Fast Matrix Based Computation of Eigenvalues and the Loewner Order in PolSAR Data

Abstract

We describe the calculation of eigenvalues of $2 \times 2$ or $3 \times 3$ Hermitian matrices as used in the analysis of multilook polarimetric synthetic aperture radar (SAR) data. The eigenvalues are calculated as the roots of quadratic or cubic equations. We also describe the pivot-based calculation of the Loewner order for the partial ordering of differences between such matrices. The methods are well suited for fast matrix-oriented computer implementation, and the speed-up over simpler calculations based on built-in eigenproblem solvers is enormous.