## 4 PePS-based feature set  
The previous section has shown that the meso-scale characteristics of ships could be described by a particular distribution of intense and strongly-polarized scatterers that may make possible the discrimination of different geometries. This section makes a step further and evaluates the possibility to exploit such information for building a _feature set_ useful for ship classification. In order to fix an unequivocally way for isolating PePS with measurable parameters, a definition is required. According to the experience gained with the scattering studies, PePS can be considered those scatterers presenting a RCS 10 dB higher than the surrounding scatterers and a stable polarimetric behavior within a solid angle of at least \(\pi/3\) steradian 2. Three issues are important in this definition: 1) stable polarimetric behavior for _PePS_\(n\) means to take the same value of \(p_{n}^{j}\) within the selected aspect angle (see Equations 4 and 5); 2) _PePS_ 3D positions, \((x_{n},y_{n},z_{n})\), are expressed with respect to a local coordinate system defined at the target and 3) the threshold in the angular aperture is selected by the response of typical canonical scatterers, like trihedrals and dihedrals. This implies that PePS locate those structures with the closest scattering behavior to canonical targets. Footnote 2: This is the solid angle subtended by a cone of 30\({}^{o}\) of aperture. With the previous definition, the PePS-based feature set can be mathematically formulated by [8]  
\[\Theta^{j}=\{\Theta_{n}^{j}\}\quad for\quad 1<n<N_{PePS} \tag{2}\]  
where \(N_{PePS}\) indicates the number of PePS used to describe the geometry of target \(j\) and  
\[\Theta_{n}^{j}=\{x_{n},y_{n},z_{n},p_{n}\}^{j} \tag{3}\]  
gathers the tri-dimensional location and polarimetric behavior of each guide scatter. On the one hand, location information is provided according to the local coordinate system defined in Fig. 7, which is centered at the center of mass of the target. On the other hand, polarimetric properties are provided by means of the Pauli mechanisms defined for the scatter \(n\) as  
**Figure 7.** Local coordinate system used to define PePS within targets. The center is located at the center of mass. \[ \begin{array}{lcl}\mbox{\bf Trihedral}&<p_{n}^{j}>=0&if&|p_{n,0}^{j}|>|p_{n,1}^{j} |,|p_{n,2}^{j}|\\ \mbox{\bf Dihedral}&<p_{n}^{j}>=1&if&|p_{n,1}^{j}|>|p_{n,0}^{j}|,|p_{n,2}^{j}|\\ \mbox{\bf Anti-symetric}&<p_{n}^{j}>=2&if&|p_{n,2}^{j}|>|p_{n,0}^{j}|,|p_{n,1}^{j}| \end{array}  \tag{4}\]  
where \(|p_{n,xx}^{j}|=p_{n,xx}^{j}/max\left\{p_{n,0}^{j},p_{n,1}^{j},p_{n,2}^{j}\right\}\) define the normalized weight with  
\[ \begin{array}{l}p_{n,0}^{j}=\frac{1}{\sqrt{2}}\frac{S_{hh}^{n}+S_{vv}^{n}}{ \left.p_{n,1}^{j}=\frac{1}{\sqrt{2}}\frac{S_{hh}^{n}-S_{vv}^{n}}{2}\right.}\\ p_{n,2}^{j}=\frac{1}{\sqrt{2}}\frac{S_{hv}^{n}}{2}\end{array}  \tag{5}\]  
being  
\[[S]^{n}=\left[ \begin{array}{cc}S_{hh}&S_{hv}\\ S_{hv}&S_{vv}\end{array} \right]^{n} \tag{6}\]  
the mono-static master scattering matrix. Note that the usage of a reference positioning system different to the SAR azimuth/slant-range one needs from incidence and sensor-to-target relative orientation angles in order to make the coordinate system transformation. The former is derived from system design whereas the latter from any of the currently available methods for estimating vessel bearings [9, 29]. The previous selection procedure of PePS and the formulae related to the feature set have been applied to the dataset presented in Section 3.. The results show that each input geometry presents a particular and different reference feature vector [8]. All of them are summarized in Table 1. They are valid for the range of simulated observation conditions fixed by \(\Phi^{SO1}=\left\{10-30^{\circ},280-350^{\circ},5.3-9.65 GHz\right\}^{SO1}\) that cover different range of values for incidence, target orientation and operating frequency.