We have seen that a pattern of activities of RFs can be made to reflect properties of the viewed object, discounting to a certain extent factors that are irrelevant to recognition. To carry out the recognition step itself, the RF activities have to be combined into a decision criterion. The goal of this stage may be, for example, to compute for each object known to the system a number between 0 and 1 that would correspond to the system's confidence as to its presence in the input. A general approach to this problem, valid in vision as well as in other domains, is to apply a standard technique for learning from examples, or, equivalently, function approximation [Poggio, 1990]. A method that is particularly suitable in the present context is approximation by radial basis functions (RBFs).
The computational reason for the feasibility of this approach is
basically the smoothness of the manifold formed by the different views
of the same object in the space of views of all possible objects
[Poggio and Edelman, 1990].
An RBF approximation module effectively constructs the manifold by
computing its ``height'' over the input measurement space as a linear
combination of the contributions of the data points (see
Figure 4). The contributions are determined by placing a
kernel (that is, a basis function) at selected points [IMAGE ],
so that
and by computing the weights [IMAGE ] that minimize the approximation error [IMAGE ] accumulated over all the data [IMAGE ]. A good choice for the shape of the kernel [IMAGE ] is the Gaussian [IMAGE ], because of the universal approximation properties of linear superpositions of Gaussians [Hartman et al., 1990], because it can be derived from a regularized solution to the approximation problem, as well as for other reasons [Poggio and Girosi, 1990]. The Gaussian kernel is especially relevant in the context of visual modeling, because it makes it possible to interpret equation 7 as a linear combination of products of activities of 2D image-based Gaussian RFs. In other words, 2D RFs can be combined multiplicatively to form the multidimensional Gaussians that serve as the basis functions in the expansion [Poggio and Edelman, 1990].
[IMAGE ]
Figure: Standard techniques for function approximation can be used
to construct a characteristic function for a given object from a
collection of its views (see section 3).
Here, radial basis function approximation in the space of all views
of an object is carried out by forming a weighted sum of responses
of RFs tuned to some of the views. The graded response of the
resulting module defines a RF in the shape space (the response
grows with increased similarity between the input and the object on
which the module has been trained). This property of the RBF
recognizer is used in section 4 to
construct a categorization mechanism for novel objects.
[IMAGE ]
Figure: A strange object (a cameleopard, left) and two more
familiar ones (center, right). See
section 4.