Voltage Reflection Matrices Based Decompositions

This type of decomposition approaches is based on the evaluation of the scattering matrix. The basic idea behind these approaches is to express the [S]-matrix as a complex sum of elementary scattering matrices, which can be assigned to certain deterministic scattering mechanisms. The Pauli basis mentioned previously (3.14) , is such an approach and the vectorization of the [S]-matrix can be interpreted as the description of a scatterer in terms of deterministic scattering mechanisms, expressed by the Pauli matrices.

$$[S] = \left[\begin{array}{cc}a+b & c -id c+id & a-b \end{array}... ...\end{array} \right]+ d\left[\begin{array}{cc}0 & -i i & 0 \end{array} \right]$$ (4.1)

where a, b, c and d are proportional to the complex elements of the scattering vector $\vec{k}_P$, of which the $4^{th}$ matrix is, due to the reciprocity theorem, not relevant for the backscattering case as discussed before. Depending on the polarization basis, physical interpretations of the basic scattering mechanisms can be given. Krogager, introduced a roll invariant decomposition interpretation where the actual scattering matrix can be regarded as a superposition of the basic scattering mechanisms isotropic surface, right wound helix and left wound helix [Krogager90]. For the case of a left wound helix, the roll-invariant formulation can be expressed in the circular basis as follows,

$$\left[ S \right]_{RL} = e^{j\phi} \left\{ k_{s}e^{j\phi_s} \left[... ...{h} \left[\begin{array}{cc} e^{j2\theta} & 0 0 & 0\end{array}\right] \right\}$$ (4.2)

For the sphere, diplane, helix approach is the components are not necessarily mutually orthogonal, but yield some other advantages. Due to the roll-invariant properties and the continuous transition of parameters in terms of invariant mechanisms, as opposed to the orthogonal Pauli-components of which two represent the same mechanism. Thus, 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. A unique advantage of the SDH representation is the property that three different elementary scatterers (mechanisms) show up in only one out of three components, although the diplane and helix components are not independent (orthogonal), which means that in the presence of both a diplane and a helix, the decomposition will in general not extract the actual strengths of the respective reflectors.

Decompositions of the complex scattering matrix are particularly suited for coherent processing and cases where the scattering is due to a few dominating scattering centers. For the linear { $\vec{e}_h,\vec{e}_v$} basis, the Pauli matrices can be interpreted in a straight forward manner, as shown in Tab. 4.1

Table: Pauli matrices and their interpretation in the { $\vec{e}_h,\vec{e}_{v}$} polarisation basis

Pauli matrix	scattering type	interpretation
$\left[\begin{array}{cc} 1 & 0 \newline 0 & 1 \end{array} \right]$	odd-bounce	surface, sphere, cornerreflectors
$\left[\begin{array}{cc} 1 & 0 \newline 0 & -1 \end{array} \right]$	even-bounce	dihedral
$\left[\begin{array}{cc} 0 & 1 \newline 1 & 0 \end{array} \right]$	even-bounce $\pi/4$ tilted	$\pi/4$ tilted dihedral
$\left[\begin{array}{cc} 0 & -i\newline i & 0 \end{array} \right]$	cross-polariser	not existent for backscattering

Analogue to the Pauli decomposition, we can derive an interpretation for the Sphere/Diplane/Helix decomposition as shown in Tab. 4.2

Table: SDH decomposition matrices and their interpretation in the { $\vec{e}_h,\vec{e}_{v}$} polarisation basis

matrix	scattering type	interpretation
$e^{j\phi_s} \left[\begin{array}{cc} 0 & j\newline j & 0\end{array} \right]$	odd-bounce	sphere, surface, cornerreflectors
$\hspace{-1.2em} \left[\begin{array}{cc} e^{j2\theta} & 0\newline 0 & -e^{-j2\theta}\end{array} \right]$	even-bounce	(tilted) dihedral
$\left[\begin{array}{cc} e^{j2\theta} & 0\newline 0 & 0\end{array} \right]$	helix	helix like scattering

Apart from the physical importance these particular mechanisms have in Radar imagery, the Pauli decomposition has the further advantage that the scattering mechanisms are orthogonal and so their separation is possible, even in the case of second order statistics where noise and depolarisation effects can be introduced.

Using the Pauli decomposition we can derive a three color composite of a fully polarimetric data set (see Fig. 4.2) for visual classification purposes.

Figure 4.2: RGB representation of a Pauli decomposition of an full polarimetric E-SAR data set from the test site Oberpfaffenhofen, Germany, blue: $S_{hh} + S_{vv}$, red: $S_{hh} - S_{vv}$, green: $2S_{x}$

From Fig. 4.2 we can clearly distinguish structures with even-bounce characteristic (buildings), areas with odd-bounce characteristics and finally areas where the scattering mechanism of the 3rd Pauli matrix is dominant.

Even though this three color composite yields helpful information for the interpretation of fully polarimetric SAR data, it is obvious that this approach is not sufficient for the interpretation of all areas in the scene. Since it is only capable to distinguish between 3 ideal scattering mechanism and their combination some ambiguous areas remain and a interpretation of the data is therefore difficult.

Polar Decomposition of the Scattering Matrix

A third approach using a polar decomposition of the scattering matrix was developped by Laura Carrea at the Chemnitz University of Technology. While the afore mentioned decompositions of the [S]-matrix are additive, this approach proposes a multiplicative decomposition of the scattering matrix. The following passage is taken from Laura's IGARSS paper [Carrea], thanks Laura ;o). As mentioned before the decomositions described above are performed as a sort of sum of more elementary quantities. An alternative idea is to consider a multiplicative decomposition which could be useful in order to try to reduce the coherent speckle noise which is multiplicative. In the coherent case, such a decomposition could be the polar decomposition of the scattering matrix.

The polar decomposition is based on a mathematical theorem [Fano] asserting that any non-singular operator is uniquely expressible in the form (the polar form):

$$S = U H,$$ (4.3)

where H is a Hermitian operator and U is a unitary one, such that:

$$H = H^{\dagger} , \qquad U^{-1}=U^{\dagger}.$$ (4.4)

Choosing a basis, operators are represented by numbers. The scattering matrix is the representation of the S operator, for example, in HV basis:

$$S=\left[ \begin{array}{cc} S_{HH} & S_{HV} S_{VH} & S_{VV} \end{array}\right],$$ (4.5)

where the S elements are complex numbers. It is always possible to "normalize" the S matrix, finding a matrix $K$ such that:

$$S = K S', \qquad \det S' = 1,$$ (4.6)

where:

$$K=\left[ \begin{array}{cc} \sqrt{\det S} & 0 0 & \sqrt{\det S} \end{array}\right].$$ (4.7)

The scattering matrix (4.3) becomes:

$$S = K U H,$$ (4.8)

where $\det (UH) = 1$. The Hermitian and the unitary matrices, in such a decomposition, result:

$$H = \sqrt{{S'}^{\dagger}{S'}}, \qquad U = S' H^{-1}$$ (4.9)

The elements of the two matrices are in function of the scattering matrix components.

The Boost and Rotation Transformations

The scattering mechanism can be interpreted as two particular kinds of transformations on the input wave: a boost $H$ and a rotation $U$. The action of such transformations is independent of the basis that is chosen to represent the operators, because $H$ and $U$ are still Hermitian and unitary whatever representation has been chosen.

Consequently, a peculiarity of such a decomposition is its independence of the polarization basis. As a result, the action of the transformation can be geometrically represented.

To have an idea about the action of a boost transformation, a pure boost [Rindler] in the z direction is considered. The boost transforms the polarization and the intensity of the wave. The visualization of the action of the boost on a polarization is realized on the Poincaré sphere. The point P moves on the Poincaré sphere along the direction of the boost, z in this case, due to the increment in the module of the boost parameter (Fig. 4.3). A boost is always a transformation along a specific direction.

The visualization of the action of the boost on the intensity could be performed in a three dimensional space, because we have been dealing only with monochromatic waves. For such waves in the four dimensional Stokes space, the relation $g_0^2-g_1^2-g_2^2-g_3^2 = 0$ holds; $g_0$, $g_1$, $g_2$, $g_3$ are the Stokes parameters. Neglecting all the points in which $g_0^2-g_1^2-g_2^2-g_3^2 \not= 0$ holds, we can collect all the points such that $g_0^2-g_1^2-g_2^2-g_3^2 = 0$ and build a three dimensional space with them.

Figure 4.3: The effect of a boost on the Poincaré sphere.

In such a space the action of a particular boost on all the points of the Poincaré sphere is shown in Fig. 4.4.

Figure 4.4: The visualization of a boost for completely polarized waves.

All the points coming out from the transformation build an ellipsoid up, and the polarization, laying on the direction of the axis of the boost, corresponds to the maximum intensity.

A unitary transformation can be visualized on the Poincaré sphere, because it doesn't influence the intensity of the wave. It results in a transformation around a direction, as shown in Fig. 4.5, for the z direction.

Figure 4.5: The effect of a rotation on the Poincaré sphere.

To any rotation is associated a quantity ("spin matrix" or "rotation operator") [Misner]:

$$U =\cos\frac{\theta}{2}-i\sin\frac{\theta}{2}(\sigma_1 n_x +\sigma_2 n_y+\sigma_3 n_z),$$ (4.10)

where $\sigma_1$, $\sigma_2$, $\sigma_3$ are the Pauli matrices. Writing explicitly the Pauli matrices, the U matrix results:

$$U = \left[ \begin{array}{cc} \cos\frac{\theta}{2}-in_{x}\sin\frac... ...ta}{2} & \cos\frac{\theta}{2}+in_{x}\sin\frac{\theta}{2} \end{array}\right],$$ (4.11)

where $\theta$ is the angle of rotation and $\hat{n}=(n_x, n_y, n_z)$ is the axis of rotation.

In the same way, it is possible to associate a quantity to any boost, which results:

$$H = \left[ \begin{array}{cc} \cosh\frac{\alpha}{2}+m_{x}\sinh\fra... ...a}{2} & \cosh\frac{\alpha}{2}-m_{x}\sinh\frac{\alpha}{2} \end{array}\right],$$ (4.12)

where $\alpha$ is the so called "boost rapidity" and $\hat{m}=(m_x, m_y, m_z)$ is the axis of the boost. Consequently, the scattering matrix, expressed in (4.8), is now expressed in function of 8 independent parameters: $\alpha$, $\hat{m}$, the boost parameters, $\theta$, $\hat{n}$, the rotation parameters, $\det S$.

The Polar Decomposition for Symmetric Scattering Matrix

The discussion is now restricted to a very important case for polarimetric radar, in which monostatic measurements predominate. If the target and the propagation medium have reciprocal constitutive relations, the reciprocity theorem holds. The consequence of it is that the scattering matrix is symmetric:

$$S_{ij}=S_{ji} \quad (i,j = H,V).$$ (4.13)

The symmetry creates a limitation on the parameters and not all the possible boosts and rotations are allowed. The imposition of the reciprocity condition on the scattering matrix, as a function of the boost and rotation parameters and of $\det S$, gives:

$$n_z=0$$ (4.14)

$$\tan\frac{\theta}{2}= -\frac{m_{z}}{n_{x}m_{y}-n_{y}m_{x}}.$$ (4.15)

The first condition (4.14) is on the rotation axis: the rotation axis must lay on the $n_z=0$ plane. The second one (4.15) is a relation between boost and rotation parameters.

It is necessary to make a choice: we choose to let the boost free and we impose both the symmetry conditions on the unitary matrix. It will be the unitary matrix which will force the symmetry. In such a case, the rotation angle given by (4.15) will produce always a rotation as shown in Fig. 4.6.

Figure 4.6: Rotation satisfying the reciprocity condition

The rotated vector has the third component in opposition to the starting vector.

As stated before, the axis of the boost corresponds to the maximum intensity. So, if the reciprocity theorem holds, and we send the polarization correspondent to the axis of the boost, we receive the polarization (canonical polarization) correspondent to the maximum intensity. In fact, the rotation under the reciprocity conditions moves the axis of the boost to the mirrored one respect to the $n_z=0$ plane (equatorial plane). After the imposition of the symmetry condition, only 5 independent parameters are left:

• $\psi$ and $\chi$, the spherical coordinates of the unit vector $\hat{m}$,
• $\alpha$, the boost rapidity,
• $\phi$, the polar coordinate of the unit vector $\hat{n}=(n_x,n_y,0)$,
• k, the modulo of $\det S$.