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Angle {deg)

If perceptions of angle magnitude are generated on the basis of past experience with the frequency of occurrence of projected angles, the angles seen should accord with their relative empirical rank of angle magnitude determined in this way. Figure 11.14A shows how the frequency of occurrence of angle projections derived from the physical geometry of the world would be expected to influence the perceptual space for angles. The empirical rank of any angle between 0° and 90° is shifted slightly in the direction of 180° compared to actual geometrical measurements, and the opposite is true for any angle between 90° and 180°. Accumulated experience with the relative frequency of angle projections on the retina generated by the geometry of the world thus predicts the psychophysical results shown in Figure 11.10 ( Figure 11.14B, C).

Figure 11.13 The physical source of two lines of the same length intersecting at or near 90° must be a larger planar surface (dashed line) than the source of the same two lines, making larger or smaller angles.

This geometrical fact explains the lower probability of 90° projections in Figure 11.12 compared to other angles. See text for further explanation. (From Howe and Purves, 2005. Copyright 2005 Springer Science+Business Media. With kind permission of Springer Science+Business Media.)

Not surprisingly, many theories have been proposed over the last century and a half to explain the anomalous perception of line lengths, angles, and related issues such as perceived object size. Attempts to rationalize these discrepancies in the nineteenth century included asymmetries in the anatomy of the eye, the ergonomics of eye movements, and cognitive compensation for the foreshortening in perspective. More recent investigators have supposed that the anomalous perceptions of geometry arise from inferences based on personal experience. For example, British psychologist Richard Gregory argued that the different line lengths seen in response to the stimulus in Figure 11.2D (the Muller-Lyer illusion) are a result of interpreting the arrow-tails and arrowheads in the stimuli as concave and convex corners, respectively. The anomalous perception of line length was taken to follow from the different distances implied by such real-world corners, with the assumption that convex corners implied by the arrowheads would be nearer to the observer than concave corners implied by arrow tails. Although such explanations are empirical, analysis of real-world geometry often contradicts intuitions that seem obvious. For example, laser scanning of buildings and rooms shows that there is no significant difference in the distance from observers of convex and concave corners. As in lightness, brightness, and color, intuitions are a poor foundation for understanding perception.

Figure 11.14 Predicting perceived angles based on the frequency of occurrence of images generated by real-world sources. A) The red curve shows the frequency of occurrence (expressed as cumulative probability) of projected angles derived from the data in Figure 11.12. For comparison, the black line shows the frequency of occurrence that would be generated if the probability of any projected angle was the same (see inset). B) The red curve shows the perception of angle magnitude predicted from the information in (A) (the dashed diagonal indicates the angle projected on the retina and the thin dashed line indicates 90°). C) The predictions in (B) compared to psychophysical measurements of angle perception taken from Figure 11.10A. (After Howe and Purves, 2005. Copyright 2005 Springer Science+Business Media. With kind permission of Springer Science+Business Media.)

Contemporary explanations of these effects have more often turned to the receptive field properties of visual neurons (see Chapters 1 and 7), suggesting, for example, that lateral inhibitory effects among orientation-selective cells in the visual cortex could underlie the anomalous percepts illustrated in Figure 11.10. In this interpretation, the perception of an angle might differ from its actual geometry because orientation-selective neurons coactivated by the two arms of the angle inhibit each other. The supposed effect would be to shift the distribution of cortical activity toward neurons whose respective selectivity would be farther apart than normal, thus explaining the perceptual overestimation of acute angles (some other interaction would be needed to explain the underestimation of obtuse angles). As with some attempts to explain lightness/brightness effects described in Chapter 8, this approach implies that perceptual discrepancies are a side effect of other goals in visual processing.

In short, a lot of explanations have been proposed for the anomalies evident in the perception of geometrical forms, and some remain in play. The advantage of the empirical explanation provided by Qing is that it covers the full range of geometrical phenomena to be accounted for (only a few examples have been described here) and provides a strong biological rationale: contending with the inverse problem as it pertains to the geometry of retinal projections. It helps that this way of accounting for geometrical percepts also accords with the empirical explanation of many aspects of lightness, brightness, and color.

Nonetheless, many people have difficulty understanding how this way of seeing apparent lengths, angles, or other aspects of geometry could possibly be helpful in guiding behavior. Surely seeing these features for what they are in the image or the world would be the most logical way to relate perception to behavior, even though the inverse problem precludes direct specification of physical sources. But to reiterate, the seeing image features would be not be useful, and the idea that they should be misses the nature of the problem that vision must solve. Only by encoding the empirical results of trial-and-error discovery from successful behavior is it possible to deal with the world effectively through the senses. Seeing geometry the way we do is a reflection of this process.

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