Estimation of hyperspectral covariance matrices

Estimation of covariance matrices is a fundamental step in hyperspectral remote sensing where most detection algorithms make use of the covariance matrix in whitening procedures. We present a simple method to improve the estimation of the eigenvalues of a sample covariance matrix. With the improved eigenvalues we construct an improved covariance matrix. Our method is based on the Marcenko-Pastur law, theory of eigenvalue bounds, and energy conservation. Our objective is to add a new method for estimating the eigenvalues of Wishart covariance matrices in scenarios where the sample size is small. Our method is simple, practical and easy to implement (it consists of a multiplication of 3 matrices). We did our study with extensive simulations and a few examples of hyperspectral remote sensing data that were measured in the long infrared wavelength region (8-12μm). We show examples of the improved eigenvalues over the sampled eigenvalues. We choose the following five figures-of-merit for evaluating our method as ratios of properties between sampled data and our solution: (i) residual (rms) that gives the improvement of the solution over the sampled data with respect to the population eigenvalues, (ii) area under the scree-plot, (iii) condition number that gives the improvement in the stability (regularization), (iv) a distance-measure that gives the average statistical improvement between the improved and the sampled eigenvalues, and (v) Kullback-Leibler distance. We show hyperspectral matched-filter detection performance (ROC curves) for TELOPS data where we use our improved covariance matrix. We compare the improved ROC to the one that are obtained with sampled (data) covariance matrix.