# Kernel Density Estimation on the Siegel Space with an Application to Radar Processing +
## 4 Application to Radar Processing
### Radar Clutter Segmentation  
Clutter refers to background Doppler signal related to meteorological conditions (e.g., wind in wooded areas, currents and breaking waves on water), which hinders detection of small and slow targets. At each range, a set of reflection coefficients are computed from the Doppler spectrum (see [31]). This set of coefficients is a point in the Poincare poly-disk. From this set of points in the poly-disk, it is possible to estimate the underlying density. Segmenting clutter, i.e., determining zones of homogeneous Doppler characteristics (see Figure 3), enables the improvement of detection algorithms on each zone. The mean-shift algorithm enables segmentation of the space according to the kernel density estimation of a set of points. It was introduced by [PERSON] and [PERSON] in 1975 (see [32]). It corresponds to a gradient ascent of the density estimator (see [33]) for a study of the statistical consistency of the gradient lines estimation. Each data point moves to a local mode of the density estimator, which yields as many clusters as modes. This algorithm has been generalized on manifolds in [34], and applied to radar images in [35]. It can thus be used to segment the set of points in the Poincare poly-disk. Unfortunately, the mean-shift algorithm requires working with a kernel depending only on the distance to its barycenter, which is not the case of the kernel defined in Equation (19). Thus, the computations are performed without the use of the corrective term \(\theta_{p}\). It is possible to solve this problem by replacing the corrective term by its average at a given radius, which leads to a kernel depending only on the distance to its barycenter. Our future work will focus on the computation of these averages. Let  
\[f_{r}^{\mathcal{K}}(x)=\frac{c_{d}}{k}\sum_{i=1}^{k}\frac{1}{r^{n}}\mathcal{K} \left(\frac{d(x_{i},x)^{2}}{r^{2}}\right),\]  
where \(c_{n}\) is a normalization constant. Let \(g=-k^{\prime}\). Figure 3: Mean and width variability of sea clutter Doppler spectrum. The mean-shift is defined by  
\[m(x)=\sum_{i=1}^{k}\frac{\frac{1}{r^{2+2}}\mathcal{S}\left(\frac{d((x,x_{i})^{2} )}{r^{2}}\right)}{\sum_{i=1}^{k}\frac{1}{r^{n+2}}\mathcal{S}\left(\frac{d(x,x_{ i})^{2}}{r^{2}}\right)}log_{x}(x_{i})\propto\frac{\
abla f_{r}^{\mathcal{K}}}{f_{r}^{ \mathcal{S}}},\]  
where \(m(x)\) is in the tangent space at \(x\). The algorithm moves from \(x\) to \(exp_{x}(m(x))\) until convergence to a local maximum. The points of the space are segmented according to the local maxima to which they converge. In order to assess the quality of unsupervised classification, we use the notion of Silhouette, see [36], which computes for each point a proximity criterion with respect to other points of the same cluster and other points of different clusters (see Figure 4). Let \(x\) be in the cluster \(A\). We respectively define \(a(x)=min_{y\in A}d(x,y)\) and \(b(x)=min_{y\
eq A}d(x,y)\), the minimum distance to points of the same (resp. other) class(es). The Silhouette of \(x\) is  
\[\frac{a(x)-b(x)}{max\{a(x),b(x)\}},\]  
which takes values between \(-1\) and \(1\), respectively, when the data point is considered \"badly\" and \"well\" clustered. The average of all the silhouettes provides an indication of the relevance of the classification. One can represent graphically the silhouette profile by plotting for each class horizontal segments of the length of the silhouette value (see Figure 5). Figure 4: Intra and inter cluster distances. Figure 5: Example of silhouette. It can be seen that, apart from a few outliers, the two clutters are well classified and that the algorithm was able to distinguish between two zones of different Doppler characteristics. We then test our algorithm on real sea clutter data (see Figures 9-11). Figure 8: Silhouettes. Figure 9: Autoregressive spectrum. The results are more difficult to interpret in that case. The Doppler spectra are varying quite a lot along the range axis.