We consider the problem of designing a near-optimal linear decision tree to classify two given point sets B and W in $\Re^n$. A linear decision tree defines a polyhedral subdivision of space; it is a classifier if no leaf region contains points from both sets. We show hardness results for computing such a classifier with approximately optimal depth or size in polynomial time. In particular, we show that unless NP = ZPP, the depth of a classifier cannot be approximated within any constant factor, and that the total number of nodes cannot be approximated within any fixed polynomial. Our proof uses a simple connection between this problem and graph coloring and uses the result of Feige and Kilian on the inapproximability of the chromatic number. We also study the problem of designing a classifier with a single inequality that involves as few variables as possible and point out certain aspects of the difficulty of this problem.

An automatic target recognition (ATR) classifier is proposed that uses modularly cascaded vector quantizers (VQs) and multilayer perceptrons (MLPs). A dedicated VQ codebook is constructed for each target class at a specific range of aspects, which is trained with the K-means algorithm and a modified learning vector quantization (LVQ) algorithm. Each final codebook is expected to give the lowest mean squared error (MSE) for its correct target class at a given range of aspects. These MSEs are then processed by an array of window MLPs and a target MLP consecutively. In the spatial domain, target recognition rates of 90.3 and 65.3 percent are achieved for moderately and highly cluttered test sets, respectively. Using the wavelet decomposition with an adaptive and independent codebook per sub-band, the VQs alone have produced recognition rates of 98.7 and 69.0 percent on more challenging training and test sets, respectively.

An automatic target recognition classifier is constructed of a set of vector quantizers (VQs) and multilayer perceptrons (MLPs) that are modularly cascaded. A dedicated VQ codebook is constructed for each target at a specific range of aspects. Each codebook is a set of block feature templates that are iteratively adapted to represent a particular target at a specific range of aspects. These templates are further trained by a modified learning vector quantization (LVQ) algorithm that enhances their discriminatory power. The mean squared errors resulting from matching the input image with the block templates in each each codebook are input to an array of window MLPs (WMLPs). Each WMLP is trained to recognize its intended-target at a specific range of aspects. The outputs of the WMLPs are manipulated and fed into a target MLP (TMLP) that produces the final recognition results. A recognition rate of 65.3 percent is achieved on a highly cluttered test set.