Eigenvalue Estimation of Hyperspectral Wishart Covariance Matrices From Limited Number of Samples

摘要

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 estimate all $p$ eigenvalues of a Wishart-distributed sampled covariance matrix (with which an improved covariance can be constructed) when the number of samples $(n)$ is small, $n/p > 1$ and less than a few tens. Our method is based on the Marcenko-Pastur (M-P) law, theory of eigenvalue bounds, and energy conservation. We compute an apparent multiplicity for each sampled eigenvalue and then shift the sampled eigenvalues according the maximum likelihood location (M-P mode). We impose energy conservation in two distinct regions; small eigenvalues and large eigenvalues, where the transition between the two regions is found by solving successive first-order regression equation for the sampled data. The method also improves the condition number of the data (small eigenvalues are shifted upward in values), hence, it is also “regularization,” where the regularization is a multiplicative vector regularization as opposed to the traditional additive scalar regularization where all eigenvalues are shifted upward by the same value.