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Qing tested the merits of this explanation by assessing the frequency of straightline projections generated by the objects in the laser-scanned scenes ( Figure 11.6). By extracting all the projected straight lines from the database that corresponded to geometrical straight lines on surfaces in the 3-D world, she could compile the frequency of occurrence of projected lines at different orientations. In effect, this analysis represents human experience with lines of different lengths and orientations in retinal images. For a line of any particular length and orientation on the retina, some percentage of projected lines in that orientation will have been shorter than the line in question, and some percentage will have been longer ( Figure 11.7). In an empirical framework, this accumulated experience will have shaped the perceptual space of line length and thus the length of the lines that observers see. If this idea is right,

the probability distributions in Figure 11.7 should predict the puzzling psychophysical results in Figure 11.5B.

Figure 11.6 The frequency of occurrence of lines in different orientations on an image plane, determined by analyzing a database of laser-scanned scenes. A) The pixels in part of an image from the database are represented diagrammatically by the grid squares; the connected black dots indicate a series of templates used to determine the frequency of occurrence of straight line projections at different orientations in images arising from straight lines in the 3-D world (note that the definition of straight lines is geometrical and not dependent on visible edges alone). B) Examples of straight-line templates overlaid on a typical image, for which the corresponding distance and direction of each pixel are known (see Figure 11.3). White templates indicate sets of points that correspond to straight lines in the world, and red templates indicate sets that do not. By repeating such sampling millions of times in different images, the frequency of occurrence of straight line projections of different lengths in different orientations on the retina can be tallied. (After Howe and Purves, 2005. Copyright 2005 Springer Science+Business Media. With kind permission of Springer Science+Business Media.)

Figure 11.7 Frequency of occurrence of straight-line projections in different orientations generated by the geometry of the world. A) The relative occurrence of projected line lengths in vertical (red) and horizontal (blue) orientations. The area under the two curves indicates accumulated human experience with projected lines of any given length in these two orientations. For a vertical line of a particular length (dashed line), humans will have experienced relatively more lines that are shorter than this length (area under the red curve to the left of the dashed line)

Figure 11.7 Frequency of occurrence of straight-line projections in different orientations generated by the geometry of the world. A) The relative occurrence of projected line lengths in vertical (red) and horizontal (blue) orientations. The area under the two curves indicates accumulated human experience with projected lines of any given length in these two orientations. For a vertical line of a particular length (dashed line), humans will have experienced relatively more lines that are shorter than this length (area under the red curve to the left of the dashed line)

compared to experience with a horizontal line of the same projected length (area under the blue curve). B) Probabilities of occurrence of lines at other orientations pertinent to explaining the psychophysical function in Figure 11.5B. See text for further explanation. (After Howe and Purves, 2005. Copyright 2005 Springer Science+Business Media. With kind permission of Springer Science+Business Media.)

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In thinking about this explanation of apparent length, recall the basic problem. Although a ruler accurately measures physical length, the visual system cannot do so because of the inverse problem (see Figure 11.1). (Note, incidentally, that the perceived length of a ruler is no more vertical than any other length we see; the markings on it vary in separation according to the orientation of its projected length on the retina, as indicated in Figure 11.5.) However, if the visual system ordered the perception of projected lengths according to feedback from accumulated experience, the inverse problem could be circumvented: Lengths on the retina would be operationally associated with behavior directed at physical lengths according to the frequency of occurrence of projected lengths. Although visually guided behavior on this basis is statistically determined, enough experience over evolutionary and individual time would make actions in the world increasingly efficient.

If this scheme is indeed being used to generate biologically useful perceptions, the puzzling variation in the apparent length of a line as a function of its orientation (see Figure 11.5B) should be predicted by the frequency of occurrence (the empirical rank) of projected line lengths as orientation changes.

When Qing used the data in Figure 11.7 to predict how the same line at different orientations would be seen on this basis, she found that the prediction given by the empirical rank of lines in different orientations matched the McDonald's arches-like function that describes the lengths people actually see ( Figure 11.8). We were all impressed with this result; it was hard to imagine that this odd psychophysical function could be explained in any other way.

Figure 11.8 Comparing the line lengths reported by subjects (A; taken from Figure 11.5B), and the perception of line length predicted from the frequency of occurrence of differently oriented straight lines in the retinal images (B). The prediction is for a particular projected length, but the same general shape is apparent for any length. (After Howe and Purves, 2005. Copyright 2005 Springer Science+Business Media. With kind permission of Springer Science+Business Media.)

Why, then, are there more sources in the world—and, thus, more projected images—of relatively short vertical lines compared to horizontal lines, and why are there even more sources that project as relatively short lines at 20°-30° away from vertical (see Figure 11.7)? Straight lines in the physical world are typically components of planes, a statement that may seem odd because we are very much aware of lines at contrast boundaries (for example, the edges of things). But whereas explicit lines generated by contrast obviously provide useful information about the edges of objects, object surfaces are by far the more frequent source of the geometrical straight lines that we experience. Thus, when considering the physical sources of straight lines that project onto the retina in different orientations, the most pertinent variable is the extension of relatively flat surfaces. Horizontal line projections in the retinal image are typically generated by the extension of planar surfaces in the horizontal axis, whereas vertical lines are typically generated by the extension of surfaces in either the vertical axis or the depth axis ( Figure 11.9A). The generation of vertical line projections from the extension of surfaces in depth, however, is inherently limited because the depth axis is perpendicular to the image plane; lines on such planes are thus foreshortened by perspective (that is, they generate shorter lines on the retina). A quick inspection of the world makes clear that the extension of surfaces in the vertical axis is also limited by gravity, adding to the prevalence of shorter vertical projected lines (overcoming gravity takes work, so objects, natural or otherwise, tend to be no taller than they have to be). Because neither of these limitations restricts the generation of horizontal line projections from real-world objects, humans experience more short vertical line projections than short horizontal ones (see Figure 11.7). As a result, a vertical line on the retina will always have a higher empirical rank than a horizontal line of the same length (that is, more lines shorter than the vertical line will have been projected on the retina, giving the vertical line a relatively higher rank in the perceptual space of apparent lengths than a horizontal line of the same length).

A different real-world bias accounts for the fact that there are more short line projections 20°-30° away from vertical than dead-on vertical, giving these oblique lines the highest empirical rank and thus the greatest apparent length (see Figures 11.8 and 11.5B). To understand this peculiarity, consider vertical and oblique lines of the same lengths superimposed on natural scenes. As illustrated in Figure 11.9B, natural surfaces provide more sources of relatively long vertical lines than oblique lines, as indicated by the data for lines at 60° in Figure 11.7B. The reason is that natural surfaces such as tree trunks tend to extend vertically instead of obliquely because of greater mechanical efficiency. As a result, relatively long linear projections somewhat away from vertical are less frequent than equally long vertical projections. Shorter projected lines oriented about 25° away from vertical are, therefore, even more common than shorter vertical lines (although shorter vertical lines are more common than horizontal ones; see above). The upshot is that their empirical rank is somewhat greater than the rank of vertical lines, causing the McDonalds-like arches in Figures 11.5B and 11.8B.

Figure 11.9 Physical bases for the biased projection of lines in different orientations. A) The projection of long vertical lines is limited by foreshortening and by the relative paucity of tall vertical objects in the world. As a result, humans will always have been exposed to more short vertical line projections than short horizontal ones. B) Image of a natural scene with superimposed vertical lines and oblique lines about 25° from the vertical axis. Despite the overall paucity of long vertical line projections compared to horizontal ones, longer vertical line projections are more likely than oblique ones because of the relative abundance of vertical compared to oblique surfaces in the world. See text for further explanation. (After Howe and Purves, 2005. Copyright 2005 Springer Science+Business Media. With kind permission of Springer Science+Business Media.)

Another challenge in rationalizing geometrical percepts on an empirical basis is perception of the apparent angle made by two lines that meet (either explicitly or implicitly) at a point. As with the apparent length of lines, an intuitive expectation about the perception of angles is that this basic feature of geometry should scale directly with the size of angles measured with a protractor. However, this is not what people see. It has long been known that observers tend to overestimate the magnitude of acute angles and underestimate obtuse ones by a few degrees ( Figure 11.10A). The anomalous perception of angles is easiest to appreciate—and most often demonstrated—in terms of a related series of geometrical stimuli that involve intersecting lines in various configurations. The simplest of these is the so-called tilt illusion, in which a vertical line in the context of an obliquely oriented line appears to be slightly rotated away from vertical in the direction opposite the orientation of the oblique "inducing line" ( Figure 11.10B). The direction of the perceived deviation of the vertical line is consistent with the perceptual enlargement of the acute angles in the stimulus and/or a reduction of the obtuse angles. This relatively small effect is enhanced in the Zöllner illusion, a more elaborate version of the tilt effect, achieved by iterating the basic features of the tilt stimulus ( Figure 11.10C). The several parallel vertical lines in this presentation appear to be tilted away from each other, again in directions opposite the oblique orientation of the contextual line segments (see also 11.2A, which depends on this same effect). The challenge for an empirical interpretation of vision is whether accumulated experience with retinal images and their sources in the world can also explain the odd way we see angles.

On empirical grounds, these differences between measured and perceived angles are expected. Just as the inverse problem makes the source of a projected line unknowable (see Figure 11.1), an angle on the retina could arise from any real-world angle. Perceiving angles on an empirical basis would allow observers to contend with this inevitable ambiguity. Understanding angles in these terms depended on much the same approach as understanding the perception of interval lengths. The frequency of occurrence of angle projections generated by the geometry of the world could be determined from laser range images, and Qing's supposition was that this information would have determined the way we see angles and should thus explain perceptual anomalies such as those in Figure 11.10. A first step was to identify regions of the laser-scanned scenes in the database of real-world geometry that contained a valid physical source of one of the two lines that form an angle (the black reference line in Figure 11.11A). After a valid reference line had been found, the occurrence of a valid second line forming an angle with it could be determined by overlaying a second straight-line template in different orientations on the image (the red lines in Figure 11.11). By systematically repeating this procedure, the relative frequency of occurrence of different projected angles on the retina could be tallied.

Figure 11.10 Discrepancies between measured angles and their perception. A) Psychophysical results showing that acute angles are slightly overestimated and obtuse ones underestimated. B) The tilt illusion. A vertically oriented test line (red) appears tilted slightly counterclockwise in the context of an oblique inducing line (black). C) The Zöllner illusion. The vertical test lines (red) appear more impressively tilted in directions opposite the orientations of the contextual lines (black) when the components of the tilt effect are repeated. (Data in A are from Nundy, et al., 2000 [Copyright 2000 National Academy of Sciences, U.S.A.]; B, C after Howe and Purves, 2005 [Copyright 2005 Springer Science+Business Media. With kind permission of Springer Science+Business Media.])

Figure 11.11 Determining the frequency of occurrence of angles generated by the geometry of the world. A) As in Figure 11.6, the pixels in an image are represented by grid squares. The black dots indicate a reference line template and the red dots indicate additional templates for sampling a second line making different angles with the reference line. B) The white line overlaid on the images indicates valid reference lines. Blowups of the boxed area show examples of the second template (red) that was overlaid on the same area of the image to sample for the presence of a second straight line making a valid angle (in each of the cases shown, the second template is also valid). (After Howe and Purves, 2005. Copyright 2005 Springer Science+Business Media. With kind permission of Springer Science+Business Media.)

Figure 11.11 Determining the frequency of occurrence of angles generated by the geometry of the world. A) As in Figure 11.6, the pixels in an image are represented by grid squares. The black dots indicate a reference line template and the red dots indicate additional templates for sampling a second line making different angles with the reference line. B) The white line overlaid on the images indicates valid reference lines. Blowups of the boxed area show examples of the second template (red) that was overlaid on the same area of the image to sample for the presence of a second straight line making a valid angle (in each of the cases shown, the second template is also valid). (After Howe and Purves, 2005. Copyright 2005 Springer Science+Business Media. With kind permission of Springer Science+Business Media.)

Figure 11.12 shows the frequency of occurrence of projected angles projected on the retina. Regardless of the orientation of the reference line (indicated by the black line in the icons under the graphs) or the type of real-world scene from which the statistics are derived, the likelihood of angle projections is always least for 90° and greatest for angles that approach 0° or 180°. In other words, the chances of finding real-world sources of an angle decrease as the two lines become increasingly perpendicular.

The cause of the bias evident in these statistical observations can, like the biased projection of line lengths in different orientations, be understood by considering the provenance of straight lines in the physical world. As with single lines, intersecting straight lines in the world are typically components of planar surfaces. Accordingly, a region of a planar surface that contains two physical lines whose projections intersect at 90° will, on average, be larger than a surface that includes the source of two lines of the same length but intersecting at any other angle (Figure 11.13). Because larger surfaces include smaller ones, smaller planar surfaces in the world are inevitably more frequent than the larger ones. Thus, other things being equal, physical sources capable of projecting angles at or near 90° are less prevalent than the sources of any other angles.

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