An Empirical Explanation: Angles
Although the difference between perceived angles and the actual subtense of their sources is only a few degrees, the perceptions predicted by the distribution of the possible sources of the stimuli accord surprisingly well with what people actually see.
The evidence for an empirical basis of vision has in other
demonstrations been restricted to the perception of light intensities (luminances)
and spectral differences (colors). A very different perceived quality is
the spatial arrangement of objects, an aspect of visual experience that
depends on a subjective sense of how the objects - lines in the simplest
case - are oriented in space with respect to the observer and to each other.
It has been known since the middle of the 19th C. that the perception of oriented lines does not always
accord with the real-world geometry of the underlying objects and their retinal projections. Thus, the
angles formed by lines making (or implying) an acute angle are seen as being a few degrees larger in
subtense than they really are, whereas obtuse angles are seen as being a few degrees smaller. Despite
a great deal of speculation about this anomaly, there has been no consensus regarding its origin. In the modern literature, these discrepancies in the perception of angles have usually been explained in terms of complex inhibitory interactions among orientation-selective neurons in the primary visual cortex.
The anomalous way we perceive angles can, however, be explained
in empirical terms, similar to the accounts of the way we perceive brightness
and color. The proximal stimuli-giving rise to perceived angles, like the
luminances or the spectral content of the returns, are profoundly ambiguous
(
Figure 1 and
Demonstration).
An angle projected onto a surface (the retina, for example) can arise from
angular objects having a variety of subtenses and arm lengths, arranged
in infinitely many orientations. In interacting with the objects that give
rise to particular retinal projections, observers will have found that the
real-world angles giving rise to the proximal stimuli vary greatly, and,
as it turns out, systematically. In consequence, the perception elicited
by an angle projected onto the retina should correspond to the frequency
distribution of the possible sources underlying the proximal stimulus in
phylogenetic and ontogenetic experience.
A particular advantage of considering the merits of a wholly empirical theory of vision vis a vis
the perception of angles (as opposed to brightness or color) is the ability to model the cumulative
visual experience on which the perceptions of angular stimuli are presumably founded. Whereas
the frequency distribution of the relevant past experience is difficult to compute for luminance or
spectral content, in the case of angles, the major features of the experience that have shaped the
relevant patterns of neural connectivity elicited by retinal stimulation can be specified by geometrical
principles, at least to a first approximation. This information can then be used to predict how angles
should be perceived according to a wholly probabilistic mechanism of vision theory proposed here,
thus providing a more rigorous test of a wholly empirical basis for vision.
The relative frequency of occurrence of all the possible three-dimensional sources of a projected angle can be assessed by analyzing all of the ways a given angular object can project onto a plane. Obviously, a line or any other object can exist in an infinite variety of orientations with respect to the observer. The simplifying assumption in the approach we used is that angular objects occupy these positions with equal probability (in fact, there is a slight bias even in natural scenes toward contours in the cardinal axes, and therefore toward right angles [see, for example, Coppola et al. 1998]). The distribution obtained in this way can be used in turn to generate the frequency distribution of the subtenses of the objects that could have given rise to any given angular distribution of the subtenses of the objects that could have given rise to any given angular projection (
Figure 2).
As indicated in
Figure 2, the most frequently occurring sources of acute angle projections are
angles larger than the subtense of the projected stimulus. Conversely, the sources of obtuse angles
projections will, by the nature of projective geometry, typically have been generated by sources that
are somewhat smaller than the projected angle. Right angle projections and straight lines, however,
will have been generated by sources that on average have the subtense of the object itself. The visual
system should, if the theory is correct, generate percepts that incorporate these statistical facts of
projective geometry, which have necessarily determined the way visual stimuli generated by angular
objects have been experienced throughout human history (and have therefore shaped the circuitry
underlying these perceptions). Although the difference between perceived angles and the actual
subtense of their sources is only a few degrees, the perceptions predicted by the distribution of the
possible sources of the stimuli accord surprisingly well with what people actually see (
Figure 3).
As expected on this basis, depicting a projected angle
such that it is more consistent with one real- world source than another
changes the perceived subtense accordingly, often quite strikingly. Thus,
whereas all the angles in this
Demonstration
subtend 90°, the stimulus is consistent with each projection having
been generated by angular objects having different subtenses and orientations.
As in the domains of brightness and color, the perceptions of the identical
stimuli vary according to their empirical significance.
A recent reexamination of angle perception using natural
scene statistics obtained empirically by laser range scanning confirms the
merits of this general approach to understanding how we perceive angles
(Howe and Purves, 2005).
