By Grant Ocean

In human visual perception, the visual angle, denoted

- or simply written in this form:
*θ = S / D*(since*tan**θ =**θ radians*for the small angle approximation; thus,*θ*can substitute for*tan**θ*).

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

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* or *R = 17 θ mm*. The line from point *O* outward through object point *B* specifies
the optical direction, *d _{B}*, of the object's base from
the eye, which is directed toward the horizon.
The line from point

The diagram above illustrates the perceived (subjective) values for a viewed object. Point

It is, however, deemed to be very hard to quantify

It is believed to be important to understand how

How the three perceived values *θ*′, *S*′, and *D*′ would be
expected to relate to each other for a given object is illustrated by the diagram above
and stated by the following equation,
which is dubbed by some as the "perceptual
size-distance invariance hypothesis":

or simply *θ*′ = *S' / D'. *Conventional "textbook" theories of "size" and distance perception do not
refer to the perceived visual angle and
some researchers even deny that it exists.
This
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 the diagram above,
but has the physical visual angle *θ* substituted for
the perceived visual angle *θ*′. The equation for the SDIH thus is

Here, *S*′ is typically called the "perceived size" or "apparent size";
more precisely it is the perceived linear size, measured in meters.
When rearranged as *S*′ = *D*′·*θ*, the equation expresses Emmert's law (which will be discussed in the next article).

The perceived visual angle has
been used to explain the Ebbinghaus illusion, for instance. In the
Ebbinghaus illusion figure on the right,
the two central circles are the same linear size *S* and the same
viewing distance *D*; so they subtend the same visual angle *θ* and
form equal-sized retinal images (see the previous article for details). But the right central circle "looks larger" than the left
one.
According to the SDIH, "looks larger" can mean only that *S*′ is
greater, and with the physical angle *θ* the same for both, the SDIH
requires that *D*′ be greater for the right circle than for the left 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".
The "paradox" completely vanishes, according to McCready, when the illusion is described,
instead, as basically a visual angle illusion:
that is, the perceived visual
angle *θ*′ is larger for the right central circle than for the left central circle. It is
as if its retinal image were larger. So, according to the "new" perceptual
invariance hypothesis
(*θ*′ = *S*′/*D*′), with *θ*′
larger for the right circle, and with *D*′ correctly the same for both
circles, then *S*′ becomes larger for the right one by the same ratio that
*θ*′ is larger.
As a consequence, the right central circle looks a larger linear
size on the page is because it looks a larger angular size than the left
one.

As already introduced,
the magnitude of an object's visual angle *θ*
is believed to determine the size *R* of its retinal image. In turn, the size of the retinal
image is believed to determine 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 is thought to harbor a
distorted but spatially isomorphic "map" of the retina, which has presumably been confirmed by Murray, Boyaci, & Kersten (2006) using functional magnetic resonance imaging.

Broerse J. et al. (1992).

http://www.psychologie.tu-dresden.de/i1/kaw/diverses%20Material/www.illusionworks.com/html/ames_room.html

http://psychology.about.com/od/sensationandperception/ig/Optical-Illusions/Ames-Room-Illusion.htm

http://www.newworldencyclopedia.org/entry/Ames_room

Appendix A: The Perceived Size and Its Mathematical Equations