# `iS-Omega` (improved S-\(\Omega\) decomposition)

This functionality computes the scattering powers for compact polarimetric SAR data. This is an improved decomposition technique based on Stokes vector(S) and the polarized power fraction (\(\Omega\)). The required input and the computed output are as follows:

    input : input_C2_folder, window_size, tau, psi, chi
    output: Ps_iSOmega.bin, Pd_iSOmega.bin,Pv_iSOmega.bin

The stokes paramters can be written in terms of the covariance matrx (C2) elements as follows:

\[\begin{split}S_0=\text{C11+C22};\qquad{}S_1=\text{C11-C22};\\ S_2=\text{C12+C21};\qquad{}S_3=\pm\text{j(C12-C21)}\end{split}\]

Then, the parameters Same-sense Circular (\(\text{SC}\)) and Opposite-sense Circular (\(\text{OC}\)) can be expressed as follows:

\[\text{SC}=\frac{S_0-S_3}{2};\qquad{}\text{OC}=\frac{S_0+S_3}{2};\]

Now, based on the ratio of \(\text{SC}\) and \(\text{OC}\) the decomposition powers can be derived as given below. Further details can be found in [[7]](#7)

\[\begin{split}\text{SC/OC}<1;\qquad{}\qquad{}\qquad{}\qquad{}\qquad{}\qquad{}\qquad{}\text{SC/OC}>1\\P_s=\Omega\left(S_{0}-\left(1-\Omega\right)\text{SC}\right);\qquad{}\qquad{}\qquad{}P_s=\Omega\left(1-\Omega\right)\text{OC}\\P_d=\Omega\left(1-\Omega\right)\text{SC};\qquad{}\qquad{}\qquad{}P_d=\Omega\left(S_{r0}-\left(1-\Omega\right)\text{OC}\right) \\P_v=S_{0}\left(1-\Omega\right)\qquad{}\qquad{}\qquad{}\qquad{}\qquad{}P_v=S_{0}\left(1-\Omega\right)\end{split}\]