Appendix B     (Posted February 05, 2007)

An Analysis of the Experiment by
Murray, S. O., Boyaci, H., & Kersten, D. (2006).
“The representation of perceived angular size
in human primary visual cortex.”

[All of the information in this Appendix B was mailed to Dr. Murray, Dr. Boyaci, Dr. Kersten and other relevant researchers in April 2006.

The experimenters measured the angular size (visual angle) illusion obtained with a flat, photo-montage that resembles a photograph of an imaginary hallway with two spheres on its brick floor at different distances with their diameters subtending the same angular size (optical angle) at the camera lens

The picture used in the study can be seen at the following link (you might open it in a new window, and click on the image there to see a larger version).

The crude sketch at the right imitates the copyrighted pattern they used. (However, for the detailed analyses here, it is best to view their original picture.)
The two disks (sphere images) on the flat screen (or page) are the same linear size so they subtend the same visual angle (V deg) at an observer’s eye, which, for the study was 6.5 degrees.
Therefore the retinal images of the disks are the same size.

(Some media reports of the experiment have mistakenly stated that the observers viewed a real hallway with real spheres! )

All 5 observers suffered the basic illusion that the upper disk’s diameter looks a larger angular size (in degrees) than the lower disk’s, so it also looks a larger linear size (in inches) on the page.

Task. The observer’s task was to change the linear size of the lower disk until both disks looked the same angular size, and thus the same linear size on the screen. That also would make the two ‘spheres’ look the same angular size.

Results. The observers made the lower disk from about 15% larger to 20% larger than the upper disk.
In other words, for a useful example here, the perceived visual angle (V’ deg) for the upper disk (and ‘sphere’) initially was, say, 17% larger than V’ deg for the lower disk (and ‘sphere’).
Also, the perceived linear size ( S’ in) for the upper disk was 17% larger than for the lower disk, while both disks had the same perceived distance (D’ in).
That is, a lower disk with a visual angle, of 7.6 deg looked the same angular size and linear size as the upper disk that subtended 6.5 deg.

Of course, most observers easily could have the “3D” pictorial illusion and perceive the flat pattern as a picture of two ‘spheres’ with the upper ‘sphere’ looking farther away and a larger linear size (behavioral size, metric size) than the lower ‘sphere’.
Moreover, a simple analysis (see later) of the brick pattern in the original picture indicates that, if it had been a photo of real set-up, the far sphere would have been 5 times farther from the camera lens and 5 times the linear diameter of the near sphere (in order to yield the equal optical angles at the lens).

So, to assign some approximate perceptual values to the original pictorial (3D) illusion, we can suggest, temporarily, that the ‘far-looking sphere’ would look 5 times farther from the viewer’s eye, and 5 times the linear diameter (in inches) than the ‘near-looking sphere’. (Let’s say, the near one looks like a 6 inch diameter ball and the far one a 30 inch diameter ball.)
However, that very large, 500% linear size illusion was not measured, mostly because it isn’t interesting. The 17% visual angle illusion is the one that has puzzled scientists for such a long time.

The authors mention that their findings relate to other flat pattern illusions such as the Ponzo illusion. The list can also include the classic Ebbinghaus illusion, and Mueller-Lyer illusion (discussed again later).
The authors also mention that their findings relate to the moon illusion (although none of the moon illusion studies they cite treated it as an angular size illusion).

Indeed, their illusion pattern is somewhat like the lower half of the figure at the right, which was discussed in Section I to illustrate that “Flat Pictures Offer Angular Size Illusions Due To Distance Cues,” and also discussed in Section IV to illustrate the “moon illusion in pictures,” as researched especially by Enright (1987a, 1987b, 1989).

Interpretation. The authors’ interpretation of their results used the conventional (most popular) theoretical approach that includes the apparent distance theory, Emmert’s law, “misapplied size-constancy scaling,” and a so-called, “scaling of retinal size.” The logic (geometry) of those approaches is the size-distance invariance hypothesis stated by, S’/D’ = tanV, which excludes a perceived visual angle concept, V’ deg. Therefore, those approaches can neither describe nor explain the angular size illusion that Murray et al. measured.

This Appendix B shows in detail how the ‘new’ general approach to size, distance and visual angle perception (McCready, 1965, 1985), (as presented in the main body of this article) describes the results of the Murray et al. experiment more fully and more accurately than do other approaches. The logic of this approach is the perceptual invariance hypothesis, stated by S’/D’ = tan V’, which includes V’ deg.
It describes how and when the perceived visual angle, V’ deg, for a target changes away from its constant visual angle V deg (and constant retinal image size).

The major challenge is to explain why those V-illusions occur.
To that end, it will be shown in detail later how the oculomotor micropsia explanation of angular size illusions can fit the illusion measured by Murray et al, and also fit other flat pattern illusions, including the moon illusion in pictures.

New Finding. The major new discovery by Murray et al, was obtained using an fMRI machine to examine and measure the activity patterns in the primary visual cortex, area V1 (Brodmann area 17) while the observer viewed the picture.
For these fMRI measures the supine observer’s head was inside the magnet’s core, so a mirror above the head was used to provide a view of a flat screen on which the picture was rear-projected.

Although the retinal images of the disks in the picture are the same size, the neural activity patterns which those equal-sized retinal images eventually generated in Area V1 were not the same size. The activity pattern for the upper disk was larger than that for the lower disk by an amount that was shown to be equivalent to the magnitude by which the perceived visual angles differed.
In other words, the perceptual magnitude, V’ deg, correlated not with the given retinal image size, but with the changed physical extent of the activity pattern in cortical area VI that corresponded with the retinal image.

The authors point out that their fMRI results do not support the currently “most popular theories” about the brain activities involved in ‘size’ illusions.
After all, those theories are based upon an assumption that the activity pattern in Area V1 that corresponds to a constant retinal image size would remain the same size when distance cues and the perceived distance for the viewed target changed.

The ‘new’ general theory advocated here makes no such assumption. Indeed, it has always been stated that a relative, visual angle illusion is as if the equal retinal images were not the same size, and that obviously implies that those equal images would generate “unequal” activity patterns at an early stage in the visual system. Indeed at a stage before the stages where brain activity patterns occur that correspond with the perceived linear values, S’ in. and D’ in.
The Murray et al. fMRI results fully support the ‘new’ theory

The present analysis has three parts.
Part 1 describes the Murray et al, method and results in detail using the ‘new’ general theory.
Part 2 offers detailed interpretations of the results.
Part 3 suggests how the oculomotor micropsia explanation (especially ‘conditioned micropsia’) can fit the Murray, et al. illusion, and other flat-pattern illusions . Detailed numerical examples are offered.


The entire article by Murray, Boyaci, & Kersten (2006) can be found at the website,

Consider first the stimulus relationships.
The two disks had the same linear size (S in) on the screen (which was not reported), so their diameters subtended the same visual angle, which was, V= 6.5 deg, or V = 0.114 radian, and Tan V = 0.113.
Therefore, the retinal images of the disks had the same diameter (about 1.9 mm).
The viewing distance (D in) to the screen was not reported, but you can approximate the viewing condition if you measure the diameter (S in) of the disk on your screen and view the picture from a distance that is 8.85 times S [because, D = S/tanV]

The instructions to the observers were phrased in two ways.
They were asked "to adjust the size of the front sphere until its angular size matched that of the back sphere."
That sounds like the well-known, ‘analytic instructions’ that foster a perceived visual angle (V’ deg) match of two viewed targets.
Then, because that instruction can be confusing, the observers also were "told to adjust the size so that the two images of the spheres [the disks] would perfectly overlap if moved to the same location on the screen."
That illustrates the well-known ‘objective instructions’ that would foster a perceived linear size (S’ in) match of the two disks.

Of course, because the disks on the screen have the same perceived distance, D’, those two quite different instructions will yield the same “size” match.

The following numerical analyses consider three different kinds of illusion that are likely to occur for the Murray et al. picture, Percepts A, B, and C.
    Percept A is a simple pictorial (3D) illusion with no visual angle illusion. The perceived spheres look the same angular size and much different linear sizes and distances. No observer had that illusion.
    Percept B is a complex pictorial (3D) illusion that adds into Percept A a 17% larger perceived visual angle for the ‘far-looking sphere’ than for the ‘near-looking sphere’. The observers had this type of illusion, which was measured indirectly.
    Percept C is the perception of the flat (2D) pattern. The disks appear at the same distance (on the same page) and the upper disk looks 17% larger than the lower disk both in angular size and in linear size.
All observers had this type of illusion. It is the illusion that was directly measured.

Now for some details.

A Simple Pictorial Illusion with No V-Illusion.
As mentioned earlier, if the picture had been a photograph of a real set-up, the upper sphere would have been 5.0 times farther from the camera lens than the lower sphere, so its linear diameter would have been 5.0.times larger than the lower sphere’s.

That is easily determined by assuming that in the supposed ‘real’ set-up, the bricks on the floor would have been the same linear size, and by noticing that a brick image on the original screen (page) under the lower disk is 5 times longer than a brick image under the upper disk,. Therefore the lower ‘sphere’ would have been at one-fifth the distance of the upper ‘sphere’.

For the present example of a simple pictorial illusion with no V-illusions, let the perceived ratios be the same as those for the supposed real set-up.
That is, let the perceived distance, D’ in, for the far sphere’ be 5.0 times D’ for the ‘near sphere’.
Hence, the perceived linear diameter, S’ in, of the ‘far sphere’ is 5.0 times S’ cm for ‘near sphere’.
For the present example, let's choose some specific perceived values for the ‘near sphere’ as follows,
Perceived angular size, V’ = 6.5 deg.
Perceived linear size, S’ = 6 inches, which makes the,
Perceived distance, D’ = 53 inches from the viewers eye.

For the ‘far sphere’ it follows that,
Its perceived angular size also is V’ = 6.5 (the same as for the ‘near sphere’).
Its perceived linear size is, S’ = 30 inches.
Its perceived distance is, D’ = 265 inches.

How those values were obtained is described below.
First, suppose those unusual floor bricks look 16 inches long (S’ = 16 in). (Which can illustrate the “familar size” or “known size” cue to linear size.)
And notice that, in the picture used , the diameter of the lower disk measures (and looks) 3/8 of the brick length where that disk sits.
So let the perceived linear size of the “near sphere’ be, S’ = 6.0 inches.

The possible perceptual outcomes should conform to the equation, S’/D’ = tan V’
We let the perceived visual angles (V’ deg) equal the visual angles of 6.5 deg.
So tanV’ = 0.113, hence S’ = 0.113 D’
And, with S’ for the ‘near sphere’ already scaled as 6 inches, its perceived distance becomes ‘calculated’ as, D’ = 53.1 inches from the viewer’s eye.

For the ‘far sphere’, with V’ also 6.5 deg, the linear values (S’ and D’) are 5.0 times greater than for ‘near sphere’, so, to a first approximation, S’ = 30 inches, and D’ = 265 inches.
That can be approximately how the 3D result looks to us.

Remember that the observers were not asked to make the perceived ‘spheres’ look the same linear diameter (the same ‘behavioral size’), which would create a linear size constancy outcome.
After all, to do that, they would have to make the ‘near sphere’ look 30 inches wide, by making the lower disk’s diameter about 5 times larger, but that huge lower disk would fill up most of the picture.
If one wanted to obtain a linear size-constancy outcome, one could make the ‘far sphere’ look about 6 inches in diameter by making its image (the upper disk) about 1/5th its size.

The instructions were to to adjust the size of the lower disk until both disks appeared the same angular size and same linear size on the screen.
For an observer who might have Percept A, (no angular size illusion, in agreement with ‘popular theories’) the two equal disks would already look the same angular size and the same linear size, so the size of the lower disk would not need to be changed.

[Notice that, for a viewing distance of D = 21.0 inches, each disk’s linear diameter on the screen would be, S = 2.37 inches. (Because S = 0.113D)]

But, all observers increased the size of the lower disk. That is, all observers had a relative visual angle illusion, as illustrated next by Percepts B and C.

Now let's add a 17% relative V-illusion into the 3D example of Percept A.

For this Percept B example let the perceived values for the ‘near sphere’ be the same as those for Percept A. So,
Its perceived angular size is V’ = 6.5 deg.
Its perceived linear size is, S’ = 6 inches.
Its perceived distance is, D’ = 53 inches.

For the ‘far sphere’ the new values become,
Its perceived angular size is V’ = 7.6 deg (17% larger than 6.5 deg).
Thus tanV’ = 0.132. And the rule becomes S’/D’ = 0.132.

Its perceived linear size is, S’ = 30 inches. (The same as in Percept A.)
Thus, D’ = 227 inches. (It looks 38 inches closer than it would in Percept A.)

The present example makes use of the fact that the 17% larger perceived angular size is occurring not just for the ‘far sphere’ but also for the ‘bricks’ on which it appears to sit.
So, the ratio of V’ deg for the ‘far brick’ to V’ deg for the ‘near brick’ is not 5.0, but (5.0)/(1.17) = 4.27.
Therefore, because those ‘bricks’ look the same linear size (S’ = 16 inches) to illustrate linear size constancy, the perceived distance of the ‘far brick’ becomes 4.27 times D’ for the ‘near brick’, so D’ for the ‘far sphere’ is 227 inches (rather than 265 inches).
It follows that the ‘calculated’ perceived linear diameter for the ‘far sphere’ is 30 inches , in accord with, S’ = 0.132D’, and also because it still looks 1-7/8 times the perceived length (16 inches) for the far brick.

In order to make the ‘near sphere’ look the same angular size as the ‘far sphere’, an observer who suffered this particular illusion would adjust the lower disk until it subtended an angular size of 7.6 deg, and the linear diameter of that lower disk on the screen would become 2.77 inches (17% larger than 2.37).
That type of pictorial (3D) illusion was measured indirectly.

Now consider how the ‘near sphere’ will look when its 'new' perceived angular size is V’ = 7.6 deg.
Its linear diameter now will look, S’ = 7 inches (17% greater than 6 inches) because it looks 17% larger than 3/8 of the ‘near brick’ length (16 inches) where it sits. (The ‘near brick’ has not been changed.)

Other starting values for S’ and/or D’ for the near sphere obviously are possible, depending upon the observer. Thus, many other specific outcomes can exist that satisfy the rule, S’/D’ = tanV’.
At any rate, the much more interesting illusion is the flat pattern illusion.

The Flat Pattern, 17% V-Illusion for the Disks.
For a convenient example let the viewing distance again be, D = 21 inches, so the disks’s diameters will be S = 2.37 inches (in accord with the rule, S/D = tanV = 0.113).

For this example, let the perceived values for the lower disk momentarily be accurate.
So, its perceived angular size is V’ = 6.5 deg (this will be modified later, in Part III).
Its perceived distance is, D’ = 21 inches.
And, its perceived linear size is, S’ = 2.37 inches.

For the upper disk, the values become,
Its perceived angular size is V’ = 7.6 deg, (17% larger than 6.5 deg, so tanV’ = 0.132.)
Its perceived distance also is, D’ = 21 inches (it looks at the same distance as the lower disk).
So, its perceived linear size is, S’ = 2.77 inches (because, S’ = 0.132D’).

Therefore, an observer who has this particular V-illusion would adjust the lower disk until it was 2.77 inches in diameter (and thus subtend 7.6 deg).
That is the type of illuson that was directly measured.

This 17% relative visual angle illusion for the disks on the page is the primary ‘size’ illusion that most needs to be explained.
As already noted, it occurs as well for the illusory 'spheres' in the complex pictorial illusion (Percept B).


To begin to interpret the results, consider first the abstract for the Murray et al. (2006) experiment, as follows.
"Two objects that project the same visual angle on the retina can appear to occupy very different proportions of the visual field if they are perceived to be at different distances. What happens to the retinotopic map in primary visual cortex (V1) during the perception of these size illusions? Here we show, using functional magnetic resonance imaging (fMRI), that the retinotopic representation of an object changes in accordance with its perceived angular size. A distant object that appears to occupy a larger portion of the visual field activates a larger area in V1 than an object of equal angular size that is perceived to be closer and smaller. These results demonstrate that the retinal size of an object and the depth information in a scene are combined early in the human visual system.”

The opening sentence is, “Two objects that project the same visual angle on the retina can appear to occupy very different proportions of the visual field if they are perceived to be at different distances.”
That sentence wholly supports the idea that there are visual angle illusions.
It also describes the normal condition while one views objects in the world (including the illusions of oculomotor micropsia/macropsia).

A later sentence is, “A distant object that appears to occupy a larger portion of the visual field activates a larger area in V1 than an object of equal angular size that is perceived to be closer and smaller.”
[This change in area V1 undoubtedly applies, as well, in oculomotor micropsia/macropsia.]

Those sentences are true for normal viewing. However, for the experiment, the two “objects” those sentences refer to are not objects. They are the illusory ‘spheres’ in the pictorial outcome, as in the Percept B example used above.

Instead, the real objects that were compared in the experiment, were, of course, the two disks on the screen (page) . And they are not perceived to be at different distances! That is, the V-illusion does not require that the objects be perceived at different distances.
Therefore, another sentence could be added to the abstract to point out that the angular size illusion and the changes in area V1 also existed for two objects (the disks) of equal angular size that were perceived to be at the same distance from the eyes!.

As pointed out in Section I of this present article, that is precisely the same problem presented by many classic flat pattern “size” illusions that have resisted explanation for such a long time. The authors mention the Ponzo illusion. There also are, for example, the Ebbinghaus illusion (Titchner’s circles), and the Mueller-Lyer illusion, all three illustrated below.

For such illusions, the crucial targets subtend the same visual angle, V deg, but the perceived visual angle , V’ deg, is slightly larger for the target that lies within a pattern that could indicate (cue) a greater perceived distance for it than for the other target, while both targets appear at the same distance (on the same frontal page).

It is important to keep in mind that the magnitude of the visual angle illusion for the two equal targets on the page depends upon how big the difference would be between the perceived distances of the illusory ‘objects’ which the flat targets might portray in a pictorial depth (3D) illusion that the pictorial distance cues could generate for the given observer.
That is, the size of the V-illusion for a particular 2D flat pattern depends upon the observer’s “ability” to convert some of the monocular distance cues into a pictorial (3D) illusion that can provide different perceived distances for the illusory ‘objects’ the flat targets may be the ‘images’ of.

For example, as discussed earlier in this present article, the Ponzo ’railroad track’ illusion (as at the right) is easily suffered by those of us who have seen such tracks, or who, at the least, have learned to use the many linear perspective cues in our ‘carpentered’ environment filled with rectangular objects (e.g., windows, doors, walls) seen at a tilt, and including the parallel edges of objects like roads, seen on the ground receding from us.

Recall that Kilbride & Leibowitz (1972) found that the Ponzo illusion was not suffered by people who lived in an environment that had no tracks or roads or other rectanglar objects.
Likewise, Rock, Shallo & Schwartz (1974) found that, the more an observer recognizes, interprets and accepts that a flat pattern indicates large depth (3D) values (large distance differences), the more V’deg increases for a target of constant V deg located at a nominally "far" place in the visual world (although they did not use the concept, V’ deg).
Also remember that, for us, such illusions usually are weakened by inverting them, which can reduce the efficacy of the distance cues (especially “height in the plane”).

Cue conflict.
In other words, the Murray et al. illusion is caused by some of the monocular distance cues in the flat ‘picture’ while other distance cues are making the targets correctly appear on the page (at the same distance).
For instance binocular cues, some monocular cues, and also one’s knowledge that one is looking at a flat picture, all indicate that the two disks have the same perceived distance, D’ cm.
Yet, ‘pictorial’ distance cues (mostly linear pespective for the brick images) are creating the V-illusion.

Given the general descriptions and examples above, the main task is to explain why changes in distance cue patterns can alter the relationship between the visual angle, V deg, and the perceived visual angle, V’ deg, which is the subjective change that Murray, et al. showed was a result of a change in the neurological relationship between a given retinal size, R mm, and its representation in cortical Area V1.
That task was addressed in the body of the present article. It will be dealt with later here, in part 3.

A difficulty appears in the Murray et al. discussion section.
The 'explanations' of the illusion use the 'popular approach', whose logic is the size-distance invariance hypothesis (S’/D’ = tanV) which omits V’ deg.
Consequently, there is much confusion between the perceived angular size (V’ deg) and the perceived linear size, (S’ inches), which is called a ‘perceived behavioral size’.
As a result, the discussion section does not explain the illusion that was measured.

For instance, it was suggested that distance cues evoke a supposed “scaling” of some entity called the viewed object's ‘retinal projection” to yield a “perceived behavioral size” for the object, "whereby retinal size is progressively removed from the representation" (p. 422).
That very old idea overlooks that the perceptual correlate of the extent between two stimulated retinal points is not a perceived linear size (S’ cm) that somehow has been "scaled" up.
For instance, the lower disk's retinal image size, R = 1.9 mm, does not have to be magically magnified to become either 2.37 inches for the disk, or 6 inches for the illusory sphere!

Instead, the flexible perceptual correlate of the extent between two stimulated retinal points is the perceived visual angle, V’ deg.
And this angular size perception is simply the perception of the different directions of two seen points from oneself. And that perceived direction difference (V' deg) certainly is not "removed from the representation."

And, as the authors clearly showed, V' deg is a perceptual correlate of the extent of the activity in area V1.

As easily predicted, other published articles already are mis-interpreting the Murray, et al. experiment in the 'popular' way.
For instance, some articles say the results illustrate Emmert’s Law, stated by S’cm = D’tanV.
And, some articles suggest that the results illustrate “misapplied size constancy scaling”, an old idea in which with the term “size constancy” refers to some sort of “scaling of the retinal size” by perceived distance.
But, as discussed in Appendix A, those conventional ideas fail to explain V-illusions and confuse linear size constancy with a supposed “visual angle constancy.”

Murray et al, point out that their fMRI results do not support ‘dominant’ theories about the probable neural activities in the brain involved in “size” illusions.
For instance, a University of Washington website offers a review of the study at,

It quotes Dr. Murray as follows (who refers to the moon illusion).
    "It almost seems like a first grader could have predicted the result. But virtually no vision or neuroscientist would have. The very dominant view is that the image of an object in the primary visual cortex is just a precise reflection of the image on the retina. I'm sure if one were to poll scientists, 99 percent of them would say the 'large' moon and the 'small' moon occupy the same amount of space in the primary visual cortex , assuming they haven't read our paper!"

That comment overlooks the writings of about two dozen reputable vision scientists who, at least since 1965, have pointed out that the angular size (visual angle) illusion for the moon, and for many classic flat-pattern illusions, is as if   the moon’s constant retinal image size changed when distance cue patterns changed.
And, because that is very basic illusion, and given what has long been known about the spatially isomorphic neural projections of the retinal surface into Brodmann area 17 (now called area V1) it has seemed quite likely that the 'size' change would already appear that early in the brain.
That is, the V’ deg experience (perceived visual angle) clearly must be related to a more basic (primitive) place in the visual system than the places that relate to the coexisting, and qualitatively much different, S’ cm experience (perceived linear size, or “apparent behavioral size”).
As discussed in Section IV of the main text, this primacy of V’ deg makes sense because the angle V’ deg is the difference between the perceived directions of objects from oneself, and it thereby guides rapid orienting responses from one object to the next.
In all animals those orienting responses are critically important for survival.
[Indeed, in many animals' visual systems, the superior colliculi are very much concerned with direction perception, so one could make a wild guess and speculate that these even more primitive brain loci are involved in the creation of angular size illusions.]

Measuring Micropsia with fMRI.
Obviously, the changes in Area V1 found by Murray et al. also would be obtained for the much larger visual angle illusion of oculomotor micropsia/macropsia, by having the observer change from strong convergence to divergence.
That easily can be done using base-out, then base-in prisms.
The magnitude of that V-illusion should be as large as 2 to 1, for small targets. It will be interesting to see how far that micropsia can be pushed for large targets.
Also, micropsia would be obtained if the observer "voluntarily" over-converged while fusing a repetitive pattern, as in the Meyer Wallpaper Illusion.

Spiral Illusion Aftereffects.
After viewing a rotating spiral that appears to be contracting, if one looks at a fixed target, its angular size looks very much larger than it does without that aftereffect. Conversely, that target's angular size looks considerably smaller after one views a rotating spiral that appears to be expanding.
Those large visual angle illusions likewise should show up in Area V1.

But, back to the flat patterns.
Currently, the best-known alternatives to the deficient “popular” explanations for the classic, flat pattern V-illusions are the visual angle contrast theory (critiqued in Section II) and the oculomotor micropsia theory (detailed in Section IV) which is applied to the Murray et al, experiment in Part 3 below.

(posted April 29, 2007)

Here it will be shown that the visual angle illusion measured by Murray et al. could be due to oculomotor micropsia in large part, if not entirely.
How the oculomotor micropsia theory may apply to flat pattern illusions has been described in general terms elsewhere (McCready, 1965, 1983, 1985, 1986) and is briefly described in Section IV of this present article.
The argument offered below gives details and examples to clarify that proposal.
It applies the simple equation for oculomotor micropsia (McCready, 1965, 1985, 1994a, 1994b, 1995), Ono (1970), Komodo & Ono (1974), Ono, Muter, & Mitson (1974) mentioned in Section IV.

Some examples for the Murray et al. picture furnish relative V-illusion values ranging from 10% up to 26% . So they fit the obtained data very well.
This analysis also provides a model of how the oculomotor micropsia explanation can apply to many other flat pattern illusions, such as the classic Ponzo illusion, Ebbinghaus Illusion, Mueller-Lyer Illusion and the moon illusion in pictures.

The analysis begins by noting, again, that, for most people, oculomotor micropsia is a natural perceptual adaptation that occurs during normal everyday viewing.
In general, the closer an object is to the eyes, the more its perceived visual angle (V’ deg) becomes less than its subtended visual angle ( V deg).
This angular size illusion tied to object distance undoubtedly serves some “purpose.”

The present assumption is that the “purpose” of the illusion is to enhance the speed of body orienting responses to viewed objects, especially the quick “emergency” rotations and other movements of the head (see Section IV again).
That assumption led to the simple equation for the amount by which this natural micropsia is expected to change the perceived visual angle (V’) for a target subtending the visual angle, V deg.

V’/V = Dc/(Dc + Tk).

Here the variable Dc, is the target’s cued distance established by some or all cues for the target’s distance.
In natural everyday viewing, Dc often equals the actual target distance (D).
It also may equal the target’s perceived distance (D’) .
But, as is especially true for classic “size and distance “ illusions, Dc may not equal either D or D’.
Indeed, in a flat picture that can create a 3D pictorial illusion, values of Dc can equal the different distances being ‘signaled’ for targets by the monocular pictorial distance cues, even when the targets correctly appear at the distance of the page.

The variable Tk, is the turn correction factor , in inches.
Theoretically, Tk is the distance from an eye rotation center to a pivot point for a head movement.
I have analyzed (McCready, 1965, 1994a, 1994b, 1995 ) published experiments that directly measured oculomotor micropsia, and several other experiments that indirectly revealed oculomotor micropsia, and found that, among observers, Tk ranges from zero (no micropsia) up to at least 6 inches (15 cm) and for horizontal head rotations Tk averages about 4 inches (10 cm).

Applying the Equation.
The first example applies the equation to the original Murray et al picture (imitated by the sketch here) and yields a 17% relative illusion by using a viewing distance of 20 inches and a Tk value of 4.5 inches. Other examples follow, and then more details of the calculations are given.

The task is to explain why the upper disk, that subtends 6.5 deg looks angularly larger than the lower one that subtends 6.5 deg, while both disks have the same perceived distance. Keep in mind that this illusion evidently will not exist (or will be very weak) for any observer who is unable to “convert” the monocular cue patterns into a 3D, pictorial illusion of a ‘receding floor” with an upper ‘sphere’ that looks, say, 5 times farther away than a lower, nearby ‘sphere’.

So, the disk illusion is linked to the ‘sphere’ illusion.

To begin, it is expected that natural micropsia will occur for the target extents on the flat page (screen) at its viewing distance.

Again, in order calculate this natural micropsia, the equation is,

V’/V = Dc/(Dc + Tk).

Suppose the viewing distance is 20 inches, and suppose the screen correctly looks 20 inches away, “because” the binocular cues and some monocular cues “assign” a cued distance of , Dc = 20 in.
With Tk = 4.5 inches and Dc = 20 inches the equation predicts a natural micropsia of, V’/V = 20/24.5 = 0.833.
That is, natural micropsia would be expected to make the perceived visual angles of extents on the screen. 22% less than their subtended visual angles.

Next, for these examples, let the lower disk be the standard.
So in the present example its perceived visual angle would be, V’ = (6.5 x 0.833) = 5.3 deg, an absolute V-illusion of 22%.
That also would be the first prediction for the upper disk, of course.
However, V’ for it is larger than V’ for the lower disk while they have the same perceived distance (D’ = 20 in).
In order to explain that relative V-illusion, the analysis uses the concept of conditioned micropsia .

In the Murray et al. pattern the pictorial distance cues (especially linear perspective for the floor bricks) can create the 3D illusion that the upper sphere and the brick under it are about 5 times farther away than the lower sphere and its brick.
In Part 1, examples were given for Percepts A and B in which the pictured 'near sphere' looks 53 inches away when it looks 6 inches in diameter.
However, binocular cues, some monocular cues and the observer's knowledge are specifying that the screen and that target are at 20 in.
So, for the present example, assume that, as far as the visual system is concerned, the real ‘near sphere’ image is in the plane of the screen at 20 inches, so Dc for it is 20 in.
And, for the ‘far sphere’ the pictorial cues now are signaling, “100 inches away”, so Dc = 100 in. for it.

For the observer in this example, oculomotor micropsia during everyday viewing often has occurred for targets at 100 inches for which Dc = 100 in, so for such a target the equation predicts that, V’/V = 100/104.5 = 0.957, a small absolute illusion of 4.5 %.
The proposal is, of course, that the monocular distance cue patterns that have been associated with a target distance of 100 in. have acquired the ability to generate micropsia appropriate for that distance when those cues appear again, even as the simple monocular pictorial cues in a flat pattern. That result would illustrate conditioned oculomotor micropsia

So, for this specific example the suggestion is that the pictorial cues are treating the ‘upper sphere’ as if it were 100 inches beyond the screen, so Dc = 100. This conditioned micropsia for the ‘upper sphere’ thus would yield for it a perceived visual angle of, V’ = (6.5 x 0.957) = 6.22 deg.
Therefore, the illusory ratio of the perceived visual angle for the ‘far sphere”to that for the “near sphere’ is, (6.22/5.3) = 1.17.

This predicted 17% relative angular size illusion agrees well with what Murray et al found both for the ‘spheres’ and the disks.
So, consider next the disks.

The Previously “Paradoxical” V-Illusion, and S-illusion for the Flat Disks.
The perceived linear size, S’ inches, of the lower disk on the page is larger than S’ for the upper one, and they also have the same perceived distance, D’ inches.
That universal finding for classic flat-pattern “size” illusions often has been referred to as the “size-distance paradox” because it cannot be explained using the dominant theory, the old apparent distance theory, stated by, S’/D’ = Tan V, (known as the size-distance invariance hypothesis) which omits V’ deg.

As can be seen, the old theory requires that, when both disks subtend the same visual angle, V deg, and D’ is the same for both, the S’ values must be equal (they cannot differ).
That “paradox” vanishes, of course, when one recognizes that the perceived visual angles differ.
So, for the present example, the reason the perceived linear size, S‘ in, for the upper disk on the screen becomes 17% larger than S’ for the lower disk is because of the more basic”size” illusion that V’ deg for the upper disk is 17% larger than V’ for the lower one.
To explain that V-illusion for the disks, it is suggested that, as far as the visual system is concerned, with Dc = 20 inches for the screen and lower disk, the monocular pictorial cues “signal” that Dc for the upper disk is 5 times greater, so Dc =100 inches, for that disk even though it correctly appears at 20 inches.
Here are more examples for the Murray et al flat pattern, using Tk values from 4 to 7 inches.

Calculations For a Viewing Distance of, D = 20 inches to the Murray et al. Pattern.
For Tk = 4.0 in. the predicted relative V-illusion is 15% .
For Tk = 5.0 in. the predicted relative V-illusion is 22%
For Tk = 6.0 in. the predicted relative V-illusion is 22%
For Tk = 7.0 in, the predicted relative V-illusion is 26%

Why the larger Tk values?
For horizontal (side to side) orientating responses of the head, the expected Tk values are about 4 inches.
For small vertical head movements (simple elevations and depressions) the pivot point for such extensions is the atlanto-occipital joint.
Diagrams that locate that joint provide an estimate of at least 6 inches for the distance, Tk, between it and the eye(s).
Moreover, Tk clearly must be even larger than 6 in. for large orienting movements that quickly aim one’s head up or down, so that one can better see (and hear) objects that demand attention from above or below the “straight ahead” horizontal direction of the erect head. For these head orientations the neck bends, so the effective pivot point is farther down the vertebral column than the C1 vertebrate.
Therefore, it seems appropriate to use even larger Tk values, say 7 inches, for illusions in which the targets lie one above and other, and the dominant pictorial distance cues are organized mostly vertically in the field of view, for instance, as in the “hallway floor” or the Ponzo “railroad tracks", or a “landscape” picture.
Don’t forget that “height in the plane” is a well-known distance cue.
And, inverting an illusion like those mentioned above weakens the relative V-illusion.

Calculations For a Greater Viewing Distance of D = 30 inches.
The viewing distances used by Murray et al., were not specified, and might have been greater than 20 in.
So consider next a distance of D = 30 inches, so Dc = 30 in. for the lower disk, and Dc = 150 in. for the upper disk. The predicted relative illusion would be less than for 20 inches.
For instance,
For Tk = 4 in. a relative V-illusion of 10%
For Tk = 5 in. a relative V-illusion of 13%
For Tk = 6 in, a relative V-illusion of 15%
For Tk = 7 in., a relative V--illusion of 18%.

The Mandelbaum Effect (See articles by Roscoe).
During the Murray et al. fMRI measures, the mirror between the observer’s eyes and the target screen would tend to make the eyes focus and converge a bit closer than the target screen. Therefore, Dc for the lower disk would be less than the target’s distance, so a slightly greater relative V’-illusion would be predicted.

Actual Accommodation and Vergence Responses.
Another likely contributor to the relative V-illusion is perspective vergence (see Enright (1987a, 1987b, 1989a) previously discussed in Section IV.
For the flat picture used by Murray et al, it would be expected that one’s accommodation and vergence responses would change slightly when one shifts fixation (or attention) between the ‘near sphere’ and ‘far sphere’.
These oculomotor changes, stimulated by the distance cue patterns, typically are larger than would be expect for ‘perfect vision” of the images on the flat page (or screen), and would induce changes in V’/V in the direction of the obtained relative illusion.
Of course, these overt changes result only when one successively views the target objects.
Murray et al., mention that such changes would have only a small effect for their picture.
At any rate, these flat pattern V’-illusions do not require overt differences in the accommodation and/or vergence responses to the two targets (disks) on the page. After all, such illusions persists with accommodation paralyzed, and with binocular viewing which (as Murrray et al note) allow only very small differences in the vergence responses for the two targets.

Conditioned Micropsia and Efference Readiness .
Changes in distance cues normally cause the brain to send neural impulses in the oculomotor nerves to the eye muscles in order to change accommodation and convergence.
At least since Helmholtz (1910, 1962) it has been known that various illusions of visual direction perception occur in special situations where this “motor command” or efference has been sent out, but does not make the muscles contract to move the eyes, say because the muscles are paralyzed.
Thus, as Helmholtz noted long ago, accurate perception of directions is controlled not by a supposed “feedback” from one’s eye muscles, but by what the eyes have been “told to do” by the efference.

As noted earlier, this efference also has been shown to control the visual direction illusions of micropsia and macropsia.
Better yet, it turns out that the crucial factor is not the efference but the neurological “preparation” to send efference, which Festinger et al, called ‘efference readiness’ (Festinger, Burnham, Ono & Bamber, 1967. Festinger, White, & Allyn, 1968).
The evidence for that includes flat pattern illusions for which a change in a monocular distance cue stimulus pattern evokes a change in micropsia without eliciting overt oculomotor changes.

Thus it can be said that an ‘efference readiness’ is established by distance cues, and it alters the relationship between V’ deg and V deg.
In turn that means that what Murray et al. showed was that the cue-established neurological activity, ‘efference readiness’, somehow modifies the afferent neural activity between the retina and area V1, such that the isomorphic representation of retinal image size (R mm) in area V1 is changed .
Moreover, it seems clear that the ‘magnitude’ of that ‘efference readiness’ is such that, if it actually were sent to the muscles as motor efference, it would adjust the eyes to the distance being cued for a given target by the relevant cues.
It is fair to suggest that this distance is what I have called the target’s cued distance, Dc cm.
That is, Dc cm is, in effect, a measure of the ‘efference readiness.’

The Moon Illusion In Pictures (see Section I and Section IV again) .
Studies of the moon illusion in pictures typically have used two disks that serve as moon images in a pattern that provides monocular distance cues that are expected to generate a pictorial illusion with a much greater D’ for the portrayed ‘horizon moon’ than for the portrayed ‘zenith moon’.
The ‘horizon’ disk on the page typically looks larger, both angularly and linearly, than the ‘zenith’ disk.

Enright (1987a, 1987b) experimented with such pictures.
Later, he reviewed such experiments (Enright, 1989a) and noted that they illustrate a visual angle illusion due to the less well-known V’-illusion of oculomotor micropsia/macropsia (as did McCready, 1965, 1983, 1985, 1986).

Enright (1989) also offered an explanation for oculomotor micropsia/macropsia that treats it as a normal perceptual adaptation for the fact that the eyes’ rotation centers lie quite a bit anterior to head rotation centers.
He offered a model of the brain systems that would yield that effect.
His rational for that adaptation refers to the VOR reflex, so it differs from mine (the orienting response).

The oculomotor micropsia explanation offered above obviously can be applied to many other flat pattern illusions that have remained unexplained, such as the Ponzo illusion, the Ebbinghaus Illusion (Titchner’s circles) and the Mueller-Lyer Illusion, as shown in the diagram below.

For many decades researchers have pointed out that the context patterns for the two equal targets in each illusion happen to be distance cues that can create a 3D pictorial illusion, with one pictured object looking farther away and a larger linear size than the other one.
So far, so good.
But the nearly universal explanation has used the apparent distance theory which , as noted earlier, requires that the targets appear the same angular size and at different distances so that they will look different linear sizes.
So, it does not (cannot) even describe the more interesting visual angle illusions.

The present approach to those visual angle illusions can apply the simple equation.
[The illusions above have been arranged to let “height in plane” be an additional cue.]

For instance, for the Ponzo illusion, it is easy to see that Dc for the upper line would be about twice Dc for the lower line.
At a viewing distance of 20 in, with Tk = 4 in. that predicts a relative illusion of 9%.
And, with Tk = 6 in, an illusion of 13%.

For the Ebbinghaus illusion, Dc for the upper (“farther”) circle also might be twice Dc for the lower (“closer’) circle (because the large context circles are twice as large as the small ones).
The predictions thus are about the same as above.

For the Mueller-Lyer illusion, one analogy is that the upper figure resembles an open book we are looking into, and the lower figure resembles an open book, but we are looking at its outer spine.
Those cues somehow can make the spine look closer than the interior edge, and that would create the relative micropsia V-illusion .
[An unpublished equation that can evaluate Dc values for that pattern is too cumbersome to derive and use here. ]

The Murray et al experiments fully support the proposal that many major classic flat pattern illusions start as visual angle illusions controlled by distance cue patterns that are able to create a 3D pictorial illusion for most observers.
Likewise for the classic moon illusion.
The task is to explain the V-illusions.

Murray et al showed that the angular size illusions result from some sort of activity that intercedes in the neural pathways from retina to cortical area V1 (Brodmann area 17) to alter the neural representation of retinal image size in Area V1.
That means it alters the perceived visual angle, V' deg, value for an otherwise constant value of V deg..
My proposal (McCready, 1965, 1983, 1985, 1986) has been that the relevant distance cues establish different efference readiness values that somehow alter the relationship between V' and V.
An explanation for such illusions is that they are examples of the more basic and ubiquitous illusion of oculomotor micropsia/macropsia.
And, an explanation for oculomotor micropsia/macropsia is that it serves a useful purpose by altering the perception of direction differences (V deg) to make them more accurate predictors of rapid orienting responses, especially of the head.

The simple equation that describes the amount by which such "corrections" of V' deg should be made, has been shown to fit the data from many published experiments very well.
That equation is shown here to fit the Murray et al. data very well.
Therefore, it can be said that the illusion that Murray et al. measured is due to oculomotor micropsia..
This present analysis can be applied to other flat pattern illusions and usually fits them quite well.

At this time (May 2007) no other published theory of such illusions can explain them satisfactorily.

The simple equation for micropsia used here, V’/V = Dc/(Dc + Tk) , describes only micropsia (V’ less than V) for all distances.
It needs to be modified so that it will also describe macropsia (V’ greater than V) for large cued distances (see the research of Higashiyama).
For instance, some published data indicate that V’ may equal V deg (no V-illusion) for objects at the resting focus distance of 1 or 2 meters.
And, V’ becomes smaller than V’ for closer targets but larger than V deg for targets beyond that distance.

Index page.
Introduction and Summary.
Section I. New Description of the Moon Illusion
Section II. Conventional Versus New Descriptions
Section III. Explaining the Moon Illusion
Section IV. Explaining Oculomotor Micropsia
Bibliography and McCready VITA
Appendix A. The (New) Theory
Appendix B. Analysis of the Murray, Boyaci & Kersten (2006) Experiment