The Poggendorff illusion explained by natural scene geometry

Abstract

One of the most intriguing of the many discrepancies between perceived spatial relationships and the physical structure of visual stimuli is the Poggendorff illusion, when an obliquely oriented line that is interrupted no longer appears collinear. Although many different theories have been proposed to explain this effect, there has been no consensus about its cause. Here, we use a database of range images (i.e., images that include the distance from the image plane of every pixel in the scene) to show that the probability distribution of the possible locations of line segments across an interval in natural environments can fully account for all of the behavior of this otherwise puzzling phenomenon.

Johann Poggendorff pointed out in the 19th century that when the continuity of an obliquely oriented line is interrupted, the positions of the line segments on either side of the interruption appear to be shifted vertically (or horizontally if the interruption is oriented horizontally) (1). When presented in the format in Fig. 1A, the right oblique line segment appears to be shifted downward relative to the left segment, even though they are actually collinear. In Fig. 1B, which is the mirror image of Fig. 1 A, the right line segment appears to be shifted but in this case upward relative to the left segment. A similar effect is elicited when the oblique line is interrupted by a space delineated by two parallel horizontal lines instead of two vertical lines, as in Fig. 1C. In this presentation, the right segment appears to be shifted to the right with respect to the left segment. The magnitude of the apparent shift of the oblique line segments in these several configurations also varies as a function of the orientation of the interrupted line and the width of the interruption. In standard presentations of the Poggendorff stimulus (Fig. 1 A and B), the effect increases as the orientation of the interrupted line becomes increasingly vertical (Fig. 1D) (2, 3); the effect also increases as the width of the interruption increases (Fig. 1E) (3). A particularly puzzling aspect of the phenomenon is that the illusion is largely abolished when only the acute components of the standard Poggendorff stimulus are shown. However, the effect remains if only the obtuse components are present (4) (Fig. 1F). Finally, when the overall orientation of the stimulus is rotated such that the interrupted line is horizontal (Fig. 1G), the effect is reduced but not completely abolished (2, 3, 5).

Fig. 1.

The Poggendorff illusion and its behavior. (A) When an obliquely oriented straight line is interrupted by a vertical occluder, the line segment on the right appears to be shifted downward with respect to the line segment on the left. (B) A similar effect occurs when the orientation of the interrupted line is reversed. In this case, the collinear extension on the right appears to be shifted upward. (C) When an oblique line is interrupted by parallel horizontal lines, the oblique line segments appear to be shifted horizontally with respect to each other. (D) The magnitude of the effect increases as the interrupted line is made more vertical. (E) The magnitude also increases as the width of the interruption increases. (F) The illusion is largely abolished when only the acute components of the stimulus are presented, but the effect is maintained when only the obtuse components are shown. (G) The illusion is diminished when the standard configuration is rotated so that the interrupted line is horizontal.

Although a variety of theories have been proposed to account for the Poggendorff illusion (see Discussion), none of these explanations rationalizes the full range of behavior illustrated in Fig. 1. Here, we test the hypothesis that the Poggendorff effect and its altered behavior in different presentations are generated by the accumulated experience of humans with the various sources of interrupted linear features in typical environments. By collecting a range image database with information about the 3D structure of a large series of natural scenes, we assessed whether these geometrical illusions are indeed explained by the probability distribution of the relative locations of line segments across a spatial interval.

Methods

The Range Image Database. The details of the range image database used have been described in ref. 6. The images were acquired by using a range imaging system (LMS-Z210 3D Laser Scanner, Riegl USA, Orlando, FL), which identified the distances of object surfaces from 2 m to ≈300 m with an accuracy of ±25 mm (Fig. 2A). By using this system, we collected ≈100 wide-field images extending 333° horizontally and 80° vertically at a resolution of 0.144°. Of these images, 25 were obtained in fully natural environments in undeveloped terrain, and the rest were obtained in outdoor or indoor settings that included human constructions (see also ref. 7). Only the midportion of the wide-field images (≈110° horizontally × 52° vertically) was used for analysis.

Fig. 2.

Determining the possible physical sources of the right line segment given the other three lines in a Poggendorff stimulus. (A) The midportion of a wide-field range image acquired with a laser range scanner that encodes the distance of each pixel in the scene (indicated by the color bar). Black areas (sky) are regions where the laser beam did not return a value. (B) Examples of the templates for sampling the lines comprising the standard Poggendorff stimulus (Fig. 1 A) in the range image database. Pixels in an image patch are represented diagrammatically by the grid squares; black pixels indicate the template for the left oblique line segment and the two vertical lines; red pixels indicate a series of templates for sampling the right line segment at various vertical locations. (C) The solid white lines indicate a valid sample of the left oblique line segment and the two vertical lines. Magnifications of the boxed portion of the scene show examples of the subsequent application of additional templates (red) to test for the presence of right oblique segments at different vertical positions. (D) The variants of the Poggendorff stimulus illustrated in Fig. 1 could be sampled by using appropriately configured templates. In these examples, the black lines are equivalent to the black template in B, and the red lines are equivalent to the red templates. (E) Definition of the relative physical location of the two segments of the interrupted line. Δ signifies the location of the right line segment relative to the location at which the left line segment, if extended (dotted red line), would intersect the right vertical line (or the lower horizontal line); Δ was designated negative if a right segment was located above (or to the left of) this point of intersection and positive if a right segment was located below (or to the right).

Empirical Framework and Hypothesis. When the continuity of a physical object is interrupted by occlusion, the location of its reappearance on the other side of the interruption is inevitably uncertain: there are many ways the object could have traversed the interval, and the interruption does not provide information about which of the possible ways actually generated the stimulus. Thus, the oblique line segments in the standard Poggendorff presentation in Fig. 1 could arise from a single object with an infinite variety of possible configurations or from two different objects altogether. In short, the stimulus reflects the fundamental uncertainty of how the physical sources of projected line segments are actually continued across a spatial interval. The hypothesis examined here is that the way humans produce useful behavioral responses in the face of this uncertainty is to generate percepts according to the accumulated experience in dealing with the physical sources of retinal stimuli. If this hypothesis is correct, the percepts elicited by the stimuli in Fig. 1 should reflect the past experience of human observers with the physical sources of straight-line segments that are in the same orientation but separated by a spatial interval. More specifically, given a line segment on one side of an implied occluder, the perceived position of a second line segment in the same orientation but on the other side should be predicted by the relative probability of occurrence of the physical sources of line segments projected in that orientation.

Sampling the Database. Fig. 2B shows examples of the geometrical templates we applied to the range images to sample the physical sources of the four lines comprising the standard Poggendorff stimulus. For present purposes, a straight line is defined as a set of points whose positions in space conform to a linear progression; thus, the lines sampled were not required to correspond to luminance contrast boundaries or to belong to a single object surface. The rationale for this purely geometrical definition in analyzing the structure of natural scenes is explained in ref. 8.

Determination of the physical sources of a Poggendorff stimulus in these terms involved several steps. We first identified a region of a scene that contained a physical source of one of the two oblique line segments (the left segment, for example), as well as the sources of the two vertical lines. Then, in the same region of the scene, we determined the frequency of occurrence of the physical sources of possible right line segments, i.e., line segments that had the same projected orientation as the left oblique segment and were located just to the right of the right vertical line. A template for sampling the left oblique line segment and the two vertical lines is shown in Fig. 2C. The points in the image underlying each straight line in the template were then examined to determine whether the corresponding physical points formed a straight line in 3D space. The underlying points were accepted as a straight line if their average deviation from a fitted line by using a least-squares method was <2% of the average distance of the physical points from the origin of the laser beam. If all of the lines identified by the template met this criterion, the set was accepted as a valid physical source of these three components of the Poggendorff stimulus (see Fig. 2C, leftmost panel). Having identified a region that contained sources of the first three lines, a template for sampling the right oblique line segment was applied just to right of the right vertical line, as shown by the red pixels in Figs. 2 B–D. This additional template was moved vertically in sequential applications to determine all of the possible physical sources of the right oblique segment at different vertical locations relative to the left oblique segment. The location of the right segment was assessed in terms of the distance (Δ, expressed in terms of the visual angle subtended) from the point of intersection of the right oblique segment with the right vertical line to the point at which the left oblique segment, if extended, intersected the right vertical line (Fig. 2E). The physical sources of the variants of the Poggendorff stimulus illustrated in Fig. 1 were similarly determined by using appropriately configured sampling templates, as indicated in Fig. 2D.

For each configuration tested, the number of occurrences of the physical sources of the right line segments at different locations relative to the left segment was tallied, and this information was used to produce the corresponding probability distribution of the sources of the right segments (≈10⁴ samples were obtained by each tested configuration). We could then ask whether these distributions predicted the perceptual phenomena elicited by the standard Poggendorff stimulus and its variants.

Results

Explanation of the Standard Poggendorff Effect. According to the empirical hypothesis being tested here, the probabilities of occurrence of the physical sources of right oblique line segments at different vertical locations relative to the left segment should determine how observers perceive the relative positions of the two segments in the Poggendorff stimulus. Fig. 3A shows that when the left oblique segment is oriented downward and to the right, as in Fig. 1 A, the probability distribution of the possible sources of the right segment is biased toward locations that correspond to negative values of Δ. In other words, the majority of the physical sources that give rise to projections of right oblique line segments in the same orientation as the left segment will, in the past experience of human observers, have projected above the point at which an extension of the left oblique line intersects the right vertical line. This probability distribution therefore predicts that a right segment that is actually collinear with the left should be seen as displaced downward.

Fig. 3.

Probability distributions of the physical sources of the right line segment in the standard Poggendorff stimuli. (A) The probabilities of occurrence of the physical sources of the right oblique line segment for the standard Poggendorff stimulus in Fig. 1 A presented as a function of the location of the right segment relative to the left oblique segment. The relative location of the two oblique line segments, Δ (see Fig. 2E), is given in terms of visual angle. (B and C) Probability distributions of the physical sources of the right line segment for the stimulus configurations in Fig. 1B (B) and Fig. 1C (C). Arrows indicate the mode of the distribution here and in subsequent figures.

To appreciate the reasoning underlying this conclusion, consider a hypothetical world in which the sources of right oblique segments at all vertical locations are equally likely. In this case, the perceived vertical location of a right line segment would not be biased one way or another by an observer's past experience. Accordingly, a right line segment that was actually collinear with the left segment would be seen as such. As shown in Fig. 3A, however, the real-world sources of the right line segment, given the left oblique segment and the vertical occluder, are biased toward locations above the point of collinearity with the left segment. Assessed against this body of past experience, then, a right line segment that is geometrically collinear with the left segment would appear shifted downward. Thus, the probability distribution derived from the fully natural scenes in the database can indeed account for the standard Poggendorff effect.

Conversely, when the orientation of the interrupted line in the stimulus is upward from the lower left, as in Fig. 1B, the probability distribution of the sources of the right line segment is biased toward locations below the point at which a continuation of the left line intersects the right vertical line (i.e., toward values of Δ > 0) (Fig. 3B). By the same reasoning, a right oblique segment that is actually collinear with the left segment will in this case be seen to be somewhat above the point where an extension of the left segment intersects the right vertical line, as indeed it is.

A similar explanation applies to the Poggendorff stimulus in which an oblique line is interrupted by two horizontal lines. When the left oblique segment is oriented downward and to the right, as in Fig. 1C, the probability distribution of the sources of the right line segment is biased toward locations to the left of the point at which Δ = 0 (Fig. 3C). Thus, the location of the right segment appears shifted to the right from the continuation of the left line, which is again consistent with what people see.

Explanation of Other Features of the Poggendorff Effect. Changing either the orientation of the interrupted line or the width of the interruption systematically also shifts the relevant probability distributions in a manner consistent with the altered magnitude of the perceptual effects in these presentations. The effects of varying the intersecting angle of the interrupted line (Fig. 1D) and of varying the width of the interruption (Fig. 1E) on the corresponding probability distributions of the physical sources of the right line segment are shown in Fig. 4 A and B, respectively. As the orientation of the interrupted line becomes closer to the orientation of the lines defining the interruption (vertical in the example in Fig. 4A) or as the interruption becomes wider, the mode of the distribution shifts progressively away from the point at which Δ = 0, in accord with the fact that the perceived shift in the apparent location of the right line segment increases as the angle of the intersection becomes smaller or as the width of the interruption increases (3).

Fig. 4.

The probability distribution of the physical sources of the right line segment changes progressively as the orientation of the interrupted line (A) or the width of the interruption (B) is altered. α represents the angle of intersection between the interrupted oblique line and the vertical lines, and w is the width of the interruption. w = 1° in (A) and α = 45° in (B); both values were measured in terms of visual angle.

Another puzzle that can be explained in these terms is the fact that the Poggendorff effect is greatly diminished when only the acute angles in the standard stimulus are presented but little affected when the presentation is restricted to the obtuse components (see Fig. 1F). As shown in Fig. 5A, when only the acute elements are used as the templates for sampling the range images (see Fig. 2D), the mode of the probability distribution (arrow) is very near the point where Δ equals zero. Accordingly, the Poggendorff effect should be largely abolished. Conversely, when only the obtuse components are used, the probability distribution is similar to that of the standard Poggendorff stimulus; thus, the illusion would be expected to retain its full magnitude.

Fig. 5.

Probability distributions of the physical sources of the right line segment when the standard Poggendorff stimulus is decomposed or rotated. (A) The probability distribution obtained when only the acute components of the stimulus are present, compared with the distribution when only the obtuse components are considered. In both cases, the orientation of the interrupted line was 45° and the width of the interruption 1°. (B) The probability distribution obtained when the standard Poggendorff stimulus is rotated such that the interrupted line is horizontal (see Fig. 1G). The distribution for the standard stimulus is replotted from Fig. 3A for comparison.

Finally, we determined the probability distribution of the sources of the right line segment when the overall orientation of the Poggendorff stimulus is rotated so that the interrupted line is horizontal, as in Fig. 1G. Compared with the standard presentation, the distribution when the stimulus is rotated 90° has a mode closer to Δ= 0, as shown in Fig. 5B. As a result, the Poggendorff effect should be reduced but not abolished when the presentation of the interrupted line is horizontal, as it is.

Statistics Derived from Different Types of Scenes. Theories purporting to explain one or more aspects of the Poggendorff illusion have often been based on intuitions about the “carpentered” world of human artifacts (9, 10). It was thus of interest to ask how the probability distributions derived from fully natural scenes (presumably representative of the human visual environment during evolution) compare with distributions derived from environments in which human construction is prevalent. We therefore computed the relevant probability distributions from the set of scenes in the database that included some or many man-made objects. As shown in Fig. 6, the pertinent probability distributions obtained from the two types of scenes are generally similar. It thus seems safe to conclude that the perceptual effects apparent in the Poggendorff illusion are not specifically dependent on interactions with the rectilinear constructions associated with human culture.

Fig. 6.

The probability distribution of the sources of the right line segment for the standard Poggendorff stimulus derived from an analysis of fully natural scenes (see Fig. 3A) compared with the distribution derived from the scenes that contained human constructions.

Physical Bases for the Statistical Biases Observed. Two questions so far deferred in considering the statistics derived from our analysis of the range images are why the observed biases in the occurrence of the physical sources of the right line segment exist and why the magnitude of these biases differs for the different configurations of the Poggendorff stimulus considered here.

A major part of the answer to these questions lies in the geometry of planar surfaces, which are the typical sources of straight-line projections on the retina. Consider, for instance, the physical sources of the standard stimulus in Fig. 1 A. The presence of a physical source of the left oblique line segment and the vertical lines would usually signify the presence of a planar surface in the corresponding region of the scene. Because a right oblique line segment above the point at which Δ = 0 is, on average, closer to the center of the plane than a line segment below this point (Fig. 7A), the set of physical points corresponding to a right line segment is statistically more likely to be in this plane when it is above the point at which Δ = 0 than when it is below this point. Thus, the likelihood of occurrence of the physical sources of the right line segment is greater for positions at which Δ< 0 (i.e., above the point of Δ= 0) than for positions at which Δ > 0. As a result, the probability distribution in Fig. 3A is biased toward Δ < 0. The same reasoning can explain the biases seen in the distributions in Fig. 3 B and C.

Fig. 7.

Physical basis for the bias in the probability distributions of the sources of Poggendorff stimuli. (A) In the standard stimulus, a right line segment above the point at which Δ = 0 (gray arrowhead) is, on average, closer to the center of the plane containing the physical source of the left segment and the vertical lines than is an otherwise comparable right line segment below the point at which Δ= 0. The distances of the right line segments from the center of the plane (black dot within the dashed line that outlines the plane) are indicated by the gray dotted lines; the difference in the lengths of the two gray dotted lines is indicated by the black bar below the diagram. (B) As the orientation of the interrupted line becomes more vertical (Left) or as the width of the interruption increases (Right), the same shift in the vertical position of the right line segment means a larger difference in the distances of the right line segments from the center of the plane. The differences in the lengths of the gray dotted lines are again indicated by the black bars.

When the orientation of the interrupted line in the standard Poggendorff stimulus is closer to vertical or when the width of the interruption increases, the same shift in the vertical position of the right line segment results in a larger difference in the distances of right line segments from the center of the plane containing the physical source of the left segment and the vertical lines (Fig. 7B). There is, accordingly, an increased bias in the probability distribution of the sources of the right line segment as a function of the orientation of the interrupted line and a function of the width of the interruption (see Fig. 4). The reason for the greater bias in the presence of only the obtuse components of the stimulus compared with only the acute components (see Fig. 5A) is also straightforward in these terms. In the stimulus comprising the acute components (see Fig. 1F), the right vertical line does not extend above the point at which Δ = 0, reducing the likelihood that its physical source (i.e., an underlying planar surface) extends above this point. This statistical change in turn reduces the likelihood of occurrence of sources of the right oblique segment above the point at which Δ = 0. In contrast, the right vertical line is present above the point at which Δ = 0 in the stimulus composed of the obtuse components; the bias toward Δ < 0 in the corresponding distribution of the sources of the right oblique segment is therefore maintained.

Finally, it is important to explain the physical basis of the reduced bias in the probability distribution of the sources of the right line segment when the standard Poggendorff stimulus is rotated such that the interrupted line is horizontal (see Fig. 5B). This reduction cannot be rationalized in terms of the relationship between the right line segment and the plane containing the source of the left segment and the vertical lines because this relationship is unchanged from that in the standard stimulus, despite the rotation. The reason in this case is the higher overall frequency of occurrence of the sources of horizontal lines in the physical world (see Fig. 8, which is published as supporting information on the PNAS web site). The greater probability of occurrence of the sources of horizontal lines compared with the sources of oblique lines makes the probability distribution of the right horizontal line segment in the rotated stimulus less susceptible to the bias caused by the presence of the other stimulus elements, much as a higher baseline makes a signal detector less sensitive to the same input.

Discussion

Despite numerous studies (1), the Poggendorff effect remains one of the most controversial of the geometrical illusions. Here we have shown that this effect and its complex behavior in different presentations can be accounted for by the relationship between the stimulus in the image plane and its possible sources in natural visual environments. From the analysis presented, it should be apparent that the biased image–source relationships that account for the Poggendorff effect and its variants are rooted in the statistical properties of the physical world in which human observers reside. Thus, this relationship has to be derived empirically by examining the natural environment and cannot be deduced from principles of projective geometry per se.

Previous Explanations of the Poggendorff Effect. The explanation most often offered in the past has been that the illusion derives from a misperception of the angles in the stimulus (11, 12). For instance, an overestimation of the acute angles in the standard stimulus in Fig. 1 A and an underestimation of the obtuse ones would presumably affect the apparent orientation of the line segments on either side of the interruption toward horizontal, thus causing them to appear noncollinear. This explanation, however, has been disputed (1) on the grounds that it cannot rationalize the paradoxical abrogation or preservation of the illusion elicited, respectively, by presentation of only the acute or only the obtuse components of the stimulus (see Fig. 1F). Nor does this explanation account for the additional features of the Poggendorff effect shown in Figs. 1D, E, and G.

Another explanation is based on the idea that the geometrical information in the retinal projection might be inappropriately “interpreted” by observers. For example, the “depth-processing” theory suggests that the oblique lines in the Poggendorff stimulus are interpreted as lines extending in depth, and are therefore perceived to be noncollinear (10, 13, 14). This category of explanation is also limited in that it considers only some aspects of the projective geometry of the stimulus and its possible sources and cannot explain the full range of the percepts elicited by the configurations in Fig. 1.

Biological Rationale. An important question not yet fully addressed is why, from the perspective of biological success, generating geometrical (and other visual) percepts on a statistical basis is useful and indeed necessary. The answer lies in the fundamentally uncertain significance of any given retinal projection for visually guided behavior. Because 3D spatial relationships in the physical world are transformed into 2D relationships in the image plane, the physical reality underlying any given geometrical relationship in the retinal image is necessarily ambiguous: a given configuration in the image plane could have been generated by many different physical sources. It seems inevitable, therefore, that this problem be addressed by taking advantage of past experience with retinal images and their physical sources. The behavioral interactions of humans and other animals with the physical sources of images in natural visual environments have presumably led to the instantiation of the relevant statistical relationships in visual system circuitry over evolutionary time. Indeed, a growing body of evidence supports this concept of vision (see, for example, refs. 15 and 16). More specifically, statistical analyses of natural geometry pertinent to the perception of surface shape and orientation (17), egocentric distance (18), spatial intervals (6), and object size (19) have successfully explained complex perceptual phenomena in each of these several categories of perceived geometry.