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).
http://faculty.washington.edu/somurray/sizedemo.html

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.

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).

POTENTIAL PICTORIAL ILLUSION NEEDED.
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.

THE POPULAR APPROACH.
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.”

REGARDING THE CHANGES IN CORTICAL AREA V1.
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,