#
Kernel Density Estimation on the Siegel Space with an Application to Radar Processing^{ †}

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. The Siegel Space

#### 2.1. The Siegel Upper Half Space

#### 2.2. The Siegel Disk

## 3. Non Parametric Density Estimation on the Siegel Space

#### 3.1. Histograms

#### 3.2. Orthogonal Series

#### 3.3. Kernels

- (i)
- ${\int}_{{\mathbb{R}}^{d}}\mathcal{K}\left(\right|\left|x\right|\left|\right)dx=1$;
- (ii)
- ${\int}_{{\mathbb{R}}^{d}}x\mathcal{K}\left(\right|\left|x\right|\left|\right)dx=0$;
- (iii)
- $\mathcal{K}(x>1)=0$;
- (iv)
- $sup\left(\mathcal{K}\right(x\left)\right)=\mathcal{K}\left(0\right)$.

**Theorem**

**1.**

**Proof.**

## 4. Application to Radar Processing

#### 4.1. Radar Data

#### 4.2. Marginal Densities of Reflection Coefficients

#### 4.3. Radar Clutter Segmentation

^{−1}, of spectral width 5 m·s

^{−1}, to which we add 125 cells of rain clutter, centered at 5 m·s

^{−1}, of spectral width 10 m·s

^{−1}. This clutter is sampled 10 times and the segmentation is performed on each simulation (see Figure 6, Figure 7 and Figure 8).

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Demonstration of Theorem 1

**Lemma**

**A1.**

**Proof.**

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**Figure 1.**$\mathcal{M}$ is a Riemannian manifold, and ${T}_{x}\mathcal{M}$ is its tangent space at x. The exponential application induces a volume change ${\theta}_{x}$ between ${T}_{x}\mathcal{M}$ and $\mathcal{M}$.

**Figure 2.**Estimation of the density of six coefficients ${\Omega}_{k}$ under rainy conditions. The expression of the used kernel is $K\left(x\right)=\frac{3}{\pi}{(1-{x}^{2})}^{2}{\mathbf{1}}_{x<1}$. Densities are rescaled for visual purposes.

**Figure 10.**Classification results for varying radii size in the density estimator (10 to 20 closest neighbours).

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**MDPI and ACS Style**

Chevallier, E.; Forget, T.; Barbaresco, F.; Angulo, J.
Kernel Density Estimation on the Siegel Space with an Application to Radar Processing. *Entropy* **2016**, *18*, 396.
https://doi.org/10.3390/e18110396

**AMA Style**

Chevallier E, Forget T, Barbaresco F, Angulo J.
Kernel Density Estimation on the Siegel Space with an Application to Radar Processing. *Entropy*. 2016; 18(11):396.
https://doi.org/10.3390/e18110396

**Chicago/Turabian Style**

Chevallier, Emmanuel, Thibault Forget, Frédéric Barbaresco, and Jesus Angulo.
2016. "Kernel Density Estimation on the Siegel Space with an Application to Radar Processing" *Entropy* 18, no. 11: 396.
https://doi.org/10.3390/e18110396