It seems logical to suppose that the perception of a line of a given length (e.g., a line drawn on a piece of paper or on a computer screen) would correspond more or less directly to the proportional length in the corresponding retinal projection. Accordingly, if a series of such stimuli having different lengths were shown to observers, one would expect their apparent lengths to scale in exact agreement with lengths of the retinal stimuli. This expectation, however, is not met.

An example of this discrepancy is the variation of the perceived length of the same line as a function of its orientation in the stimulus (Figure 1A). As investigators have repeatedly shown over the last 150 years, a line that is oriented more or less vertically in the retinal image appears to be somewhat longer than a horizontal line of the same length, the maximum length being seen, oddly enough, when the stimulus is oriented about 30° from vertical (Figure 1B).

This effect is evidently a particular manifestation of a general tendency to perceive the extent of any spatial interval differently as a function of its orientation in the retinal image. For instance, the apparent distance between a pair of dots varies systematically with the orientation of an imaginary line between them (as Wundt first showed in 1862), and a perfect square or circle appears to be slightly elongated along its vertical axis. Indeed, numerous observers have pointed out that measurements made with rulers or protractors of a variety of simple visual stimuli are often at odds with the perceptions they elicit, frequently in a most engaging way. Constructing stimuli that produce a particularly intriguing geometrical illusion was something of a cottage industry in the 19th and early 20th centuries; although some of these demonstrations were described by the pre-eminent vision scientists of the time, a good geometrical illusion has provided eponymous immortality to a number of investigators whose names would otherwise not be known today.

Some of the best known geometrical illusions - and the ones whose etiology has been most hotly debated - are illustrated in Figure 2. Perhaps the simplest of these is the "vertical-horizontal" or "T-illusion" attributed to J. J. Oppel, in which the vertical line appears longer than the horizontal line, despite the fact that they of equal length (Figure 2A). Oppel is also credited with having coined the term "geometrical illusion". A more elaborate example attributed to Ewald Hering shows two parallel lines (indicated in red) that appear bowed away from each other when presented on a background of converging lines (Figure 2B). In the Ponzo illusion (created by Mario Ponzo in 1928) the upper horizontal line appears longer than the lower one, despite the fact that they are again identical (Figure 2C). In the more complex Müller-Lyer illusion (created by Franz Müller-Lyer in 1889), the line terminated by arrow tails looks longer than the same line terminated by arrowheads (Figure 2D). The final example, also dating from the 19th C., was devised by Johann Poggendorff (Figure 2E). In this stimulus, the continuation of a line interrupted by a bar appears to be displaced vertically, even though the two line segments are actually collinear. Many other instances of the discrepancies between the objective geometrical features of a stimulus and the percept it gives rise to can be found in various popular books on illusions. Most of these, however, are variations on the basic themes illustrated in Figure 2.

It would be a mistake to conclude that, because the stimuli in Figure 2 are simple geometrical figures that bear little resemblance to real-world objects, these discrepancies between stimulus and percept are not significant for behavior in natural environments. Even when such stimuli are presented more realistically, the perceptual effects elicited by the simpler versions in Figure 2 persist.

Despite extensive study of these phenomena, no generally accepted explanation has been forthcoming. In work aimed at explaining the geometrical illusions, we have taken more or less the same approach as already described for brightness and color. Just as illumination, reflectance and other factors are conflated in the retinal image, the parameters that define the location and arrangement of the sources of light - the size, distance and orientation of objects - are inextricably intertwined in projected images (see Figure 2 under the introductory section called "The Problem". As a result, the spatial relationships of the sources of visual stimuli are always uncertain. A great deal of evidence now suggests that the human visual system contends with this further aspect of stimulus ambiguity by generating geometrical percepts according to the probability distributions of the possible real-world sources of retinal stimuli. The signature of this solution with respect to the perception of geometry in a visual scene is a wealth of subtle (and sometimes not so subtle) discrepancies between spatial percepts and the metrics of the stimuli that generate them (i.e., the geometrical illusions in Figure 2).

In terms of this general hypothesis about the genesis of geometrical illusions, the variation in the perceived length of lines illustrated in Figure 1 should reflect the statistical relationship between the projected images of lines and the their real-world sources. Recall that the underlying rationale for this way of understanding perception is a biologically effective way of contending with the inherent ambiguity of visual stimuli.

To test the merits of this explanation, we sampled the physical sources of straight-line projections in a range image database determined by laser-scanning natural scenes. The observed variation in the apparent length of lines as a function of their projected orientation agrees remarkably well with the percentile ranks of lines on the relevant empirical scales of line length derived from the probability distributions of the physical sources of line projections. Thus, this otherwise puzzling peculiarity about one of the simplest aspects of perceived geometry can be neatly explained as a particular manifestation of a broader visual strategy that generates percepts according to the probability distributions of the possible sources of inherently ambiguous stimuli. Importantly, other geometrical illusions such as the Ponzo, Müller-Lyer and Poggendorff illusion can be explained in the same general way based on statistical analyses of natural scene geometry.