Four-Component Scattering Power Decomposition of Remainder Coherency Matrices Constrained for Nonnegative Eigenvalues

The motivation of this letter is to resolve the nonnegative eigenvalue constraint (NNEC) problem of four-component decomposition (FCD). It is analyzed that the NNEC is an essential requirement for remainder coherency matrices in the FCD, however the measured polarimetric synthetic aperture radar (POLSAR) data experiment shows there exits the NNEC problem that some remainder coherency matrices of the FCD do not satisfy the NNEC, which means these matrices are not positive semi-definite. In addition, it is analyzed that the scheme using the nonnegative eigenvalue decomposition (NNED) for three-component decomposition (TCD) cannot be directly extended to the FCD to overcome the NNEC problem, so a scheme using the NNED for the FCD is proposed as follow. From matrix theory, we draw a conclusion that if the last remainder coherency matrix satisfies the NNEC, then all remainder coherency matrices also satisfy the NNEC; we successively analyze that the NNEC problem of the last remainder coherency matrices results from the overestimation of scattering powers. Then a shrinkage coefficient is used to depress all possible overestimations of scattering powers, and the overestimation case with the minimum remainder power is chosen to resolve the NNEC problem. Moreover, we have simplified the solution to NNED, which is used to calculate the shrinkage coefficient. The measured POLSAR data experiment shows that the proposed FCD can further enhance double-bounce scattering and depress volume scattering for urban areas.