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The Bounds of Perspective: Marginal Distortions (page 2)

The Bounds of Perspective: Marginal Distortions

Fig.8.5 Four displays and response keys used by Sanders (1963).


A somewhat different procedure, developed by A. Sanders (1963, Experiment 3, pp. 49-52) required a subject to look at a fixation point where a column consisting of either tour or five lights would appear, while simultaneously, to the right of the fixated column of lights, another of column of lights would appear, also consisting of four or five lights (Figure 8.5). The angular distance between the two displays varied from 19 to 94°. Furthermore, there were two viewing conditions: one in which subjects were allowed to move their eyes to scan the display, and one in which they were instructed to keep their eyes on the location of the left column. The subject's task was to press one of four keys as quickly as possible after the two columns of lights were turned on. One key meant that both columns consisted of four lights, a second key meant that both had five, and the remaining two keys covered the remaining two possibilities of unequal numbers in the two columns. The median reaction times of two subjects are shown in Figure 8.6 First, look at the reaction times represented by the filled circles and summarized by the broken curve (condition 1: eye movements forbidden). The larger the display angle, the longer the reaction time; beyond 34°, the task was impossible. Second, look at the reaction times represented by unfilled circles and summarized by the solid curve (condition II: eye movements required). Up to about 30° , reaction times were longer than those obtained in the absence of eye movements, suggesting that eye movements were not necessary to see the right-hand column for smaller visual angles. This then is an estimate of the size of the field encompassed by the stationary eye. This estimate of the field normally captured by a glance is not inconsistent with Olmer's normal visual field.

Fig.8.6 Median reaction time (in seconds) as a function of display angle and fixation conditions: Condition l: Eyes immobile, fixating left column (filled circles, broken line). Condition II: Eye movements required (blank circles, solid line). Panels a and b represent data of two subjects.
Fig.8.7 Display used by Finke and Kurtzman (1981) to measure extent of visual field in imagery and perception.

The most impressive confirmation of our attempt to link the extent of Olmer's normal visual field for perspective drawings with the extent of what we can encompass in a single glance is provided by an experiment done by Finke and Kurtzman (1981). Imagine that you are looking at Figure 8.7 and that you are handed a pointer with a red dot on its tip and are asked to move it up along the diagonal line while keeping your eyes on the red dot. As you move the pointer and your gaze away from the center of the circle, it becomes gradually more difficult to discriminate the two sets of stripes, until you cannot tell that there are two distinct sets. The distance from the center at which this loss occurs is taken as an estimate of the boundary of the visual field. If the pattern is turned 45° clockwise and the observer is asked to move the pointer and his eyes rightward along the horizontal line, the boundary is found somewhat further from the center of the circle. If the procedure is repeated six more times, once for each remaining radial line, a rough estimate of the shape of the visual field can be obtained.

The size of the visual field estimated by this procedure varies with changes in the number of bars per inch: The higher the density of bars in the central pattern, the sooner the observer will report that the pattern has melted into a blur. The widest patterns used gave a field of 35 by 28°, gratifyingly close to Olmer's estimate.1 This correspondence suggests that we are comfortable with perspective drawings only if the scene they encompass does not subtend a visual angle greater than we would normally encompass in our visual field.

To find what it is in perspective pictures subtending a large visual angle that causes us to reject them, let us look back at Olmer's figures, which subtend large visual angles (Figures 8.3 and 8.4): Not all the cubes that fall outside the interior frame that bounds Olmer's normal visual field (between the two points D/3 in Figure 8.3 and within the rectangle xyx¢y¢ in Figure 8.4) look equally distorted. In Figure 4, for instance, compare the cube just below the linex1 y1 to the cube just to the left of x1 . The former looks considerably more distorted than the latter; it violates Perkins's law for forks, one of the angles of the fork being less than 90°. Only the cubes that violate Perkins's laws look distorted; the others do not. Therefore, perhaps it is not the wide angle of the view per se, but rather local features of the depictions, that cause these pictures to look distorted.

We are now in a position to understand the connection between Perkins's laws and the limited size of our visual field. We have seen from Olmer's drawings that the perspective drawings of rectangular objects are likely to violate Perkins's laws only when they fall outside a field that subtends 37° by 28 degrees. We have also seen that, because our visual field subtends about 37° by 28°, we are unlikely to perceive objects in our environment that fall outside such a field. In other words, the projections of objects that fall within our field of view all obey Perkins's laws. Because Perkins's laws are very simple, we may notice their violation in pictures only because they constitute a striking deviation from what we are accustomed to see and not because of the relation of Perkins's laws to parallel projection, which we observed in Chapter 7.2

1 Finke and Kurtzman went further. They also trained observers to imagine the grating and then asked them to move their eyes away from the position of the imagined pattern until it was too blurred to be seen by the mind's eye. The results were extremely close to the results obtained for perceived patterns.

2 Hagen and Elliott (1976) have made unwarranted claims in favor of the hypothesis that parallel projection is more natural than central projection (and predates it by about two millennia). They showed subjects pictures of 7 different objects (2 cubes and 5 regular pentagonal right prisms) using 6 different degrees of "perspective convergence front conical (traditional linear perspective) to axonometric (parallel) projection" (p. 481). They claim that "for a given object of fixed dimension observed from a fixed station point, a family of perspective views may be generated ... " (p. 481). Among the problems that invalidate this experiment and the authors' interpretation of it, I will mention four: (1) Changes in convergence are equivalent to changes in the location of the center of projection (the station point). It is meaningless to speak of a change in convergence without a concomitant change in the center of projection. (2) In their experiment, not all the pictures that were meant to depict different projections of one object showed the same number of the object's faces, and many of these pictures were degenerate to the extent that they precluded the recognition of the object (for example, one picture Of a cube was a rectangle divided into two rectangles by a vertical line). At least to of the 42 pictures suffered from such extreme degeneracy, and 4 of the 7 objects depicted had at least r degenerate picture. Because the purported differences in "degree of perspective convergence were inextricably confounded with large variations in the amount of visual information these pictures conveyed, it is impossible to interpret r subjects' preferences for some of the representations. (3) Of the 3 objects whose 6 pictures did not include cases of extreme degeneracy, 1 (the most convergent central projection of a cube, labeled A in their Figure 1) was a borderline violation of Perkins's law; hence it was fated to be rejected by subjects, but not because of their putative preference for parallel projection. (4) Among the 3 objects whose pictures did not include cases of degeneracy, only 1 yielded data unequivocally in support of Hagen and Elliott's conclusion that parallel-perspective drawings were the most natural or realistic drawings.

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