Theories of Object Identification 201factors such as differences in lightness, color, texture,
binocular disparity, and other low-level sensory features.
A green square on a yellow ground is seen as having the
same shape as a blue square on a red ground, for example,
even though they will require separate templates. A gen-
eral square template would thus have to be the disjunction
of a huge number of very specific square templates.2.Spatial transformations:Shape is largely invariant over the
similarity transformations—translations, rotations, dila-
tions, reflections, and their various combinations (Palmer,
1989)—yet comparing template representations that differ
by such transformations will not generally produce good
matches. Three ways to solve this problem for template
representations are replication, interpolation, and normal-
ization.Replicationrefers to the strategy of constructing a
different template for each distinct shape in each position,
orientation, size, and sense (reflection), as the visual sys-
tem does for receptive field structures in area V1. This is
feasible only if the set of template shapes is very small,
however.Interpolationis a way of reducing the number of
templates by including processes that can construct inter-
mediate representations between a pair of stored templates,
thus reducing the number of templates, but at the expense
of increasing the complexity of the matching process.
Normalization postulates processes that transform (or
normalize) the input image into a canonical position, orien-
tation, size, and sense prior to being matched against the
templates so that these factors do not matter. How to nor-
malize effectively then becomes a further problem.
3.Part structure:People perceive most objects as having a
complex hierarchical structure of parts (see this chapter’s
section entitled “Perceptual Organization”), but templates
have just two levels: the whole template and the atomic
elements (receptors) that are associated within the tem-
plate. This means that standard templates cannot be
matched on a partwise basis, as appears to be required
when an object is partly occluded.
4.Three dimensionality:Templates are intrinsically two-
dimensional, whereas most objects are three-dimensional.
There are just two solutions to this problem. One is to
make the internal templates three-dimensional, like the
objects themselves, but that means that three-dimensional
templates would have to be constructed by some complex
process that integrates many different two-dimensional
views into a single three-dimensional representation
(e.g., Lowe, 1985). The other solution is to make the in-
ternal representations of three-dimensional objects two-
dimensional by representing two-dimensional projections
of their shapes. This approach has the further problem that
different views of the same object would then fail to match
any single template. Solving this problem by replication
requires different templates for each distinct perspective
view, necessitating hundreds or thousands of templates for
complex three-dimensional objects. Solving it by inter-
polation requires additional processes that generate inter-
mediate views from two stored views (e.g., Poggio &
Edelman, 1990; Ullman & Basri, 1991). Normalization is
not feasible because a single two-dimensional view sim-
ply does not contain enough information to specify most
objects from some other viewpoint.Feature ListsA more intuitively appealing class of shape representation is
feature lists:symbolic descriptions consisting of a simple set
of attributes. A square, for example, might be represented by
the following set of discrete features:is-closed, has-four-
sides, has-four-right-angles, is-vertically-symmetrical, is-
horizontally-symmetrical, etc. The degree of similarity
between an object shape and that of a stored category can then
be measured by the degree of correspondence between the
two feature sets.
In general, two types of features have been used for repre-
senting shape: global properties,such as symmetry, closure,
and connectedness, and local parts,such as containing a
straight line, a curved line, or an acute angle. Both types of
properties can be represented either as binary features(e.g., a
given symmetry being either present or absent) or ascontin-
uous dimensions(e.g., the degree to which a given symmetry
is present). Most classical feature representations are of the
discrete, binary sort (e.g., Gibson, 1969), but ones based on
continuous, multidimensional features have also been pro-
posed (e.g., Massaro & Hary, 1986).
One reason for the popularity of feature representations is
that they do not fall prey to many of the objections that so crip-
ple template theories. Feature representations can solve the
problem of concreteness simply by postulating features that
are already abstract and symbolic. The feature list suggested
for a square at the beginning of this section, for example,
made no reference to its color, texture, position, or size. It is an
abstract, symbolic description of all kinds of squares. Features
also seem able to solve the problem of part structure simply by
including the different parts of an object in the feature list, as
in the previously mentioned feature list for squares. Similarly,
a feature representation of a human body might include
the following part-based features:having-a-head, having-a-
torso, having-two-legs,and so forth. The features of a head
would likewise include having two-eyes, having-a-nose,
having-a-mouth,etc. Features also seem capable of solving