Eyes that follow you
Solution 1:
The Differential Rotation Effect
The phrase is not in common use but the 'following eyes' are due to the differential rotation effect.
One famous example is the iconic British World War I recruitment poster, picturing Field Marshal Horatio Herbert Kitchener, 'hero' of the Second Boer War and a potent symbol of the rejuvenated imperialist militarism of the late Victorian period, replete with an appropriately bristling officer's moustache.
A forward-facing, pointing image of Kitchener with a singularly piercing and unyielding stare, was pressed into service to encourage ordinary British men and boys to sign away their lives for "the Old Lie: Dulce et Decorcum Est // Pro patria mori".1
A 2014 blog post on the UK Post Office website, entitled The Recruitment Poster That Stood the Test of Time, marks the image's centenary and offers some explanation as to its longevity:
One observation as to just why the advert was so effective and indeed has remained so iconic to this day, is that it has the ‘differential rotation effect’ whereby Kitchener’s eyes and foreshortened arm appear to follow the viewer no matter of the viewer’s orientation to the artwork itself.
(Emphasis added.)
Kitchener's accusatory pose and staring eyes has 'inspired' numerous other recruitment posters, perhaps most notably James Montgomery Flagg's 1917 "I Want You For U.S. Army", featuring Uncle Sam, but also Zionist, Stalinist and Nazi tributes, among others... as well as imitation in other fields, such as the US Forest Service's Smokey Bear, who has a more benign and reasonable authoritarian message placing emphasis on personal responsibility:
For those unfamiliar with Smokey Bear, the complete admonition is: Only you (can prevent forest fires). The paw-brandished shovel is, I presume, to throw suffocatory soil (or, perhaps, dirt, as is more likely in Smokey Bear territory) over your dying campfire ashes.
To return to the phrase differential rotation effect and its usage, a few examples with context follow.
Rogers, Lunsford et al., "The Mona Lisa Effect: Perception of Gaze Direction in Real and Pictured Faces" in
Sheena Rogers, Judith Effken, Studies in Perception and Action VII (2003), p19
The eyes in a portrait often seem to follow observers as they pass (the Mona Lisa effect). All 3-D objects in a picture, not only gaze, will rotate in virtual space as the observer moves past the picture (Rosinski & Farber, 1980). This phenomenon is predicted by the geometry of pictorial space (See Rogers, 1995, for a review) but it may also be due to limits in our ability to perceive the direction of another's gaze even in the real world, or to general inaccuracies in picture perception.
Sedgewick's (1991) analysis shows that the virtual orientation of objects is affected both by the objective orientation of the gaze (towards to the station point or away to one side) and by the degree to which the picture is slanted relative to the observer. According to the geometry, objective gaze direction should be increasingly mis-perceived (distorted) as the angle of gaze increases away from the station point (or center) (a differential rotation effect).
Stephen Ellis, Pictorial Communication In Real And Virtual Environments (2003), p497
When the picture is viewed at an angle of 20o (far to the right side of the picture plane), the relationship between the two orientations is diffeerent than when it is viewed at 160o (far to the left side of the picture plane). These differences are manifestations of the differential rotation effect—the fact that pictured objects oriented more parallel to the picture plane rotate less in response to an observer's change in viewing angle than do pictured objects that are oriented more perpendicular to the picture plane.
And, perhaps most importantly, from a journal article abstract, which credits an originator:
Cutting, James E., "Affine distortions of pictorial space: Some predictions of Goldstein (1987) that La Gournerie (1859) might have made" in
Journal of Experimental Psychology: Human Perception and Performance, Vol 14(2), May 1988, pp305-311
Goldstein (1987) studied the perception of pictures seen from the front and the side. Several distinctions arose from his results and analysis, but only one is central to the reanalysis presented here: The perceived orientation of objects within a picture with respect to the external world is a function of viewer position in front of the picture. For example, the eyes of a portrait subject appear to follow an observer who moves around a gallery. Viewed from many positions, such objects can be said to rotate, following a mobile viewer. Goldstein called this the differential rotation effect because those objects that point directly out of the picture (at 90°) rotate most; those pointing at other angles rotate in decreasing amounts. Goldstein offered no theoretical model and little in the way of explanation for this effect...
(Emphasis added.)
A few caveats regarding the differential rotation effect as a phrase:
It isn't common and is highly technical.
It isn't easy to understand: its meaning isn't obvious and immediately apparent without context or previous acquaintance, and, even in the limited technical contexts in which it is used, the phrase isn't introduced without some kind of explanation. So, even when writing for specialists, it cannot be assumed the phrase will be understood.
It has another meaning: the phrases "differential rotation" and even "differential rotation effect" are more commonly found in reference to astronomy and mechanics (in the physics sense) than in relation to forward-facing portraits whose eyes follow.
Depending on the context, there may well be better alternatives as the phrase differential rotation effect is of a technical register, potentially a little confusing, and uncommon to boot - not a great combination if, for instance, you're writing journalism for a general audience. In that instance, you might well prefer the Mona Lisa effect, which one of the quotes above refers to, even going so far as to use it in the article title as an appropriate way to set the subject, and, more importantly, which EL&U user Cascabel has offered as an answer.
And a final warning: never be too quick to attribute to the differential rotation effect that which might be better explained by another cause:
Solution 2:
Mona Lisa Effect
The Mona Lisa effect is the phenomenon in which certain portraits appear to direct their gaze at observers almost regardless of where observers are located with respect to the picture. This phenomenon has been well known for almost 2,000 years, yet it has not been studied much. The effect does not essentially depend on the motion of the observer. Rather, the effect is due to the fact that perception of gaze direction of a “looker” does not depend only on the position of the irises within the sclera but also on the orientation of the looker’s head with respect to the observer.
-Oxford Scholarship Online
There are differing views on whether this is even a real effect in the Mona Lisa.
Since the 1960s, scientific teams have studied this perception of being watched, and the Mona Lisa Effect is generally accepted to be a legitimate phenomenon. But according to a team from Bielefeld University in Germany, no one has ever done a rigorous test to see if it is present when viewing the Mona Lisa. After conducting their own tests, the team has published a paper that concludes, “There is no doubt about the existence of the Mona Lisa effect—it just does not occur with Mona Lisa herself.”
-Gizmodo
Also reported in the Daily Mail