The role of angles in inducing misalignment in the Poggendorff figure

The Effects of Adding Pictorial Depth Cues to the Poggendorff Illusion

We tested if the misapplication of perceptual constancy mechanisms might explain the perceived misalignment of the oblique lines in the Poggendorff illusion. Specifically, whether these mechanisms might treat the rectangle in the middle portion of the Poggendorff stimulus as an occluder in front of one long line appearing on either side, causing an apparent decrease in the rectangle's width and an apparent increase in the misalignment of the oblique lines. The study aimed to examine these possibilities by examining the effects of adding pictorial depth cues. In experiments 1 and 2, we presented a central rectangle composed of either large or small bricks to determine if this manipulation would change the perceived alignment of the oblique lines and the perceived width of the central rectangle, respectively. The experiments demonstrated no changes that would support a misapplication of perceptual constancy in driving the illusion, despite some evidence of perceptual size rescaling of the central rectangle. In experiment 3, we presented Poggendorff stimuli in front and at the back of a corridor background rich in texture and linear perspective depth cues to determine if adding these cues would affect the Poggendorff illusion. The central rectangle was physically large and small when presented in front and at the back of the corridor, respectively. The strength of the Poggendorff illusion varied as a function of the physical size of the central rectangle, and, contrary to our predictions, the addition of pictorial depth cues in both the central rectangle and the background decreased rather than increased the strength of the illusion. The implications of these results with regards to different theories are discussed. It could be the case that the illusion depends on both low-level and cognitive mechanisms and that deleterious effects occur on the former when the latter ascribes more certainty to the oblique lines being the same line receding into the distance.

Saccades to Explicit and Virtual Features in the Poggendorff Figure Show Perceptual Biases

Human participants made saccadic eye movements to various features in a modified vertical Poggendorff figure, to measure errors in the location of key geometrical features. In one task, subjects (n = 8) made saccades to the vertex of the oblique T-intersection between a diagonal pointer and a vertical line. Results showed both a small tendency to shift the saccade toward the interior of the angle, and a larger bias in the direction of a shorter saccade path to the landing line. In a different kind of task (visual extrapolation), the same subjects fixated the tip of a 45° pointer and made a saccade to the implicit point of intersection between pointer and a distant vertical line. Results showed large errors in the saccade landing positions and the saccade polar angle, in the direction predicted from the perceptual Poggendorff bias. Further experiments manipulated the position of the fixation point relative to the implicit target, such that the Poggendorff bias would be in the opposite direction from a bias toward taking the shortest path to the landing line. The bias was still significant. We conclude that the Poggendorff bias in eye movements is in part due to the mislocation of visible target features but also to biases in planning a saccade to a virtual target across a gap. The latter kind of error comprises both a tendency to take the shortest path to the landing line, and a perceptual error that overestimates the vector component orthogonal to the gap.

Geometrical features underlying the perception of collinearity

The magnitude of the Poggendorff bias in perceived collinearity was measured with a 2AFC task and roving pedestal, and was found to be in the region of 6-8deg, within the range of previous estimates. Further measurements dissected the bias into several components: (1) The small (∼1deg) repulsion of the orientation of the pointer from the parallel, probably localized in the part of the line near the intersection (2) A small (<1deg) location bias affecting the intersection of pointers and inducing lines; and (3) A larger (>1deg) bias in the orientation of virtual lines crossing the gap between two parallels, towards the orientation of the parallels, or equivalently (4) An orthogonal bias in actively constructing a virtual line across the gap. We conclude that orientation repulsion by itself is an inadequate explanation of the Poggendorff effect, and that a full explanation must take account of the way in which observers construct virtual lines in visual space in order to carry out elementary geometrical tasks such as extrapolation.

Applying Emmert's Law to the Poggendorff illusion

The Poggendorff illusion was approached with a novel perspective, that of applying Emmert's Law to the situation. The extensities between the verticals and the transversals happen to be absolutely equal in retinal image size, whereas the registered distance for the verticals must be smaller than that of the transversals due to the fact that the former is assumed to occlude the latter. This combination of facts calls for the operation of Emmert's Law, which results in the shrinkage of the occluding space between the verticals. Since the retinal image shows the transversals to be in contact with the verticals, the shrinkage must drag the transversals inwards in the cortical representation in order to eliminate the gaps. Such dragging of the transversals produces the illusory misalignment, which is a dictation of geometry. Some of the consequences of this new explanation were tested in four different experiments. In Experiment 1, a new illusion, the tilting of an occluded continuation of an oblique line, was predicted and achieved. In Experiments 2 and 3, perceived nearness of the occluding entity was manipulated via texture density variations and the predicted misalignment variations were confirmed by using a between-subjects and within-subjects designs, respectively. In Experiment 4, tilting of the occluded segment of the transversal was found to vary in the predicted direction as a result of being accompanied by the same texture cues used in Experiments 2 and 3.

The Poggendorff illusion affects manual pointing as well as perceptual judgements

Pointing movements made to a target defined by the imaginary intersection of a pointer with a distant landing line were examined in healthy human observers in order to determine whether such motor responses are susceptible to the Poggendorff effect. In this well-known geometric illusion observers make systematic extrapolation errors when the pointer abuts a second line (the inducer). The kinematics of extrapolation movements, in which no explicit target was present, where similar to those made in response to a rapid-onset (explicit) dot target. The results unambiguously demonstrate that motor (pointing) responses are susceptible to the illusion. In fact, raw motor biases were greater than for perceptual responses: in the absence of an inducer (and hence also the acute angle of the Poggendorff stimulus) perceptual responses were near-veridical, whilst motor responses retained a bias. Therefore, the full Poggendorff stimulus contained two biases: one mediated by the acute angle formed between the oblique pointer and the inducing line (the classic Poggendorff effect), which affected both motor and perceptual responses equally, and another bias, which was independent of the inducer and primarily affected motor responses. We conjecture that this additional motor bias is associated with an undershoot in the unknown direction of movement and provide evidence to justify this claim. In conclusion, both manual pointing and perceptual judgements are susceptible to the well-known Poggendorff effect, supporting the notion of a unitary representation of space for action and perception or else an early locus for the effect, prior to the divergence of processing streams.

Biases and sensitivities in the Poggendorff effect when driven by subjective contours

PURPOSE

A consensus in the existing literature suggests that the Poggendorff effect (a perceptual misalignment of two collinear transversal segments when separated by a pair of parallel contours) persists when the parallels are defined by Kanizsa-like subjective contours. However, previous studies have often been complicated by a lack of quantitative measures of effect size, statistical tests of significance, appropriate measures of baseline and control biases, or stringent definition of subjective contours. The aim of this study was thus to determine whether subjective contours are capable of driving the Poggendorff effect once other factors are accounted for.

METHODS

Twenty participants were tested on a number of test and control figures incorporating first-order (luminance-defined) and subjective parallels using the method of adjustment. All figures were tested at two different orientations, and observer sensitivities and observer biases were assessed.

RESULTS

A systematic response bias (in the direction of the classical effect) was found for Poggendorff figures that incorporated subjective parallels. The effect was highly significant and greater than for control figures. There was no concomitant change in judgment sensitivity (positional certainty). Finally, there was a positive correlation between the effect size for figures incorporating first-order and subjective parallels.

CONCLUSIONS

The findings reported demonstrate conclusively that true Kanizsa-like subjective contours are capable of driving the Poggendorff effect. Further, the data are consistent with a growing body of evidence that suggests both first-order and subjective contours are processed at early loci in the visual pathways when position is encoded.

3-D processing in the Poggendorff illusion

In the Poggendorff illusion two collinear oblique lines, separated by two vertical lines, appear to be misaligned. 3-D processing of the oblique but not the vertical lines is considered to cause this apparent misalignment. We investigated whether more explicitly triggering 2-D versus 3-D interpretations of the different parts of Poggendorff-like displays would influence the apparent misalignment. In Experiment 1, we found that compared to 2-D controls, 3-D interpretations of the vertical parts did not influence apparent misalignment, while for the oblique parts 3-D processing resulted in more apparent misalignment than 2-D controls. In Experiment 2, the amount of contour convergence of the oblique parts was manipulated resulting in the 3-D blocks, but not the 2-D line patterns, to be perceived as receding in depth. Now, apparent misalignment increased the more the 3-D blocks were perceived as receding in depth. We conclude that apparent misalignment in Poggendorff-like displays can be influenced by different interpretations of its separate parts, while keeping the local junctions between the different elements the same.

Learned irrelevance and retrospective correlation learning

In 1973 Mackintosh reported an interference effect that he called learned irrelevance in which exposure to uncorrelated (CS/US) presentation of the unconditional stimulus (US) and the conditioned stimulus (CS) interfered with future Pavlovian conditioning. It has been argued that there is no specific interference effect in learned irrelevance; rather the interference is the sum of independent CS and US exposure effects (CS + US). We review previous research on this question and report two new experiments. We conclude that learned irrelevance is a consequence of a contingency learning and a specific learned irrelevance mechanism. Moreover even the independent exposure controls, used in previous experiments to support the CS and US exposure account, provide support for the correlation learning process.

Characterisation of the misalignment and misangulation components in the Poggendorff and corner-Poggendorff illusions

In the Poggendorff illusion, two colinear segments abutting obliquely on an intervening configuration (often consisting of two long parallel lines) appear misaligned. We report here the results of a component analysis of the illusion and several of its variants, including in particular the 'corner-Poggendorff' illusion, and variants with a single arm. Using a nulling method, we determined an 'orientation profile' of each configuration, that is, how the illusions varied as the configuration was rotated in the plane of the display. We were able to characterise a pure-misalignment component (having peaks and dips around the +/- 22.5 degrees and +/- 67.5 degrees orientations of the arms) and a pure misangulation component of constant sign, having peaks at the +/- 45 degrees orientations of the arms. Both these components were present in both the classic and the corner-Poggendorff configurations. Thus, the misangulation component appears clearly in the classic Poggendorff illusion, once the misalignment component is partitioned out. Similarly, the corner-Poggendorff configuration, which essentially estimates a misangulation component, contains a misalignment component which becomes apparent once the misangulation is nulled. While our analysis accounts for much of the variability in the shapes of the profiles, additional assumptions must be made to explain the relatively small misangulation measured in the corner-Poggendorff configuration (1.5 degrees, on average, at peak value), and the relatively large illusion measured in the configurations with a single arm (above 6 degrees, on average, at peak values). We invoke the notion that parallelism and colinearity detectors provide counteracting cues, the first class reducing misangulation in the corner-Poggendorff configuration, and the second class reducing the illusion in the Poggendorff configurations with two arms.

The Poggendorff illusion: a bias in the estimation of the orientation of virtual lines by second-stage filters

The veridical perception of collinearity between two separated lines is distorted by two parallel lines in the space between them (the Poggendorff illusion). This paper tests the conjecture that the perception of collinearity of separated lines is based on a two-stage mechanism. The first stage encodes the orientation of the virtual line between the proximal terminators of the target lines. The second stage compares this virtual orientation with the orientation of the target lines themselves. Errors can and do arise from either process. Two parallel lines, abutting against the target lines, cause the classical Poggendorff misalignment bias. The magnitude of the bias is increased by Gaussian blur, as is a version of the Poggendorff figure containing only acute angles. In the obtuse-angle figure, on the other hand, blur decreases the misalignment bias. We argue that the acute- and obtuse-angle biases depend upon different mechanisms, and that the obtuse-angle effect is more related to the obtuse-angle version of the Muller-Lyer illusion, which is also decreased by blur. If observers attempt to match the orientation of the virtual line between the two line intersections in the Poggendorff figure they make an error in the same direction as the Poggendorff bias. The orientation of the target lines in the figure, however, is veridically matched to a Gabor-patch probe, unless the target lines are very short, in which case the error is in the same direction as the Poggendorff bias. A small bend in the target lines where they abut the parallels increases the Poggendorff bias if it makes the line more orthogonal to the parallel, but has little effect in the opposite direction. The Poggendorff bias is unlikely to depend upon biases in first-stage linear filters because (a) it still exists in figures composed of short, luminance-balanced lines which are defined by contrast only; and (b) it also exists if the parallels are replaced by grating patches with the same mean luminance as the background. The orientation of the grating in the latter case affects the magnitude of the bias, but even an orientation which should reverse the Poggendorff bias by the mechanism of cross-orientation inhibition fails to do so. The Poggendorff bias is a complex effect arising from several sources. Blurring in second-stage filters with large receptive fields can explain many aspects of the phenomenon.

The half-Zöllner illusion

The Zollner figure contains stacks of short parallel segments oriented obliquely to the direction of the stack. Adjacent parallel stacks of opposite polarity seem to diverge where their top segments form an arrowhead. To probe whether or not the opposite polarities are necessary to the illusion, three 'half-Zollner' configurations were designed, containing stacks of a single polarity. The 'orientation profile' of these configurations was studied, that is, the way the strength of the perceived illusion varies with the orientation of the stacks. The subjects had to align two stacks or align stacks with target segments situated at a slight distance from them. All three half-Zollner configurations produced errors that could be assimilated to global-orientation misjudgments. These errors were of opposite sign for the two types of stacks and varied with the orientation of the stacks as in the standard Zollner illusion. A further study was conducted in which the effect of several configurational parameters was explored for a single observer. The standard Zollner illusion increases with the separation of the stacks. The illusion is also increased when the orientations of the segments in different stacks are orthogonal, independently of the particular longitudinal orientations of the stacks. When the ends of the short segments are curved so that at their endpoints they become precisely perpendicular to the axis of the stacks, the standard and half-Zollner illusions are reduced, but not abolished. Therefore, they cannot be entirely accounted for by a mechanism of alignment of illusory contours generated at these endpoints. The results are consistent with the existence of a single common mechanism at work in both the standard and the half-Zollner illusion. It is suggested that the illusion itself is not a rotation of the stacks but either a shear deformation in which the segments of a stack slide with respect to one another, or an expansion of the stacks orthogonally to the segments.

The orthogonal orientation shift and spatial filtering

A line abutting two tilted flanks is apparently shifted towards the orientation orthogonal to the flanks and at the same time is reduced in its apparent length. It has been suggested that both effects are caused by band-pass spatial filtering, followed by location of the end points of the line at the peaks in the filtered image. Here implications of the filtering explanation of these effects are explored further. In the first experiment, it was predicted that orientation thresholds (as opposed to biases) would be increased for short line lengths, and would be further increased by abutting bars. The predictions were confirmed. It was shown in experiment 2 that the orientation shift was reduced by a small (4 min arc) gap between target lines and orthogonal flanks. In experiment 3 the threshold elevations and the orientation shift produced by orthogonal and tilted flanks were compared. Last, in experiment 4, the threshold elevations and orientation shift produced by orthogonal and tilted flanks, at different retinal eccentricities varying from 0 to 3.2 deg were compared, and the prediction that the magnitude of the orientation shift would decrease with line length and increase with eccentricity was confirmed. The connection is explored between the orientation shift and the Zöllner illusion, and demonstrations are presented of the Zöllner effect in which target and inducing lines are of opposite contrast on a gray background. It is concluded that the Judd and Zöllner illusions do not depend upon a single mechanism.

On apparent misalignment of collinear edges and boundaries

In two experiments, subjects adjusted various pairings of the top and bottom boundaries of two obliquely oriented outline bars (Experiment 1) and those of two similarly oriented complete and incomplete squares (Experiment 2) to apparent alignment. The data from the first experiment showed that the misalignment effects were determined jointly by the directional properties of the bar ends (vertical, oblique, and semicircular) and the pair of boundaries that were aligned (both top boundaries, top of upper bar with bottom of lower bar, bottom of upper bar with top of lower bar). The results from the second experiment showed that the misalignment effects were the same for the oblique boundaries of solid and outline squares and persisted when the squares were reduced to two parallel lines. The effect was undiminished when the ends of the parallels were aligned, but was markedly reduced when pairs of parallels themselves were aligned. The outcomes of the two experiments are explained in terms of the apparent positions of the oblique boundaries. It is proposed that these vary with the positions of the elements (bar or square) relative to the visual field, the position of the boundaries relative to the stimulus elements, and the positions of the boundaries relative to axes that are delineated by the parallel adjacent ends of bars and sides of squares. This relative-position basis for apparent misalignment is held to be the basis of misalignment effects in other figures.

Structure and strategy in encoding simplified graphs

Tversky and Schiano (1989) found a systematic bias toward the 45 degrees line in memory for the slopes of identical lines when embedded in graphs, but not in maps, suggesting the use of a cognitive reference frame specifically for encoding meaningful graphs. The present experiments explore this issue further using the linear configurations alone as stimuli. Experiments 1 and 2 demonstrate that perception and immediate memory for the slope of a test line within orthogonal "axes" are predictable from purely structural considerations. In Experiments 3 and 4, subjects were instructed to use a diagonal-reference strategy in viewing the stimuli, which were described as "graphs" only in Experiment 3. Results for both studies showed the diagonal bias previously found only for graphs. This pattern provides converging evidence for the diagonal as a cognitive reference frame in encoding linear graphs, and demonstrates that even in highly simplified displays, strategic factors can produce encoding biases not predictable solely from stimulus structure alone.

Spatial filtering and spatial primitives in early vision: an explanation of the Zöllner-Judd class of geometrical illusion

The apparent length and orientation of short lines is altered when they abut against oblique lines (the Zöllner and Judd illusions). Here we present evidence that the length and orientation biases are geometrically related and probably depend upon the same underlying mechanism. Measurements were done with an 'H' figure, in which the apparent length and orientation of the cross-bar was assessed by the method of adjustment while the orientation of the outer flanking lines was varied. When the flanking lines are oblique the apparent length of the central line is reduced and its orientation is shifted so that it appears more nearly at right-angles to the obliques than is in fact the case. Measurements of the orientation and length effects were made in three observers, over a range of flanking-line angles (90, 63, 45, 34 and 27 deg) and central line lengths (9, 17, 33 and 67 arc min). The biases increased with the tilt of the flanking-lines, and decreased with central line length. The extent of the length bias could be accurately predicted from the angular shift by simple trigonometry. We describe physiological and computational models to account for the relation between the orientation and length biases.

How far can attraction-caused misalignment account for the Morinaga misalignment effect?

When a line (the pointer) is collinear with a dot, the addition of a second line (the induction line) contiguous with the dot or near it may cause the pointer to appear to be collinear with a point further along or nearer to the induction line. The geometrical relations upon which this effect (which we call attraction-caused misalignment) depends, have been studied with the Obonai and Wundt-Loeb (Hotopf, 1981; Hotopf & Brown, 1988) figures. Drawing upon the studies of misalignment in the Morinaga figure carried out by Restle (1976), Day, Bellamy, and Norman (1983), and Day and Kasperczyk (1985), as well as upon two new experiments, we show that misalignment in the Morinaga figure is also attraction-caused misalignment, as previously defined. We conclude with a discussion of a number of theories that aim at accounting for attraction misalignment.

Constant errors in judgements of collinearity due to the presence of neighbouring objects

If a line (the pointer) is aligned with a dot (the target) that stands on another line (the induction line) which is at an angle to the pointer, the pointer and the dot may no longer appear collinear. Whether they do or not depends upon the angle formed by the pointer with the induction line: the smaller the angle, the greater the misalignment effect. Misalignment is always in the direction of the induction line, which is why this alignment illusion is called attraction-caused misalignment (attraction misalignment for short). Three experiments are described in which this illusion is explored further. In the first it is shown that the induction line can exert its influence even when not contiguous with the target, though the size of the effect varies inversely with the distance of the induction line from the target. In the second experiment it is demonstrated that a dot as well as a line can induce attraction misalignment and that similarity between the induction and target items increases misalignment. Evidence in support of the theory that the termination of the induction line, as well as the part contiguous with the target dot, may induce attraction misalignment is provided in the third experiment.