Formal Theories of Visual Information
The structure o( light m a point of obsermtion can be described mathematimll\· bv analrzing how the visual solid angles intersect with a hypothetical projection surf'ace. Although all existing analyses of visual information are based on some variation of this assumption , there are important distinctions among different theories involving the shape of the projection surface and the relative orientations of the incoming light rays. Consider the issue of shape. For many theorists the most intuitive way of analyzing visual information is to adopt a hemispherical projection surface, because that is the approximate shape of the human retina. It is important to keep in mind , however, that the structure of light does not depend on the specific anatomical structure of the human eye , and could be analyzed just as easily using some other form of projection surface such as a plane (e.g., Longuet-Higgens & Prazdny . 1980) or a cylinder (e.g .. Lee , 1974). For the purposes of a formal analysis , the choice of a projection surface is purely a matter of mathematical convenience , since a description of visual information in terms of one projection surface can always be uniquely transformed into a description in terms of any other. A more important distinction among existing theories involves the relative orientations of the incoming light rays as they intersect the projection surface toward the point of observation. Consider the two diagrams depicted in Fig. 5.2. In Fig. 5.2A. the incoming light rays intersect the projection surface at oblique angles to one another as they converge on the point of observation a relatively short distance away. This is called a polar or central projection. In Fig. 5.28. on the other hand , the light rays converge at a poim that is infinitely far from the projection surface so that they all travel along parallel trajectories. This is called a parallel or orthographic projection . Although a perfect parallel projection can never be achieved under natural viewing conditions, it is closely approximated whenever the size of an object is smaller than one tenth its distance from the point of observation . Thus. analyses based on parallel projection are only relevant to human vision when an object is relatively small or is viewed from a relatively long distance. Analyses based on polar projection have no restrictions on viewing distance from a purely mathematical
point of view. However, because the tolerance for measurement error becomes smaller and smaller as viewing distance increases . they are only useful in practice for objects that are relatively large or a small distance from the point of observation. There is an overwhelming amount of evidence that human observers can accurately perceive the layout of the environment under either parallel or polar projection . but a theoretical explanation for this high degree of llexability remains a mystery.