Visual Angle Illusion

The visual angle, V degrees, subtended by a viewed object sometimes looks larger or smaller than its actual value, creating a visual angle illusion (V-illusion).

These V-illusions have been explicitly described by many vision researchers, including Joynson (1949), McCready (1963, 1965, 1985), 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).

V-illusions are most obvious as relative V-illusions in which two objects that subtend the same visual angle appear different angular sizes; it is " as if" their equal-sized images on the retina were of different sizes.

A Relatively "New" Idea

Specifically, the researchers cited above have advocated the relatively new idea that many of the best-known "size" illusions (see Note 3) clearly demonstrate that, for most observers the (subjective) perceived visual angle, V' deg, can change for a viewed target that subtends a constant (physical) visual angle, V deg.

Indeed, various experiments have revealed most of the factors responsible for these V-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, a very old viewpoint (hypothesis) that the visual angle is not perceivable. That's why they do not (cannot) properly describe or explain the illusions that most people suffer. In order to clarify the "new" idea (paradigm) which wholly 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, V deg, is the difference between two real (optical) directions in the field of view , while the perceived visual angle, V' deg, is the difference by which the directions of two viewed points from oneself appear to differ in the visual field. Consider some definitions.

The Physical Measures, S, D, V, R, and d.

Figure 1 illustrates an observer's eye looking at a frontal extent, AB, that has a linear size, S meters (also called its "metric size" or "tape-measure size"). The extent's lower endpoint at B lies D meters from point O, which for present purposes can represent the center of the eye's entrance pupil and the two nodal points (See visual angle)

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, V deg, which can be calculated using Equation 1.

tanV = S/D (Equation 1)

The retinal images at b and a, are separated by the distance, R mm, given by the equation,

R/n = tanV, 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 Optical Directions.The line from point O outward through object point B, specifies the optical direction, d_B, of the object's base from the eye, let's say toward the horizon (zero degrees elevation).The line from point O through point A specifies that endpoint's optical direction, d_A, toward some specific elevation value (say, 18 degrees). The difference between those real directions (d_A - d_B) is, again, the visual angle, V deg.

The Perceived Measures, S', D', d', and V'.

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 ¹ (Ono, 1970, Ono, Mapp & Howard, 2002.)

The Perceived Linear Values, D' and S'

In Figure 2, D' meters is the perceived distance of subjective point, B' from O'. The observer might simply say how far away Point B looks, in inches or meters or miles. And, S' meters 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, (or apparent linear size).

The Perceived Visual Angle, V' deg.

The perceived endpoint at B' has the perceived direction, d'_B, and the observer might simply say "it looks straight ahead and toward the horizon."

This concept of the (subjective) visual direction is very old ². However, as Wade, Ono & Mapp (2006) noted, it unfortunately has been ignored in many current theories of "size" perception, and "size" illusions.

The object's other perceived endpoint, A', has a perceived direction, d'_A; about which the observer might say "it appears toward a higher elevation than point B." The difference between the two perceived directions (d'_A - d'_B) is the perceived visual angle, V' degrees.

For a viewed object's width, height or diameter. V' deg also is called its perceived angular size. .

Measuring The Perceived Visual Angle, V' deg.

It is not easy to quantify V' deg. For instance, a well-trained observer might say that point A "looks about 25 degrees higher" than B, but most of us cannot reliably say how large a direction difference looks. After all, we don't practice that skill because it is easier to use pointing gestures (Ono, 1970): For example, we often tell someone about the change in the directions we see for two viewed points by pointing something , say a finger or our eyes from one point to the other.

Therefore, in some experiments the observers aimed a pointer from one viewed point to the other, so the angle through which the pointer rotated was the measure of V' deg, (Komodo, 1970, Komodo & Ono, 1974, Ono, Muter, & Mitson, 1974, Gogel & Eby, 1997).

Also, because V' deg, specifies the amount by which one should rotate one's eye to quickly look from one seen point to another eye tracking, saccade , observers in other experiments shifted their gaze from one object endpoint to the other, and the angle the eye rotated through was measured as V' deg for that object (Yarbus (1967).

The Difference Between V' and S'

It is important to understand how V' deg differs from S' meters. Consider an example illustrated by the sketch at the right.Suppose we are looking through a window at a 30 foot wide house 240 feet away so it subtends a visual angle of about 7 degrees. The 30 inch wide window opening is 10 feet away so it subtends a visual angle of 14 degrees. We can say the house "looks larger and farther away" than the window, meaning that the perceived linear size, S', for the house's width is much larger than S' for the window; for instance a person might say the house "looks about 40 feet wide" and the window "looks about 3 feet wide."

We also can say the house "looks smaller and farther away" than the window, and that does not contradict the other statement because now we mean that the amount (V' deg) by which directions of the house's edges appear to differ is, say, about half the apparent direction difference for the window edges. Notice that we experience both the linear size and the angular size comparisons at the same time, along with the distance comparison (Joynson, 1949). Thus any report merely that one object "looks larger" than another object is ambiguous. It needs to specify whether "looks larger" refers to the perceived angular size (V' deg) or to the perceived linear size ( S' m) or to both of those qualitatively different "size" experiences (Joynson, 1949, McCready, 1965, 1985, Ono, 1970). Notice that in everyday conversations "looks larger" often refers to an angular size comparison rather than a linear size comparison. Additional confusion has resulted from widespread use of the ambiguous terms, "apparent size" and "perceived size" because they sometimes have referred to V' degrees and sometimes to S' meters without clarification, so the reader must try to ascertain what they mean.

(Also, in astronomy, "apparent size" refers to the physical angle, V deg, rather than to the subjective angle, V' deg.)

The Perceptual Size-Distance Invariance Hypothesis.

How the three perceived values, V' deg, S' meters, and D' meters would be expected to relate to each other for a given object is illustrated by Figure 2 and stated by Equation 2 (McCready, 1965, 1985, Ono, 1970, Komoda and Ono, 1974, Reed, 1989).

S'/D' = tan V' Equation 2 Ross & Plug (2002, Page 31) dubbed this new rule the "perceptual size-distance invariance hypothesis".

Retinal Size, "Cortical Size" and V' deg.

As already noted, the magnitude of an object's visual angle, V deg, determines the size, R mm, of its retinal image. And, the size of the retinal image normally determines the extent of the neural activity pattern the retina's neural activity eventually generates in the primary visual cortex, area V1 or Brodmann area 17. This cortical area harbors a distorted but spatially isomorphic "map" of the retina (see Retinotopy). This neurological relationship recently was confirmed by Murray, Boyaci, & Kersten (2006) using Functional magnetic resonance imaging, fMRI

The retinal image is not perceived or sensed. That is, experimental psychologists long ago rejected any idea that we "sense" a proximal stimulus such as the retinal image. As Gogel (1969, 1997) has repeatedly emphasized, there is no "sensation" which could be called the "perceived retinal image size," R' mm. Also rejected is a popular idea that an object's "perceived size" results from a "scaling of retinal size;" an illogical process that somehow "magnifies" the very small "retinal size" to yield the viewed object's much larger perceived linear size, S' meters.

Instead, the physical retinal extent, R mm, normally determines the magnitude of the perceived visual angle, V' deg. But, as already noted, "other factors" can intervene to slightly change V' deg for a target forming a constant sized retinal image (and thereby create a visual angle illusion). Indeed, the major discovery by Murray et al (2006) concerns this flexible relationship between R mm and V' deg, as described below.

V-Illusions and Area V1.

The Murray, et al (2006) observers viewed a flat picture with two disks that subtended the same visual angle (V deg) and formed retinal images of the same size (R mm), but the perceived angular size, V' deg , for one disk was larger than V' deg for the other (say, 17% larger) due to differences in their background patterns. And, in cortical Area V1, the sizes of the activity patterns related to the disks were unequal, despite the fact that the retinal images were the same size. The difference between these "cortical sizes" in Area V1 for the illusion disks was essentially the same as the difference produced by two non-illusory disks whose retinal image sizes differed by, say, 17% .

The researchers pointed out that their findings dramatically disagree with the hypothetical models of neural events being proposed in nearly all current theories of visual spatial perception. Murray, et al (2006) also noted that the flat illusion pattern they used can represent other classic "size' illusions, such as the Ponzo illusion and, as well, the moon illusion which is a visual angle illusion for most observers, (McCready, 1965, 1986, Restle 1970, Plug & Ross, 1989, p.21, Ross & Plug, 2002).

A detailed meta-analysis of the Murray et al. ( 2006) results is available in McCready (2007, Appendix B))

The Old "Size-Distance Invariance Hypothesis".

Conventional "textbook" theories of "size" and distance perception do not refer to the perceived visual angle (e.g., Gregory, 1963, 1970, 1998, 2008 ) and some researchers even deny that it exists (Kaufman & Kaufman, 2002). This (mistaken) idea that one does not see the different directions in which objects lie from oneself is a basis of the so-called "size-distance invariance hypothesis" (SDIH).

That old SDIH logic (geometry) is typically illustrated using a diagram that resembles Figure 2, but has the physical visual angle, V deg, substituted for the perceived visual angle , V' deg. The equation for the SDIH thus is ,

S'/D' = TanV (The SDIH)

Here, S' is called merely "perceived size" or "apparent size," but it obviously must be the perceived linear size, in meters. When rearranged as, S' = D'TanV, the equation expresses Emmert's law.

However, at least since 1962, researchers have pointed out that many classic "size" and distance illusions ³ can be neither described nor explained using the SDIH, so a new hypothesis is needed (Boring 1962, Gruber, 1956, McCready, 1965, Baird, 1970, Ono 1970). For instance, consider the simple Ebbinghaus illusion.

Ebbinghaus Illusion Example.

The two central circles are the same linear size, S mm, so, at the same viewing distance, D mm, so they subtend the same visual angle, V deg, and form equal sized retinal images. But the lower one "looks larger" than the upper one.

According to the SDIH, "looks larger" can mean only that S' mm is greater, and with the physical angle, V deg , the same for both, the SDIH requires that D' mm be greater for the lower one than for the upper one. However, for most observers, both circles appear unequal while also appearing at the same distance (on the same page). This commonly found disagreement between published data and the SDIH is known as the "size-distance paradox" (Gruber, 1956, Ono, et al. 1974).

The "paradox" completely vanishes, however, when the illusion is described, instead, as basically a visual angle illusion: That is, the perceived visual angle, V' deg, is larger for the lower circle than for the upper circle: It is as if its retinal image were larger. So. according to the "new" perceptual invariance hypothesis, (S'/D' = tan V'), with V' deg larger for the lower circle, and with D' cm correctly the same for both circles, then S' mm becomes larger for the lower one by the same ratio that V' deg is larger. That is, the reason the lower one looks a larger linear size on the page is because it looks a larger angular size than the upper one.

Explaining V-illusions Remains Difficult.

The new hypothesis that includes V' deg along with S' meters describes the Ebbinghaus Illusion and many other classic "size" illusions ³ more completely and more logically than does the popular SDIH. What still needs to be explained, however, is why the basic V-illusion occurs in each example.

Describing the few existing explanations for V-illusions is beyond the scope of this present entry. The most recent theories have been presented mostly in articles concerning the moon illusion (Baird et al., 1990, Enright, 1989a, 1989b, Hershenson, 1982, 1989b, Higashiyama, 1992, McCready 1986, 1999-2007, Plug & Ross, 1989, Reed, 1989, Roscoe, 1989, and especially in two "moon illusion" books (Hershenson, 1989; Ross & Plug, 2002) which make it quite clear that vision scientists have not yet agreed upon any particular theory of V-illusions.

There also is the lesser-known, but evidently the largest V-illusion of oculomotor micropsia (convergence micropsia) for which a few different explanations are being considered (McCready, 1965, 2007, Ono, 1970, Komoda & Ono, 1974, Ono, et al. 1974, Enright, 1987b, 1989a, 1989b).

NOTES

1. In some theories the cyclopean eye is, in effect, approximately midway between where one feels one's eye are located in one's body image of one's head (Ono, 1970, Ono, Mapp, & Howard, 2002). Some other theories define the place from which one feels one is viewing the world as the visual egocenter (Roelofs, 19xx, McCready, 1964, 1965, Sakuma & Pfaff, 1979, ) which, among observers, ranges from about midway between the eyes to at least as far back as the center of the head, about 4 inches behind the eyes, approximately midway between the two ears, on the axis for horizontal head rotations .

2. The subjective experiences of visual directions were fully researched by Hering (1942/1879) and by Helmholtz (1962/1910) who distinguished between the perceived oculocentric directions and the perceived egocentric directions.They, and other theorists, have pointed out that the an egocentric direction (d'B and d'A here) is determined by a process that necessarily combines the position of the point's image on the retina with information about the position of the eye with respect to the head (and body). 3. Here is a partial list of "size and distance" illusions that begin as visual angle illusions (angular size illusions) for most observers, but which still need to be described as such in Wikipedia and other articles that offer only the incomplete SDIH descriptions and explanations.

The Moon Illusion -+-+Oculomotor Micropsia (convergence micropsia) -+-+ The Ebbinhaus Illusion (Titchner's Circles) -+-+ The Hering Illusion -+-+The Ponzo Illusion -+-+The Müller-Lyer illusion -+-+The Rotating Spiral "size aftereffect " -+-+The Orbison illusion -+-+The Jastrow illusion -+-+The Wundt illusion -+-+The Meyer wallpaper illusion -+-+The Curvature of the apparent fronto-parallel plane (AFPP).

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