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4 The robustness of perspective

We have seen that Brunelleschi's peepshow, by placing the
viewer's eye at the center of projection, can give rise to a
compelling illusion of depth. Some students of perspective
have thought that it can also protect the viewer from distortions
one might expect to experience while viewing a
picture from a point other than the center of projection.
One of these was Leonardo:

Why did Leonardo expect most paintings to “look wrong”
when viewed from somewhere other than the center of
projection? Because he was thinking about perspective in
geometric terms. As we saw in Chapter 1, if it is known
(or assumed) that a picture such as Masaccio's Trinity (Figure

1-1) was generated according to the laws of central
projection, it is possible, by making some assumptions
about the scene, to reconstruct the scene.

Before a geometer can solve the inverse problem of perspective,
the location of the center of projection must be
determined. If an error is made in locating this center, the
reconstructed scene will be distorted. For instance, in Figure
4-1, panel 97 is the inferred plan of the scene shown
in panel 98 if the center of projection is assumed to be at
point o. An observer standing at point o as specified in
panel 97 would see a rectangular nave, as La Gournerie's
plan shows. But if the center of projection is assumed to
have moved to the left, as in panel 96, a geometer cannot
solve the inverse perspective problem posed in panel 98
and still reconstruct a building whose ground plan is based
on right angles. The ground plan in panel 96 is a shear
transformation of the one in panel 97: Points of the plan
in panel 97 are shifted laterally, parallel to the picture plane;
the greater the distance of a point from the picture plane,

the greater the lateral displacement. (To visualize a shear
transformation, imagine yourself holding a pack of cards
and tapping the edge of the pack against the surface of a
table, while the cards are at an oblique angle to the surface
of the table.) If the assumed center of projection is moved
laterally as much as in panel 96, but is also moved further
from the picture plane, the shear transformation is combined
with a magnification, as shown in panel 95. You may
notice that the plan in panel 95 looks less distorted than
does the one in panel 96. There are two reasons for this:
First, the amount of shear is smaller in panel 95; p' is closer
to p in panel 95 than in panel 96. Second, the greater the
magnification, the smaller the angle at which the nave
intersects the picture plane.

If perception solved the problem of inverse perspective
in the same way as the geometer would, and if perception
assumed that the center of projection always coincides with
the perceiver's current point of view,[1] then an observer
standing at point o as specified in panels 95 and 96 would
see an oblique nave in accord with La Gournerie's plan.[2]
As the reader can ascertain by moving in front of panel
98, no such striking distortions are experienced. I call this
violation of our geometric expectations by our perceptual
experience the robustness of perspective.

Such claims about the robustness of perspective have
been made before, but not everyone agrees with the way
the problem has been formulated and about the nature of
the evidence in favor of robustness. For instance, Rosinski
and Farber write:

I vehemently disagree. The contrast between Leonardo's
geometric expectations and our experience is the very issue
at hand, the issue we wish to understand. No one has
claimed that “the fact observers are not consciously aware
of distortions in virtual space” implies “that the nature of
virtual space is unregistered by the visual system.” On the
contrary, most theoreticians of picture perception (including
Rosinski and Farber) believe that observers are not
aware of distortions in virtual space because a part of the
visual system (whose workings are unconscious) registers
both the nature of the virtual space and the orientation of
the surface of the picture, and corrects the former in the
light of the latter.

Furthermore, Rosinski and Farber are wrong when they
say that “to comment on whether a picture seems distorted
is to assess the correspondence between virtual and environmental
space.” I think that to comment on whether a
picture seems distorted entails a far richer implicit cognitive
process: One must first mentally reconstruct the scene that
the painter had in mind and then assess whether — within
the conventions of the genre — the representation is correct.
Take, for example, the exercise in perspective by the early
seventeenth-century designer of architectural and ornamental
pattern books Jan Vredeman de Vries (1968) shown

[ILLUSTRATION]

Figure 4-2. Jan Vredeman de Vries,
architectural perspective

in Figure 4-2. Although technically in accord with the rules
of central projection, the steles on the left are clearly distorted.
We know this without ever having seen the architectural
structures depicted and without being in a position
to assess the correspondence between virtual and environmental
space.

Returning to Leonardo's recommendation to artists, we
can say that on the whole his worries were unfounded. In
general, it is not necessary to view a picture from the center
of projection to see an undistorted version of the scene it
represents. Although it is true (as we will see later in this
chapter) that certain types of objects seen under certain
special points of view (such as eyes looking at the viewer,
and the barrel of a gun or a finger pointing at the viewer)
seem to follow us when we move in front of the picture,
these are not the distortions Leonardo was worried about,
and they are not true violations of the robustness of
perspective.

And yet Leonardo was not entirely mistaken; there do
exist conditions under which the geometer's expectations
are confirmed and the robustness of perspective fails. We
are fortunate to know about these conditions because they
provide us with a clue to understanding what makes the
robustness of perspective possible under most circumstances.
The robustness of perspective fails when “the spectator
is unable to see the painted surface, qua surface”

(Pirenne, 1970, p. 84); for example, in Pozzo's ceiling discussed
in Chapter 3. Here is Pirenne's description:

To prove Pirenne's thesis, one must show that (a) when
the surface of a picture is hard to perceive, the virtual space
of the picture is perceived in accordance with geometric
expectations; and (b) when the surface of the picture can
be seen, the virtual space of the picture is perceived to be
invariant despite changes in the observer's vantage point.

Rosinski and his colleagues performed two experiments
that provide exactly these sorts of data (Rosinski et al.,
1980; see also Rosinski and Farber, 1980). Figure 4-3 shows
how the stimuli were created. In panel 1, we see the object.
Thirteen different photographs of this object were taken,
as shown in panel 2; each was taken at a different angle of
slant. In panel 3, we see one of these photographs appropriately
cropped and mounted on flat black matte board.
In Figure 4-4, we can see the apparatus used in the two
experiments.

In the first experiment, Rosinski et al. simulated Brunelleschi's
peepshow. They minimized the amount of information
the observer would receive regarding the location
of the picture plane by using a latter-day perspective-cabinet

[ILLUSTRATION]

Figure 4-3. Preparation of stimuli in
the Rosinski et al. (1980) experiments.
(1) Frontal view of photographed
object. (2) Top view of
object, at 60-degree slant, and of
camera. (3) Frontal view of perspective
photograph of object at 60-degree
slant.

with two peepholes: One peephole afforded a line of
sight perpendicular to the surface of the photograph displayed
in it; the other peephole shifted the observer's line
of sight so that it formed a 45-degree angle with the surface
of the photograph. To further reduce the visibility of the
surface of the picture, Rosinski et al. put cross-polarized
filters into the viewing box to minimize the amount of
glare by diffusing the light reflected by the surface of the

[ILLUSTRATION]

Figure 4-4. Presentation of stimuli
in the Rosinski et al. (1980) experiments.
(1) Experiment 1: Information
regarding picture surface is
minimized. (2) Experiment 2: Information
regarding picture surface is
not reduced.

photograph. The observer would be asked to view the
photograph through a peephole at the center of projection,
so that his or her line of sight would be orthogonal to the
picture plane, or through a peephole from which the observer's
line of sight would form a 45-degree angle with
the picture plane (both were 50 cm or just under 20 in.
from the picture plane). The observer's task was to adjust
a palm board that could be rotated about a vertical pivot
to indicate the perceived slant of the plane represented in
the photograph.

In the second experiment, Rosinski et al. made no attempt
to conceal the location of the picture plane, and
therefore no viewing box was used: Observers were positioned
45 degrees to the right or 45 degrees to the left of

[ILLUSTRATION]

Figure 4-5. Data for Experiment 1
of Rosinski et al. Angle to which
palm board was adjusted to match
apparent slant of surface in photograph
for the two points of view.
Values of independent variable are
determined by assumption that center
of projection coincides with 90-degree
point of view.

the center of projection, 50 cm away from the picture
plane. The picture was viewed binocularly and the frame
of the picture was visible.

Let us look at the results of the first experiment, shown
in Figure 4-5. Look first at the line labeled “90.” It presents
the data for the adjustments made when the observers
looked at the picture from the center of projection. Had
the observers been able to correctly match the angle of the
palm board to the slant of the surface in the photograph,
the data points would fall on the dotted line. Because the
data points deviate systematically from the line, we conclude
either that subjects underestimated the extremity of
the deviation of surface slants from the frontal plane or
that they overestimated the extremity of the settings of the
palm boards. The data do not allow us to decide which of
these two interpretations is correct. Furthermore, the palm-board
settings were invariably higher when the photographs
were viewed from the oblique vantage point (labeled
“135”) than when they were viewed from the center
of projection. This means that the photographs looked
different when the observers viewed them from the two
different vantage points, but not necessarily that the observers
failed altogether to compensate for the change in
vantage point. As we saw in Chapter 1, we can use geometry

[ILLUSTRATION]

Figure 4-6. Modified data for Experiment
1 of Rosinski et al. Angle
to which palm board was adjusted to
match apparent slant of surface in
photograph for the two points of
view. Values of independent variable
are determined by assumption
that center of projection coincides
with observer's point of view.

to calculate what an observer who assumes that
his or her eye is at the center of projection can legitimately
infer about the represented scene. If we do that for the data
shown in the curve for the eccentric point of view, the
shape of that curve changes somewhat; on the whole, it
shifts to the right and, as may be seen in Figure 4-6, appears
to coincide with the curve for the data obtained for the
view from the center of projection. In other words, the
difference between the data obtained for the two vantage
points is eliminated if one assumes that observers who
viewed the picture through a peephole were unable to compensate
for the change in vantage point, and that they
perceived the photographs as if they assumed that the center
of projection coincided with their vantage point.

Now we should turn to the results of the second experiment
(shown in Figure 4-7). Here Rosinski et al. had
made no attempt to reduce the perceptibility of the picture
plane. The settings of the palm board appear to be no more
accurate than in the first experiment; but the evidence regarding
the robustness of perspective is unequivocal: There
is no difference between the settings of the palm board for
the two vantage points, thus demonstrating that the perceived
slant of the plane represented in the photograph was
independent of vantage point. So, Rosinski et al. 's experiment

[ILLUSTRATION]

Figure 4-7. Data for Experiment 2
of Rosinski et al. Angle to which
palm board was adjusted to match
apparent slant of surface in photograph
for the two points of view.
Values of independent variable are
determined by assumption that center
of projection coincides with 90-degree
point of view.

confirms Pirenne's hypothesis: If the subjects can
see the picture plane, perspective is robust; if they cannot,
perspective is not robust. In other words, the availability
of information regarding the location and the orientation
of the picture plane is necessary and sufficient for the robustness
of perspective.[4]

Although in general perspective is robust, certain pictures
are an interesting exception to robustness. I am referring
to an illusion of “following” that we experience
when we move in front of some paintings. Hans Wallach
writes:

This observation has been confirmed experimentally by E.
Bruce Goldstein (1979), who affixed a black-and-white
photograph of the painting by Theodore Rousseau mentioned
by Wallach to an upright panel that could be turned
right or left about a vertical axis. Rotating the panel was
a convenient substitute for having the viewer walk around
the reproduction. Just below the panel was a pointer that
could turn independently of the panel about the same axis.
The panel was shown to the viewer at several different
viewing angles. For each angle, the observer was asked to
adjust the pointer so that it would point in the same direction
as the road. At all angles (ranging from 15 degrees,
the right side of the painting turned toward the observer
so that it was seen almost edge on, to 165 degrees, i.e.,
the left side of the painting turned toward the observer so
that it was seen almost edge on), each observer set the

[ILLUSTRATION]

Figure 4-9. Schematic maps completed
by two observers, A and B,
at two viewing angles, 15 degrees
and 165 degrees: To the two parallel
lines representing road, they were
asked to add a short line to represent
rut in road, a rectangle to represent
house, and two dots to represent
trees.

pointer to point directly at him or her.[5] Because this result
appears to run counter to the robustness of perspective,
Goldstein performed a further experiment: The observers
were shown the picture at various orientations and were
given a schematic map with two parallel lines to represent
the road. On this map, they were asked to mark the location
of the closest house (on the left), of the two closest
trees, and of a rut cutting across the road in the foreground.
The maps (see Figure 4-9) were unaffected by the rotation
of the picture, lending support to the robustness of perspective.
I will explain the exceptions to the robustness of
perspective in the latter part of the next chapter.