Block-diagonal representations for covariance-based Anomalous change detectors

We use singular vectors of the whitened cross-covariance ma- trix of two hyper-spectral images and the Golub-Kahan per- mutations in order to obtain equivalent tridiagonal represen- tations of the coefficient matrices for a family of covariance- based quadratic Anomalous Change Detection (ACD) algo- rithms. Due to the nature of the problem these tridiagonal matrices have block-diagonal structure, which we exploit to derive analytical expressions for the eigenvalues of the coefficient matrices in terms of the singular values of the whitened cross-covariance matrix. The block-diagonal struc- ture of the matrices of the RX, Chronochrome, symmetrized Chronochrome, Whitened Total Least Squares, Hyperbolic and Subpixel Hyperbolic Anomalous change detectors are re- vealed by the white singular value decomposition and Golub- Kahan transformations. Similarities and differences in the properties of these change detectors are illuminated by their eigenvalue spectra.