Probability density estimation on the hyperbolic space applied to radar processing

28/10/2015
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14311

Résumé

The two main techniques of probability density estimation on symmetric spaces are reviewed in the hyperbolic case. For computational reasons we chose to focus on the kernel density estimation and we provide the expression of Pelletier estimator on hyperbolic space. The method is applied to density estimation of reflection coefficients derived from radar observations.

Probability density estimation on the hyperbolic space applied to radar processing

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application/pdf Probability density estimation on the hyperbolic space applied to radar processing Emmanuel Chevallier, Jesús Angulo, Frédéric Barbaresco

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The two main techniques of probability density estimation on symmetric spaces are reviewed in the hyperbolic case. For computational reasons we chose to focus on the kernel density estimation and we provide the expression of Pelletier estimator on hyperbolic space. The method is applied to density estimation of reflection coefficients derived from radar observations.

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Probability density estimation on the hyperbolic space applied to radar processing October 28, 2015 Emmanuel Chevalliera, Frédéric Barbarescob, Jesús Anguloa a CMM-Centre de Morphologie Mathématique, MINES ParisTech; France b Thales Air Systems, Surface Radar Domain, Technical Directorate, Advanced Developments Department, 91470 Limours, France emmanuel.chevallier@mines-paristech.fr 1/20 Probability density estimation on the hyperbolic space Three techniques of non-parametric probability density estimation: histograms kernels orthogonal series The Hyperbolic space of dimension 2 Histograms, kernels and orthogonal series in the hyperbolic space Density estimation of radar data in the Poincaré disk 2/20 Probability density estimation on the hyperbolic space Three techniques of non-parametric probability density estimation Histograms: partition of the space into a set of bins counting the number of samples per bins 3/20 Probability density estimation on the hyperbolic space Kernels: a kernel is placed over each sample the density is evaluated by summing the kernels 4/20 Probability density estimation on the hyperbolic space Orthogonal series: the true density f is studied through the estimation of the scalar products between f and an orthonormal basis of real functions. Let f be the true density f , g = f gdµ let {ei } is a orthogonal Hilbert basis of real functions f = ∞ i=−∞ f , ei ei , since fI , ei = fI ei dµ = E (ei (I)) ≈ 1 n n j=1 ei (I(pj )) we can estimate f by: f ≈ N i=−N  1 n n j=1 ei (I(pj ))   ei = ˆf . 5/20 Probability density estimation on the hyperbolic space Homogeneity and isotropy consideration non homogeneous bins non istropic bins Absence of prior on f : the estimation should be as homogeneous and isotropic as possible. → choice of bins, kernels or orthogonal basis 6/20 Probability density estimation on the hyperbolic space Remark on homogeneity and isotropy Figure: Random variable X ∈ Circle. The underlying space is not homogeneous and not isotropic, the density estimation can not consider every points and directions in an equivalent way. 7/20 Probability density estimation on the hyperbolic space The 2 dimensional hyperbolic space and the Poincaré disk The only space of constant negative sectional curvature The Poincaré disk is a model of hyperbolic geometry ds2 D = 4 dx2 + dy2 (1 − x2 − y2)2 Homogeneous and isotropic 8/20 Probability density estimation on the hyperbolic space Density estimation in the hyperbolic space: histograms A good tilling: homogeneous and isotropic There are many polygonal tilings: There is no homotetic transformations for all λ ∈ R Problem: not always possible to scale the tiling to the studied density 9/20 Probability density estimation on the hyperbolic space Density estimation in the hyperbolic space: orthogonal series Standard choice of basis: eigenfunctions of the Laplacian operator ∆ In Rn: (ei ) = Fourier basis → characteristic function density estimator. f , [a, b] → R, f = ∞ i=−∞ f , ei ei , f , R → R, f = ∞ ω=−∞ f , eω eωdω, Compact case: estimation of a sum Non compact case: estimation of an integral 10/20 Probability density estimation on the hyperbolic space Density estimation in the hyperbolic space: orthogonal series On the Poincaré disk D, solutions of ∆f = λf are known for f , D → R but not for f , D ⊂ D → R with D compact Computational problem: the estimation involves an integral, even for bounded support functions 11/20 Probability density estimation on the hyperbolic space Kernel density estimation on Riemannian manifolds K : R+ → R+ such that: i) Rd K(||x||)dx = 1, ii) Rd xK(||x||)dx = 0, iii) K(x > 1) = 0, sup(K(x)) = K(0). Euclidean kernel estimator: ˆfk = 1 k i 1 rd K ||x, xi || r Riemannian case: K ||x − xi || r → K d(x − xi ) r 12/20 Probability density estimation on the hyperbolic space Figure: Volume change θxi induced by the exponential map θx : volume change (TM, Lebesgue) expx −→ (M, vol) Kernel density estimator proposed by Pelletier: ˆfk = 1 k i 1 rd 1 θxi (x) K d(x, xi ) r 13/20 Probability density estimation on the hyperbolic space θx in the hyperbolic space θx can easily be computed in hyperbolic geometry. Polar coordinates at p ∈ D: at p ∈ D, if the geodesic of angle α of length r leads to q, (r, α) ↔ q In polar coordinates: ds2 = dr2 + sinh(r)2 dα2 thus dvolpolar = sinh(r)drdα and θp((r, θ)) = sinh(r) r 14/20 Probability density estimation on the hyperbolic space Density estimation in the hyperbolic space: kernels Kernel density estimator: ˆfk = 1 k i 1 rd d(x, xi ) sinh(d(x, xi )) K d(x, xi ) r Formulation as a convolution Fourier−Helgason ←→ 0rthogonal series Reasonable computational cost 15/20 Probability density estimation on the hyperbolic space Radar data Succession of input vector z = (z0, .., zn−1) ∈ Cn z: background or target? Assumptions: z = (z0, .., zn−1) is a centered Gaussian process. Centered → dened by its covariance Rn = E[ZZ∗ ] =       r0 r1 . rn−1 r1 r0 r1 . rn−2 . . . r1 rn−1 . r1 r0       Rn ∈ T n: Toeplitz (additional stationary assumption) and SPD matrix 16/20 Probability density estimation on the hyperbolic space Auto regressive model Auto regressive model of order k: ˆzl = − k j=1 ak j zl−j k-th reection coecient : µk = ak k Dieomorphism ϕ: ϕ : T n → R∗ + × Dn−1 , Rn → (P0, µ1, · · · , µn−1) (z0, ..., zn−1) ↔ (P0, µ1, · · · , µn−1) 17/20 Probability density estimation on the hyperbolic space Geometry on T n ϕ : T n → R∗ + × Dn−1 , Rn → (P0, µ1, · · · , µn−1) metric on T n: product metric on R∗ + × Dn−1 Multiple acquisitions of an identical background: distribution of the µk? Potential use: identication of a non-background objects 18/20 Probability density estimation on the hyperbolic space Application of density estimation to radar data µ1, N = 0.007 µ2, N = 1.61 µ3, N = 14.86 µ1, N = 0.18 µ2, N = 2.13 µ3, N = 4.81 Figure: First row: ground, second row: Rain 19/20 Probability density estimation on the hyperbolic space Conclusion The density estimation on the hyperbolic space is not a fundamentally dicult problem Easiest solution: kernels Future works: computation of the volume change in kernels for Riemannian manifolds deepen the application for radar signals Thank you for your attention 20/20 Probability density estimation on the hyperbolic space