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POLARIMETRIC DETECTION OF TARGETS ININHOMOGENEOUSCLUTTER MartinHurtadoand AryeNehorai DepartmentofElectricaland SystemsEngineering WashingtonUniversityinSt. Louis OneBrookingsDrive St. Louis,MO63130 {mhurta3, nehorai}@ese.wustl.edu ABSTRACT fromtheschemepresentedin[4]hasaclosed-formexpres- siononlyforthespecialcaseoftwopolarimetricchannels, Polarization diversity in radar is useful to detect targets, and the detector proposedin [5] doesnotsupportthe con- particularly when Dopplerdiscriminationisnot possible. stant false-alarm rate (CFAR) property with respect to the Weintroduceanewpolarimetricradarmodelthatincludes cluttercovariance. the realistic dependence of the clutter reflections on the Inthispaper,wedevelopapolarimetricdetectorwhose transmittedsignal.Wethendevelopastatisticalpolarimet- decision-making capability is based on the data collected ric detectiontest, robustto heavyinhomogeneousclutter. onlyfromtherangecellundertest,withoutresortingtosec- We prove that the detector’s false-alarm rate is invariant ondarydataorpriorknowledgeofthetargetandclutter. In tothespaceandtimevariabilitiesoftheclutter; hence,it ordertoderivethedetector,wefirstintroduceanewpolari- has the constantfalse-alarm(CFAR) property, while still metric radar model that states the dependence of the clut- maintaining a good probability of detection. We demon- ter reflections on the transmitted signal. We prove that it stratetheimprovedperformanceoftheproposeddetector isrobustagainstinhomogeneousandnon-stationaryclutter; incomparisonwithexistingdetectorsusingnumericalex- i.e.thedetector’sfalse-alarmrateisinsensitivetospaceand amplesandrealdata. timevariationsoftheclutter,whilestillmaintainingagood probability of detection. Moreover, this test is CFAR. In 1. INTRODUCTION thatsense,thisdetectorresemblesthatof[6];however,our work is distinctive in several ways: (i) The detector in [6] The detection of static or slowly moving targets in heavy isdevelopedundertheassumptionthatthetargetispresent cluttered environments is considered a challenging prob- only in one of the multiple scans of the radar dwell; thus, lem, mainly because it is not possible to discriminate the itwillnotdetecta static target. Ourmethodisspecifically target from the clutter using the Doppler effect. Polariza- aimed at detecting static or slow targets. (ii) Our detector tiondiversityprovidesadditionalinformationthatenhances hasbeenextendedtoconsideranyarbitrarytransmitpolar- thedetectionoftargets,particularlyundertheconditionsde- ization. (iii) Our detector can also be applied to advanced scribedabove. polarimetric receivers, such as tripoles and vector-sensor Earlier work in the field of polarimetric detection has antennas,inadditiontoconventionalverticalandhorizontal addressedthedesignofdetectorsundertheassumptionthat (orleftandrightcircular)polarizedarraysofantennas. (iv) trainingdataisavailable[1]-[3]. Theperformanceofthese Our detector is tested with real radar data. An additional detectors can be severely degraded in the presence of in- advantageof our detection algorithmis thatit is computa- homogeneousandnon-stationaryclutter. Toovercomethis tionallyless intensebecauseit doesnotrequireprocessing problem, the application of compound-Gaussian distribu- oftrainingdata. tionsformodelingtheglobalbehaviorofnonhomogeneous clutterhasbeenproposed.However,theuseofnon-Gaussian modelsincreasesthedifficultyofdevelopingefficientdetec- 2. POLARIMETRICRADARMODEL tionalgorithms. Forinstance,thedetectionstatisticderived Weconsideramono-staticradarcapableoftransmittingwave- This work was supported by the Department of Defense under forms with any arbitrary polarization on a pulse-by-pulse the Air Force Office of Scientific Research MURI Grant FA9550-05- basis. Inaddition,weassumethattheradarsystemillumi- 0443,AFOSRGrantFA9550-05-1-0018andDARPAunderNRLGrant N00173-06-1G006. natesapointtarget. Therecordeddatafromtherangecell Report Documentation Page Form Approved OMB No. 0704-0188 Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. 1. REPORT DATE 3. DATES COVERED 2007 2. REPORT TYPE 00-00-2007 to 00-00-2007 4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER Polarimetric Detection of Targets in Inhomogeneous Clutter 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION Washington University in St. Louis,Department of Electrical and Systems REPORT NUMBER Engineering,One Brookings Drive,St. Louis,MO,63130 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S) 11. SPONSOR/MONITOR’S REPORT NUMBER(S) 12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release; distribution unlimited 13. SUPPLEMENTARY NOTES Proc. Workshop Adaptive Sensor Array Processing (ASAP), Lincoln Laboratory, Lexington, MA, Jun. 5-6, 2007 14. ABSTRACT Polarization diversity in radar is useful to detect targets, particularly when Doppler discrimination is not possible. We introduce a new polarimetric radarmodel that includes the realistic dependence of the clutter reflections on the transmitted signal. We then develop a statistical polarimetric detection test, robust to heavy inhomogeneous clutter. We prove that the detector’s false-alarm rate is invariant to the space and time variabilities of the clutter; hence, it has the constant false-alarm (CFAR) property, while still maintaining a good probability of detection. We demonstrate the improved performance of the proposed detector in comparison with existing detectors using numerical examples and real data. 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF ABSTRACT OF PAGES RESPONSIBLE PERSON a. REPORT b. ABSTRACT c. THIS PAGE Same as 6 unclassified unclassified unclassified Report (SAR) Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18 under test consist notonly of the targetechoes but also of • Thevectore(t)representsthethermalnoise. theundesiredreflectionsfromthetargetenvironment(clut- Equation(1)canbewrittenas ter). y(t)=s(t)Bξ¯(µ+x)+e(t), (5) 2.1. PolarimetricData where the scattering coefficient vectors of the target and The output of a diversely polarized array of Q sensors re- clutter are µ = [sthh,stvv,sthv]T and x = [schh,scvv,schv]T, ceivingtheechoesfromthecellundertestcanbeexpressed respectively, which have dimensionP = 3. The polariza- as tionmatrixξ¯is y(t)=B St+Sc ξ(t)+e(t), t=1,...,N, (1) ξ¯= ξ1 0 ξ2 . (6) (cid:0) (cid:1) (cid:20)0 ξ2 ξ1(cid:21) where Thetimesamplescanbestackedinonevectorofdimension • TheQ×1vectory(t)isthecomplexenvelopeofthe NQ×1: measurements. y = s⊗Bξ¯ (µ+x)+e, (7) wheres=[s(1),...(cid:0),s(N)]T(cid:1)and⊗istheKroneckerprod- • TheQ×2matrixB istheresponseofthediversely uct. Pilingtogetherthedatacorrespondingtopulsesofdif- polarizedsensorarray.Ifthereceiverarrayisavector ferentpolarizationyields sensor[7],thearrayresponseisgivenby y =Aµ+Ax+e, (8) −sinφ −cosφsinψ  cosφ −sinφsinψ where A is a M ×P (M = KNQ) complex matrix that 0 cosψ representsthesystemresponse: B = , (2) −cosφsinψ sinφ    s⊗Bξ¯ −sinφsinψ −cosφ  1   .  cosψ 0  A= .. , (9)   whereφ andψ are the azimuthand elevationangles s⊗Bξ¯K ofthecellundertest,respectively.Foraconventional andξ¯ isthepolarizationmatrixofeachdiverselypolarized k polarizedradarmeasuringthehorizontalandvertical pulse(k =1,...,K). componentsof the electric field and assuming these twosensorsareorthogonaltothedirectionthatpoints 2.2. StatisticalModel towardthe cell under test, the array response matrix isB =I ,whereI isthe2×2identitymatrix. Weassumethatthetargetisasmallman-madeobject;hence, 2 2 µisadeterministicvector.Ontheotherhand,theclutterin • The complex scattering matrix S represents the po- therangecellundertestcanbeconsideredasalargecollec- larization change of the transmitted signal upon its tion of point scatterers producingincoherentreflections of reflectiononthetargetorclutter: theradarsignal. Then,xisazero-meancomplexGaussian random vector with covariance matrix Σ. The noise e is s s S = 11 12 , (3) azero-meancomplexGaussianrandomvectorwithcovari- (cid:20)s21 s22(cid:21) ancematrixσI ,whereI istheM ×M identitymatrix. M M wherethe variabless ands are co-polarscatter- In addition, we assume that the clutter reflections and the 11 22 ingcoefficientsands ands arecross-polarcoef- thermalnoisearestatisticallyindependent. 12 21 ficients. For the mono-static radar case, s = s . The radar dwell consists of a series of pulses that can 12 21 Thesuperscriptstandcrefertothetargetandclutter. be seen as “snapshots” of the range cell under test. If the pulse duration is short with respect to the dynamic of the • Thevectorξ(t)isthenarrowbandtransmittedpolar- target and its environment, it is reasonable to assume that izedsignalwhichcanberepresentedby theirscatteringcoefficientsareconstantduringeachpulse. ξ cosα sinα cosβ However,frompulsetopulse,weconsidertheclutterscat- ξ(t)= 1 s(t)= s(t), (cid:20)ξ2(cid:21) (cid:20)−sinα cosα(cid:21)(cid:20)jsinβ(cid:21) tering coefficients as independent realizations of the same (4) randomprocess.Then,thedistributionofeachsnapshotis whereξ1andξ2arethesignalcomponentsonthepo- y ∼CN(Aµ,AΣAH +σI ), d=1,...,D, (10) larizationbasisofthetransmitter,αandβaretheori- d M entationand ellipticity angles, respectively, and s(t) where D is the total number of snapshots or pulses in the isthecomplexenvelopeofthetransmittedsignal. radardwellandH denotestheconjugatetransposeoperator. 2.3. KnownandUnknownModelParameters TheMLEofΣis We assume the system response matrix A is known, since Σˆ =A+S A+H −σ(AHA)−1, (15) 0 0 we consider that the receiver antenna array has been cali- brated. In addition, we assume that the power of the ther- where A+ = (AHA)−1AH is the generalized matrix in- mal noise σ is known, because it can be easily estimated verse. UnderhypothesisH1,thelikelihoodfunctionis from the recorded data when no signal has been transmitted. However,wesupposethatwehavenopriorknowledge lnf1(µ,Σ)=−D Mlnπ+ln|C|+tr(C−1C˜1) , h i aboutthe targetand the clutter, nor do we counton a sec- (16) ondary data set for estimating the statistical properties of where the clutter. Hence, the vector µ and the matrix Σ are the D unknownparametersofthestatisticaldatamodel(10). C˜ = 1 (y −Aµ)(y −Aµ)H. (17) 1 D d d Xd=1 3. DETECTIONTEST TheMLEoftheunknownparametersare We aim to decide whether a target is present or not in the µˆ = A+y¯, (18) 1 rangecellundertest,basedontherecordeddata. Then,the Σˆ = A+S A+H −σ(AHA)−1, (19) decisionproblemconsistsofchoosingbetweentwopossible 1 1 hypotheses:thenullhypothesisH (target-freehypothesis) 0 wherey¯ isthesamplemeanvector orthealternativehypothesisH (target-presenthypothesis) 1 D 1 H : µ=0, Σ y¯ = y , (20) 0 (11) D d (cid:26) H1 : µ6=0, Σ Xd=1 andS isthesamplecovariancematrix where the matrixΣ is consideredasa nuisanceparameter. 1 Next,wederivethegeneralizedlikelihoodratio(GLR)test D 1 [8] and study its performance. We omit derivation details S = (y −y¯)(y −y¯)H. (21) inthispaperduetospacelimitation;referto[9]foranex- 1 D Xd=1 d d tendedversionofthispaper. Then, substituting the MLEs and likelihood functions in (12),andaftersomemathematicalsimplifications,theGLR 3.1. GLRTest testcanbewrittenas TheGLRtestdecidesH1 if L = 1+y¯HA AHS A −1AHy¯ D. (22) GLR 1 f (y ,...,y ;µˆ ,Σˆ ) h (cid:0) (cid:1) i L = 1 1 D 1 1 >γ, (12) GLR f (y ,...,y ;Σˆ ) Since(22)isamonotonicallyincreasingfunctionofthesec- 0 1 D 0 ond term inside the brackets, an equivalent detection test wheref andf arethelikelihoodfunctionsunderH and statisticcanbedefinedas 0 1 0 H1,Σˆ0andΣˆ1arethemaximumlikelihoodestimates(MLE) T =y¯HA AHS A −1AHy¯. (23) ofΣunderH andH ,µˆ istheMLEofµunderH ,andγ GLR 1 0 1 1 1 (cid:0) (cid:1) isthedetectionthreshold.Forsimplicityofnotationwewill omitreferencestothedataintheargumentsofthefunctions 3.2. DetectionPerformance f andf intherestofthepaper. 0 1 ApplyingCorollary5.2.1from[10],itisstraightforwardto We startthe derivationoftheGLR testbydetermining verify that the detection statistic (23) is distributed as fol- thelikelihoodfunctionsandMLEsoftheunknownparam- lows eters. UnderhypothesisH ,itisassumedthatµ=0;then 0 lnf0(Σ)=−D Mlnπ+ln|C|+tr(C−1S0) , (13) TGLRDP−P ∼(cid:26) FF202PP,2,2(D(D−−PP))(λ) uunnddeerrHH01 (24) (cid:2) (cid:3) mwhaterriexCof =theAdΣatAa,Hde+finσeIdMinis(1t0h)e,tahnedorSet0iciasltchoevasarimanpclee wgrheeerseoFfνf1r,eνe2ddoemn,oatensdaFn0Fν1,dνi2s(tλri)budteinonotwesitahnνo1na-ncednνtr2aldFe- covariancematrix distributionwithν1andν2degreesoffreedomandnon-centrality parameterλ. Thenon-centralityparameterisgivenby D 1 S0 = D ydyHd . (14) λ=2DµH Σ+σ AHA −1 −1µ. (25) Xd=1 h (cid:0) (cid:1) i inhomogeneousclutter. The clutter echoes are assumed to 1 follow a compound-Gaussian distribution with covariance Analytical 0.9 Monte Carlo matrixτΣ. Thetextureτ hasageneralizedGammaproba- 0.8 CNR=10dB bilitydensityfunction[4]: M=8 on0.7 P=3 ecti D=10 p(τ)= 1 ν ντν−1exp −ντ , (27) Det0.6 Γ(ν)(cid:16)δ(cid:17) (cid:16) δ (cid:17) of 0.5 bility 0.4 whereν istheorderparameterandδ istheaveragepower. Proba0.3 PFA=10−1 PFA=10−2 P =10−3 Nsiaontedtihsatrtibνu=tio∞n, bcuotrrweshpeonnνdsdteoccreluatsteesr,etchheoyedsewviitahteGfaroums- FA 0.2 Gaussian.Additionalinformationaboutthesimulationsetup 0.1 isasfollows.Thetransmittedsignalsarerectangularpulses, which are transmitted in a sequence of alternating vertical 0 −20 −15 −10 −5 0 5 Target to Clutter Ratio (dB) (V) and horizontal (H) polarization (K = 2). The radar dwellconsistsof10repetitionsofthispolarimetricsequence Figure 1: Detection performance of the GLR test statistic (D = 10). On the receiver side, samples of the V and H T asafunctionofthetarget-to-clutterratio,fordifferent electric field are recorded (Q = 2) at a rate of one sam- GLR valuesofP . ple perpulse(N = 1). Sincethe simulationsareintended FA to compute the P , no target is considered. The average FA powerofthetextureisδ = 50andthespecklecomponent Thus,thedetectionperformancebecomes ofthecovarianceisΣ=I . 3 For comparisonpurposes, we also computethe perfor- PFA = QF2P,2(D−P)(γ) manceofotherdetectorsthatusesecondarydata: PD = QF02P,2(D−P)(λ)(γ), (26) • Polarization-space-timeGLR(PST-GLR):Thisdetec- whereP istheprobabilityofdetection,P istheproba- torwasderivedundertheassumptionofhomogeneous D FA bilityoffalsealarm,Qistheright-tailprobabilityfunction Gaussianclutter[1]. [8,Chap.2]andγisthedetectionthresholdfortherequired • Texturefree GLR (TF-GLR):This detector was for- probability of false alarm. In particular, note that the ex- mulatedassumingthattheclutterfollowsacompound pressionforP doesnotdependonthecovarianceofclut- FA Gaussiandistribution[4]. terandthermalnoise,noronthetransmittedsignal;hence, equation(23)isaCFARtest. • HybridUtest: Thisdetectorwasdevelopedforasin- We validated the analytical performance of the devel- glepolarimetricchannelassumingthattheclutterfol- oped test by computer simulations. We performed Monte lowsaWeibulldistribution[11]. Carlosimulationsbasedon100/P independenttrails.Fig- FA ure 1 showscurvesof P asa functionof target-to-clutter InFig.2,theP ofourdetectiontestT ,aswellas D FA GLR ratio (TCR) for different values of P and a clutter-to- ofthePST-GLR,TF-GLR,andHybridUtests,isplottedas FA noise ratio (CNR) of 10dB. The definitions of TCR and a functionoftheorderparameterofthecluttertexturedis- CNRaregivenintheAppendix.Theresultsdepictedinthe tribution. Forthelatterthreetests,weconsider80adjacent figureconsistoftheaverageof100differentcases. Foreach range cells for generatingthe secondarydata. We observe case,theentriesofµ,Σ,andAareindependentrealization thattheperformanceofourdetectorT andtheTF-GLR GLR ofarandomvariablewithdistributionCN(0,1);thenthose testremainsconstantatthedesignedfalse-alarmrate.How- vectorsand matricesare scaled to fulfill the requiredTCR ever,thereisadiscrepancybetweenthedesignandtheac- andCNR. tual clutterthatincreasesthe probabilityof false alarm for thePST-GLRtestastheclutterprocessdepartsfrombeing 3.3. DetectorPerformanceinInhomogeneousClutter Gaussian. The Hybrid U test behaves similar to the PST- GLR;butithaslargerP duetothemismatchbetweenthe FA Asignificantcharacteristicofthedetectorpresentedinthis modelusedtogeneratetheclutterdataandtheoneassumed paper is that the test statistic (23) is computed using only bythealgorithm. data from the cell under test. No assumption is required abouttherangehomogeneityoftheclutter. Thus,thedetec- 3.4. DetectorPerformanceUsingRealData torisrobustagainstspatiallyfluctuatingclutter. To support this assertion, the P is computed using To evaluate the performance of the polarimetric detectors FA 5·105 runsofMonteCarlosimulationsinthepresenceof in a real applicationsituation, we use data collected at the Our Pol Detector HH Channel PST−GLR TF−GLR Hybrid U m ar CNR=10dB Al10−2 M=4 > < e P=3 Fals D=10 VH Channel of y bilit a ob > < Pr ge 10−3 Ran VV Channel 10−1 100 101 102 > < ν HV Channel Figure2: Probabilityoffalsealarmasafunctionoftheor- derparameterν ofthecluttertexturedistribution. > < Time OsborneHeadGunneryRange(OHGR),Dartmouth,Nova Scotia, Canada, with the McMaster University IPIX radar [12]. Specifically, we use the data recordedon November Figure3: MagnitudeinrangeandtimedomainoftheIPIX 11, 1993. Fig. 3 shows the magnitudeof the echoes from radardatasetstare1,collectedonNovember11,1993. The asmalltargetininhomogeneousseaclutterforthefourpo- targetlocationisindicatedbythemarkers“>”and“<”. larimetricchannels:HH,VH,VV,andHV. Fig. 4 shows detection maps for the T , PST-GLR, GLR TF-GLR, and Hybrid U tests. The detection threshold is again set for P = 10−3, and other parameters are also Our Pol Detector FA as in the former subsection. For the latter three tests, a guard region of two range cells on either side of the cell undertest isused to avoidthe leakageof thetargetenergy > < intothesecondarydata. Weobservethatourdetectiontest Baseline: PST−GLR finds the true position of the target with only a few false detections despite the strong clutter echoes, which can be appreciatedfrom Fig. 3. On the otherhand, the PST-GLR > < hashighfalse-alarmrateduetothepresenceofinhomoge- e g neous clutter, as was predicted in the previoussubsection. an Baseline: TF−GLR R TheHybridUandTF-GLRtestsreducethefalse-alarmrate withrespecttothePST-GLRtest; however,itstill remains considerablyhigh. > < One possible way of summarizing the performance of Baseline: Hybrid U (Ch VV) thesedetectiontestsinrelationtorealdataistoplottheem- piricalreceiveroperatingcharacteristic(ROC)curves. The empirical ROC is obtained by dividing the detection map > < intworegions. Oneregioncorrespondstotherangewhere Time thetargetislocatedandtheotherregioncorrespondstothe range where there is only clutter. For different values of the detection threshold γ, the empirical P and P are Figure4: Detectionmapin rangeandtime domainforthe D FA computedbycountingthepixelsoneachrespectiveregion. IPIXradardatasetstare1,collectedonNovember11,1993. Fig.5showsthattheempiricalROCcurveoftheT test Blackpixelsindicatethatthedetectionstatisticislargerthan GLR isonthetop,outperformingtheotheralgorithms. thethreshold. Thetargetlocationisindicatedbythemark- WeacknowledgethatcomparingtheHybridUandTF- ers“>”and“<”. GLRtestswiththeothertwoisnotstrictlyfair,sincethese 6. REFERENCES 1 0.9 [1] H. Park, J. Li, and H. Wang, “Polarization-space-time domain generalized likelihood ratio detection of radar 0.8 n targets,” SignalProcessing, Vol. 41, pp. 153-164,Jan. o0.7 cti 1995. e et0.6 D of 0.5 [2] D.Pastina,P.Lombardo,andT.Bucciarelli,“Adaptive y bilit0.4 polarimetrictargetdetectionwithcoherentradar.PartI: ba Detection against Gaussian background,” IEEE Trans. Pro0.3 Aerosp.Electr.Syst.,Vol.37,pp.1194-1206,Oct.2001. Our Pol Detector 0.2 TF−GLR PST−GLR [3] A. De Maio and G. Ricci, “A polarimetric adaptive 0.1 Hybrid U (Ch HH) Hybrid U (Ch VV) matched filter,” Signal Processing, Vol. 81, pp. 2583- 0 0 0.2 0.4 0.6 0.8 1 2589,Dec.2001. 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