Scattering Purity and Complexity in Radar Polarimetry

The generalized degree of polarimetric purity is a vital descriptor widely studied and interpreted for electromagnetic wave characterization. It is invariant under the rotation of the reference frame. In this work, we first propose an alternate expression of this purity measure using the mean $(m)$ and standard deviation $(s)$ of the real positive eigenvalues of a Hermitian positive semidefinite matrix. We then use this expression to propose a polarimetric scattering purity and scattering complexity measure. To obtain these expressions, we use certain inequalities on the bounds of the condition number for Hermitian positive definite matrices defined in terms of $m$ and $s$ . The polarimetric scattering purity parameter characterizes the overall polarization structure in the scattered wave. In contrast, the polarimetric scattering complexity parameter describes the mixture of orthogonal polarized pure components in the scattered wave. First, we demonstrate the two proposed measures by analyzing two cases: 1) multiple scattering and 2) a mixture of canonical targets. Then, we utilize full-polarimetric C- and L-band synthetic aperture radar (SAR) data to describe the variation of these measures over various land cover classes. We compare their spatial variations over the ocean surface, built-up areas, and vegetation. We observe notable contrasts in the purity and the complexity parameters over a diverse mixture of targets in the scene. Finally, we critically interpret the variation of the two measures over the temporal scene of rice crop acquired by C-band full-polarimetric SAR data. These analyses affirm the importance of these measures for explicit target characterization. The open-source version of the code is available at https://github.com/Subho07/scattering-purity- and-complexity