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 sparse, band-diagonal form. This matrix is used in the quadratic form of the discriminant function of a maximum likelihood classifier (MLC). It can be written in a formal way as a function of partial and multiple correlation coefficients. This allows one to interpret the sparse form of the inverse covariance matrix to show where the important inter-band information lies in a hyperspectral image. Using these results, MLC is 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 full MLC are developed which capture most of the information it uses but which are much more efficient both to train and to evaluate during classification of a whole image. The essence of the new methods is to approximate the inverse covariance matrix by an exactly band-diagonal matrix. A theoretical result about matrices is used to evaluate bounds on the errors in the quadratic form that these approximations induce.