Bayes error estimation has long been an interesting topic in the pattern recognition area. For data with small dimensionality and large sample size, it is easy to estimate the upper and lower bounds of the Bayes error using global metrics in nonparametric classifiers. However, for the SAR (Synthetic Aperture Radar) data, the data dimensionality is very large (n=64x64=4,096) and the sample size is very small (Nz1,OOO). No conventional methods are able to estimate the Bayes error of the SAR data. Our solution to this problem is to hlly use the property that the intrinsic dimensionality of the SAR data is much smaller than its data dimensionality. Another important issue related to the SAR data is the design of parametric classifiers. For such large n and small N, computing the inverse of the covariance matrix is very difficult and time-consuming. Many classifier designers have to approximate the covariance matrix by a diagonal one. This motivates us to figure out a better way of designing a quadratic classifier which can give a much better performance and run in real time. In this thesis, We have derived an algorithm to estimate local metrics. This algorithm has been tested on many artificial and real data sets and has been proven to work very well. We have also developed a new way, called "window-shifting", for measuring distances between samples. This method can boost the performance of quadratic classifiers over the current ones, and it is very suitable for real-time applications.
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