Generalised-structure-tensor-based-infrared-small-

Generalised-structure-tensor-basedinfraredsmalltargetdetectionCh.-Q.Gao,J.-W.TianandP.WangAnefficientmethodbasedongeneralisedstructuretensorispresentedtodetectinfraredsmalltargetsundercomplexskybackgrounds.Thesecondconjugatesymmetryderivativeofa2DGaussianisappliedtotheorientationfieldtomodelthesmalltargetandthefinaltargetdetectionisaccomplishedbyadaptivesegmentation.Experimentsshowthatthemethodisrobust.Introduction:Infrared(IR)smalltargetdetectionisakeytechniqueinareaslikespacesurveillancesystems,early-warningsystemsandmissiletrackingsystemsetc.ItisachallengingtasktodetectIRsmalltargetsundercomplexskybackgroundbecauseofcloudclutter.Theconventionalmethodsareusuallybasedoncluttersuppression,suchasthe2Dleastmeansquare(TDLMS)adaptivefilter[1],the2Dnormal-isedLMS(TDNLMS)filter[2],themediansubtractionfilter[3],etc.InthisLetter,weusethesecondconjugatesymmetryderivativeofa2DGaussian(SCSDTDG)tomodelthesmalltargetpatternbasedonthegeneralisedstructuretensor(GST).Experimentsshowthattheproposedmethodisrobust.Foundationoftheory:AneighbourhoodfL(x,y)ina2Dcontinuousimagehasperfectlinearsymmetry(LS)propertyifitsisocurveshaveacommondirection,i.e.thereexistsascalarfunctionofonevariablegsuchthatfL(x,y)¼g(kTr)wherekisatwo-dimensionalrealvectorthatisconstantwithregardtotheco-ordinatevectorr¼(x,y)T[4].WhetherornotaneighbourhoodfL(x,y)hasperfectLSpropertycanbedeterminedbyeigenanalysisofthestructuretensor(ST),whichisdefinedasfollows[4]:SðfLÞ¼ÐÐðDxfLÞ2dxdyÐÐðDxfLÞðDyfLÞdxdyÐÐðDxfLÞðDyfLÞdxdyÐÐðDyfLÞ2dxdy!ð1ÞwhereDxandDyarepartialderivativeoperators.ThecomplexscalarImn(k)¼ÐÐ(xþiy)m(x–iy)nk(x,y)dxdywithmandnbeingnonnegativeintegers,isthecomplexmomentm,nofthefunctionk.Theordernumberofacomplexmomentreferstomþn.Forconvenienceofanalysis,theSTisrepresentedbysecond-ordercomplexmoments,I20andI11,asfollows[4]:I20ðjFj2Þ¼ðlmaxlminÞei2f¼ðððvxþivyÞ2jFj2dvxdvy¼ððððDxþiDyÞfLÞ2dxdyð2ÞI11ðjFj2Þ¼lmaxþlmin¼ðððvxþivyÞðvxivyÞjFj2dvxdvy¼ðððDxþiDyÞfL2dxdyð3ÞwherejFj2isthepowerspectrumoftheimagefL,lmaxandlminarethemaximumandminimumeigenvaluesofS(fL),respectively,andwisthedirectionofthemajoreigenvector.TheSchwartzinequalityjI20jjI11jholdswithequalityifandonlyiffLhasperfectLS,wherethecommondirectionoftheisocurvesisdeterminedbyw.Foranintricatepattern,theimagecanbedeformedviaco-ordinatetransformation(CT)withcorrespondingharmonicfunctionpairj¼j(x,y)andh¼h(x,y)sothatithastheLSpropertyunderanewco-ordinate.Therefore,theGSTholdinginharmonicco-ordinatesisdefinedasfollows[4]:I020¼ðlmaxlminÞei2w¼ððððDjþiDhÞf0LÞ2djdhð4ÞI011¼lmaxþlmin¼ðððDjþiDhÞf0L2djdhð5Þwheref0Listhetransformedlocalimage,lmax;lminandwaretheeigen-valuesandthedirectionofthemajoreigenvectoroftheST,respectively.OurinterestisindiscretecomputationoftheGSTandthedetaileddeductionofalternatingbetweenitscontinuousversionanddiscreteversioncanbefoundin[4].Infact,theGSTdoesnotnecessitateexplicitCT.Instead,theCTisimplicitlyencodedinacomplex-valuedfilter.ThediscreteschemeoftheGSTforallneighbourhoodsinanimageisdefinedasfollows[4]:I0020¼CnhðG1;s1fgkfkÞ2;I0011¼CnhjjG1;s1fgkfk2ð6Þwherethesymbolindicates2Dconvolution,fkistheoriginalimage,Gkf1,s1gisthenormaliseddiscreteversionofthefirstsymmetryderivativeof2DGaussian[4],whichiscomputedthroughG1;s1fgkðxk;ykÞ¼Gs1gðxk;ykÞPiPjGs1gði;jÞ;whereGs1gðxk;ykÞ¼ðxkþiykÞ2ps41eðx2kþy2kÞ2s21ð7ÞCnisaconstantamountingtoauniformscalingofthegreyvaluegamutoftheoriginalimage;histhediscretefilterencodingCTimplicitlyandthusisadetectorforcorrespondingpattern.GST-basedIRsmalltargetdetection:Inourcase,sincetheIRsmalltargetisclosetoaspotstructure,wechoosetheSCSDTDGasthefilterh,whichcorrespondstoLSinapolarco-ordinateandgivesameasureofcircularsymmetry(CS).Thenormalisedversionofhiscom-putedthroughhðxk;yk;s2Þ¼hs2gðxk;ykÞPiPjhs2gði;jÞ;wherehs2gðxk;ykÞ¼ðxkiykÞ22ps62eðx2kþy2kÞ2s22ð8ÞIn(7)and(8),elementsofeachfilteraredividedbythe1-normofthecorrespondingmatrixtoensurethatmagnitudesofelementsareintherange[0,1].TodetectIRsmalltargets,thecertaintyimageCTiscomputedthroughCTðxk;ykÞ¼ðl2maxl2minÞð1þcosuÞ¼I0020ðxk;ykÞI0011ðxk;ykÞð1þcosuÞð9ÞwhereuisthephaseofthecomplexnumberI0020(xk,yk).Equation(9)hasamaximumwhenlmin¼0andu¼0,whichcorrespondstoperfectLSinpolarco-ordinates,i.e.thepatternhasperfectCSinCartesianco-ordi-nates,inotherwords,theremostprobablyexistsatargetatthepoint(xk,yk)intheoriginalimagefk.Thedetectionalgorithmisdescribedasfollows:Step1:ComputetheorientationfieldZ¼(DxfkþiDyfk)2byconvolu-tion,Gkf1,s1gfk,andthenpixel-wisecomplexsquaring,andcomputethecomplexfilterH¼Cnh.Step2:ComputethecompleximageI0020,(6),byconvolution,HZ.Step3:ComputethescalarimageI0011,(6),byconvolvingthemagnitudeoftheorientationfieldZwiththemagnitudeofthecomplexfilterHowingtoGf1;s1gkfk2¼ðGf1;s1gkfkÞ2.Step4:ComputethecertaintyimageCT,(9),usingtheimageI0020andI0011.Step5:Assign0totheboundaries(5pixelswidth)oftheimageCTforeliminatingtheinfluenceoftheboundaries.Step6:SegmenttheimageCTbyadaptivethresholddeterminedbyT¼maxðTmin;0:67TmaxÞ,whereTmaxisthemaximalvalueoftheimageCT,andTminisaconstantdeterminedexpe