DTIC ADA475588: Polarimetric Detection of Targets in Inhomogeneous Clutter

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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
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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
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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 transmit-
ted. 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
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4. CONCLUSIONS
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2005.
5. APPENDIX
[12] S. Haykin, C. Krasnor, T. J. Nohara, B. W. Currie,
Thetarget-to-clutterratioandclutter-to-noiseratioare andD.Hamburger,“Acoherentdual-polarizedradarfor
studyingtheoceanenvironment,”IEEETrans.Geosci.
||µ||2 tr(Σ)
TCR= , CNR= , RemoteSensing,Vol.29,pp.189-191,Jan.1991.
tr(Σ) σ
where||·||isthenormofthevector.