## 5 Vessel Classification Algorithm  
VCA algorithm consists basically on estimating, from POLInSAR data, possible feature sets according to the formulae of previous section. The resulting vectors are then correlated with the reference ones and a similarity value is provided. The geometry which reference set provides the highest similarity is used to describe the observed target. The number of reference feature vectors (or patterns) should accomplish a trade-off between geometry description accuracy and method robustness. On the one hand, a low number of vectors could lead to an overestimation of the classification rate providing a wrong knowledge about the real performance expected for VCA. On the other hand, an excessive number of patterns would make quite similar geometries to be classified in different classes. In this case, little \begin{table}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|} \hline \multicolumn{4}{|c|}{\(\Theta^{SPA}\)} & \multicolumn{4}{|c|}{\(\Theta^{ICE}\)} \\ \hline \(\Theta_{i}^{SPA}\) & \(x_{i}\) & \(y_{i}\) & \(z_{i}\) & \(p_{i}\) & \(\Theta_{i}^{ICE}\) & \(x_{i}\) & \(y_{i}\) & \(h_{i}\) & \(p_{i}\) \\ \hline
1 & 3.5 & -10 & 0 & 0 & 1 & 3 & -2 & 0 & 1 \\
2 & 3.5 & -5 & 0 & 0 & 2 & -6 & -14 & 6.5 & 1 \\
3 & -0.5 & -8 & 2.5 & 1 & 3 & -1 & 1 & 6.5 & 1 \\
4 & -0.5 & -2 & 4.5 & 1 & - & - & - & - & - \\ \hline \multicolumn{4}{|c|}{\(\Theta^{FER}\)} & \multicolumn{4}{|c|}{-} \\ \hline \(\Theta_{i}^{FER}\) & \(x_{i}\) & \(y_{i}\) & \(z_{i}\) & \(p_{i}\) & & - & \\ \hline
1 & 12 & -25 & 0 & 0 & & - & \\
2 & 10 & -25 & 0.5 & 1 & & - & \\
3 & 9 & 8 & 2.5 & 1 & & - & \\
4 & 9 & -8 & 2.5 & 1 & & - & \\ \hline \end{tabular}
\end{table} 
Table 1: Feature vectors for targets \(j=SPA\), \(j=ICE\) and \(j=FER\) and \(\Phi^{SO1}=\{10-30^{o},280-350^{o},5.3-9.65 GHz\}^{SO1}\). Heights (z) are normalized with respect to the lowest value in order to match interferometric height conventions. modifications of target environment may generate unpredictable results and, hence, the same ship would not be classified with the same class. For retrieving estimates of the feature vector, VCA analyzes the input POLInSAR data set with the quad-pol Pauli vector 2 This leads to three different interferograms (one per each Pauli mechanism) from which local maxima are isolated. All these \(m\) maxima are combined with the \(M\) PePS associated with pattern \(p\) generating \(P=m\cdot\frac{M!}{(M-m)! }\) estimated feature vectors [7]. For the permutation \(1<\kappa<P\), the similarity parameter \(0<S_{p}^{\kappa}<1\) is [7, 8]  
Footnote 2: Due to the Pauli decomposition is a complete representation, the same simple mechanisms are isolated in any basis, for instance linear or circular. \[S_{p}^{\kappa}\doteq\frac{r}{M}\cdot(1-\sum_{j=1}^{4}e_{j}^{p}\cdot W_{j}) \tag{9}\]  
where \(0\geq e_{j}^{p}\geq 1\)  
\[e_{j}^{p}=\frac{1}{r}\sum_{n=1}^{r}e_{j,n}^{p} \tag{10}\]  
are the four errors related to the four elements of the PePS vector (\(x\), \(y\) and \(z\) coordinate plus \(p\) polarimetric mechanism). In both formulas, \(r\) indicates the number of PePS with a corresponding scatter in the image and helps to better discriminate those patterns with no relation with the measured structure. In this sense, each permutation applies a _suitability scatter_ step that discards, in the comparison between an estimated and reference feature set, those local maxima that induce a value of 1 in at least two of the four errors of Equation 9.  
For \(x\) (\(j_{n}=x_{n}\)) and \(y\) (\(j_{n}=y_{n}\)) coordinates, the components error of Equation 10 are  
\[ \begin{array}{ccc}e_{j,n}^{p}=1&if&\frac{|<j_{n}>-j_{n}|}{\Delta j}\geq 1\\ e_{j,n}^{p}=0&if&\frac{|<j_{n}>-j_{n}|}{\Delta j}<1\end{array}  \tag{11}\]  
where \(j_{n}\) indicates the coordinate of the selected dimension for the position of PePS n. \(<\cdot>\) is the expectation operator and indicates the value of such coordinates for the scatter that has been associated with the PePS spot.