Figure 3: A 3D object and a pair of RFs.
The output of a bank of RFs that process images of an object
undergoing rotation in 3D changes with viewpoint. To increase the
invariance of RF-based representations in the face of such
changes, RFs may be paired according to a criterion described in
the Pairs of RFs section; this pairing may be supported
by lateral connections.
The applicability of the particular view-based model described in
[4] to stimuli other than wireframe objects is limited
because it encodes shapes by the coordinates of the wire's vertices.
A natural way to extend that model to deal with realistic shapes is to
provide it with a preprocessing stage consisting of a bank of
receptive fields which would transduce the input images into points
in 
(see Figure 3). As argued in
[8],
the criterion for the choice of RFs in that case would be faithful
representation of similarity: points corresponding to views that
belong to the same object should be situated closer to each other than
points corresponding to views of different objects (cf.
[49]). In other words, the task of a model that
involves view-based representations can be facilitated by making the
similarity relationships in input space reflect as closely as possible
the true similarities between objects prevailing in the world, even
when complete invariance with respect to viewpoint is unattainable.
The rest of this section shows how to construct an RF-based
representation that is relatively stable under consistent changes in
the object's attitude with respect to the observer (see
[7] for details). Consider a rigid object undergoing
rotation around an arbitrary but fixed axis in depth. Pick at random
two patches, 
and 
, on the object's surface, and let
and 
be the corresponding patches
after a small rotation around a fixed axis. Assume that there is a
distant point light source in the direction 
, that the
object's surface is Lambertian, and that the mean albedo at 
and 
is, respectively, 
and 
. Then the
intensities at the two patches before rotation are
where 
and 
are the surface normals at
and 
. Following the rotation, the intensities are
where the assumption of a distant light source was used
to equate 
with 
. Taking the difference
between intensities of the two patches, one obtains
where 
(
) is the angle between 
and
before (after) the
rotation. Because the object was assumed rigid, we have
This means that the magnitude of the vector 
that expresses the difference of orientation
between patches 
and 
is invariant under the rotation.
Thus, if the quantity 
changes following rotation (that is,
if 
), this could be only due to a
change in the orientation of the vector 
with respect to the direction of the
illumination 
.
In the special case when the vector 
is parallel to the axis of rotation, the
angle 
will not change, and, consequently, the difference in
intensity between the two patches, 
, will remain invariant
under rotation. Consider now a set of locally averaged measurements of
intensity such as the one provided by the set of receptive fields in
Figure 3. To obtain an invariant representation of an
object by a subset of those measurements, one should pick pairs
of RFs for which the difference in activity is stable over small
rotations of the object. For any such pair of RFs, and for a fixed
axis of rotation, 
will then remain stable. A snapshot of
activities of the chosen set of RF pairs can be used to represent the
object (for a different object, another set of RF pairs will have to
be picked). As suggested in
[7], the pairing of RFs
necessary for obtaining invariance under the specified conditions can
be supported by lateral connections.
