On the use of covariance and correlation matrices in hyperspectral detection

Many standard multivariate detection algorithms used in hyperspectral image analysis (e.g., Mahalanobis distance, matched filter) rely on the inverse covariance matrix, C^-1. The inverse covariance matrix has a clear geometrical interpretation as a “whitening” operator that transforms the original measurement coordinates such that the background data distribution is uncorrelated with equal variance along all axes. Occasionally, the inverse correlation matrix, R^-1, is used instead-such as in constrained energy minimization (CEM) or R-RX [1]-usually with the justification of improved performance on a particular data set. To our knowledge there is no clear theoretical or geometrical justification for substituting R^-1 for C^-1, and no theory has been developed to explain when R^-1 would be a better choice than C^-1. In this paper we investigate the differences in performance of anomaly detection (i.e., Mahalanobis distance) using both the covariance and correlation matrices as a function of three parameters: the magnitude of the background mean, the magnitude of the target vector, and the cosine angle between them. Detection performance is measured by comparing the area under the receiver operating characteristic (ROC) curves for the two methods, assuming normally distributed hyperspectral measurements and an additive signal-plus-noise model. We show that using R^-1 can offer improved performance; however, this only occurs at a relatively small fraction of the parameter space and the potential performance gain is modest, whereas the potential performance loss can be catastrophic. Use of R^-1 in the anomaly detector only seems justified if some knowledge of a particular problem restricts the parameter space to regions of improved performance. In this paper we also analyze the performance difference of matched filter (MF) detection using the covariance matrix (standard MF) a- d the correlation matrix (CEM), and show theoretically that the CEM algorithm can never improve detection performance of the matched filter.