. Consider, for example, pattern orientation. Tyler and Nakayama (1984) proposed that the shift in apparent axis of short columns of obliques without explicit axial lines comes about through the mutual inhibition of cells sensitive to stimulus orientation, but at two very different scales. They pointed out that cells late in visual processing that integrate information from earlier stages appear to be optimised for orientation in areas of the visual field, but at a wide range of scales. There could thus be inhibition between cells optimised for such orientation at the scale of the length of individual oblique lines, and on the scale of the overall orientation of the virtual axis of the whole stack. Some such process could also involve the virtual axes in Zo«llner illusions as well. Recent studies have explored circumstances in which pattern orientation may be detected in poststriate processing, presumably by summation of the output of pattern-selective channels earlier in the visual pathway. Olzack and Wickens's study (1997) includes responses to orientation generally at such a proposed stage of processing, and Wenderoth et al (1993) studied it explicitly in a situation involving the contribution of virtual pattern axes to inhibition between all the axes in certain pattern configurations. Parlangeli and Roncato (1995, page 511) discuss Wenderoth et al's findings in relation to the Zo«llner illusion. It is also suggestive that, in various studies noted by Wenderoth and Welsh in a more recent paper (1998), the angles at which single symmetry axes in patterns are most readily salient vary inversely with the orientations at which the Zo«llner illusion appears strongest. Processes like these at a late stage in processing are therefore also candidates as partners or opponents for inference scaling, in the enhancement and suppression of apparent divergence in experiment 1.

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