Page - 357 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 396 4.3. RadarClutterSegmentation Clutter refers to backgroundDoppler signal related tometeorological conditions (e.g., wind inwooded areas, currents and breakingwaves onwater), which hinders detection of small and slowtargets.Ateachrange,asetof reflectioncoefficientsarecomputedfromtheDopplerspectrum (see [31]). This set of coefficients is a point in the Poincaré poly-disk. From this set of points in thepoly-disk, it ispossible toestimate theunderlyingdensity. Segmentingclutter, i.e.,determining zonesofhomogeneousDopplercharacteristics (seeFigure3), enables the improvementofdetection algorithmson each zone. Themean-shift algorithmenables segmentation of the space according to thekerneldensityestimationofasetofpoints. Itwas introducedbyFukunagaandHostetler in 1975 (see [32]). It corresponds toagradientascentof thedensityestimator (see [33]) fora studyof thestatistical consistencyof thegradient linesestimation. Eachdatapointmoves toa localmodeof thedensityestimator,whichyieldsasmanyclustersasmodes. Thisalgorithmhasbeengeneralized onmanifolds in [34], andapplied to radar images in [35]. It can thusbeused tosegment thesetof points in thePoincarépoly-disk. Unfortunately, themean-shift algorithmrequiresworkingwitha kerneldependingonlyonthedistance to itsbarycenter,which isnot thecaseof thekerneldefinedin Equation(19). Thus, thecomputationsareperformedwithout theuseof thecorrective termθp. It is possible tosolve thisproblembyreplacingthecorrective termbyitsaverageatagivenradius,which leads toakerneldependingonlyonthedistance to itsbarycenter.Our futureworkwill focusonthe computationof theseaverages. Let fˆKr (x)= cd k k ∑ i=1 1 rn K ( d(xi,x)2 r2 ) , where cn isanormalizationconstant. Letg=−k′. Figure3.MeanandwidthvariabilityofseaclutterDopplerspectrum. 357