Refinement and generalization of the extension method of covariance matrix inversion by regularization for spectral filtering optimization

Many spectral signature detection algorithms depend on numerically inverting covariance matrices. Hyperspectral data rarely span the full band space because of factors such as sensor noise, numerical round-off, sparse sampling, and band correlation inherent in the data or introduced by data processing. Processing the full order of the covariance matrix without regard to its useful rank leads to reduced detection performance. It was previously shown that the performance of inverse-covariance based detection algorithms can be improved by regularizing the covariance matrix inversion through extension of an optimally chosen eigenvalue. The extension method provides a robust way to optimize signal to clutter ratio (SCR) on data collected with a detector of uniform gain. The method of trusted eigenvalue extension has now been applied to data collected with a sensor with multiple gain regions. Multiple gain regions are used on wide spectral range sensors such as HYDICE and complicate the inversion of the covariance matrix over the full range of spectral bands. Further optimization of the trusted eigenvalue is presented and compared against traditional regularization methods. Since the extension method is particularly intended for sparsely sampled data with high dimensionality, a comparison is presented between the extension method and band coaddition.