Structured Gaussian Components for Hyperspectral Image Classification

The large number of bands in hyperspectral images leads to a large number of parameters to estimate. It has been argued in the literature that class-conditional distributions of hyperspectral images are non-Gaussian; thus, multiple components might be needed to describe the classes accurately. In this paper, we propose to represent the Gaussian components in the classifier with a smaller number of parameters by allowing some or all component distributions to share eigenstructure by decomposing the covariance matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Sigmaₖ</tex> of each of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> components into a product of three parameters, namely: 1) scalar <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">lambdaₖ</tex> measuring volume; 2) diagonal matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Aₖ</tex> of normalized eigenvalues measuring shape; and 3) matrix of eigenvectors <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Dₖ</tex> measuring orientation. Any combination of these parameters can be common for any subset of the covariance matrices, allowing a flexible set of possible configurations that can be used to approximate the true covariance using fewer parameters. A simple bottom-up algorithm for searching for possible parameter-sharing models is developed. Experiments on three data sets were performed: one concerned with woodland classification and two on urban mapping. Results from these experiments indicate that the method outperforms conventional classifiers and performs comparably with state-of-the-art classifiers such as support vector machines.