SPARSE INVERSE COVARIANCE MATRICES AND EFFICIENT MAXIMUM-LIKELIHOOD CLASSIFICATION OF HYPERSPECTRAL DATA

The inverse covariance matrix of a block of Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) hyperspectral data tends towards a spars e, band-diagonal form. This matrix is used in the quadratic form of th e discriminant function of a maximum likelihood classifier (MLC). It c an be written in a formal way as a function of partial and multiple co rrelation coefficients. This allows one to interpret the sparse form o f the inverse covariance matrix to show where the important inter-band information lies in a hyperspectral image. Using these results, MLC i s related to multiple linear regression, and one finds that the noise in each band becomes an important factor. With the understanding this theoretical analysis engenders, three families of approximations to fu ll MLC are developed which capture most of the information it uses but which are much more efficient both to train and to evaluate during cl assification of a whole image. The essence of the new methods is to ap proximate the inverse covariance matrix by an exactly band-diagonal ma trix. A theoretical result about matrices is used to evaluate bounds o n the errors in the quadratic form that these approximations induce.