不变矩阵视觉模式识别英文文献翻译.docx

不变矩阵视觉模式识别英文文献翻译.docx

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不变矩阵视觉模式识别英文文献翻译

VisualPatternRecognitionbyMomentInvariants

MING-KUEIHU,SENIORMEMBER,IRE

Summary

Inthispaperatheoryoftwo-dimensionalmomentinvariantsforplanargeometricfiguresispresented.Afundamentaltheoremisestablishedtorelatesuchmomentinvariantstothewellknownalgebraicinvariants.Completesystemsofmomentinvariantsundertranslation,similitudeandorthogonaltransformationsarederived.Somemomentinvariantsundergeneraltwo-dimensionallineartransformationsarealsoincluded.

Boththeoreticalformulationandpracticalmodelsofvisualpatternrecognitionbaseduponthesemomentinvariantsarediscussed.ASimplesimulationprogramtogetherwithitsperformancearealsopresented.Itisshownthatrecognitionofgeometricalpatternsandalphabeticalcharactersindependentlyofposition,sizeandorientationcanbeaccomplished.Itisalsoindicatedthatgeneralizationispossibletoincludeinvariancewithparallelprojection.

I.INTRODUCTION

Recognitionofvisualpatternsandcharactersindependentofposition,size,andorientationinthevisualfieldhasbeenagoalofmuchrecentresearch.Toachievemaximumutilityandflexibility,themethodsusedshouldbeinsensitivetovariationsinshapeandshouldprovideforimprovedperformancewithrepeatedtrials.Themethodpresentedinthispapermeetsalltheseconditionstosomedegree.

Ofthemanyingeniousandinterestingmethodssofardevised,onlytwomaincategorieswillbementionedhere:

1）Theproperty-listapproach,and2）Thestatisticalapproach,includingboththedecisiontheoryandrandomnetapproaches[1].Theproperty-listmethodworksverywellwhenthelistisdesignedforaparticularsetofpatterns.Intheory,itistrulyposition,size,andorientationindependent,andmayalsoallowforothervariations.Itsseverelimitationisthatitbecomesquiteuseless,ifadifferentsetofpatternsispresentedtoit.Thereisnoknownmethodwhichcangenerateautomaticallyanewproperty-list.Ontheotherhand,thestatisticalapproachiscapableofhandlingnewsetsofpatternswithlittledifficulty,butitislimitedinitsabilitytorecognizepatternsindependentlyofposition,sizeandorientation.

Thispaperreportsthemathematicalfoundationoftwodimensionalmomentinvariantsandtheirapplicationstovisualinformationprocessing[2].Theresultsshowthatrecognitionschemesbasedontheseinvariantscouldbetrulyposition,sizeandorientationindependent,andalsoflexibleenoughtolearnalmostanysetofpatterns.

Inclassicalmechanicsandstatisticaltheory,theconceptofmomentsisusedextensively;centralmoments,sizenormalization,andprincipalaxesarealsoused.Totheauthor’sknowledge,thetwo-dimensionalmomentinvariants,absoluteaswellasrelative,thataretobepresentedhavenotbeenstudied.Inthepatternrecognitionfield,centroidandsizenormalizationhavebeenexploited[3]-[5]for“preprocessing.”Orientationnormalizationhasalsobeenattempted[5].Themethodpresentedhereachievesorientationindependencewithoutambiguitybyusingeitherabsoluteorrelativeorthogonalmomentinvariants.Themethodfurtheruses“momentinvariants”（tobedescribedinIII）orinvariantmoments（momentsreferredtoapairofuniquelydeterminedprincipalaxes）tocharacterizeeachpatternforrecognition.

SectionIIgivesdefinitionsandpropertiesoftwodimensionalmomentsandalgebraicinvariants.Themomentinvariantsundertranslation,similitude,orthogonaltransformationsandalsounderthegenerallineartransformationsaredevelopedinSectionIII.TwospecificmethodsofusingmomentinvariantsforpatternrecognitionaredescribedinIV.Asimulationprogramofasimplemodel（programmedforanLGP-30）,theperformanceoftheprogram,andsomepossiblegeneralizationsaredescribedinSectionV.

II.MOMENTSANDALGEBRAICINVARIANTS

A.AUniquenessTheoremConcerningMoments

Inthispaper,thetwo-dimensional（p+q）thordermomentsofadensitydistributionfunctionρ（x,y）aredefinedintermsofRiemannintegralsas

Ifitisassumedthatρ（x,y）isapiecewisecontinuousthereforeboundedfunction,andthatitcanhavenonzerovaluesonlyinthefinitepartofthexyplane;thenmomentsofallordersexistandthefollowinguniquenesstheoremcanbeproved.

UniquenessTheorem:

Thedoublemomentsequence{mpq}isuniquelydeterminedbyρ（x,y）;andconversely,ρ（x,y）isuniquelydeterminedby{mpq}.

Itshouldbenotedthatthefinitenessassumptionisimportant;otherwise,theaboveuniquenesstheoremmightnothold.

B.CharacteristicFunctionandMomentGeneratingFunction

Thecharacteristicfunctionandmomentgeneratingofρ（x,y）maybedefined,respectively,as

Inbothcases,uandvareassumedtobereal.Ifmomentsofallordersexist,thenbothfunctionscanbeexpandedintopowerseriesintermsofthemomentsmpqas