Sphere/Diplane/Helix Decomposition

This decomposition is a bit more challenging due to the basis transformation. On the other hand it holds some advantages in terms of invariant mechanisms, as opposed to the orthogonal Pauli-components of which two represent the same mechanism. In spite of the mathematical advantage and elegancy of orthogonal sets, a practical disadvantage of the Pauli representation of polarimetric radar data is that a double-bounce reflector shows up in two different components, one representing an unrotated diplane, another representing a 45$^o$ tilted diplane. Please note, that the diplane and helix components are not independent (orthogonal), which means that in the presence of both a diplane and a helix. Therfore, the decomposition will in general not extract the actual strengths of the respective reflectors. In the following an example for the case of a left wound helix is described. If your data set is provided in the linear HV-basis you first have to transform the data into the circular RL- basis (R = right hand circular, L = left hand circular)

$$S_{RR} = i S_X+\frac{S_{HH}+S_{VV}}{2}$$ (B.3)

$$S_{LL} = i S_X+\frac{S_{HH}-S_{VV}}{2}$$

$$S_{RL} = i \frac{S_{HH}+S_{VV}}{2}$$

The surface component is now given by the absolute value of the right left circular component

$$S=\vert S_{RL}\vert$$ (B.4)

for the two others we have to distinguish between the areas where the absolute value of the right right circular component is bigger than the absolute value of the left left circular component (area A), and for areas where the absolute value of the right right circular component is lower than the absolute value of the left left circular component (area B). The diplane component is given by the absolute values of the right right circular component from area A and from the absolute values of the left left circular component from area B. (i.e. for all pixels from area A you use the first criteria for all pixel belonging to area B you use the second) The Helix component is given for pixels belonging to area A by the difference of the absolute values of the right right circular component and the absolute values of the left left circular component and for all pixels belonging to area B difference of the absolute of the left left circular component and the absolute values of the right right circular component. then, like above, you calculate from each of the 3 components the absolute value and convert it into byte array (8 bit per pixel), assign to each of the layers a color (usually surface = red , diplane = green, and helix = blue and combine them in a 3 color composite with a program or some image manipulation software. In order to improve the contrast you might want so scale the images not linear. I usually use a square root like scaling or scaling with an exponent of 0.7. For details look in the source code of the program decomp.pro on http://epsilon.nought.de.