To obtain an `appearance-based' representation, one must first transform the image into a low-dimensional coordinate system that preserves the general perceptual quality of the target object's image. This transformation is necessary in order to address the `curse of dimensionality'. The raw image data has so many degrees of freedom that it would require millions of examples to learn the range of appearances directly.
Typical methods of dimensionality reduction include Karhunen-Loève transform (KLT) (also called Principal Components Analysis (PCA)) or the Ritz approximation (also called `example-based representation'). Other dimensionality reduction methods are sometimes also employed, including sparse filter representations (e.g., Gabor Jets, Wavelet transforms), feature histograms, independent components analysis, and so forth.
These methods have in common the property that they allow efficient characterization of a low-dimensional subspace with the overall space of raw image measurements. Once a low-dimensional representation of the target class (face, eye, hand, etc.) has been obtained, standard statistical parameter estimation methods can be used to learn the range of appearance that the target exhibits in the new, low-dimensional coordinate system. Because of the lower dimensionality, relatively few examples are required to obtain a useful estimate of either the PDF or the inter-class discriminant function.
An important variation on this methodology is discriminative models, which attempt to model the differences between classes rather than the classes themselves. Such models can often be learned more efficiently and accurately than when directly modeling the PDF. A simple linear example of such a difference feature is the Fisher discriminant. One can also employ discriminant classifiers such as Support Vector Machines (SVM) which attempt to maximize the margin between classes.