Local invariant feature histograms for texture cla

InternalReport3/97,Albert-Ludwigs-UniversitatFreiburg,IIF-LMB,Germany,December1997LocalInvariantFeatureHistogramsforTextureClassicationS.Siggelkow,H.BurkhardtAlbert-Ludwigs-UniversitatFreiburgInstitutfurInformatik79085Freiburgi.Br.,[e-mail]fortextureclassicationbasedoninvari-antgrayscalefeatures.Thesefeaturesremainconstantiftheimagesaretransformedaccordingtotheactionofatransformationgroup.Thebasicmethodappliedforex-tractinginvariantfeatures,isgivenbyanintegrationoverthetransformationgroup.ForthetransformationgroupofplanarorEuclideanmotion(translationandrota-tion)onecanshow,thattheintegrationcanbesplitintotwoparts:Therstistheevaluationofanonlinearlocalfunctionforeverypixeloftheimage,andthesecondthesummingoftheresultsoftheselocalcomputations.Insteadofthesecondstepwecalculateahistogramofthelocalcomputationswhichpreservestheinvariancepropertyandismorerobusttorealtexturedeviationsthanasinglefeature.Fur-thermoreinamultidimensionalhistogramapproachthecombinationofdierentfeaturescanbeperformed,thusincreasingthediscriminationpower.1IntroductionTextureisanimportantcharacteristicofmanytypesofimages.Neverthelesstheredoesn’texistaformalapproachorexactdenitionoftexture.Howeverthefollowingroughdef-initionwillbeusefulinthecontextofthispaper:Atextureisdescribedbytwoparts[1]:graylevelprimitiveelementsandaspatialorganisationofthese.Thegraylevelprimitivesmaybesinglepixelsorregionswithgraylevelproperties.Thespatialorganisationmayberandom,dependononeoneormoreprimitives,periodicetc.ThereportedworkwassupportedbytheEuropeanESPRIT/ITProject20229NOBLESSE-Non-linearModel-BasedAnalysisandDescriptionofImagesforMultimediaApplications1Thereexistseveralapproachesfortextureclassication:Amongotherstherearemethodsbasedoncoocurrencematrices,autocorrelation,digitaltransformmethods(e.g.[2]),tex-turaledgeness,morphologicalmethods(e.g.[3]),fractalsignatures,andMarkovrandomeldmodels.Agoodoverviewovertheseexistingmethodsisgivenin[1].Wewanttopresentanewmethodbasedonlocalinvariantgrayscalefeatures.Bycon-structingtranslationandrotationinvariantfeatureswecircumventtheproblemofsomeoftheabovementionedmethodsthatneedtoextractthetexturefeaturesforseveraldirections.Intheconstructionofourinvariantfeaturesweexplicitlyregardontheonehandthegrayvalueinformationandontheotherhandthestructuralinformation.Thusthisapproachconsidersbothimportantpartsoftheabovegivendenitionoftexture.Inthenextsectionwewanttosummarizethekeypointsofamethodfortheconstruc-tionofinvariantimagefeaturesbyanintegrationoverthetransformationgroupofimagerotationsandtranslations.Insection3thesefeatureswillbemodiedtoconstructinvari-antfeaturehistogramswhicharemorerobustthantheoriginalfeatures.Furthermoreweintroduceanewmultidimensionalhistogramwithaweightedassignmentruleinordertoavoidtheunsteadyassignmentatthehistogrambinboundariesthatisgivenfortradi-tionalhistograms.Insection4wethenperformexperimentalresults,ontheonehandtoverifytheinvarianceproperty,andontheotherhandwetestourmethodwithseveralrealfeatures,thatnaturallydeviatefromtheidealmodelofapuretranslationandrotation.Theconclusiongiveninsection5summarizesthekeypointsofourtechniqueandgivesanoutlookforfurtherexperiments.2InvariantimagefeaturesInthissectionwewanttosumupkeypointsofamethodfortheconstructionofinvariantfeaturesforgrayscaleimagesdescribedin[4].Wewilluseespeciallyfeatureswhichareinvariantwithrespecttoimagerotationsandtranslations.2.1InvariantfeaturesforgrayscaleimagesLetM=fM(i;j)g,0iN;0jMbeagrayscaleimage.ThevalueM(i;j)representsthegrayvalueatthepixelcoordinate(i;j).WithGwedenoteatransformationgroupwithelementsg2Gactingontheimages.ForanimageMandagroupelementg2GthetransformedimageisdenotedbygM.GivenagrayscaleimageMandanelementg2Gofthegroupofimagerotationsandtranslations,anangle’2[0;2)andatranslationvectort=(t0;t1)T2IR2existssothat(gM)(i;j)=M(k;l)with(1)kl!=cos’sin’sin’cos’!ij!t0t1!:(2)AllindicesareunderstoodmoduloNorM.Notethatduetothisconventiontherangeofthetranslationvectort=(t0;t1)T2IR2canberestrictedto0t0N;0t1M.2AninvariantfeatureisacomplexvaluedfunctionF(M)whichisinvariantwithrespecttotheactionofthetransformationgroupontheimages,i.e.F(gM)=F(M)8g2G:(3)2.2ConstructionofinvariantfeaturesForagivengrayscaleimageMandacomplexvaluedfunctionf(M)itispossibletoconstructaninvariantfeatureA[f](M)byintegratingf(gM)overthetransformationgroupG:A[f](M)=ZGf(gM)dg:(4)Thisaveragingtechniqueforconstructinginvariantfeaturesisexplainedindetailin[5]forgeneraltransformationgroups.ForthegroupofimagerotationandtranslationtheintegraloverthegroupcanbewrittenasA[f](M)=12NMNZt0=0MZt1=02Z’=0f(gM)d’dt1dt0:(5)Togivesomeintuitiveinsightinequation(5)considerthefollowingexample:Forthefunctionf(M)=M(0;0)M(0;1)thegroupaverageisA[f](M)=12NMNZt0=0MZt1=02Z’=0M(t0;t1)M(sin’t0;cos’t1)d’dt1dt0:(6)Letusrstconsidertheinnerintegralover’.Thecoordinate(sin’t0;cos’t1)describesfor0’2acircleofradiusonearound(t0;t1).Thismeanswehavetodetermineallpixelswhichhavedistanceonefromt