Novel multistatic adaptive microwave imaging methods for early breast cancer detection

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Novel multistatic adaptive microwave imaging methods for early breast cancer detection
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EURASIP Journal on Applied Signal Processing
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Xie, Yao
Guo, Bin
Li, Jian
Stoica, Petre
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Multistatic adaptive microwave imaging (MAMI) methods are presented and compared for early breast cancer detection. Due to the significant contrast between the dielectric properties of normal and malignant breast tissues, developing microwave imaging techniques for early breast cancer detection has attracted much interest lately. MAMI is one of the microwave imaging modalities and employs multiple antennas that take turns to transmit ultra-wideband (UWB) pulses while all antennas are used to receive the reflected signals. MAMI can be considered as a special case of the multi-input multi-output (MIMO) radar with the multiple transmitted waveforms being either UWB pulses or zeros. Since the UWB pulses transmitted by different antennas are displaced in time, the multiple transmitted waveforms are orthogonal to each other. The challenge to microwave imaging is to improve resolution and suppress strong interferences caused by the breast skin, nipple, and so forth. The MAMI methods we investigate herein utilize the data-adaptive robust Capon beamformer (RCB) to achieve high resolution and interference suppression.We will demonstrate the effectiveness of our proposed methods for breast cancer detection via numerical examples with data simulated using the finite-difference time-domain method based on a 3D realistic breast model.
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Publication of this article was funded in part by the University of Florida Open-Access publishing Fund. In addition, requestors receiving funding through the UFOAP project are expected to submit a post-review, final draft of the article to UF's institutional repository, IR@UF, (www.uflib.ufl.edu/ufir) at the time of funding. The Institutional Repository at the University of Florida (IR@UF) is the digital archive for the intellectual output of the University of Florida community, with research, news, outreach and educational materials

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HindawiPublishingCorporation EURASIPJournalonAppliedSignalProcessing Volume2006,ArticleID91961,Pages 1 – 13 DOI10.1155/ASP/2006/91961NovelMultistaticAdaptiveMicrowaveImagingMethods forEarlyBreastCancerDetectionYaoXie,1BinGuo,1JianLi,1andPetreStoica21DepartmentofElectricalandComputerEngineering,UniversityofFlorida,P.O.Box116200,Gainesville,FL32611-6200,USA2SystemsandControlDivision,DepartmentofInformationTechnology,UppsalaUniversity,P.O.Box337, 75105Uppsala,Sweden Received19October2005;Accepted21December2005 Multistaticadaptivemicrowaveimaging(MAMI)methodsarepresentedandcomparedforearlybreastcancerdetection.Dueto thesignicantcontrastbetweenthedielectricpropertiesofnormalandmalignantbreasttissues,developingmicrowaveimaging techniquesforearlybreastcancerdetectionhasattractedmuchinterestlately.MAMIisoneofthemicrowaveimagingmodalities andemploysmultipleantennasthattaketurnstotransmitultra-wideband(UWB)pulseswhileallantennasareusedtoreceive thereectedsignals.MAMIcanbeconsideredasaspecialcaseofthemulti-inputmulti-output(MIMO)radarwiththemultiple transmittedwaveformsbeingeitherUWBpulsesorzeros.SincetheUWBpulsestransmittedbydi erentantennasaredisplaced intime,themultipletransmittedwaveformsareorthogonaltoeachother.Thechallengetomicrowaveimagingistoimprove resolutionandsuppressstronginterferencescausedbythebreastskin,nipple,andsoforth.TheMAMImethodsweinvestigate hereinutilizethedata-adaptiverobustCaponbeamformer(RCB)t oachievehighresolutionandinterferencesuppression.Wewill demonstratethee ectivenessofourproposedmethodsforbreastcancerdetectionvianumericalexampleswithdatasimulated usingthenite-di erencetime-domainmethodbasedona3Drealisticbreastmodel. Copyright2006HindawiPublishingCorporation.Allrightsreserved.1.INTRODUCTION Breastcancertakesatremendoustollonoursociety.Onein eightwomenintheUSwillgetbreastcancerinherlifetime [ 1 ].Eachyearmorethan200000newcasesofinvasivebreast cancerarediagnosedandmorethan40000womendiefrom thediseaseintheUSalone[ 1 ].Earlydiagnosisiscurrently thebesthopeofsurvivingbreastcancer. Currently,X-raymammographyisthestandardroutine breastcancerscreeningtool.However,thee ectivenessofXraymammographyhasbeenquestionedbycertainsources inrecentyearsandissomewhatcurrentlyunderdebatedue toitsinherentlimitationsinresolvingbothlow-andhighcontrastlesionsandmassesinradiologicallydenseglandularbreasttissues.Breasttissuesofyoungerwomentypically presentahigherratioofdensetofattytissues,limitingthe e ectivenessofX-raymammography.Hencemammographypresentsitsmajorlimitationinthesectorofthepopulationofhighestpublichealthinterestandcriticality.Some techniquessuchasmagneticresonanceimaging(MRI)and Positronemissiontomography(PET)haveledtoanincrease intheidenticationofsmallabnormalitiesinthehuman breast,butthewidespreaduseofMRIandPETforroutine breastcancerscreeningisunlikelyduetotheirhighcosts. Ultra-wideband(UWB)confocalmicrowaveimaging (CMI)isoneofthemostpromisingandattractivenew screeningtechnologiescurrentlyunderdevelopment:itis nonionizing(safe),noninvasive(comfortable),sensitive(to tumors),specic(tocancers),andlow-cost[ 2 ].Itsphysical basisliesinthesignicantcontrastinthedielectricpropertiesbetweennormalandmalignantbreasttissues[ 3 – 7 ].In CMI,UWBpulsesaretransmittedfromantennasatdi erentlocationsnearthebreastsurfaceandthebackscattered responsesfromthebreastarerecorded,fromwhichtheimageofthebackscatteredenergydistributionisreconstructed coherently. Thedataacquisitionapproachesandtheassociatedsignal processingmethodsa ecttheCMIimagingquality.There arethreemajordataacquisitionschemes:monostatic[ 8 ], bistatic[ 9 10 ],andmultistatic[ 11 ].FormonostaticCMI, thetransmitterisalsousedasareceiverandismovedacross thebreasttoformasyntheticaperture.ForbistaticCMI, onetransmittingandonereceivingantennaareusedasa pairandmovedacrossthebreasttoformasyntheticaperture.FormultistaticCMI,arealaperturearray(see Figure1 ) isusedfordatacollection.Eachantennainthearraytakes turnstotransmitaprobingpulse,andallantennas(insome cases,allexceptthetransmittingantenna)areusedtoreceive

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2EURASIPJournalonAppliedSignalProcessing Tumor Antennaarray x y z Figure 1:Antennaarrayconguration.thebackscatteredsignals.MultistaticCMIcanbeconsideredasaspecialcaseofthewidebandmulti-inputmultioutput(MIMO)radar[ 12 – 14 ]withthemultipletransmittedwaveformsbeingeitherUWBpulsesorzeros.Sincethe UWBpulsestransmittedbydi erentantennasaredisplaced intime,themultipletransmittedwaveformsareorthogonal toeachother.Themonostaticandbistaticschemesexploit thetransmitterspatialdiversity,andthemultistaticscheme takesadvantageofthetransmitter-and-receiverspatialdiversity.Themultistaticapproachcangivebetterimagingresults thanitsmono-orbistaticcounterpartswhenthesynthetic apertureformedbythelattertwoapproachesissimilarto therealaperturearrayusedbytheformer.Anintuitiveexplanationwouldbethatthemultistaticapproachutilizesthe receiverdiversityaswell,bysimultaneouslyrecordingmultiplereceivedsignalsthatpropagateviadi erentroutesand henceaccruesmoreinformationaboutthetumor. ThechallengetoCMIimagingistodevisesignalprocessingalgorithmstoimproveresolutionandsuppressstrong interferencescausedbythebreastskin,nipple,andso forth.Signalprocessingalgorithmscanbeclassiedasdatadependent(data-adaptive)anddata-independentmethods. Formono-andbistaticultra-widebandCMI,thesimple data-independentdelay-and-sum(DAS)[ 8 11 ],thedataindependentmicrowaveimagingspace-time(MIST)beamforming[ 15 ],thedata-adaptiverobustCaponbeamforming (RCB)[ 9 10 ],aswellasthedata-adaptiveamplitudeand phaseestimation(APES)[ 9 10 ]methodshavebeenconsideredforimageformation.Formultistaticultra-wideband CMI,theDAS-[ 11 ]andRCB-basedadaptive[ 16 ]methodshavebeenconsidered.Thedata-adaptivemethodscan havebetterresolutionandmuchbetterinterferencesuppressioncapabilityandcansignicantlyoutperformtheirdataindependentcounterparts. Inthispaper,weconsidermultistaticadaptivemicrowave imaging(MAMI)methodstoformimagesofthebackscatteredenergyforearlybreastcancerdetection.Foralocation ofinterest(orfocalpoint) r withinthebreast,thecomplete recordedmultistaticdatacanberepresentedbyacube,as shownin Figure2 .In[ 16 ],weproposedaMAMIapproach, iM Transmitter index t0NŠ1 Time index Receiverindexslicing MAMI-1 MAMI-2 Receiver index M Figure 2:MultistaticCMIdatacubemodel.InStageI,MAMI-1 slicesthedatacubeforeachtimeindex,whereasMAMI-2slicesthe datacubeforeachtransmitterindex.ThenRCBisappliedtoeach dataslicetoobtainmultiplewaveformestimates.referredtoMAMI-1herein,whichisatwo-stagetimedomainsignalprocessingalgorithmformultistaticCMI.In StageI,MAMI-1slicesthedatacubecorrespondingtoeach timeindex,andprocessesthedataslicebytherobustCapon beamformer(RCB)[ 17 – 19 ]toobtainbackscatteredwaveformestimatesateachtimeinstant.Basedontheseestimates, inStageIIascalarwaveformisretrievedviaRCB,theenergyofwhichisusedasanestimateofthebackscatteredenergyforthefocalpoint.MAMI-1hasbeenshowntohave betterperformancethanotherexistingmethods.AnalternativewayofslicingthedatacubeinStageIbeforeapplying RCBistoselectaslicecorrespondingtoeachtransmitting antennaindex(see Figure2 ).Theso-obtainedapproachis referredtoasMAMI-2herein.WewillshowthatMAMI2tendstoyieldbetterimagesthanMAMI-1forhighinput signal-to-interference-noiseratio(SINR),butworseimages atlowSINR.WewillalsoshowthatcombiningMAMI-1and MAMI-2yieldsgoodperformanceinallcasesofSINR.We refertothecombinedmethodasMAMI-Cherein. WewilldemonstratetheperformanceoftheMAMI methodsusingdatasimulatedwiththenitedi erencetime domain(FDTD)method.Thesimulatedbreastmodelsconsideredintheliteratureincludeatwo-dimensional(2D) modelbasedonabreastMRIscan[ 8 15 ],simplethreedimensional(3D)andplanarmodels[ 20 ],andthemorerealistic3Dmodel[ 9 10 21 ].Oursimulationsarebasedon the3Dhemisphericalbreastmodel.Thetumorresponsefor therealistic3Dmodelismuchsmallerthanthatforthe2D (or3Dcylindrical)modelduetotumorbeingassumedinnitelylonginthelattermodel.TheMAMImethodscandetecttumorsassmallas4mmindiameterbasedontherealistic3Dmodel.Basedon2Dmodels,theMAMImethodscan detecttumorsassmallas1.5mmindiameter.Wehaveonly

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YaoXieetal. 3 includedtherealistic3D-model-basedexampleshereinsince theconclusionsdrawnfrom2Dbasedmodelsaresimilar. Thefollowingnotationwillbeused:()Tdenotesthe transpose, RmnstandsfortheEuclideanspaceofdimension mn B0meansthat B ispositivesemidenite,bold lowercasesymbolsrepresentvectors,andboldcapitalletters representmatrices. 2.DATAMODEL Weconsideramultistaticimagingsystem,where K antennas arearrangedonahemisphererelativelyclosetothebreast skin,atknownlocations.Thecongurationofthearrayis shownin Figure1 .Theantennasarearrangedon P layers with Q antennasperlayer,where K=PQ .Eachantenna takesturnstotransmitanUWBprobingpulsewhileallof theantennasareusedtorecordthebackscatteredsignals.Let xi j( t ), i=1, ... K j=1, ... K t=0, ... NŠ1,denotethe backscatteredsignalgeneratedbytheprobingpulsesentby the i thtransmittingantennaandreceivedbythe j threceiving antenna,where t denotesthetimesample.The31vector r denotesthefocalpoint(i.e.,animaginglocationwithinthe breast).Inouralgorithms,thelocation r isvariedtocoverall gridpointsofthebreastmodel. Ourgoalistoforma3Dimageofthebackscatteredenergy E ( r )onagridofpointswithinthebreast,withthescope ofdetectingthetumor.Thebackscatteredenergyisestimated fromthecompletereceiveddata{xi j( t )}foreachlocation r ofinterest. Beforeimageformation,wepreprocessthereceivedsignals{xi j( t )}toremove,asmuchaspossible,backscatteredsignalsotherthanthetumorresponse,toalignallthe recordedsignalsfrom r bytime-shifting,andtocompensate forthepropagationlossofthesignalamplitude.(See[ 16 ]for details.)Thepreprocessedsignals yi j( t )obtainedfrom xi j( t ) canbedescribedas yi j( t )=si j( t )+ ei j( t ), i j=1, ... K t=0, ... NŠ1, (1) where si j( t )representsthetumorresponseand ei j( t )representstheresidualterm.Theresidualterm ei j( t )includesthe thermalnoiseandtheinterferenceduetoundesiredreectionsfromthebreastskin,nipple,andsoforth.Tocast( 1 )in aformsuitablefortheapplicationofRCB[ 17 ],weapproximatethedatamodel( 1 )bymakingdi erentassumptions.In thefollowingweuse t ( t=0, ... NŠ1)todenoteageneric giventimeindex,and i ( i=1, ... K )todenoteageneric giventransmitterindex. MAMI-1approximatesthedatamodel( 1 )as yi( t )=a ( t ) si( t )+ ei( t ),(2) where yi( t )=[ yi ,1( t ), ... yi K( t )]Tand ei( t )=[ ei ,1( t ), ... ei K( t )]T.Thescalar si( t )denotesthebackscatteredsignal (fromthefocalpointatlocation r )correspondingtothe probingsignalfromthe i thtransmittingantenna.Thevector a ( t )in( 2 )isreferredtoasthearraysteeringvector.Notethat a ( t )isapproximatelyequalto 1K1sinceallthesignalshave beenalignedtemporallyandtheirattenuationscompensated forinthepreprocessingstep. Therearethreeassumptionsmadetowritethemodelin ( 2 ).First,thesteeringvectorisassumedtovarywith t ,but benearlyconstantwithrespectto i (theindexofthetransmittingantenna).Second,weassumethatthebackscattered signalwaveformdependsonlyon i butnoton j (theindexof thereceivingantenna).Thetruth,however,isthatthesteeringvectorisnotexactlyknownanditchangesslightlywith both t and i duetoarraycalibrationerrorsandotherfactors. Thesignalwaveformcanalsovaryslightlywithboth i and j ,duetothe(relativelyinsignicant)frequency-dependent lossymediumwithinthebreast.Thetwoaforementioned assumptionssimplifytheproblemslightly.Theycauselittle performancedegradationswhenusedwithourrobustadaptivealgorithms.Third,weassumethattheresidualtermis uncorrelatedwiththesignal. MAMI-2approximatesthedatamodel( 1 )di erentlyas follows: yi( t )=aisi( t )+ ei( t ),(3) where aidenotesthesteeringvector,whichisagainapproximately 1K1.Thesecondandthirdassumptionsusedtoobtain( 2 )arealsomadetoobtain( 3 ).However,MAMI-2assumesthatthesteeringvectorvarieswith i ,butisconstant withrespectto t Inpractice,thesteeringvectors a ( t )and aimaybeimprecise,inthesensethattheirelementsmaydi erslightly from1.Thisuncertaintyinthesteeringvectormotivatesus toconsiderusingRCBforwaveformestimation.Becausethe steeringvectorsin( 2 )and( 3 )arebothapproximately 1K1, weassumethatthetruesteeringvector a ( t )or ailiesinuncertaintyspheres,thecentersofwhicharetheassumedsteering vector a=1K1.(Forthemoregeneralcaseofellipsoidaluncertaintysets,see[ 19 ]andthereferencestherein.)Theonly knowledgeweassumeabout a ( t )and aiis,respectively,that a ( t )Š a 2 1, aiŠ a 2 2, (4) where1and2areusedtodescribetheamountofuncertaintyin a ( t )and ai,respectively. Thechoiceoftheuncertaintysizeparameters,1and2, aswellasoftheircounterpartsinStageIIofMAMI-1and MAMI-2(seebelow),isdeterminedbyseveralfactorssuch asthesamplesize N andthearraycalibrationerrors[ 17 18 ]. First,theyshouldbemadeassmallaspossible.Otherwisethe abilityofRCBtosuppressaninterferencethatisclosetothe signalofinterestwillbelost.Second,Thesmallerthe N or thelargerthesteeringvectorerrors,thelargershouldthey bechosen.Third,toavoidtrivialsolutiontotheoptimizationproblemofRCB,theyshouldbelessthanthesquareof

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4EURASIPJournalonAppliedSignalProcessing thenormoftheassumedsteeringvector[ 17 18 ].Suchqualitativeguidelinesareusuallysu cientforthechoiceofthe uncertaintysizeparameters,sincetheperformanceofRCB doesnotdependverycriticallyonthem(aslongastheytake on“reasonablevalues”)[ 19 ].Inournumericalexamples,we choosecertainreasonableinitialvaluesforthemandthen makesomeadjustmentsempiricallybasedonimagingqualities(i.e.,makingthemsmallerwhenthecurrentresultedimageshavelowresolutionorlotsofclutter,ormakingthem largerwhenthetargetinthecurrentresultedimagesappears tobesuppressedtoo). 3.MAMI-1ANDMAMI-2 InStageI,bothMAMI-1andMAMI-2obtain K signalwaveformestimatesviaRCB.InStageIofMAMI-2,forthe i th probingpulse,thetruesteeringvector aicanbeestimated viathecovariancettingapproachofRCB: max2 i, ai2 isubjecttoRYiŠ2 iaiaT i0, aiŠ a 2 2, (5) where 2 iisthepowerofthesignalofinterest,andRYi=1 N YiYT i(6) isthesamplecovariancematrixwith Yi= yi(0), yi(1), ... yi( NŠ1), YiRKN. (7) ByusingtheLagrangemultipliermethod,thesolutiontothis optimizationproblemisgivenby[ 17 ]ai= aŠ I + RYiŠ1 a ,(8) where 0isthecorrespondingLagrangemultiplierthat canbesolvede cientlyfromthefollowingequation(e.g., usingtheNewtonmethod): I + RYiŠ1 a 2= ,(9) sincetheleft-handsideof( 9 )ismonotonicallydecreasingin (see[ 17 ]formoredetails).Afterdeterminingthemultiplier ,aiisdeterminedby( 8 ).Toeliminateascalingambiguity (see[ 17 ]),wescaleaitomake ai2=M .Thenwecanapply thefollowingweightvectortothereceivedsignals(see[ 17 ] fordetails):w2, i= ai K1 / 2 RYi+(1 / ) IŠ1 a aT RYi+(1 / ) IŠ1RYi RYi+(1 / ) IŠ1 a (10) toobtainthecorrespondingsignalwaveformestimate.Note that( 10 )hasadiagonalloadingform,whichcanbeused evenwhenthesamplecovariancematrixisrank-decient. Thebeamformeroutputcanbewrittenasthevectorsi= wT 2, iYiT,siRN1,(11) whichisthewaveformestimateofthebackscatteredsignal (fromthexedlocation r )forthe i thprobingsignal.Repeatingtheaboveprocessfor i=1through i=K ,weobtain thecompletesetof K waveformestimatesS2=[s1, ... ,sK]T,S2RKN. Similarly,inStageIofMAMI-1,weobtainasetofwaveformestimatesS1=[s (0), ... ,s ( NŠ1)],S1RKN(see [ 16 ]fordetails). NotethatStageIofbothMAMI-1andMAMI-2yields K waveformestimatesofthebackscatteredsignals(one foreachtransmittingantenna).Let{ s1( t )}t=0, ... NŠ1,and{ s2( t )}t=0, ... NŠ1denotethecolumnsofthematricesS1andS2,respectively.Sinceallprobingsignalshavethesamewaveform,weassumethatthetruebackscatteredsignalwaveformsare(nearly)identical.Thismeansthat,forexample, forMAMI-2,theelementsofthevectors2( t )areallapproximatelyequaltoanunknown(scalar)signal s ( t ).SoinStage II,wecanemployRCBtorecoverascalarwaveforms ( t ) from{ s1( t )}or{ s2( t )}(see[ 16 ]formoredetailsonStage IIofMAMI-1;StageIIofMAMI-2issimilar).Finally,the backscatteredenergy E ( r )iscomputedas E ( r )=NŠ1t=0s2( t ) (12) Itiswellknownthattheerrorsinsamplecovariancematrices(e.g.,theRYiabove)andthesteeringvectorscauseperformancedegradationsinadaptivebeamforming[ 22 23 ]. Notethat,ononehand,MAMI-2usesmoresnapshots (namely, N )thanMAMI-1(namely, K )toestimatethesamplecovariancematrix.Therefore,thesamplecovariancematrixofMAMI-2ismoreprecisethanthatofMAMI-1.On theotherhand,MAMI-1employsRCB N times,whereas MAMI-2usesRCB K times(recallthat N>K ),sothere ismore“room”forrobustnessinMAMI-1thaninMAMI-2, whichmeansthatMAMI-1shouldbemorerobusttosteeringvectorerrors.Insummary,MAMI-2usesamoreprecisesamplecovariancematrix,whereasMAMI-1ismorerobustagainststeeringvectormismatch.Therefore,according towhatwassaidabove,athighinputSINR(whenthesample covariancematrixerrorsaremoreimportant)wecanexpect MAMI-2toperformbetterthanMAMI-1,andviceversaat lowinputSINR(whentheerrorsinthesteeringvectorare critical). 4.MAMI-C ThepreviousintuitivediscussionsonMAMI-1andMAMI-2 andthenumericalexamplespresentedlateronimplythat

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YaoXieetal. 5 MAMI-2hasbetterperformanceathighSINR,whileMAMI1usuallyoutperformsMAMI-2atlowSINR.ThisfactmotivatesustoconsidercombiningMAMI-1andMAMI-2to achievegoodperformanceinallcasesofSINR.Inthecombinedmethod,whichisreferredtoasMAMI-C,weuse thetwosetsof K waveformestimatesyieldedbyStageIof MAMI-1andStageIofMAMI-2simultaneouslyinStageII (notethatMAMI-1andMAMI-2haveasimilarStageII).In thiswaythecombinedmethodincreasesthenumberof“ctitious”arrayelementsfrom K to2 K ThecombinedsetofestimatedwaveformsisdenotedbySC=[ST 1ST 2]T,SCR2 KN,wherethesubscript()Cstands forMAMI-C.Letthe2 K1vectors{ s ( t )}t=0, ... NŠ1denotethe columnsofSC.StageIIofMAMI-Cconsistsofrecoveringa scalarwaveformfrom{ s ( t )}. Thevectors ( t )istreatedasasnapshotfroma2 K element(ctitious)“array”:s ( t )=aCs ( t )+ eC( t ), t=0, ... NŠ1,(13) where aCisassumedtobelongtoanuncertaintysetcentered ata=12 K1,and eC( t )representstheestimationerror.UsingRCB,weestimate aCandthenobtaintheadaptiveweight vectorviaanexpressionsimilarto( 10 ):wC= aC K1 / 2 RC+(1 / ) IŠ1a aT RC+(1 / ) IŠ1RC RC+(1 / ) IŠ1a (14) where isthecorrespondingLagrangemultiplier(see[ 17 ] formoredetails),andRCisthefollowingsamplecovariance matrix:RC=1 NNŠ1t=0s ( t )sT( t ) (15) Thebeamformeroutputgivesanestimateofthesignalof interest:s ( t )= wT Cs ( t ) (16) Finally,thebackscatteredenergyatlocation r iscomputed using( 12 ). Remark1. ItisnaturaltocomeupwithathirdwayofslicingthedatacubeinStageIbeforeapplyingRCB:toselect aslicecorrespondingtoeachreceivingantennaindex(see Figure2 ).OurnumericalexamplesshowthattheperformanceofthismethodissimilartothatofMAMI-2.Moreover,wecanusethewaveformestimatesfromthisapproach togetherwiththoseestimatedinStageIofMAMI-1and StageIofMAMI-2toestimateascalarwaveform.However, numericalexamplesshowthatsuchacombinationprovides nosignicantimprovementoverMAMI-C,butthecomputationalcomplexitiesincreaseduetotheincreaseddatadimensioninStageII.Therefore,wewillnotconsiderthisoptionanyfurtherhereafter. 5.NUMERICALEXAMPLES Weconsidera3Dbreastmodelasin[ 16 ]inournumericalexamples.Themodelincludesrandomlydistributedfatty breasttissue,glandulartissue,2mm-thickbreastskin,aswell asthenippleandchestwall.Toreducethereectionsfrom thebreastskin,thebreastmodelisimmersedinalossless liquidwithpermittivitysimilartothatofthebreastfattytissue[ 24 ].Thebreastmodelisahemispherewith10cmin diameter.Atumorthatis6mm(or4mm)indiameterislocated2 7cmundertheskin(at x=70mm, y=90mm, z=60mm).Twocross-sectionsofthe3Dmodelareshown in Figure3 Weassumethatthedielectricproperties(permittivity andconductivity)ofthebreasttissuesareGaussianrandom variableswithameanequaltotheirnominalvaluesanda varianceequalto0.1timestheirmeanvalues.Thisvariation representsanupperboundonreportedbreasttissuevariabilities[ 4 5 ].Thenominalvaluesarechosentobethetypical valuesreportedintheliterature[ 3 – 7 ],asshownin Table1 SinceUWBpulsesareusedasprobingsignals,thedispersive propertiesofthefattybreasttissueandthoseofthetumorare alsoconsideredinthemodel.Thefrequencydependencies ofthepermittivity ( )andconductivity ( )aremodelled accordingtoasingle-poleDebyemodel[ 8 ].Therandomly distributedbreasttissueswithvariabledielectricproperties representthephysicalnonhomogeneityofthehumanbreast. Asshownin Figure1 ,theantennaarrayconsistsof K=72elementsthatarearrangedonahemisphere,whichis1cm awayfromthebreastskin,on P=6layersinthe z -axisdimension.Thelayersofantennasarearrangedalongthe z -axis between5.0cmand7.5cm,with0.5cmspacing.Withineach layer, Q=12antennasareplacedonacross-sectionalcircle withuniformspacing.TheUWBsignalusedisaGaussian pulsegivenby G ( t )=exp Š tŠ0 2,(17) where 0=25 s, =10 s,andthepulsewidthisroughly 120ps.Eachantennaofthearraytakesturnstotransmitthe Gaussianprobingpulse,andall72antennasareusedtoreceivethebackscatteredsignals. FDTD[ 25 26 ]isusedtoobtainthesimulateddata.The gridcellsizeusedis1mm1mm1mmandthetimestep is1 667ps.Themodelisterminatedaccordingtoperfectly matchedlayerabsorbingboundaryconditions[ 27 ].The Z transform[ 28 ]isusedtoimplementtheFDTDmethod whenevermaterialswithfrequency-dependentpropertiesare

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6EURASIPJournalonAppliedSignalProcessing 20406080100120140160180 x (mm) Model: xŠy planeat z=6cm 20 40 60 80 100 120 140 160 180y (mm)Glandular tissue Fattissue Tumor Skin Immersion liquid (a) 20406080100120140160180 x (mm) Model: xŠz planeat y=9cm 20 40 60 80 100 120z (mm)Glandular tissue Fattissue Tumor Skin Immersion liquid Chestwall (b)Figure 3:Cross-sectionsofa3Dhemisphericalbreastmodelat(a) z=60mmand(b) y=90mm.involved.Finally,thetimewindowinthepreprocessingstep consistsof150samples,whichmeansthat N=150foreach ofthepreprocessedsignals. TheperformancecomparisonsofMAMI-1withother existingmethodscanbefoundin[ 16 ].Inthefollowing, wefocusoncomparingMAMI-1withtheothertwoMAMI methods. Inthefollowingexamples,weaddwhiteGaussiannoise withzero-meananddi erentvariancevalues 2 0tothereceivedsignals.WedeneSNR(signal-to-noiseratio)as SNR=10log10 1 /K2 K i=1K j=1(1 /N )NŠ1 t=0 x2 i j( t ) 2 0dB, (18) andSINRas SINR=10log10 1 /K2 K i=1K j=1(1 /N )NŠ1 t=0 x2 i j( t ) 1 /K2 K i=1K j=1(1 /N )NŠ1 t=0 I2 i j( t )+ 2 0dB (19) The xi j( t )in( 19 )isthereceivedsignalduetothetumoronly, and Ii j( t )isduetotheinterferencefrombreastskin,nipple, andsoforth(withouttumorresponse),bothofwhichare notavailableinpractice.TocomputeSNRandSINR,weperformedthesimulationtwice,withandwithoutthetumor,regardedthesecondsetofreceivedsignalsasinterferenceonly,Table 1:Nominaldielectricpropertiesofbreasttissues. Tissues Dielectricproperties Permittivity(F/m)Conductivity(S/m) Immersionliquid 90 Chestwall 507 Skin 364 Fattybreasttissue 90.4 Nipple 455 Glandulartissue 11–150.4–0.5 Tumor 504 andusedthedi erencebetweenthetwosetsofreceivedsignalstoapproximate xi j( t ).Alltheimagesaredisplayedon alogarithmicscalewithadynamicrangeof40dB(notethat herethedynamicrangeusedislargerthanthe20dBdynamic rangein[ 16 ]). Figures 4 and 5 showtheCMIimagesofa6mm-diameter tumor,atlowandhighthermalnoiselevels,respectively. Atthelownoiselevel(SNR=12 1dB,SINR=Š1 4dB), theimagesproducedbyMAMI-2havemuchmorefocused tumorresponsesthanthoseofMAMI-1.Theimagesof MAMI-ChavesimilarqualitiestothoseofMAMI-2.In Figure5 ,atthehighnoiselevel(SNR=Š13 8dB,SINR= Š14 1dB),MAMI-1yieldsbetterimagesthanMAMI-2,and thatMAMI-CisslightlybetterthanMAMI-1.ThisexampledemonstratesthatMAMI-Cinheritsthemeritsofboth MAMI-1andMAMI-2. Figures 6 and 7 showtheimagesofa4mm-diametertumorwithdi erentthermalnoiselevels.Thebackscattered microwaveenergy,whichisproportionaltothesquareofthe tumordiameter,ismuchlessinthiscasethanintheprevious

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YaoXieetal. 7 20406080100120140160180 x (mm) Image: xŠy planeat z=6cm 20 40 60 80 100 120 140 160 180y (mm)Š40Š35Š30Š25Š20Š15Š10Š5 0 (a) 20406080100120140160180 x (mm) Image: xŠz planeat y=9cm 20 40 60 80 100 120z (mm)Š40Š35Š30Š25Š20Š15Š10Š5 0 (b) 20406080100120140160180 x (mm) Image: xŠy planeat z=6cm 20 40 60 80 100 120 140 160 180y (mm)Š40Š35Š30Š25Š20Š15Š10Š5 0 (c) 20406080100120140160180 x (mm) Image: xŠz planeat y=9cm 20 40 60 80 100 120z (mm)Š40Š35Š30Š25Š20Š15Š10Š5 0 (d) 20406080100120140160180 x (mm) Image: xŠy planeat z=6cm 20 40 60 80 100 120 140 160 180y (mm)Š40Š35Š30Š25Š20Š15Š10Š5 0 (e) 20406080100120140160180 x (mm) Image: xŠz planeat y=9cm 20 40 60 80 100 120z (mm)Š40Š35Š30Š25Š20Š15Š10Š5 0 (f)Figure 4:Thecross-sectionimagesofthe6mm-diametertumor,atlownoiselevel(SNR=12 1dB,SINR=Š1 4dB).(a)and(b)MAMI-C; (c)and(d)MAMI-2with2=7;(e)and(f)MAMI-1with1=3.(Inallofourexamples,thegiven2and1areusedforbothstages.)example.Thatis,ifthethermalnoiseleveliskeptthesameas inthe6mm-diametertumorcase,boththeSNRandSINR willbemuchlowerinthe4mm-diametercase,whichpresentsachallengetoanyimageformationalgorithm.In Figure 6 ,atalownoiselevel(SNR=1 5dB,SINR=Š12 5dB), MAMI-2andMAMI-CyieldimagesofcomparablequalitiesandtheyoutperformMAMI-1. Figure7 showstheimagesproducedviatheMAMImethodsatahighnoiselevel (SNR=Š24 5dB,SINR=Š24 8dB).Onceagain,MAMICyieldsthebestimages.

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8EURASIPJournalonAppliedSignalProcessing 20406080100120140160180 x (mm) Image: xŠy planeat z=6cm 20 40 60 80 100 120 140 160 180y (mm)Š40Š35Š30Š25Š20Š15Š10Š5 0 (a) 20406080100120140160180 x (mm) Image: xŠz planeat y=9cm 20 40 60 80 100 120z (mm)Š40Š35Š30Š25Š20Š15Š10Š5 0 (b) 20406080100120140160180 x (mm) Image: xŠy planeat z=6cm 20 40 60 80 100 120 140 160 180y (mm)Š40Š35Š30Š25Š20Š15Š10Š5 0 (c) 20406080100120140160180 x (mm) Image: xŠz planeat y=9cm 20 40 60 80 100 120z (mm)Š40Š35Š30Š25Š20Š15Š10Š5 0 (d) 20406080100120140160180 x (mm) Image: xŠy planeat z=6cm 20 40 60 80 100 120 140 160 180y (mm)Š40Š35Š30Š25Š20Š15Š10Š5 0 (e) 20406080100120140160180 x (mm) Image: xŠz planeat y=9cm 20 40 60 80 100 120z (mm)Š40Š35Š30Š25Š20Š15Š10Š5 0 (f)Figure 5:Thecross-sectionimagesofthe6mm-diametertumor,athighnoiselevel(SNR=Š13 8dB,SINR=Š14 1dB).(a)and(b) MAMI-C;(c)and(d)MAMI-2with2=7;(e)and(f)MAMI-1with1=3.Finally, Figure8 presentsthe3Dimagesofthe6mmaswellasthe4mm-diametertumor.The3Dimages,althoughnotasclearvisuallyasthecross-sectionalimages,illustratethereconstructedbackscatteredenergyoutsidethe twocross-sectionalplanes.Hereweonlyshowthe3Dimages forthelow-noise-levelcases.Intheseguresthetruetumor locationsaremarkedwithsmall“+”s.InFigures 8(a) and 8(d) ,whichcorrespondtotheimagesproducedbyMAMI-C,

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YaoXieetal. 9 20406080100120140160180 x (mm) Image: xŠy planeat z=6cm 20 40 60 80 100 120 140 160 180y (mm)Š40Š35Š30Š25Š20Š15Š10Š5 0 (a) 20406080100120140160180 x (mm) Image: xŠz planeat y=9cm 20 40 60 80 100 120z (mm)Š40Š35Š30Š25Š20Š15Š10Š5 0 (b) 20406080100120140160180 x (mm) Image: xŠy planeat z=6cm 20 40 60 80 100 120 140 160 180y (mm)Š40Š35Š30Š25Š20Š15Š10Š5 0 (c) 20406080100120140160180 x (mm) Image: xŠz planeat y=9cm 20 40 60 80 100 120z (mm)Š40Š35Š30Š25Š20Š15Š10Š5 0 (d) 20406080100120140160180 x (mm) Image: xŠy planeat z=6cm 20 40 60 80 100 120 140 160 180y (mm)Š40Š35Š30Š25Š20Š15Š10Š5 0 (e) 20406080100120140160180 x (mm) Image: xŠz planeat y=9cm 20 40 60 80 100 120z (mm)Š40Š35Š30Š25Š20Š15Š10Š5 0 (f)Figure 6:Theimagesofthe4mm-diametertumor,atlownoiselevel(SNR=1 5dB,SINR=Š12 5dB).(a)and(b)MAMI-C;(c)and(d) MAMI-2withS=8 5;(e)and(f)MAMI-1withM=5.andin 8(b) and 8(e) ,whichcorrespondtotheimagesproducedbyMAMI-2,besidesthetumorresponses,noclutteris clearlyvisible.Figures 8(c) and 8(f) showtheMAMI-1images;particularlyinthelatterimage,clutteraboundswithin thebreastvolume. 6.CONCLUSIONS Wehavepresentedandcomparedseveralmultistaticadaptive microwaveimaging(MAMI)methodsforearlybreastcancerdetection.TheMAMImethodsutilizethedata-adaptive

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10EURASIPJournalonAppliedSignalProcessing 20406080100120140160180 x (mm) Image: xŠy planeat z=6cm 20 40 60 80 100 120 140 160 180y (mm)Š40Š35Š30Š25Š20Š15Š10Š5 0 (a) 20406080100120140160180 x (mm) Image: xŠz planeat y=9cm 20 40 60 80 100 120z (mm)Š40Š35Š30Š25Š20Š15Š10Š5 0 (b) 20406080100120140160180 x (mm) Image: xŠy planeat z=6cm 20 40 60 80 100 120 140 160 180y (mm)Š40Š35Š30Š25Š20Š15Š10Š5 0 (c) 20406080100120140160180 x (mm) Image: xŠz planeat y=9cm 20 40 60 80 100 120z (mm)Š40Š35Š30Š25Š20Š15Š10Š5 0 (d) 20406080100120140160180 x (mm) Image: xŠy planeat z=6cm 20 40 60 80 100 120 140 160 180y (mm)Š40Š35Š30Š25Š20Š15Š10Š5 0 (e) 20406080100120140160180 x (mm) Image: xŠz planeat y=9cm 20 40 60 80 100 120z (mm)Š40Š35Š30Š25Š20Š15Š10Š5 0 (f)Figure 7:Theimagesofthe4mm-diametertumor,athighnoiselevel(SNR=Š24 5dB,SINR=Š24 8dB).(a)and(b)MAMI-C;(c)and (d)MAMI-2with2=8 5;(e)and(f)MAMI-1with1=5.robustCaponbeamformer(RCB)toachievehighresolution andinterferencesuppression.WehavedemonstratedtheeffectivenessoftheMAMImethodsforearlybreastcancerdetectionvianumericalexampleswithdatasimulatedusingthe nitedi erencetimedomainmethodbasedona3Drealisticbreastmodel.WehaveshownthattheMAMI-Cmethod candetecttumorsassmallas4mmindiameterbasedonthe realisticallysimulated3Dbreastmodel.

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YaoXieetal. 11 180 140 100 60 20 x (mm) 20 40 60 80 100 120z (mm)50 100 150 y (mm) (a) 180 140 100 60 20 x (mm) 20 40 60 80 100 120z (mm)50 100 150 y (mm) (b) 180 140 100 60 20 x (mm) 20 40 60 80 100 120z (mm)50 100 150 y (mm) (c) 180 140 100 60 20 x (mm) 20 40 60 80 100 120z (mm)50 100 150 y (mm) (d) 180 140 100 60 20 x (mm) 20 40 60 80 100 120z (mm)50 100 150 y (mm) (e) 180 140 100 60 20 x (mm) 20 40 60 80 100 120z (mm)50 100 150 y (mm) (f)Figure 8:The3Dimagesofthe6mm-diametertumor,atlownoiselevel(SNR=12 1dB,SINR=Š1 4dB),obtainedvia(a)MAMI-C,(c) MAMI-2,(e)MAMI-1.Also,the3Dimagesofthe4mm-diametertumor,atlownoiselevel(SNR=1 5dB,SINR=Š12 5dB),obtained via(b)MAMI-C,(d)MAMI-2,(f)MAMI-1.Theshadedhemisphereisthecontourofthebreast,andthedottedshadeswithinthebreast correspondtohighbackscatteredenergy.Thesmall“+”marksthetruelocationofthetumor.ACKNOWLEDGMENT ThisworkwassupportedinpartbytheNationalInstitutesof Health(NIH)Grantno.1R41CA107903-1andtheSwedish ScienceCouncil(VR). REFERENCES[1]S.J.Nass,I.C.Henderson,andJ.C.Lashof, Mammography andBeyond:DevelopingTechniquesfortheEarlyDetectionof BreastCancer ,InstituteofMedicine,NationalAcademyPress, Washington,DC,USA,2001. [2]E.C.Fear,S.C.Hagness,P.M.Meaney,M.Okoniewski,and M.A.Stuchly,“Enhancingbreasttumordetectionwithneareldimaging,” IEEEMicrowaveMagazine ,vol.3,no.1,pp.48– 56,2002. [3]C.Gabriel,R.W.Lau,andS.Gabriel,“Thedielectricpropertiesofbiologicaltissues:II.Measurementsinthefrequency range10Hzto20GHz,” PhysicsinMedicineandBiology vol.41,no.11,pp.2251–2269,1996. [4]S.S.Chaudhary,R.K.Mishra,A.Swarup,andJ.M.Thomas, “Dielectricpropertiesofnormalandmalignanthumanbreast tissuesatradiowaveandmicrowavefrequencies,” IndianJournalofBiochemistryandBiophysics ,vol.21,pp.76–79,1984.

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12EURASIPJournalonAppliedSignalProcessing [5]W.T.Joines,Y.Zhang,C.Li,andR.L.Jirtel,“Themeasured electricalpropertiesofnormalandmalignanthumantissues from50to900MHz,” MedicalPhysics ,vol.21,no.4,pp.547– 550,1994. [6]A.J.Surowiec,S.S.Stuchly,J.R.Barr,andA.Swarup,“Dielectricpropertiesofbreastcarcinomaandthesurrounding tissues,” IEEETransactionsonBiomedicalEngineering ,vol.35, no.4,pp.257–263,1988. [7]A.Swarup,S.S.Stuchly,andA.J.Surowiec,“DielectricpropertiesofmouseMCA1brosarcomaatdi erentstagesofdevelopment,” Bioelectromagnetics ,vol.12,no.1,pp.1–8,1991. [8]X.LiandS.C.Hagness,“Aconfocalmicrowaveimagingalgorithmforbreastcancerdetection,” IEEEMicrowaveandWirelessComponentsLetters ,vol.11,no.3,pp.130–132,2001. [9]B.Guo,Y.Wang,J.Li,P.Stoica,andR.Wu,“Microwaveimagingviaadaptivebeamformingmethodsforbreastcancerdetection,”in ProceedingsofProgressinElectromagneticsResearch Symposium(PIERS’05) ,Hangzhou,China,August2005. [10]B.Guo,Y.Wang,J.Li,P.Stoica,andR.Wu,“Microwaveimagingviaadaptivebeamformingmethodsforbreastcancerdetection,” JournalofElectromagneticWavesandApplications vol.20,no.1,pp.53–63,2006. [11]R.Nilavalan,A.Gbedemah,I.J.Craddock,X.Li,andS.C. Hagness,“Numericalinvestigationofbreasttumourdetection usingmulti-staticradar,” IEEElectronicsLetters ,vol.39,no.25, pp.1787–1789,2003. [12]E.Fishler,A.Haimovich,R.Blum,D.Chizhik,L.Cimini,and R.Valenzuela,“MIMOradar:anideawhosetimehascome,” in ProceedingsofIEEERadarConference ,pp.71–78,Philadelphia,Pa,USA,April2004. [13]E.Fishler,A.Haimovich,R.Blum,D.Chizhik,L.Cimini,and R.Valenzuela,“Spatialdiversityinradars—modelsanddetectionperformance,”toappearin IEEETransactionsonSignal Processing [14]L.Xu,J.Li,andP.Stoica,“RadarImagingviaAdaptiveMIMO Techniques,”in Proceedingsof14thEuropeanSignalProcessing Conference(EUSIPCO’06) ,Florence,Italy,September2006, ftp://www.sal.u.edu/xuluzhou/EUSIPCO2006.pdf [15]E.J.Bond,X.Li,S.C.Hagness,andB.D.VanVeen,“Microwaveimagingviaspace-timebeamformingforearlydetectionofbreastcancer,” IEEETransactionsonAntennasand Propagation ,vol.51,no.8,pp.1690–1705,2003. [16]Y.Xie,B.Guo,L.Xu,J.Li,andP.Stoica,“Multi-staticadaptivemicrowaveimagingforearlybreastcancerdetection,”in Proceedingsof39thASILOMARConferenceonSignals,Systems andComputers ,PacicGrove,Calif,USA,October2005. [17]J.Li,P.Stoica,andZ.Wang,“OnrobustCaponbeamforming anddiagonalloading,” IEEETransactionsonSignalProcessing vol.51,no.7,pp.1702–1715,2003. [18]P.Stoica,Z.Wang,andJ.Li,“RobustCaponbeamforming,” IEEESignalProcessingLetters ,vol.10,no.6,pp.172–175,2003. [19]J.LiandP.Stoica,Eds., RobustAdaptiveBeamforming ,John Wiley&Sons,NewYork,NY,USA,2005. [20]E.C.Fear,X.Li,S.C.Hagness,andM.A.Stuchly,“Confocal microwaveimagingforbreastcancerdetection:localizationof tumorsinthreedimensions,” IEEETransactionsonBiomedical Engineering ,vol.49,no.8,pp.812–822,2002. [21]E.C.FearandM.Okoniewski,“Confocalmicrowaveimagingforbreastcancerdetection:Applicationtohemisphericalbreastmodel,”in ProceedingsofIEEEMTT-SInternational MicrowaveSymposiumDigest ,vol.3,pp.1759–1762,Seattle, Wash,USA,June2002. [22]R.A.MonzingoandT.W.Miller, IntroductiontoAdaptiveArrays ,JohnWiley&Sons,NewYork,NY,USA,1980. [23]D.D.FeldmanandL.J.Gri ths,“Aprojectionapproachfor robustadaptivebeamforming,” IEEETransactionsonSignal Processing,vol.42,no.4,pp.867–876,1994. [24]P.M.Meaney,“Importanceofusingareducedcontrastcouplingmediumin2Dmicrowavebreastimaging,” Journalof ElectromagneticWavesandApplications ,vol.17,no.2,pp.333– 355,2003. [25]D.M.Sullivan, ElectromagneticSimulationUsingFDTD Method ,Wiley/IEEEPress,NewYork,NY,USA,1stedition, 2000. [26]A.TaoveandS.C.Hagness, ComputationalElectrodynamics:TheFinite-Di erenceTime-DomainMethod ,ArtechHouse, Boston,Mass,USA,3rdedition,2005. [27]S.D.Gedney,“AnanisotropicperfectlymatchedlayerabsorbingmediumforthetruncationofFDTDlattices,” IEEE TransactionsonAntennasandPropagation ,vol.44,no.12,pp. 1630–1639,1996. [28]D.M.Sullivan,“ Z -transformtheoryandtheFDTDmethod,” IEEETransactionsonAntennasandPropagation ,vol.44,no.1, pp.28–34,1996. YaoXie receivedtheB.S.degreeinelectrical engineeringandinformationsciencefrom theUniversityofScienceandTechnologyof China(USTC),Hefei,China,in2004.She iscurrentlypursuingthePh.D.degreewith theDepartmentofElectricalandComputer EngineeringattheUniversityofFlorida, Gainesville.SheisaMemberofTauBetaPi andEttaKappaNu.Shewastherst-place winnerintheStudentBestPaperContest atthe2005AnnualAsilomarConferenceonSignals,Systems,and Computers,forherworkonbreastcancerdetection.Herresearch interestsincludesignalprocessing,medicalimaging,andarraysignalprocessing. BinGuo receivedtheB.E.andM.S.degreesinelectricalengineeringfromXian JiaotongUniversity,Xian,China,in1997 and2000,respectively.FromApril2002to July2003,hewasanAssociateResearchScientistwiththeTemasekLaboratories,NationalUniversityofSingapore,Singapore. SinceAugust2003,hehasbeenaResearch AssistantwiththeDepartmentofElectrical andComputerEngineering,Universityof Florida,Gainesville,whereheispursuingthePh.D.degreeinelectricalengineering.Hiscurrentresearchinterestsincludebiomedicalapplicationsofsignalprocessing,microwaveimaging,andcomputationalelectromagnetics. JianLi receivedtheM.S.andPh.D.degrees inelectricalengineeringfromtheOhio StateUniversity,Columbus,in1987and 1991,respectively.FromJuly1991toJune 1993,shewasanAssistantProfessorwith theDepartmentofElectricalEngineering, UniversityofKentucky,Lexington.Since August1993,shehasbeenwiththeDepartmentofElectricalandComputerEngineering,UniversityofFlorida,Gainesville, wheresheiscurrentlyaProfessor.Hercurrentresearchinterests includespectralestimation,statisticalandarraysignalprocessing,

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YaoXieetal. 13 andtheirapplications.Dr.LiisaFellowofIEEEandaFellowof IEE.Shereceivedthe1994NationalScienceFoundationYoungInvestigatorAwardandthe1996O ceofNavalResearchYoungInvestigatorAward.ShehasbeenaMemberoftheEditorialBoard ofSignalProcessing,apublicationoftheEuropeanAssociationfor SignalProcessing(EURASIP),since2005.SheispresentlyaMemberoftwooftheIEEESignalProcessingSocietytechnicalcommittees:theSignalProcessingTheoryandMethods(SPTM)Technical CommitteeandtheSensorArrayandMultichannel(SAM)TechnicalCommittee. PetreStoica receivedtheM.S.andPh.D.degreesinautomaticcontrolfromthePolytechnicInstituteofBucharest,Bucharest, Romania,in1972and1979,respectively. HeiscurrentlyaProfessorofsystemmodelingatUppsalaUniversity,Uppsala,Sweden.Otherdetailsabouthimareavailable at http://user.it.uu.se/ps/ps.html .