Principal Component Analysis (PCA) is one of the most popular techniques for dimensionality reduction of multivariate data points
with application areas covering many branches of science. The fundamental technique has been extended in a variety of ways including non-linear variants of PCA, combination of local linear PCA, mixture models for PCA, and probabilistic models for PCA. However, conventional PCA handles the multivariate data in a discrete manner only, i.e., the covariance matrix represents only sample data points rather than higher-order data representations.

In this paper we extend conventional PCA by proposing techniques for constructing the covariance matrix of uniformly sampled continuous regions in parameter space. These regions include polytops defined by convex combinations of sample data, and polyhedral regions defined by the intersection of half spaces. The application of these ideas in practice are simple and shown to be very effective in providing much superior generalization properties than conventional PCA for appearance-based recognition applications.