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Article KernelDensityEstimationontheSiegelSpacewith anApplicationtoRadarProcessing† EmmanuelChevallier 1,*,ThibaultForget 2,3,FrédéricBarbaresco2 andJesusAngulo3 1 DepartmentofComputerScienceandAppliedMathematics,WeizmannInstituteofScience, Rehovot7610001, Israel 2 ThalesAirSystems,SurfaceRadarBusinessLine,AdvancedRadarConceptsBusinessUnit, VoiePierre-GillesdeGennes,Limours91470,France; thibault.forget@mines-paristech.fr (T.F.); frederic.barbaresco@thalesgroup.com(F.B.) 3 CMM-CentredeMorphologieMathématique,MINESParisTech,PSL-ResearchUniversity, Paris75006,France; jesus.angulo@mines-paristech.fr * Correspondence: emmanuelchevallier1@gmail.com;Tel.: +972-58-693-7744 † Thispaper isanextendedversionofourpaperpublishedin the2ndconferenceonGeometricScienceof Information,Paris,France,28–30October2015. AcademicEditors:AryeNehorai,SatyabrataSenandMuratAkcakaya Received: 13August2016;Accepted: 31October2016;Published: 11November2016 Abstract:ThispaperstudiesprobabilitydensityestimationontheSiegel space. TheSiegel space is ageneralizationof thehyperbolic space. ItsRiemannianmetricprovidesan interesting structure to theToeplitz blockToeplitzmatrices that appear in the covariance estimation of radar signals. The main techniques of probability density estimation on Riemannian manifolds are reviewed. Forcomputational reasons,wechose to focusonthekerneldensityestimation. Themainresultof thepaper is theexpressionofPelletier’skerneldensityestimator. Thecomputationof thekernels ismadepossiblebythesymmetric structureof theSiegel space. Themethodisapplied todensity estimationof reflectioncoefficients fromradarobservations. Keywords:kerneldensityestimation;Siegel space; symmetric spaces; radarsignals 1. Introduction Various techniquescanbeusedtoestimate thedensityofprobabilitymeasure in theEuclidean spaces, such as histograms, kernelmethods, or orthogonal series. Thesemethods can sometimes beadaptedtodensities inRiemannianmanifolds. Thecomputationalcostof thedensityestimation dependsonthe isometrygroupof themanifold. In thispaper,westudythespecial caseof theSiegel space. TheSiegel space is ageneralizationof thehyperbolic space. Ithasa structureof symmetric Riemannianmanifold,which enables the adaptation of different density estimationmethods at a reasonablecost. Convergenceratesof thedensityestimationusingkernelsandorthogonal serieswere graduallygeneralizedtoRiemannianmanifolds (see [1–3]). TheSiegel space appears in radarprocessing in the studyofToeplitz blockToeplitzmatrices, whoseblocks representcovariancematricesofa radarsignal (see [4–6]). TheSiegelalsoappears in statisticalmechanics, see[7]andwasrecentlyusedinimageprocessing(see[8]). Informationgeometry is nowa standard framework in radar processing (see [4–6,9–13]). The information geometry on positivedefiniteTeoplitzblockTeoplitzmatrices isdirectlyrelatedto themetricontheSiegel space (see [14]). Indeed,ToeplitzblockToeplitzmatricescanberepresentedbyasymmetricpositivedefinite matrixandapoint layinginaproductofSiegeldisks. ThemetricconsideredonToeplitzblockToeplitz matrices is inducedbytheproductmetricbetweenametriconthesymmetricpositivedefinitematrices andtheSiegeldisksmetrics (see [4–6,9,14]). Entropy2016,18, 396 347 www.mdpi.com/journal/entropy