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Three-dimensional symmetric shapes
are discriminated more
efficiently than asymmetric ones
Zili Liu
Department of Psychology, University of California, Los Angeles, 1285 Franz Hall, Box 951563, Los Angeles,
California 90095
Daniel Kersten
Department of Psychology, University of Minnesota, Elliott Hall, 75 East River Road, Minneapolis, Minnesota 55455
Received September 19, 2002; revised manuscript received February 10, 2003; accepted March 10, 2003
Objects with bilateral symmetry, such as faces, animal shapes, and many man-made objects, play an important
role in everyday vision. Because they occur frequently, it is reasonable to conjecture that the brain may be
specialized for symmetric objects. We investigated whether the human visual system processes three-
dimensional (3D) symmetric objects more efficiently than asymmetric ones. Human subjects, having learned
a symmetric wire object, discriminated which of two distorted copies of the learned object was more similar to
the learned one. The distortion was achieved by adding 3D Gaussian positional perturbations at the vertices
of the wire object. In the asymmetric condition, the perturbation was independent from one vertex to the
next. In the symmetric condition, independent perturbations were added to only half of the object; perturba-
tions on the other half retained the symmetry of the object. We found that subjects’ thresholds were higher in
the symmetric condition. However, since the perturbation in the symmetric condition was correlated, a stimu-
lus image provided less information in the symmetric condition. Taking this into consideration, an ideal-
observer analysis revealed that subjects were actually more efficient at discriminating symmetric objects.
This reversal in interpretation underscores the importance of ideal-observer analysis. A completely opposite,
and wrong, conclusion would have been drawn from analyzing only human discrimination thresholds. Given
the same amount of information, the visual system is actually better able to discriminate symmetric objects
than asymmetric ones. © 2003 Optical Society of America
OCIS codes: 330.4060, 330.5020, 330.5510, 330.6100.
1. INTRODUCTION
How is the shape of a three-dimensional (3D) object rep-
resented in the visual system, and how does its represen-
tation in the brain differ from the physical shape in the
world? One way of formalizing the question of represen-
tation is to ask how the physical shape space is mapped to
the perceptual shape space.1,2 The visually perceived
world is not a replica of the physical world but rather re-
flects ecological and functional requirements. We would
not expect such a mapping to be uniform, but rather we
expect that certain physical details may be exaggerated
and others reduced. We can experimentally address the
nature of this mapping by testing whether two similar ob-
jects can be discriminated and what determines such per-
ceptual discrimination.
Because of their biological, adaptive, and social signifi-
cance, objects with bilateral symmetry have been consid-
ered an important subclass of objects3–15 (see Ref. 16 for a
review). This raises the possibility that the human vi-
sual system is specialized in some way for the processing
of bilaterally symmetric objects. If true, such specializa-
tion could be manifest in two ways (Fig. 1). First, the vi-
sual system may be particularly good at detecting depar-
tures from symmetry. In fact, most studies have shown
that human observers are sensitive to small deviations
from perfect symmetry, with the interpretation that vi-
sion has mechanisms specialized for symmetry. On a
closer look, however, it appears that this good sensitivity
is not appropriately compared.17–20 That is to say, the
evidence so far has shown that a small deviation from
perfect symmetry can be detected; but whether the same
amount of deviation can be better detected from an asym-
metric object is unknown.
This paper addresses the second sense in which sym-
metry may be special. That is, special in the sense that
deformations of a symmetric object may be detected more
easily than deformations of an asymmetric object. In
other words, if the same amount of deviation is applied to
a symmetric object in such a way that the object remains
symmetric after the deviation, can this deviation be bet-
ter detected? For reasons that will become clear below,
the question will be addressed in the context of discrimi-
nation. Two objects, both deformed from the same 3D
symmetric object, one with more deformation than the
other, are presented to a human subject. The subject de-
cides which of the two is more similar to the original sym-
metric object that has been learned. There are two ways
to accomplish the deformation. (1) Symmetric condition:
The deformation is applied in such a way that the de-
formed objects remain symmetric. (2) Asymmetric condi-
Z. Liu and D. Kersten Vol. 20, No. 7/ July 2003/ J. Opt. Soc. Am. A 1331
1084-7529/2003/071331-10$15.00 © 2003 Optical Society of America
tion: The two objects become asymmetric after the defor-
mation. The question is which condition is easier to
detect?
To answer this question, we must ensure that the de-
formations in the two conditions are equal or, more pre-
cisely, the signal-to-noise ratio of the stimulus in the two
conditions are identical. At issue is the correlated nature
of the deformation applied to the symmetric condition.
For a symmetric object to remain symmetric after defor-
mation, the deformation itself has to be symmetric. In
contrast, for a symmetric object to become asymmetric af-
ter deformation, the deformation has to be asymmetric.
For this reason, we use the ideal observer to equalize the
stimulus signal-to-noise ratio in both conditions.21–24
Calculating the performance of ideal observers is often
theoretically difficult; however, when an appropriate task
and object representation are chosen, ideal observer
analysis can be made tractable. In Subsection 1.A we
will elaborate on the ideal observer approach.
The second question we ask is the following. If per-
ceiving a symmetric object is different from perceiving an
asymmetric one, then to what extent does the assumption
that an object is symmetric per se facilitate ascertaining
the 3D structure of the object? We know that a shaded
thick wire object, as compared with its silhouette or a thin
wire-frame counterpart, offers the following additional in-
formation about its 3D shape: shading, self-occlusion
that provides relative depth ordering between the occlud-
ing cylinder and the occluded, and the shape of the
creases between two neighboring cylinders. Our ques-
tion is if we replace the shaded objects with their silhou-
ettes, does an assumption of bilateral symmetry provide
sufficient information to make discriminations of accu-
racy comparable with the shaded case? In other words,
can human subjects achieve similar discrimination per-
formance if the only 3D shape information is the knowl-
edge that the object has bilateral symmetry?
Previously, we have found that human subjects can
take advantage of the information from shading, self-
occlusion, and the shape of creases between two neighbor-
ing cylinders.25 This is evident from their better dis-
crimination performance for asymmetric thick wire
objects than for the thin wire-frame counterparts, which
lack the aforementioned shape information. We have
also found that when an image of a wire object can be in-
terpreted as projected from either a symmetric or an
asymmetric object, human subjects prefer the symmetric
interpretation.26 In other words, when a 3D object can
be interpreted as symmetric, it will be. However, it re-
mains an open question whether such preference for a
symmetric object interpretation translates into a more
precise specification of the symmetric object’s shape, just
as shading, self-occlusion, and crease shapes do. The last
of the three experiments in this paper will address this
question.
A. Ideal-Observer Approach
For clarity, we use the same example of symmetry dis-
crimination as above to motivate the ideal-observer ap-
proach. To answer under which of the conditions it is
easier to discriminate, it is straightforward to set up a
psychophysical task to measure the thresholds of a sub-
ject’s performance in the two conditions. However, the
threshold difference cannot tell us whether better perfor-
mance in one condition is due to richer information in the
stimulus or to superior specialization in the brain or to
both. More specifically, in condition (1), the fact that
both objects remain symmetric implies that stimulus in-
formation is redundant. When one half of an object is de-
formed, the other half has to be deformed accordingly to
retain the symmetry. In theory, therefore, one half of the
object and the plane of symmetry completely specify the
entire object. In condition (2), on the other hand, since
the deformation to one half of the object is completely in-
dependent of the other half, deformation at all vertices of
an object is informative for the discrimination task.
To make a fair comparison, the stimulus information in
the two conditions has to be equalized. The first step in
achieving this is to quantify the input stimulus informa-
tion in the following way. Given a specific task, an ideal
observer is defined to have unlimited computational ca-
pacity and achieve the best possible performance.21,27,28
The only limitation of the ideal observer’s performance is
the amount of stimulus input information, so the ideal ob-
server can achieve as best as the input information al-
lows. Consequently, the ideal observer’s performance is
a measure of the input stimulus information. The more
input information, the better it performs.
Now if we give the same task to human subjects, they
will also achieve a certain performance level, typically
measured by the threshold at a fixed proportion correct.
If we combine the thresholds of the ideal observer and hu-
man observers appropriately (see Appendix A), we have
normalized the stimulus information and obtained a mea-
sure called statistical efficiency. It means the proportion
of stimulus information that a human observer has effec-
tively used in the given task. A simple example is detect-
ing which of two patterns of a large number of random
dots was drawn from a distribution with a higher mean
number.29 The ideal observer could count all the dots,
whereas a human subject could not, and the human’s per-
formance would be lower than the ideal observer’s. We
Fig. 1. Schematic illustration of a physical shape subspace,
where along one dimension (horizontal) the degrees of asymme-
try of an object varies and along an orthogonal dimension (verti-
cal) shape varies while symmetry is unchanged. In this paper
we ask whether two shapes are equally discriminable along the
two axes, respectively, if their physical difference is kept constant
(by an ideal observer).
1332 J. Opt. Soc. Am. A/Vol. 20, No. 7/ July 2003 Z. Liu and D. Kersten
can interpret the result as if the human subject can see
only a fraction of the dots but otherwise use them opti-
mally in the decision.
Coming back to the symmetry-discrimination task, we
measure the threshold and statistical efficiency in the fol-
lowing way. The 3D symmetric object consists of a chain
of four cylinders that have five vertices total. The defor-
mation is done by adding 3D positional Gaussian pertur-
bations at each vertex. In the symmetric condition, ver-
tices 1–3 have independent perturbations, whereas
vertices 4 and 5 are perturbed such that the resultant
shape remains symmetric. In the asymmetric condition,
perturbations at all five vertices are independent of each
other. In any one trial, there will be two deformed ob-
jects, one drawn from a Gaussian distribution in which
the variance is constant, whereas the other is drawn from
a distribution with a larger variance that is adjustable in
order to find the discrimination threshold. We call the
first object the target and the second object the distractor.
The experiment is to find out, with a staircase procedure,
how much larger the second variance has to be for an ob-
server to be 75% correct, and the resultant standard de-
viation will be the measure of the threshold. Defining
the task in this way ensures the existence of a simple and
provably optimal ideal observer in the signal-detection
sense, which we will elaborate in the following.
The ideal observer is assumed to have knowledge of the
3D structure of each object, namely, the (x,y,z) coordi-
nates of the five vertices and their ordering. From each
input object image, the ideal observer is assumed to know
the two-dimensional (2D) (x,y) coordinates of the verti-
ces and their ordering except that it does not know which
end of the cylinder chain is the head and which is the tail.
The ideal observer also knows the value of the constant
variance of the Gaussian added to the target. Conse-
quently, the ideal observer’s job is to consider all possible
2D images of the learned 3D object, calculate the prob-
ability that an input image is generated from each of the
2D images, and integrate the probability (as a function of
viewing angle) over all 2D images to obtain the likelihood
of an input image being generated from the learned ob-
ject. Since there are two input images in any trial, the
ideal observer obtains two likelihood measures and se-
lects the one more likely to be more similar to the learned
object. We have thus summarized the operations of the
ideal observer; interested readers are referred to Liu
et al.25 for more technical details. As outlined in Appen-
dix A of the present paper, the statistical efficiency Ecan
be expressed as23
E⫽
共
dI兲2⫺
t2
共
dH兲2⫺
t2, (1)
where
tis the standard deviation for the target;
dIis
the ideal observer’s threshold, namely, the standard de-
viation for the distractor at 75% correct; and
dHis the
threshold of human subjects.
For clarity, we will specify in more detail the computa-
tions of the ideal observer after specifying the experimen-
tal task in Subsection 1.B.
B. Rationale of the Experiments
We describe three experiments. Experiment 1 addresses
the question of whether it is easier to discriminate one
asymmetric object from another or one symmetric object
from another. In the asymmetric condition, independent
positional perturbations are added to all vertices of a
symmetric wire object, so that the resultant object be-
comes asymmetric. In contrast, in the symmetric condi-
tion, independent perturbations are added to only half of
an object, whereas the other half are adjusted to retain
symmetry. Discrimination thresholds for human and
ideal observers are measured, and statistical efficiencies
are calculated.
Experiment 2 addresses the question of whether any ef-
fect in experiment 1 is due to the difference in the number
of independent noise sources. That is to say, in the asym-
metric condition in experiment 1, five vertices are added
with independent perturbations, whereas in the symmet-
ric condition, only three vertices are. So it might be that
any difference in experiment 1 is due to subjects’ limited
ability in dealing with more noise sources in the asym-
metric condition. Therefore, in this experiment, indepen-
dent noise is added to only three vertices to make a sym-
metric object asymmetric. The prediction is that if the
difference in experiment 1 is only due to this difference of
the number of noise sources, then no difference should be
found between the new condition and the symmetric con-
dition in experiment 1.
Experiment 3 addresses the putative specialness of bi-
lateral symmetry from a different perspective. We ask
whether the information that an object is symmetric (or
nearly so, with asymmetric perturbations) can also pro-
vide 3D shape information in the following sense. Since
there is no depth information when a silhouette image is
presented, the 3D pose of the corresponding 3D object, or,
in other words, the object’s 3D shape, is ambiguous.
However, if the object is known or assumed to be symmet-
ric, then the uncertainty of the 3D pose of the object and
the object’s 3D shape is reduced. In this experiment we
compare symmetric wire objects that are either silhou-
ettes or shaded. Our hypothesis is that if human sub-
jects’ performance does not deteriorate after we remove
from a symmetric object shape information, such as shad-
ing, self-occlusion, and the shape of the creases between
two neighboring cylinders, it will be evidence that the
knowledge of symmetry alone is sufficient to reduce the
uncertainty of the object’s pose and shape, just as the re-
moved information did for asymmetric objects.
2. EXPERIMENT 1
A. Stimuli
Each symmetric object consisted of five spheres linked by
four cylinders in a chain in three dimensions. The diam-
eter of each sphere and that of each cylinder were identi-
cal (0.22 cm). The length of each cylinder was 2.54 cm.
Figure 2 shows one example object from multiple views.
The objects were rendered with Lambertian shading, with
a point light source at infinity. The illumination was di-
rected toward the camera from the upper front, with an
angle of 26.6° relative to the vertical direction. Ambient
light with an intensity of 3% relative to the directed light
was also present.
Z. Liu and D. Kersten Vol. 20, No. 7/ July 2003/ J. Opt. Soc. Am. A 1333
The images were rendered with parallel projection on a
Silicon Graphics computer with the Open Inventor soft-
ware (Silicon Graphics, Inc.). The viewing distance was
57 cm. An object was approximately 5 cm in size, which
was approximately 5° visual angle. Subjects viewed the
stimuli binocularly in a darkened room.
B. Method
Each subject was tested with six symmetric objects; three
became asymmetric after structural distortion was added,
and the other three remained symmetric after structural
distortion. Each object was tested in a single block. The
order of the test and which objects were symmetric and
which asymmetric were counterbalanced among the sub-
jects. Within each block the experimental procedure was
as follows.
1. Learning
A symmetric image of the object was shown with x⫽0
being the symmetric plane. The subject rotated the ob-
ject by pressing the middle button of the computer mouse.
Each rotational step was 60°. The object was rotated
first around the yaxis and then around the xaxis (when
all the images were symmetric). Hence the object was
shown from 11 viewing angles. Figure 2 shows the se-
quence of these images for one example object. A subject
looked through this image sequence twice.
2. Practice
Two images were presented side by side to the subject,
who decided which was the learned object. Feedback
was provided by a computer beep for correct responses.
One image was chosen randomly from one of the 11
learned views. The other image was generated either
from a different symmetric object or from the same
learned object by adding distortions. The distortion was
created by adding Gaussian positional noise
N(0, 0.254 cm) in three dimensions to one symmetric half
of the object; the other half of the object was distorted in
such a way that symmetry was preserved.
At any time during the practice, the subject could press
the middle mouse button to review the learned object.
The image of the learned object would change into the
first image of the object and rotate around the yaxis and
then around the xaxis under the subject’s mouse button
control. Meanwhile, the image of the distractor object re-
mained unchanged.
The subject needed 100 trials to finish this practice, in
addition to the review trials. This practice was the same
no matter if the object would be distorted symmetrically
or asymmetrically during the next stage, the test stage, so
that the subject was trained to learn both the symmetry
and the geometric details of the object. Figure 3 shows
an example of the object when the subject was tested un-
der the symmetric and asymmetric conditions, respec-
tively.
3. Test
Two object images were presented side by side to the sub-
ject, who decided which was more similar to the learned
object. No feedback was provided. Both objects were
distorted versions of the same learned symmetric object
and rendered from the same viewing angle. Both objects
were generated by adding Gaussian positional distortions
in three dimensions to the vertices. For one object, the
standard deviation was always 0.254 cm. For the other,
the standard deviation was larger and could be adjusted
to find a threshold. We used a staircase procedure to find
the threshold value of the larger standard deviation that
gave rise to 75% correct performance.30 At any time dur-
ing the test, the subject could press the middle mouse but-
ton and review the learned object from the 11 viewpoints.
Discounting these review trials, there were 500 test tri-
als.
The total number of 500 test trials was divided into
three conditions, which were randomly interleaved. In
200 of the trials, the viewing position was from one of the
11 learned positions. We call this the learned view con-
dition. In another 200 trials, the viewing position was
randomly chosen from the viewing sphere, with a random
rotation around the viewing direction. We call this the
novel view condition. The remaining 100 trials served as
Fig. 2. Three-dimensional symmetric object from multiple views
by rotation around the yaxis (top two rows, left to right) and
then around the xaxis (bottom two rows). These 11 views (the
top left, second row right, and bottom right are identical) are
called the learned views.
Fig. 3. Example stimuli in the test stage when objects were dis-
torted versions of the learned symmetric object. (a) Both objects
were symmetric after the distortion. (b) Both were asymmetric.
1334 J. Opt. Soc. Am. A/Vol. 20, No. 7/ July 2003 Z. Liu and D. Kersten
refresh trials that were identical to those in the practice
stage, and feedback was provided.
C. Subjects
Twelve subjects, naı
¨ve to the purpose of the experiment,
participated; each was with six prototype objects, three
for the symmetric condition and three for the asymmetric
one, in a counterbalanced manner.
D. Results
1. Thresholds
Within-subjects analysis of variance (ANOVA) revealed a
significant main effect of learned versus novel viewing
angles 关F(1, 11) ⫽7.92, p⬍0.02兴. As expected, the
threshold for the learned views was smaller than for the
novel views (0.71 cm versus 0.84 cm). ANOVA also re-
vealed a significant main effect of symmetric versus
asymmetric conditions 关F(1, 11) ⫽4.72, p⫽0.05兴. The
threshold for the asymmetric condition was smaller than
for the symmetric condition (0.71 cm versus 0.85 cm); this
means that the performance was better for the asymmet-
ric than for the symmetric objects. The interaction was
not significant (F⬍1 ) . Figure 4 shows the threshold
performance when the subjects were correct 75% of the
time.
2. Ideal Observer Computation
We start by describing the computation for the asymmet-
ric condition (see Liu et al.25 for more details). The ques-
tion is, given two input images [i.e., for each image, the
(x,y) coordinates of each of the five vertices and their or-
der] of two deformed objects, which is more likely to come
from the learned object? The 3D structure of the learned
object [i.e., the (x,y,z) coordinates of each of the five ver-
tices and their order] is known. Unknown are the projec-
tion angle from which both images are obtained and
which end vertex is the head and which is the tail.
The ideal observer tries out all possible projection
angles and, for each angle, tries all possible angles of ro-
tation (in 360°). For each particular projection angle and
rotation angle, a Euclidean distance between vertices of
an input image and the newly projected image is com-
puted [the origin (0, 0, 0) is assumed known to the ideal
observer]. The likelihood is then computed by using the
Gaussian with the known variance, which is the constant
variance of the target object. Because of the head–tail
ambiguity, the same computation is repeated with a
flipped head–tail correspondence between the two images
above. The likelihood values for all the projection angles
and rotation angles are integrated with equal weighting.
Finally, the input image with the larger total likelihood
will be chosen as the one more similar to the learned ob-
ject.
Numerically, 32,776 points on the viewing sphere were
chosen in such a way that their distribution was maxi-
mally uniform on the sphere (the enclosed convex hull
had the maximum volume). Each point represented a
projection angle. For each projection angle, 360 rota-
tional angles were chosen (1° each).
For the symmetric condition, the computation was
similar except for the following two differences. The first
was that, since independent noise was added only to three
vertices on one side of an object, the Euclidean distance
was calculated between two images accordingly, as well
(because of the head–tail ambiguity, the head–tail flip
was still needed). The second difference was that, since
an object after deformation remained symmetric, the in-
formation regarding the plane of symmetry was not com-
pletely lost, so the ideal observer used the information as
follows.
For any bilaterally symmetric object, a line connecting
any two corresponding points is perpendicular to the
plane of symmetry. Therefore all such lines are parallel
to each other. In our case, the lines connecting vertices 1
and 5 and vertices 2 and 4 were parallel to each other.
Since the object remained symmetric after deformation,
these two lines remained parallel to each other as well.
Now, since the 3D symmetric object was projected to its
2D image orthographically, these two lines were also par-
allel to each other. Therefore, although the projection
angle was still not completely known, we do know that
the rotational angle around the projection direction had
to be such that the two new lines had to be parallel with
the old lines. More specifically, there were only two such
rotational angles (differing by 180°). Again, the head–
tail ambiguity remained.
3. Statistical Efficiencies
We calculated the statistical efficiencies for the symmetric
and asymmetric conditions separately. For the symmet-
ric condition, we simulated 20 objects, each with the
learned view condition and novel view condition, and each
condition was simulated with the staircase procedure for
2000 trials. Because the ideal observer had knowledge of
the 3D structure of each object, its performance for the
learned view and the novel view conditions should be the
same. We simulated both conditions to self-check that
the thresholds of both conditions for each object were
identical, and they indeed were. We obtained an average
threshold of 0.379 cm with a standard deviation of 0.013
cm.
For the asymmetric condition, we needed to simulate
only nine objects with just the novel condition to obtain
an average threshold with a much smaller standard de-
viation than the symmetric ones: 0.321 ⫾0.007 cm.
Fig. 4. Discrimination threshold of 12 subjects when they were
correct 75% of the time. The error bars are standard errors be-
tween subjects. Note that these error bars cannot directly re-
flect the statistical significance, since within-subjects analysis is
used (and the same for Figs. 5–7 and 9).
Z. Liu and D. Kersten Vol. 20, No. 7/ July 2003 / J. Opt. Soc. Am. A 1335
We then calculated the statistical efficiencies using Eq.
(1). ANOVA demonstrated, as expected, a better perfor-
mance for the learned view than for the novel view
[22.50% versus 16.90%, F(1, 11) ⫽12.82, p⬍0.01].
The ANOVA also indicated better performance for the
symmetric condition than for the asymmetric condition
[23.04% versus 16.36%, F(1, 11) ⫽15.00, p⬍0.01],
which was unexpected from the discrimination-threshold
results. The interaction was not significant ( F⬍1).
Figure 5 shows the result.
3. EXPERIMENT 2
There was a potential confound in experiment 1.
Namely, subjects’ threshold difference between the sym-
metric and the asymmetric conditions might have had
nothing to do with symmetry but rather might be due to
the different number of noise sources. That is, there
were only three independent noise sources for the sym-
metric condition and five for the asymmetric condition.
To test this possibility, we conducted experiment 2 un-
der the same conditions as experiment 1, except that in-
dependent noise was added to only three vertices, 1, 3,
and 5, and no noise was added at vertices 2 and 4. The
resultant object therefore would be asymmetric.
If the result in experiment 1 is completely due to the
difference of the number of noise sources, then we expect
no difference between the results of experiment 2 and the
symmetric condition of experiment 1. On the other hand,
if symmetry versus asymmetry has a genuine effect, then
we expect a difference in thresholds and statistical effi-
ciencies between this experiment and the symmetric con-
dition in experiment 1.
A. Subjects
Twelve fresh subjects, naı
¨ve to the purpose of the experi-
ment, participated, each with three prototype objects.
B. Results
1. Thresholds
Two-way ANOVA was used to compare the thresholds
from this experiment and those of the symmetric condi-
tion in the last experiment, with the main factors of view
(learned versus novel) and condition (asymmetric versus
symmetric). As expected, the main effect of view was sig-
nificant; it was easier to discriminate the learned views
than the novel views [0.84 cm versus 1.13 cm, F(1, 22)
⫽14.01, p⬍0.001]. The main effect of condition was
not significant [symmetric: 0.85 cm; asymmetric: 1.12
cm, F(1, 22) ⫽1.17, p⫽0.29]. Of particular interest
was the interaction, which was significant, meaning that
discrimination dropped more quickly from the learned to
novel views for the asymmetric condition in the current
experiment (0.87 to 1.36 cm) than for the symmetric con-
dition (0.80 to 0.90 cm) 关F(1, 22) ⫽5.96, p⬍0.02, Fig.
6]. This means that, although noise had been added to
the same number of vertices, discrimination is different
for symmetric versus asymmetric objects.
2. Ideal Observer Computation
We describe here the ideal observer for the task in experi-
ment 2. Since no noise was added to vertices 2 and 4 and
since the origin (0, 0, 0) was always known, the 2D coor-
dinates of vertices 2 and 4 (four knowns) completely de-
termine the projection angle (3 unknowns), except that
we still have to consider the head–tail ambiguity. We
used 20 objects and each object again with 2000 trials.
The average threshold and standard deviation were
0.339 ⫾0.004 cm.
3. Statistical Efficiencies
We computed the statistical efficiencies using Eq. (1) for
the learned and novel views. We then performed a two-
way between-subjects ANOVA, comparing efficiencies in
Fig. 5. Statistical efficiencies for the symmetric versus asym-
metric conditions and learned versus novel views. Note that the
pattern of symmetric versus asymmetric comparison is com-
pletely reversed here in contrast to the threshold performance.
Fig. 6. Discrimination thresholds for the learned and novel
views of the symmetric condition in experiment 1 and of the
asymmetric condition in experiment 2.
Fig. 7. Statistical efficiencies of the symmetric condition in ex-
periment 1 and of the asymmetric condition in experiment 2 for
the learned and novel views.
1336 J. Opt. Soc. Am. A/ Vol. 20, No. 7/ July 2003 Z. Liu and D. Kersten
this experiment and those in experiment 1 of the symmet-
ric condition. Both main effects were significant. Effi-
ciencies for the learned views were higher than for the
novel ones (19.95% versus 13.50%), F(1, 22) ⫽12.93,
p⬍0.002. Efficiencies for the symmetric objects were
higher than for asymmetric ones (23.04% versus 10.41%),
F(1, 22) ⫽9.21, p⬍0.006. The interaction was not sig-
nificant 关F(1, 22) ⬍1兴. Figure 7 shows the result.
4. EXPERIMENT 3
So far, the computations of the ideal observer have as-
sumed that only 2D information, namely, the (x,y) coor-
dinates of an object’s vertices, is available from an input
image. However, shading, self-occlusion, and the shape
of creases formed at the joints between neighboring cylin-
ders may provide potential 3D information about the
length and depth orientation of each of an object’s
cylinders.25 Although the computations of the ideal ob-
server, so far, are all valid, since no bias is introduced in
the comparisons, we would still like to know to what ex-
tent such 3D information is used by human subjects.
To address this question, we changed the objects into
silhouettes so that an object had the same luminance ev-
erywhere and all three potential sources of 3D informa-
tion were absent (Fig. 8). We predict that if subjects’ per-
formance became worse, then it would be evidence that
the 3D information was used. Otherwise, it means that
these sources of 3D information play no significant role
when the prototype objects are symmetric in three dimen-
sions.
A. Subjects
Four subjects from experiment 1 plus eight fresh subjects
participated; each was with six prototype objects, three in
the symmetric condition and three in the asymmetric, in a
counterbalanced manner.
B. Results
We compared the results with those in experiment 1 in a
three-way ANOVA (silhouette versus shaded, learned
view versus novel view, and symmetric versus asymmet-
ric). We first report here the within-subjects analysis
with the four subjects. Only two main effects were sig-
nificant, both confirming the results from experiment 1
[Fig. 9(a)]. The thresholds for the symmetric condition
were higher than for the asymmetric condition (0.765 cm
versus 0.667 cm): F(1, 3) ⫽15.30, p⬍0.03. Thresh-
olds for the learned views were lower than for the novel
views (0.622 cm versus 0.810 cm): F(1, 3) ⫽13.03,
p⫽0.037. Importantly, the difference between the sil-
houette and the shaded conditions was not significant
(0.767 cm versus 0.665 cm): F(1, 3) ⫽2.53, p⫽0.21.
We now look at the between-subjects ANOVA between
the eight subjects in the current experiment and the re-
maining eight subjects in experiment 1. The results
were consistent with the within-subjects analysis above
[Fig. 9(b)]. Perhaps owing to between-subjects varia-
tions, the thresholds for the symmetric condition were
now only marginally significantly worse than for the
asymmetric condition (0.926 cm versus 0.784 cm):
F(1, 14) ⫽3.16, p⫽0.097. Thresholds for the learned
views were better than for the novel views (0.735 cm ver-
sus 0.975 cm): F( 1, 14) ⫽11.24, p⬍0.005. Thresh-
olds between the silhouette and the shaded conditions
were not significantly different (mean ⫾standard error:
0.875 ⫾0.075 cm versus 0.835 ⫾0.080 cm) : F(1, 14)
⫽0.05 ⬍1, p⫽0.828.
5. DISCUSSION
We have demonstrated that it is harder, in terms of
threshold, to discriminate between two deformed objects
when the deformation retains the symmetry. However,
when we take into consideration that a symmetric defor-
mation has only half of the degrees of freedom as an
asymmetric deformation, we found that human observers
are better able to discriminate symmetric deformations
when the input image information is equalized. We
stress that without the ideal-observer analysis such a
Fig. 8. Experimental stimuli in the test phase when all the ob-
jects were silhouettes.
Fig. 9. Threshold comparisons between experiment 1 and the
silhouette condition of experiment 3. (a) Within-subjects com-
parison (four subjects). (b) Between-subjects comparison (eight
subjects each). There was no significant difference between the
silhouette and the shaded conditions.
Z. Liu and D. Kersten Vol. 20, No. 7/ July 2003 / J. Opt. Soc. Am. A 1337
conclusion cannot be reached. This suggests that the vi-
sual system is indeed better equipped to process symmet-
ric objects.3,16,20,31–33
In addition, these results suggest that symmetry
within a 2D image does not account for all of the advan-
tages for symmetric objects. Otherwise, we should not
expect any difference for the novel views between the
symmetric and the asymmetric conditions, since a novel
view was almost always an asymmetric image. Within-
image symmetry, on the other hand, does suggest an ad-
vantage. This is because, for the learned views, the sym-
metric condition is more efficient than the asymmetric
condition. Note that in half of the trials of the symmetric
condition, the images were symmetric. Whereas for the
asymmetric condition, practically no image was symmet-
ric. We remark that, within the learned views of the
symmetric condition, we cannot directly compare the
symmetric images with asymmetric ones because we used
a single staircase procedure to obtain a single threshold.
Therefore we cannot rule out the possibility that this ad-
vantage is completely due to the 3D symmetry and has
nothing to do with the 2D symmetry in an image.
Although our results show that object discrimination is
more efficient for symmetric than for asymmetric objects,
we emphasize that this does not imply distinct physiologi-
cal mechanisms for symmetric versus asymmetric objects.
In Bayesian terms, symmetric objects may be handled
more efficiently as a consequence of being more common
or more important. Yet, somehow priors and task rel-
evance have to be instantiated in the brain. This kind of
specialization may not be completely based on familiarity
with many near-symmetric objects in the world, natural
and artificial alike, because the objects used in the cur-
rent study are novel. High efficiencies may have to do
with the way the visual system codes for near-symmetric
objects, where symmetry is a type of regularity that the
visual system exploits to form economic object
representations.12,34
The results in the third experiment indicated that, for
symmetric prototype objects, shading, self-occlusion, and
the shape of creases between two neighboring cylinders
do not provide more information about the 3D shape of an
object than the silhouette of the object does already. This
is, in fact, consistent with subjects’ impressions that a sil-
houette of a symmetric object already appears to be a 3D
symmetric object. This is in stark contrast with the pre-
vious finding in Liu et al.25 In that study, shaded (with
self-occlusion and creases at the junctions), but asymmet-
ric, objects gave rise to better (i.e., smaller) thresholds
than objects (called tinker toys) of the same 3D configu-
rations but with all the cylinders shrunk to a line, so that
no shading, self-occlusion, and creases were available.
Moreover, in that study the statistical efficiency for the
shaded wire objects was also higher than for the tinker
toys when the ideal observer, for the former, was assumed
to be able to recover the full 3D structure of an object [the
(x,y,z) coordinates of the vertices] from its single 2D
image and, for the latter, the ideal observer had only the
2D (x,y) coordinates of the vertices. It was a strong re-
sult because the ideal observer for the shaded objects was
a super-ideal observer, who was able to recover a 3D
shape from shading from a single image. Such a super-
ideal observer gave rise to the lower bound of statistical
efficiency for the shaded wire objects, which was still
higher than for the tinker toys. The fact that no statis-
tically significant difference has been found here between
the shaded and the silhouette symmetric objects is, we be-
lieve, a strong testament that the knowledge of symmetry
alone provides information about the 3D shape of the ob-
jects.
So how may the knowledge of 3D symmetry be obtained
from a 2D image? The following theoretical result may
offer a clue. A necessary condition for a 2D image to be
the parallel projection of a 3D symmetric object is this:
All lines, each of which connects two corresponding points
of a 3D symmetric object, have to be parallel in the image
plane.35 By extension, in a perspective projection, all
these lines have to converge to a single point. (The par-
allel lines in the orthographic projection case are, of
course, a special case, since they converge at infinity.)
There is also a theoretical possibility that our result of
the symmetry advantage may be explained, at least in
part, by the following. When symmetry is detected, it
may produce a compelling percept to the extent that
symmetry makes other differences less distinguish-
able.19,20,36,37 We cannot rule out or confirm this possi-
bility but speculate that, unless the image itself is sym-
metric, the symmetry percept is not as strong.
Human efficiencies are less than 100%. What are the
possible sources of the inefficiencies? Specifically, such
inefficiencies can be attributed to three categories: input
stimulus encoding, representation of an object model, and
the matching process (please see Section 4.1 in Ref. 25
and Refs. 23 and 38 for a discussion in a broader context).
First, unlike the ideal observer who codes the (x,y) posi-
tions of the vertices precisely, humans are bound to be in-
accurate, encoding the shape of an input stimulus, e.g., ei-
ther by the vertex positions or by the lengths of the
cylinders and their relative angles. We remark that as-
suming that the ideal observer always knows the origin of
the coordinate system, which humans may not, results in
a lower efficiency, since the uncertainty for the ideal ob-
server is reduced. Second, even if humans can represent
an input stimulus perfectly, they may not be able to rep-
resent the 3D structure of the object precisely, in contrast
to the ideal observer. By precisely representing the 3D
structure, we mean that the information contained in the
representation can accurately describe the 3D shape of
the object, no matter what the format of this representa-
tion is. Third, even if the human visual system can en-
code an input stimulus and represent the 3D structure of
the object perfectly, it may still be imperfect in matching
an internal object model with an input stimulus. For in-
stance, humans may not be able to consider and integrate
over all possible relative orientations of the model relative
to the input. The visual system may not be able to effec-
tively rotate the 3D model with a large angle to match
with a 2D input.
APPENDIX A
We now outline the assumptions used in deriving the sta-
tistical efficiency, while leaving the derivation in Ref. 23.
Statistical efficiency Eis defined as the squared ratio of
1338 J. Opt. Soc. Am. A/ Vol. 20, No. 7/ July 2003 Z. Liu and D. Kersten
discrimination index d⬘for the human observer d⬘Hver-
sus for the ideal observer d⬘I, where d⬘I⫽x/
, with xas
the strength of the signal and
as the standard deviation
of the Gaussian noise present,
E⫽
冉
d⬘H
d⬘I
冊
2
.(A1)
As in the standard analysis, we assume that the human
observer is ideal except that there is Gaussian internal
noise
Hadded to the input signal and noise. Therefore
d⬘H⫽x/
冑
2⫹
H2. We now need another mathemati-
cal result that we proved in Ref. 23 in order to complete
the derivation. This result says that the percentage cor-
rect for the ideal observer is determined by the ratio of
standard deviations of target and of distractor noise,
t
and
d. In our experimental setting, Pthres ⫽f(
t/
d).
[In fact, Pthres(
t/
d,n) is also a function of n, the num-
ber of object vertices with independent noise. This is rel-
evant to the issue of different number of noise sources
that is discussed in Subsection 1.B.] We assumed, in de-
riving this result, that an ideal observer knows about a
prototype object in terms of a vector corresponding to the
object’s vertices. It also knows the vectors of an input
target object [prototype plus noise N(0,
t)] and of an in-
put distractor object [prototype plus noise N(0,
d)].
This ideal observer also knows the relative position and
orientation between the prototype object and an input ob-
ject. Note that this is an approximation to the ideal ob-
server used in this paper. The ideal observer in this pa-
per does not know the relative orientation of the
prototype object relative to an input object. It instead
needs to integrate over all possible orientation angles.
Here the optimal strategy is to choose between the two in-
put vectors the one that is closer in Euclidean distance to
the prototype vector. The input target and distractor
each have a probability distribution of this Euclidean dis-
tance, from which the percentage correct for the ideal ob-
server can be derived as Pthres ⫽f(
t/
d). Having ob-
tained this relationship, we then know that when the
human and ideal observers have the same percentage cor-
rect,
t2
共
dI兲2⫽
t2⫹
H2
共
dH兲2⫹
H2,(A2)
which leads to Eq. (1).
ACKNOWLEDGMENTS
We thank the two reviewers for helpful comments. This
research was previously reported at the European Confer-
ence on Visual Perception, Trieste, Italy, 1999. Z. Liu is
supported in part by a National Science Foundation grant
IBN-9817979 and a National Eye Institute grant R03
EY14113. D. Kersten is supported by National Institutes
of Health grant EY02857. We thank Sara L. van Driest,
Chris Wainman, and Ingrid Wu for their assistance in
conducting the experiments.
Z. Liu can be reached by fax, 310-206-5895, or e-mail,
zili@psych.ucla.edu. D. Kersten can be reached by fax,
612-626-2079, or e-mail, kersten@umn.edu.
REFERENCES
1. J. J. Koenderink, A. J. van Doorn, and A. M. L. Kappers,
‘‘Depth relief,’’ Perception 24,115–126 (1995).
2. J. J. Koenderink, A. J. van Doorn, and A. M. L. Kappers,
‘‘Pictorial surface attitude and local depth comparisons,’’
Percept. Psychophys. 58, 163–173 (1996).
3. M. Leyton, Symmetry, Causality, Mind (MIT, Cambridge,
Mass., 1992).
4. H. I. Bailit, P. L. Workman, J. D. Niswander, and C. J.
MacLean, ‘‘Dental asymmetry as an indicator of genetic and
environmental conditions in human populations,’’ Hum.
Biol. 42, 626–638 (1970).
5. P. A. Parsons, ‘‘Fluctuating asymmetry: an epigenetic
measure of stress,’’ Biol. Rev. 65, 131–145 (1990).
6. A. P. Møller, ‘‘Female swallow preference for symmetrical
male sexual ornaments,’’ Nature 357, 238–240 (1992).
7. R. Thornhill, ‘‘Fluctuating asymmetry and the mating sys-
tem of the Japanese scorpionfly Panorpa japonica,’’ Anim.
Behav. 44, 867–879 (1992).
8. A. L. R. Thomas, ‘‘On avian asymmetry: the evidence of
natural selection for symmetrical tails and wings in birds,’’
Proc. R. Soc. London Ser. B 252, 245–251 (1993).
9. A. L. R. Thomas, ‘‘The aerodynamic costs of asymmetry in
the wings and tails of birds: asymmetric birds can’t fly
round tight corners,’’ Proc. R. Soc. London Ser. B 254, 181–
189 (1993).
10. K. Grammer and R. Thornhill, ‘‘Human (Homo sapiens)fa-
cial attractiveness and sexual selection: the role of sym-
metry and averageness,’’ J. Comp. Psychol. 108, 233–242
(1994).
11. P. J. Watson and R. Thornhill, ‘‘Fluctuating asymmetry and
sexual selection,’’ Trends Ecol. Evol. 9,21–25 (1994).
12. M. Enquist and A. Arak, ‘‘Symmetry, beauty and evolution,’’
Nature 372,169–172 (1994).
13. R. Thornhill and S. W. Gangestad, ‘‘Human fluctuating
asymmetry and sexual behavior,’’ Psych. Sci. 5, 297–302
(1994).
14. G. Rhodes, F. Profitt, J. M. Grady, and A. Sumich, ‘‘Facial
symmetry and the perception of beauty,’’ Psychon. Bull. Rev.
5, 150–163 (1998).
15. A. P. Møller and R. Thornhill, ‘‘Bilateral symmetry and
sexual selection: a metaanalysis,’’ Am. Nat. 151, 174–192
(1998).
16. C. W. Tyler, ed., Human Symmetry Perception and Its Com-
putational Analysis [VSP (VNU Science Press) BV, Utrecht,
The Netherlands, 1996].
17. D. P. Carmody, C. F. Nodine, and P. J. Locher, ‘‘Global de-
tection of symmetry,’’ Percept. Mot. Skills 45,239–249
(1977).
18. B. Jenkins, ‘‘Component processes in the perception of bi-
laterally symmetric dot textures,’’ J. Exp. Psychol. 9, 258–
269 (1983).
19. B. S. Tjan and Z. Liu, ‘‘Symmetry discrimination of faces,’’
in ARVO Abstracts. Investigative Ophthalmology and Vi-
sual Science, Vol. 39, Suppl. 4 (Association for Research in
Vision and Ophthalmology, Fort Lauderdale, Fla., 1998), p.
s170.
20. Z. Liu and B. S. Tjan, ‘‘Near-bilateral symmetry impedes
symmetry discrimination,’’ in Proceedings of the European
Conference on Visual Perception, Vol. 27 (Suppl.) (Pion Ltd.,
London, 1998), p. 6.
21. D. C. Knill and D. Kersten, ‘‘Ideal perceptual observers for
computation, psychophysics and neural networks,’’ in Pat-
tern Recognition by Man and Machine,R.Watt,ed.,Vol.14
of Vision and Visual Dysfunction, J. Cronly-Dillon, gen. ed.
(MacMillan, London, 1991), Chap. 7.
22. R. A. Eagle and A. Blake, ‘‘Two-dimensional constraints on
three-dimensional structure from motion tasks,’’ Vision Res.
35, 2927–2941 (1995).
23. Z. Liu, D. C. Knill, and D. Kersten, ‘‘Object classification for
human and ideal observers,’’ Vision Res. 35, 549–568
(1995).
24. B. S. Tjan, W. L. Braje, G. E. Legge, and D. Kersten, ‘‘Hu-
man efficiency for recognizing 3-D objects in luminance
noise,’’ Vision Res. 35, 3053–3070 (1995).
Z. Liu and D. Kersten Vol. 20, No. 7/ July 2003 / J. Opt. Soc. Am. A 1339
25. Z. Liu, D. Kersten, and D. C. Knill, ‘‘Dissociating stimu-
lus information from internal representation—a case
study in object recognition,’’ Vision Res. 39, 603–612
(1999).
26. W. Knecht, Z. Liu, and D. Kersten, ‘‘Symmetry detection for
2D projections of 3D wire objects,’’ in ARVO Abstracts: In-
vestigative Ophthalmology, Vol. 34, Suppl. 4 (Association for
Research in Vision and Ophthalmology, Rockville, Md.,
1993), p. s1867.
27. D. M. Green and J. A. Swets, Signal Detection Theory and
Psychophysics (Krieger, Huntington, N.Y., 1974).
28. A. Burgess and H. B. Barlow, ‘‘The precision of numerosity
discrimination in arrays of random dots,’’ Vision Res. 23,
811–820 (1983).
29. H. B. Barlow, ‘‘The efficiency of detecting changes of
density in random dot patterns,’’ Vision Res. 18, 637–650
(1978).
30. A. B. Watson and D. G. Pelli, ‘‘QUEST: a Bayesian adap-
tive psychometric method,’’ Percept. Psychophys. 33,113–
120 (1983).
31. H. B. Barlow and B. C. Reeves, ‘‘The versatility and abso-
lute efficiency of detecting mirror symmetry in random dot
displays,’’ Vision Res. 19, 783–793 (1979).
32. T. Vetter, T. Poggio, and H. H. Bu
¨lthoff, ‘‘The importance of
symmetry and virtual views in three-dimensional object
recognition,’’ Curr. Biol. 4,18–23 (1994).
33. P. Wenderoth, ‘‘The effects on bilateral-symmetry detection
of multiple symmetry, near symmetry, and axis orienta-
tion,’’ Perception 26, 891–904 (1997).
34. R. A. Johnstone, ‘‘Female preference for symmetrical males
as a by-product of selection for mate recognition,’’ Nature
372,172–175 (1994).
35. T. Poggio and T. Vetter, ‘‘Recognition and structure from one
2D model view: observations on prototypes, objects classes
and symmetries,’’Tech. Rep., Artificial Intelligence Memo
1347 and Center for Biological Information Processing Pa-
per 69 (MIT, Cambridge, Mass., 1992).
36. P. Verghese and S. L. Stone, ‘‘Perceived visual speed con-
strained by image segmentation,’’ Nature 381, 161–163
(1996).
37. I. Bu
¨lthoff, H. H. Bu
¨lthoff, and P. Sinha, ‘‘Top-down influ-
ences on stereoscopic depth-perception,’’ Nat. Neurosci. 1,
254–257 (1998).
38. Z. Liu and D. Kersten, ‘‘2D observers for human 3D object
recognition?’’ Special issue on Models of Recognition, Vi-
sion Res. 38, 2507–2519 (1998).
1340 J. Opt. Soc. Am. A/ Vol. 20, No. 7/ July 2003 Z. Liu and D. Kersten
