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Figure 6. Classical Karcher flow in Poincaré’s unit disk

Figure 6 Classical Karcher flow in Poincaré’s unit disk

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Figure 1. Fréchet-Karcher Flow on Cartan- Hadamard Manifold But, center of mass is not useful for robust statistic. Replacing L” square geodesic distance by L’ geodesic distance, we can extend this approach to estimate a Median in metric meM Of the geodesic from m to a : exp((a)eT,M- the “exponential map” and exp-'(a) is the tangent vector at strictly convex (its restriction to any geodesic is strictly convex as a function of one variable) and achieves a unique minimum at a point called the center of mass of A for the distribution da. Moreover, this point is characterized by being the unique Ze10 of the gradient vector field: yp __ Jexp;'(a) ada where exp(.) is
Figure 1. Fréchet-Karcher Flow on Cartan- Hadamard Manifold But, center of mass is not useful for robust statistic. Replacing L” square geodesic distance by L’ geodesic distance, we can extend this approach to estimate a Median in metric meM Of the geodesic from m to a : exp((a)eT,M- the “exponential map” and exp-'(a) is the tangent vector at strictly convex (its restriction to any geodesic is strictly convex as a function of one variable) and achieves a unique minimum at a point called the center of mass of A for the distribution da. Moreover, this point is characterized by being the unique Ze10 of the gradient vector field: yp __ Jexp;'(a) ada where exp(.) is
Figure 4. Fourier Heat Equation on a 1D graph of covariance matrices: anisotropic diffusion Figure 3. Fourier Heat Equation on a 1D graph of covariance matrices: isotropic diffusion
Figure 4. Fourier Heat Equation on a 1D graph of covariance matrices: anisotropic diffusion Figure 3. Fourier Heat Equation on a 1D graph of covariance matrices: isotropic diffusion
From each Time covariance matrix, we can compute Doppler Spectrum. In the following exemple, we give image with range on X axis and Doppler frequency on Y axis. Fourier heat Diffusion is applied on covariance matrices and then, we draw associated Doppler spectrum of results :
From each Time covariance matrix, we can compute Doppler Spectrum. In the following exemple, we give image with range on X axis and Doppler frequency on Y axis. Fourier heat Diffusion is applied on covariance matrices and then, we draw associated Doppler spectrum of results :
Figure 5. Modified Karcher Flow in Poincaré disk by homeomorphism
Figure 5. Modified Karcher Flow in Poincaré disk by homeomorphism
Figure 7. Dual Karcher Flow centered at the origin We will extend previous approach to space-time covariance matrix for STAP. Based on generalization of Trench Algorithm [5], if we consider Toeplitz-block-Toeplitz Hermitian Positive Definite matrix:
Figure 7. Dual Karcher Flow centered at the origin We will extend previous approach to space-time covariance matrix for STAP. Based on generalization of Trench Algorithm [5], if we consider Toeplitz-block-Toeplitz Hermitian Positive Definite matrix:
From Burg-like parameterization [11], we can deduced this inversion of Toeplitz-Block-Toeplitz matrix :
From Burg-like parameterization [11], we can deduced this inversion of Toeplitz-Block-Toeplitz matrix :
Siegel Upper-half Plane sy_ is transformed in unit Siegel disk Figure 8. Geometry of Siegel Upper Half-Plane
Siegel Upper-half Plane sy_ is transformed in unit Siegel disk Figure 8. Geometry of Siegel Upper Half-Plane
Figure 9. Frechet Median Barycenter = Busemann Barycenter association is given by the limit point when t tends to infinity along the geodesic from the barycenter toward w..
Figure 9. Frechet Median Barycenter = Busemann Barycenter association is given by the limit point when t tends to infinity along the geodesic from the barycenter toward w..
Figure 10. Doppler Processing, Antenna Processing, Space-Time Processing & Polar processing
Figure 10. Doppler Processing, Antenna Processing, Space-Time Processing & Polar processing
Figure 11. (up) Non-regularized & (bottom) Regularized Doppler AR Spectrum
Figure 11. (up) Non-regularized & (bottom) Regularized Doppler AR Spectrum
Figure 13. OS-HDR-CFAR Algorithm with illustration of two first reflection coefficients Figure 12. (top figure) Classical OS-CFAR after filter banks, (bottom figure) OS-HDR-CFAR
Figure 13. OS-HDR-CFAR Algorithm with illustration of two first reflection coefficients Figure 12. (top figure) Classical OS-CFAR after filter banks, (bottom figure) OS-HDR-CFAR
Figure 15. ROC curves for 3 approaches : OS-HDR-CFAR, OS- CFAR/Doppler-Filters, & method based on Wasserstein barycenter/distance (optimal transport theory)
Figure 15. ROC curves for 3 approaches : OS-HDR-CFAR, OS- CFAR/Doppler-Filters, & method based on Wasserstein barycenter/distance (optimal transport theory)
Figure 16. Probability of detection versus SNR for fixed Pfa=10°, with o relative position in Doppler of the target normalized by Doppler Clutter Spectrum Width (a=1 means that the target is positioned on the edge in Doppler of the ground clutter)
Figure 16. Probability of detection versus SNR for fixed Pfa=10°, with o relative position in Doppler of the target normalized by Doppler Clutter Spectrum Width (a=1 means that the target is positioned on the edge in Doppler of the ground clutter)
Figure 17. Comparison of “Mean” in red and “Median” in blue for clutter level estimation (top) and through Pd/Pfa curves (down).
Figure 17. Comparison of “Mean” in red and “Median” in blue for clutter level estimation (top) and through Pd/Pfa curves (down).
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