In human visual perception, the visual angle, denoted θ, subtended by a viewed object sometimes looks larger or smaller than its actual value. One approach to this phenomenon posits a subjective correlate to the visual angle: the perceived visual angle or perceived angular size. An optical illusion where the physical and subjective angles differ is then called a visual angle illusion or angular size illusion. Angular size illusions are most obvious as relative angular size illusions, in which two objects that subtend the same visual angle appear to have different angular sizes; it is as if their equal-sized images on the retina were of different sizes. Angular size illusions are contrasted with linear size illusions, in which two objects that are the same physical size do not appear so. An angular size illusion may be accompanied by (or cause) a linear size illusion at the same time. The perceived visual angle paradigm begins with a rejection of the classical size–distance invariance hypothesis (SDIH), which states that the ratio of perceived linear size to perceived distance is a simple function of the visual angle. The SDIH does not explain some illusions, such as the Moon illusion, in which the Moon appears larger when it is near the horizon. It is replaced by a perceptual SDIH, in which the visual angle is replaced by the perceived visual angle. This new formulation avoids some of the paradoxes of the SDIH, but it remains difficult to explain why a given illusion occurs. This paradigm is not universally accepted; many textbook explanations of size and distance perception do not refer to the perceived visual angle, and some researchers deny that it exists. Some recent evidence supporting the idea, reported by Murray, Boyaci and Kersten (2006), suggests a direct relationship between the perceived angular size of an object and the size of the neural activity pattern it excites in the primary visual cortex.
A relatively new idea Visual angle illusions have been explicitly described by many vision researchers, including Joynson (1949), (McCready 1963, 1965, 1985, 1999), Rock & McDermott (1964), Baird (1970), Ono (1970), Roscoe (1985, 1989), Hershenson (1982, 1989), Reed (1984, 1989), Enright (1989), Plug & Ross (1989, 1994), Higashiyama & Shimono (1994), Gogel, & Eby (1997), Ross & Plug (2002), and Murray, Boyaci & Kersten (2006). Specifically, these researchers cited have advocated a relatively new idea: that many of the best-known size illusions demonstrate that for most observers the (subjective) perceived visual angle, θ′, can change for a viewed target that subtends a constant (physical) visual angle θ. Indeed, various experiments have revealed most of the factors responsible for these visual angle illusions, and a few different explanations for them have been published (Baird, Wagner, & Fuld, 1990, Enright, 1987, 1989, Hershenson, 1982, 1989, Komoda & Ono, 1974, McCready, 1965, 1985, 1986, 1994, Ono, 1970, Oyama, 1977, Reed, 1984, 1989, Restle, 1970, Roscoe, 1985, 1989). On the other hand, nearly all discussions (and explanations) of those classic size illusions found in textbooks, the popular media, and on the internet use, instead, an older hypothesis that the visual angle is not perceivable (Gregory, 2008, Kaufman & Kaufman, 2002). They can describe and explain only a linear size illusion, which is why they do not properly describe or explain the illusions that most people experience. In order to clarify the new paradigm which replaces the old one, it helps to keep in mind that an angle is the difference between two directions from a common point (the vertex). Accordingly, as described below, the visual angle θ is the difference between two real (optical) directions in the field of view, while the perceived visual angle θ′, is the difference by which the directions of two viewed points from oneself appear to differ in the visual field.
Physical measures S, D, R, and θ
Figure 1 illustrates an observer's eye looking at a frontal extent AB that has a linear size S (also called its "metric size" or "tape-measure size"). The extent's lower endpoint at B lies at a distance D from point O, which for present purposes can represent the center of the eye's entrance pupil. The line from B through O indicates the chief ray of the bundle of light rays that form the optical image of B on the retina at point b, let's say, on the fovea. Likewise, endpoint A is imaged at point a. The optical (physical) angle between those chief rays is the visual angle θ which can be calculated:
tan θ = S / D {\displaystyle \tan \theta =S/D\,}
The retinal images at b and a are separated by the distance R, given by the equation
R / n = tan θ {\displaystyle R/n=\tan \theta \,}
in which n is the eye's nodal distance that averages about 17 mm. That is, a viewed object's retinal image size is approximately given by R = 17 S/D mm. The line from point O outward through object point B specifies the optical direction, dB, of the object's base from the eye, let's say toward the horizon. The line from point O through point A specifies that endpoint's optical direction, dA, toward some specific elevation value (say, 18 degrees). The difference between those real directions (dA − dB) is, again, the visual angle θ.
Perceived measures Figure 2 diagrams the perceived (subjective) values for a viewed object.
Point O′ represents the place from which the observer feels that he or she is viewing the world. For present purposes, O′ can represent the cyclopean eye (Ono, 1970, Ono, Mapp & Howard, 2002).
Perceived linear values D′ and S′ In Figure 2, D′ is the perceived distance of the subjective point B′ from O′. The observer might simply say how far away point B′ looks, in inches or meters or miles. Similarly, S′ is the perceived linear extent by which the subjective point A′ appears directly above point B′. The observer could simply say how many inches or meters that vertical distance looks. For a viewed object, S′ thus is its perceived linear size in meters, (or apparent linear size).
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