The Poggendorff illusion explained by natural scene geometry - PNAS

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The Poggendorff illusion explained by natural scene geometry - PNAS
The Poggendorff illusion explained by natural
                                        scene geometry
                                        Catherine Q. Howe, Zhiyong Yang, and Dale Purves*
                                        Center for Cognitive Neuroscience and Department of Neurobiology, Duke University, Durham, NC 27708

                                        Contributed by Dale Purves, April 7, 2005

                                        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.

                                        geometrical illusions 兩 natural scene statistics 兩 range images 兩 visual
                                        perception

                                        J   ohann 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.
                                                                                                                   Fig. 1. The Poggendorff illusion and its behavior. (A) When an obliquely
                                        1 A, the right line segment appears to be shifted but in this case
                                                                                                                   oriented straight line is interrupted by a vertical occluder, the line segment on
                                        upward relative to the left segment. A similar effect is elicited          the right appears to be shifted downward with respect to the line segment on
                                        when the oblique line is interrupted by a space delineated by two          the left. (B) A similar effect occurs when the orientation of the interrupted line
                                        parallel horizontal lines instead of two vertical lines, as in Fig. 1C.    is reversed. In this case, the collinear extension on the right appears to be
                                        In this presentation, the right segment appears to be shifted to           shifted upward. (C) When an oblique line is interrupted by parallel horizontal
                                        the right with respect to the left segment. The magnitude of the           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
                                        apparent shift of the oblique line segments in these several
                                                                                                                   is made more vertical. (E) The magnitude also increases as the width of the
                                        configurations also varies as a function of the orientation of the         interruption increases. (F) The illusion is largely abolished when only the acute
                                        interrupted line and the width of the interruption. In standard            components of the stimulus are presented, but the effect is maintained when
                                        presentations of the Poggendorff stimulus (Fig. 1 A and B), the            only the obtuse components are shown. (G) The illusion is diminished when
                                        effect increases as the orientation of the interrupted line be-            the standard configuration is rotated so that the interrupted line is horizontal.
                                        comes 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               Methods
                                        illusion is largely abolished when only the acute components of            The Range Image Database. The details of the range image
                                        the standard Poggendorff stimulus are shown. However, the                  database used have been described in ref. 6. The images were
                                        effect remains if only the obtuse components are present (4)               acquired by using a range imaging system (LMS-Z210 3D Laser
                                        (Fig. 1F). Finally, when the overall orientation of the stimulus is        Scanner, Riegl USA, Orlando, FL), which identified the dis-
                                        rotated such that the interrupted line is horizontal (Fig. 1G), the        tances of object surfaces from 2 m to ⬇300 m with an accuracy
                                                                                                                                                                                                        NEUROSCIENCE

                                        effect is reduced but not completely abolished (2, 3, 5).                  of ⫾25 mm (Fig. 2A). By using this system, we collected ⬇100
                                           Although a variety of theories have been proposed to account            wide-field images extending 333° horizontally and 80° vertically
                                        for the Poggendorff illusion (see Discussion), none of these               at a resolution of 0.144°. Of these images, 25 were obtained in
                                        explanations rationalizes the full range of behavior illustrated in        fully natural environments in undeveloped terrain, and the rest
                                        Fig. 1. Here, we test the hypothesis that the Poggendorff effect           were obtained in outdoor or indoor settings that included human
                                        and its altered behavior in different presentations are generated          constructions (see also ref. 7). Only the midportion of the
                                        by the accumulated experience of humans with the various                   wide-field images (⬇110° horizontally ⫻ 52° vertically) was used
                                        sources of interrupted linear features in typical environments. By         for analysis.
                                        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 proba-             Freely available online through the PNAS open access option.
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                                        bility distribution of the relative locations of line segments across      *To whom correspondence should be addressed. E-mail: purves@neuro.duke.edu.
                                        a spatial interval.                                                        © 2005 by The National Academy of Sciences of the USA

                                        www.pnas.org兾cgi兾doi兾10.1073兾pnas.0502893102                                                   PNAS 兩 May 24, 2005 兩 vol. 102 兩 no. 21 兩 7707–7712
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. ⌬ 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); ⌬ 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                         percepts according to the accumulated experience in dealing
                                        physical object is interrupted by occlusion, the location of its                     with the physical sources of retinal stimuli. If this hypothesis is
                                        reappearance on the other side of the interruption is inevitably                     correct, the percepts elicited by the stimuli in Fig. 1 should
                                        uncertain: there are many ways the object could have traversed                       reflect the past experience of human observers with the physical
                                        the interval, and the interruption does not provide information                      sources of straight-line segments that are in the same orientation
                                        about which of the possible ways actually generated the stimulus.                    but separated by a spatial interval. More specifically, given a line
                                        Thus, the oblique line segments in the standard Poggendorff                          segment on one side of an implied occluder, the perceived
                                        presentation in Fig. 1 could arise from a single object with an                      position of a second line segment in the same orientation but on
                                        infinite variety of possible configurations or from two different                    the other side should be predicted by the relative probability of
                                        objects altogether. In short, the stimulus reflects the fundamen-                    occurrence of the physical sources of line segments projected in
                                        tal uncertainty of how the physical sources of projected line                        that orientation.
                                        segments are actually continued across a spatial interval. The
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                                        hypothesis examined here is that the way humans produce useful                       Sampling the Database. Fig. 2B shows examples of the geometrical
                                        behavioral responses in the face of this uncertainty is to generate                  templates we applied to the range images to sample the physical

                                        7708 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0502893102                                                                                                       Howe et al.
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 progres-
                                        sion; 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 seg-
                                        ments 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 (⌬, 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).    Fig. 3. Probability distributions of the physical sources of the right line
                                        The physical sources of the variants of the Poggendorff stimulus        segment in the standard Poggendorff stimuli. (A) The probabilities of occur-
                                                                                                                rence of the physical sources of the right oblique line segment for the standard
                                        illustrated in Fig. 1 were similarly determined by using appro-
                                                                                                                Poggendorff stimulus in Fig. 1 A presented as a function of the location of the
                                        priately configured sampling templates, as indicated in Fig. 2D.        right segment relative to the left oblique segment. The relative location of the
                                           For each configuration tested, the number of occurrences of          two oblique line segments, ⌬ (see Fig. 2E), is given in terms of visual angle. (B
                                        the physical sources of the right line segments at different            and C) Probability distributions of the physical sources of the right line
                                        locations relative to the left segment was tallied, and this            segment for the stimulus configurations in Fig. 1B (B) and Fig. 1C (C). Arrows
                                        information was used to produce the corresponding probability           indicate the mode of the distribution here and in subsequent figures.
                                        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          therefore predicts that a right segment that is actually collinear
                                        elicited by the standard Poggendorff stimulus and its variants.         with the left should be seen as displaced downward.
                                                                                                                   To appreciate the reasoning underlying this conclusion, con-
                                        Results                                                                 sider a hypothetical world in which the sources of right oblique
                                        Explanation of the Standard Poggendorff Effect. According to the        segments at all vertical locations are equally likely. In this case,
                                        empirical hypothesis being tested here, the probabilities of            the perceived vertical location of a right line segment would not
                                                                                                                                                                                                    NEUROSCIENCE

                                        occurrence of the physical sources of right oblique line segments       be biased one way or another by an observer’s past experience.
                                        at different vertical locations relative to the left segment should     Accordingly, a right line segment that was actually collinear with
                                        determine how observers perceive the relative positions of the          the left segment would be seen as such. As shown in Fig. 3A,
                                        two segments in the Poggendorff stimulus. Fig. 3A shows that            however, the real-world sources of the right line segment, given
                                        when the left oblique segment is oriented downward and to the           the left oblique segment and the vertical occluder, are biased
                                        right, as in Fig. 1 A, the probability distribution of the possible     toward locations above the point of collinearity with the left
                                        sources of the right segment is biased toward locations that            segment. Assessed against this body of past experience, then, a
                                        correspond to negative values of ⌬. In other words, the majority        right line segment that is geometrically collinear with the left
                                        of the physical sources that give rise to projections of right          segment would appear shifted downward. Thus, the probability
                                        oblique line segments in the same orientation as the left segment       distribution derived from the fully natural scenes in the database
                                        will, in the past experience of human observers, have projected         can indeed account for the standard Poggendorff effect.
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                                        above the point at which an extension of the left oblique line             Conversely, when the orientation of the interrupted line in the
                                        intersects the right vertical line. This probability distribution       stimulus is upward from the lower left, as in Fig. 1B, the probability

                                        Howe et al.                                                                                      PNAS 兩 May 24, 2005 兩 vol. 102 兩 no. 21 兩 7709
Fig. 4. The probability distribution of the physical sources of the right line      Fig. 5. Probability distributions of the physical sources of the right line
                                        segment changes progressively as the orientation of the interrupted line (A)        segment when the standard Poggendorff stimulus is decomposed or rotated.
                                        or the width of the interruption (B) is altered. ␣ represents the angle of          (A) The probability distribution obtained when only the acute components of
                                        intersection between the interrupted oblique line and the vertical lines, and       the stimulus are present, compared with the distribution when only the
                                        w is the width of the interruption. w ⫽ 1° in (A) and ␣ ⫽ 45° in (B); both values   obtuse components are considered. In both cases, the orientation of the
                                        were measured in terms of visual angle.                                             interrupted line was 45° and the width of the interruption 1°. (B) The prob-
                                                                                                                            ability distribution obtained when the standard Poggendorff stimulus is
                                                                                                                            rotated such that the interrupted line is horizontal (see Fig. 1G). The distri-
                                                                                                                            bution for the standard stimulus is replotted from Fig. 3A for comparison.
                                        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 ⌬ ⬎ 0) (Fig.             acute angles in the standard stimulus are presented but little
                                        3B). By the same reasoning, a right oblique segment that is actually                affected when the presentation is restricted to the obtuse com-
                                        collinear with the left segment will in this case be seen to be                     ponents (see Fig. 1F). As shown in Fig. 5A, when only the acute
                                        somewhat above the point where an extension of the left segment                     elements are used as the templates for sampling the range images
                                        intersects the right vertical line, as indeed it is.
                                                                                                                            (see Fig. 2D), the mode of the probability distribution (arrow) is
                                           A similar explanation applies to the Poggendorff stimulus in
                                                                                                                            very near the point where ⌬ equals zero. Accordingly, the
                                        which an oblique line is interrupted by two horizontal lines.
                                                                                                                            Poggendorff effect should be largely abolished. Conversely,
                                        When the left oblique segment is oriented downward and to the
                                                                                                                            when only the obtuse components are used, the probability
                                        right, as in Fig. 1C, the probability distribution of the sources of
                                                                                                                            distribution is similar to that of the standard Poggendorff
                                        the right line segment is biased toward locations to the left of the
                                        point at which ⌬ ⫽ 0 (Fig. 3C). Thus, the location of the right                     stimulus; thus, the illusion would be expected to retain its full
                                        segment appears shifted to the right from the continuation of the                   magnitude.
                                        left line, which is again consistent with what people see.                             Finally, we determined the probability distribution of the
                                                                                                                            sources of the right line segment when the overall orientation of
                                        Explanation of Other Features of the Poggendorff Effect. Changing                   the Poggendorff stimulus is rotated so that the interrupted line
                                        either the orientation of the interrupted line or the width of the                  is horizontal, as in Fig. 1G. Compared with the standard
                                        interruption systematically also shifts the relevant probability                    presentation, the distribution when the stimulus is rotated 90°
                                        distributions in a manner consistent with the altered magnitude                     has a mode closer to ⌬ ⫽ 0, as shown in Fig. 5B. As a result, the
                                        of the perceptual effects in these presentations. The effects of                    Poggendorff effect should be reduced but not abolished when
                                        varying the intersecting angle of the interrupted line (Fig. 1D)                    the presentation of the interrupted line is horizontal, as it is.
                                        and of varying the width of the interruption (Fig. 1E) on the
                                        corresponding probability distributions of the physical sources of                  Statistics Derived from Different Types of Scenes. Theories purport-
                                        the right line segment are shown in Fig. 4 A and B, respectively.                   ing to explain one or more aspects of the Poggendorff illusion
                                        As the orientation of the interrupted line becomes closer to the                    have often been based on intuitions about the ‘‘carpentered’’
                                        orientation of the lines defining the interruption (vertical in                     world of human artifacts (9, 10). It was thus of interest to ask how
                                        the example in Fig. 4A) or as the interruption becomes wider, the                   the probability distributions derived from fully natural scenes
                                        mode of the distribution shifts progressively away from the point                   (presumably representative of the human visual environment
                                        at which ⌬ ⫽ 0, in accord with the fact that the perceived shift                    during evolution) compare with distributions derived from en-
                                        in the apparent location of the right line segment increases as the                 vironments in which human construction is prevalent. We there-
                                        angle of the intersection becomes smaller or as the width of the                    fore computed the relevant probability distributions from the set
                                        interruption increases (3).                                                         of scenes in the database that included some or many man-made
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                                           Another puzzle that can be explained in these terms is the fact                  objects. As shown in Fig. 6, the pertinent probability distribu-
                                        that the Poggendorff effect is greatly diminished when only the                     tions obtained from the two types of scenes are generally similar.

                                        7710 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0502893102                                                                                                    Howe et al.
plane when it is above the point at which ⌬ ⫽ 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 ⌬ ⬍ 0 (i.e., above the point of ⌬ ⫽ 0) than for positions
                                                                                                                            at which ⌬ ⬎ 0. As a result, the probability distribution in Fig.
                                                                                                                            3A is biased toward ⌬ ⬍ 0. The same reasoning can explain the
                                                                                                                            biases seen in the distributions in Fig. 3 B and C.
                                                                                                                               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
                                        Fig. 6. The probability distribution of the sources of the right line segment       containing the physical source of the left segment and the
                                        for the standard Poggendorff stimulus derived from an analysis of fully             vertical lines (Fig. 7B). There is, accordingly, an increased bias
                                        natural scenes (see Fig. 3A) compared with the distribution derived from the        in the probability distribution of the sources of the right line
                                        scenes that contained human constructions.                                          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
                                        It thus seems safe to conclude that the perceptual effects                          components of the stimulus compared with only the acute
                                        apparent in the Poggendorff illusion are not specifically depen-                    components (see Fig. 5A) is also straightforward in these terms.
                                        dent on interactions with the rectilinear constructions associated                  In the stimulus comprising the acute components (see Fig. 1F),
                                        with human culture.                                                                 the right vertical line does not extend above the point at which
                                                                                                                            ⌬ ⫽ 0, reducing the likelihood that its physical source (i.e., an
                                        Physical Bases for the Statistical Biases Observed. Two questions so                underlying planar surface) extends above this point. This statis-
                                        far deferred in considering the statistics derived from our                         tical change in turn reduces the likelihood of occurrence of
                                        analysis of the range images are why the observed biases in the                     sources of the right oblique segment above the point at which
                                        occurrence of the physical sources of the right line segment exist                  ⌬ ⫽ 0. In contrast, the right vertical line is present above the
                                        and why the magnitude of these biases differs for the different                     point at which ⌬ ⫽ 0 in the stimulus composed of the obtuse
                                        configurations of the Poggendorff stimulus considered here.                         components; the bias toward ⌬ ⬍ 0 in the corresponding
                                           A major part of the answer to these questions lies in the                        distribution of the sources of the right oblique segment is
                                        geometry of planar surfaces, which are the typical sources of                       therefore maintained.
                                        straight-line projections on the retina. Consider, for instance, the                   Finally, it is important to explain the physical basis of the
                                        physical sources of the standard stimulus in Fig. 1 A. The                          reduced bias in the probability distribution of the sources of the
                                        presence of a physical source of the left oblique line segment and                  right line segment when the standard Poggendorff stimulus is
                                                                                                                            rotated such that the interrupted line is horizontal (see Fig. 5B).
                                        the vertical lines would usually signify the presence of a planar
                                                                                                                            This reduction cannot be rationalized in terms of the relationship
                                        surface in the corresponding region of the scene. Because a right
                                                                                                                            between the right line segment and the plane containing the
                                        oblique line segment above the point at which ⌬ ⫽ 0 is, on
                                                                                                                            source of the left segment and the vertical lines because this
                                        average, closer to the center of the plane than a line segment                      relationship is unchanged from that in the standard stimulus,
                                        below this point (Fig. 7A), the set of physical points correspond-                  despite the rotation. The reason in this case is the higher overall
                                        ing to a right line segment is statistically more likely to be in this              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 sus-
                                                                                                                            ceptible 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
                                                                                                                                                                                                  NEUROSCIENCE

                                        Fig. 7. Physical basis for the bias in the probability distributions of the         between the stimulus in the image plane and its possible sources
                                        sources of Poggendorff stimuli. (A) In the standard stimulus, a right line          in natural visual environments. From the analysis presented, it
                                        segment above the point at which ⌬ ⫽ 0 (gray arrowhead) is, on average,             should be apparent that the biased image–source relationships
                                        closer to the center of the plane containing the physical source of the left seg-   that account for the Poggendorff effect and its variants are
                                        ment and the vertical lines than is an otherwise comparable right line segment      rooted in the statistical properties of the physical world in which
                                        below the point at which ⌬ ⫽ 0. The distances of the right line segments from       human observers reside. Thus, this relationship has to be derived
                                        the center of the plane (black dot within the dashed line that outlines the         empirically by examining the natural environment and cannot be
                                        plane) are indicated by the gray dotted lines; the difference in the lengths of
                                                                                                                            deduced from principles of projective geometry per se.
                                        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     Previous Explanations of the Poggendorff Effect. The explanation
                                        position of the right line segment means a larger difference in the distances       most often offered in the past has been that the illusion derives
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                                        of the right line segments from the center of the plane. The differences in the     from a misperception of the angles in the stimulus (11, 12). For
                                        lengths of the gray dotted lines are again indicated by the black bars.             instance, an overestimation of the acute angles in the standard

                                        Howe et al.                                                                                             PNAS 兩 May 24, 2005 兩 vol. 102 兩 no. 21 兩 7711
stimulus in Fig. 1 A and an underestimation of the obtuse ones                       fundamentally uncertain significance of any given retinal pro-
                                        would presumably affect the apparent orientation of the line                         jection for visually guided behavior. Because 3D spatial rela-
                                        segments on either side of the interruption toward horizontal,                       tionships in the physical world are transformed into 2D rela-
                                        thus causing them to appear noncollinear. This explanation,                          tionships in the image plane, the physical reality underlying any
                                        however, has been disputed (1) on the grounds that it cannot                         given geometrical relationship in the retinal image is necessarily
                                        rationalize the paradoxical abrogation or preservation of the                        ambiguous: a given configuration in the image plane could have
                                        illusion elicited, respectively, by presentation of only the acute or                been generated by many different physical sources. It seems
                                        only the obtuse components of the stimulus (see Fig. 1F). Nor                        inevitable, therefore, that this problem be addressed by taking
                                        does this explanation account for the additional features of the                     advantage of past experience with retinal images and their
                                        Poggendorff effect shown in Figs. 1D, E, and G.                                      physical sources. The behavioral interactions of humans and
                                           Another explanation is based on the idea that the geometrical                     other animals with the physical sources of images in natural
                                        information in the retinal projection might be inappropriately                       visual environments have presumably led to the instantiation of
                                        ‘‘interpreted’’ by obser vers. For example, the ‘‘depth-                             the relevant statistical relationships in visual system circuitry
                                        processing’’ theory suggests that the oblique lines in the Poggen-                   over evolutionary time. Indeed, a growing body of evidence
                                        dorff stimulus are interpreted as lines extending in depth, and                      supports this concept of vision (see, for example, refs. 15 and 16).
                                        are therefore perceived to be noncollinear (10, 13, 14). This                        More specifically, statistical analyses of natural geometry perti-
                                        category of explanation is also limited in that it considers only                    nent to the perception of surface shape and orientation (17),
                                        some aspects of the projective geometry of the stimulus and its                      egocentric distance (18), spatial intervals (6), and object size (19)
                                        possible sources and cannot explain the full range of the percepts                   have successfully explained complex perceptual phenomena in
                                        elicited by the configurations in Fig. 1.                                            each of these several categories of perceived geometry.
                                        Biological Rationale. An important question not yet fully ad-                        We thank F. Long, S. Nundy, D. Schwartz, and J. Voyvodic for useful
                                        dressed is why, from the perspective of biological success,                          comments and criticisms. This work was supported by the National
                                        generating geometrical (and other visual) percepts on a statis-                      Institutes of Health, the Air Force Office of Scientific Research, and the
                                        tical basis is useful and indeed necessary. The answer lies in the                   G. B. Geller Endowment.

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                                        7712 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0502893102                                                                                                           Howe et al.
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