Handlingofimprecisenessingraylevelcornerdetectionusingfuzzysettheoreti - PDF document

HandlingofimprecisenessingraylevelcornerdetectionusingfuzzysettheoreticapproachMinakshiBanerjee,MalayK.KunduMachineIntelligenceUnit,CenterforSoftComputingResearch,IndianStatisticalInstitute,203,B.T.Road,Kolkata700108,IndiaReceived20March2007;receivedinrevisedform3April2007;accepted13September2007Availableonline12February2008Reliablecornerdetectionisanimportanttaskindeterminingshapeofdifferentregionsinanimage.Todetectcornersinagraylevelimageunderimpreciseinformation,analgorithmbasedonfuzzysettheoreticmodelisproposed.Theuncertaintiesarisingduetovarioustypesofimagingdefectssuchasblurring,illuminationchange,noise,etc.,usuallyresultinmissingofsigniÞcantcurvaturejunctions(corners).Fuzzysettheor A vailable online at www.sciencedirect.com AppliedSoftComputing8(2008)1680Ð1691 *Correspondingauthor.+913325753108;fax:+9133257833.E-mailaddresses:minakshi_@isical.ac.in(M.Banerjee),rmalay@isical.ac.in(M.K.Kundu).MinakshiBanerjeeisgratefultotheDepartmentofScienceandTechnol-ogy,NewDelhi,India,forsupportingtheworkundergrant(SR/WOS-A/ET-1568-4946/$Ðseefrontmatter2008ElsevierB.V.Allrightsreserved.10.1016/j.asoc.2007.09.001 Cornerisanimportantfeatureusedinvariousimageanalysisapplications[12,13].Detectionofcorners,usingclassiÞcationbasedapproach[14,15]andfuzzyreasoningbasedmethod[16,17]bothforcolorimagessandgraylevelimagesisanimportantresearchissue.Reallifeimagedataarealwaysimpreciseduetoinherentuncertaintiesthatmayarisefromtheimagingprocess(suchasdefocusing,noise,widevariationsofilluminations,etc.).AsaresultpreciselocalizationanddetectionofcornersbecamedifÞcultundersuchimperfectsituations.Fuzzylogicgiccanbeusedforpossibleprecisemeasurementwithimprecisedata[20,21]Inthispaper,wehaveproposedanalgorithmtoextractsigniÞcantgraylevelcornerpointsbasedonfuzzysettheoreticapproach.Thehighcurvaturepointslocatedatthediscontinuitiesbetweendifferentuniformintensitysurfaces,constitutethefuzzycornerset.Themeasureofcornernessvarieswithfuzzyedgestrengthandgradientdirection.Differentsetoffuzzycornersareobtainedusingdifferentvaluesofthresholdonthefuzzyedgemap.Theuncertaintiesinlocatingthecornerspointswhichmayarise,duetodiscretization,noiseandotherimagingdefects,arehandledwithfuzzymodel.TherobustnessoftheproposedalgorithmisexperimentallyveriÞedusingbothsimulatedimagedataandnaturalimages,tojustifythesuitabilityofthealgorithm.Thepaperisorganizedasfollows:Sectiondescribesthemathematicalmodelusedinthiswork.Sectiondescribesthefeaturesextractionprocess.Sectiondescribesthefuzzycornerextractionprocess.Sectiondescribestheexperimentalresults.Sectiongivesaconclusion.2.MathematicalmodelingofgraylevelcornersImageasfuzzysets:animageofsize,withLgraylevelscanbeconsideredasafuzzysubset()inaspaceofwithacontinuumgradeofmembership.WhereeachpointinXcanbecharacterizedbyamembershipfunction¼fðwhere0Thiskindofimagerepresentationisusefultohandletheuncertaintiesarisingoutofgraylevelaswellasspatialdigitizationtion.Afuzzysubset()isdeÞnedintermsofthemembershipvaluesbetween[0Ð1].OneofthemostwidelyusedmappingfunctiontodofuzziÞcationforconvertingadigitalimagetocorrespondingfuzzysubset,isthestandardfunction,deÞnedas ðxaÞðcaÞ;axb12 Fig.1showsitsgraphicalrepresentation,wheretheparameteristhecrossoverpoint,i.e.,SimilarlyisdeÞnedastheshoulderpointatwhich0andisthefeetpointi.e.fuzzyalphacut:Afuzzysubsetcanbedividedbysuitablethresholdingofmembershipvaluesaroundtherangeofinterest.Thefuzzyalpha-cut,comprisesallelementsofdegreeofmembershipinisgreaterorequaltowhere0Plateautop,plateaubottom:Inanimage,edgesarethetransitionsbetweentwouniformintensitysurfacesdeÞnedasPlateauss.Letdenotethesetofallpixelsinanimage.Thepixels.Byaplateauin,ismeantamaximumconnectedsubsetonwhichtheintensity()hasaconstantvalue.InotherwordsisaPlateauifisconnected.(ii)forallforallpairofneighboringpoints,i.e.,wherebelongstooneplateau.APlateauisatop,ifitsgrayvalueisalocalmaximumforallpairsofneighboringpointi.e,.Similarlywecallabottom,ifitsgrayvalueisalocalminimum.Thepixelsinborderregion)canbedeÞnedasthepointswhichareeightneighborsofatleastoneelementof.ThepixelsarelabeledaspixelsofaPlateauTop,BottomandBorder,considering33neighborhoododaroundeachpixel.3.ExtractionoffuzzyedgemapandcharacteristiclocalGraylevelimagesareinherentlyfuzzyinnature.Evenforperfectlyhomogeneousobjectsthecorrespondingimageswillhavegradedcompositionofgraylevelsduetoimperfectionofimaging.Thebasicnotionbehindtheproposedalgorithmisthat,adigitalimagecanbethoughtofas2Dplane,wherethereareridgesorvalleys[25,26].Thisistrue,whentherearesimplyconnectedsequenceofpixelshavinggraytoneintensityvalues Fig.1.S-typemembershipfunction.M.Banerjee,M.K.Kundu/AppliedSoftComputing8(2008)1680–1691 signiÞcantlyhigher(lower)inthesequencethantheneighboringpixels.Desiredfeaturescanthereforebeobtainedbyextractingandassemblingtopographiccharacteristicsofintensitysurfaces.Thebasicassumptionisthat,cornerpointsarehighcurvaturepointsandshouldlieongrayleveledges.ItshouldhavesigniÞcantchangeinedgedirectionwithlineararmsupportofconsiderablelengthonbothsides.3.1.FeatureextractionThefeaturecomputationprocessconsistsoftwophases.IntheÞrstphase,thepossiblecandidateedgepixels()areextractedfromtheborderregionsbetweentheuniformintensitysurfaces,asexplainedintheearliersection,whicharedeÞnedintermsofPlateauTopandBottom.Thesearesimilartoridgesandvalleysofgraylevelimages.Theedgecandidates(whichbelongtotheborderregionsareassignedgradientmembershiprshipbasedontheirrespectivegradientstrength.Afuzzyedgeset()comprisingoffortheborderpointsisformed,asdeÞnedin.Inthenextstep,twomembershipfunctions()arecomputedtoestimatethefuzzyconnectivitystrengthalongapath,intheforwardandbackwarddirectionwithrespecttothecandidatepixel.ThebasicstepsareexplainedinFig.2.Thedetailedimplementationofthestepsaredescribedinthefollowingsubsections.¼fð3.2.EstimationofgradientstrengthTheinputimageisconvolvedwiththeGaussianfunction,toobtaintheGaussiansmoothenedimagematrixwhereeffectivelydeterminesthedegreeofsmoothing.GaussianÞltering,hasbeenchosentoperformeffectivesmoothingofsmalldistortionscausedbynoiseandtoobtainblurboundaries.ThesizeoftheGaussiansmoothingÞlterisÞxedto33pixelsandvalueofto1.5.Themembershipforthepixelsareestimatedasfollows.Foreveryedgepixelwhereisthegrayvalueofpixel(),a33windowisconsideredasshowninFig.3.InFig.3thesymbolsrepresentthegrayvaluesatdifferentneighborhoodsof.Thedifferencebetween(),(),()aretakenasgrayleveldifferencesinfour Fig.2.Blockdiagramoftheproposedalgorithm.M.Banerjee,M.K.Kundu/AppliedSoftComputing8(2008)1680–1691 differentdirections.Theratioofgraylabelchanges()arecomputedfromtwomutuallyperpendicularsetofpixelpairswithintheneighborhood.Consideringthemutuallyperpendi-cularpair(),(),thecomputedratiosare,are,.Similarly,(),()areconsidered.Thefourvaluesofpixelcontrastratios()asobtainedfromtheneighborhoodofeachcandidateedgepixelareshownin 11a2j11c2j; 11c2j11a2j; 11b2j11d2j; Inawindowofeightneighborhood,anedgepixelwillhavemaximumgrayleveldifferenceinadirection,perpendiculartoitstrueedgedirection().Theedgedirection()shouldpointalongtheminimumdifferencedirectionection.Theminimumpixelcontrastratio(istheparameter(x)usedforcomputingthegradientmembershipwithatypefunction,asshownEq.isusedtorepresenttheuncertaintiesofedgestrengthandlocationoftrueedgepoint.Thechoiceofmembershipfunctionisproblemdependent.Hereamonotonictypefunctionhasbeenchosenforsuitablerepresentationoftheambiguitiesoftheset,computedfrompixelcontrastratios.Wehavecomputedthefeetandtheshoulderpointusingmax()andmin()valuesofthecontrastratios(),overwhichthemembershipcomputed.ThehistogramplotsofpixelcontrastratioareshownFig.6(a)and(b)fortheimagesFig.5(a)and(b)respectively.Thevalueofdeterminestheedgestrength.Highervaluesofgradientmemberships,i.e.5correspondtomediumandstrongedgepoints.Lowervaluesofcorrespondtoweakornoisyedgepoints.Thefuzzygradientmap()asshowninisobtained.3.3.EstimationofconnectivitystrengthThetwomembershipvalues()arecomputedonaselectedsubsetof()showninobtainedbythresholding().ThemembershipscomputedfromthedifferenceinedgedirectionsbetweentheconnectedpixelswithinaÞxedwindow.Theactualcomputa-tionofaremadeasfollows:representtheedgedirectionofasequenceofpixelsonanedgesegment.Thepresentapproach,dealswiththechangesinedgedirections.Fourrelative(theanglesubtendedbetweentwosuccessivepixels),directionsareconsideredina33window.Thedirections()alongthehorizontallinei.e.(0and180arelabeledas(0),similarlyalongtheverticallinesas()andalongthediagonallinesasasshowninFig.3.Asaresult,theedgesalongdifferentdirectionsmaybelabelledasshowninThechangeofdirectionswithrespectto()betweenthesuccessiveedgepixelsmayhavevalues(4),(2),(2)inaneightneighborhood.Howeverduetoblurringoftheimages,thesharpchangeslike(2)betweenthesuccessivepixelsareconvertedtogentlechangeshavingvalueslessthan2.Asaresult,thechangesatastepof45areconsidered.Ifthedirectionofthecandidatepixel,then4isconsideredasrelativeforwarddirectionand4isconsideredastherelativebackwarddirectionwithrespecttowindowiscenteredaroundtheselectedcandidateedgepixelsandthenumberofsimplyconnectededgepixelsof()whichhavedirectionsarecounted.Ifthelabelofis(0)then,thelabelsrepresentstherespectively.Similarlyifthelabelofis(1),thelabels(0)representthecountsrespectivelyandsoon.Thiscountisexpectedtovarywiththesharpnessofthecurvaturetype.Thevalues()arerepresentedwiththeformofmembershipfunction, isdeÞnedby Fig.3.33neighborhoodofapixel. Fig.4.Determinationofcornerness.M.Banerjee,M.K.Kundu/AppliedSoftComputing8(2008)1680–1691 whereisaconstantmultiplier.Itissoselectedthatthevalueofshouldlieinbetween0and1.0fromtheÞnitecountsofoftheimage.Eachcandidateedgepixel()selectedforcornernesstesting,isthusrepresentedbyathree-dimensionalfeature.Detectionofpossiblefuzzycornersfromtheinputedgemap()willbediscussedinthenextsection.4.MultilevelfuzzycornerextractionThefuzzyedgemap()isrepresentedassetofpoints.Intheinitialstage,asuitablethresholdvalueofgradientmembership,hastobedecidedtoselectasubset,andonlythosepointsareusedforcomputationof,fordetectionoffuzzycorners.4.1.MembershiptransformationAnynaturalimageconsistsofdifferenthomogeneousregions,wheretheshapeofeachregionischaracterizedbyitsboundinglines.ButinmanypracticalsituationstheboundariesaresofaintthatitbecomesdifÞculttodistinguishbetweentworegions.Moreoverduetonoiseandnonuniformillumination,spuriousedgesmayalsoappear.ItisalsodifÞculttodiscriminatebetweenspuriousedgesandweakedges.Undersuchsituation,thegradientinformation(bothedgestrength,anddirectioninformation)mayberequiredtocutofwhereisverysmall.TolocatepointsfromsigniÞcantportionsontheimage,acontrasttransformationmaybeusedasapreprocessingstep.Theextractionofprobableedgecandidates,isachievedbythresholdingthroughnon-lineartransformationofmembershipvaluessuchthat,thepointshavingvaluesgreaterthan0.5arestretchedandthosebelow0.5aresqueezed.ApixelcontrasttransformationoperationtionisrepresentedTheresultsbeforeandaftertransformationofmembershipvaluesareshowninFig.7(a)Ð(c).AsseenfromFig.7thenumberofinsigniÞcantcandidatepointsarereducedatthesamethresholdvalue.Thresholdingthetransformededgemap)abovedifferentmembershipvaluesmaybeobtainedbyusingproper((asmentionedinsection.Asaresult,weobtaintheedgemapsatdifferentlevels Fig.5.(a)Originalimageofhouse.(b)Imagehavingprominentcurvaturejunctions. Fig.6.(a)Pixelcontrasthistogramof:Fig.5(a).(b)PixelcontrasthistogramofFig.5M.Banerjee,M.K.Kundu/AppliedSoftComputing8(2008)1680–1691 ,asshowninwhere00ThecandidatesofcanberepresentedbythelocalfeaturesBysuchthresholdingof,multilevelfuzzyedgemapsmaybegenerated,wherethepixelsmaybesegregatedas(strong,medium,weak)edgepixelsbasedontheirgradientmembershipvaluesasshowninFig.10(b)Ð(d).Ifthelocalcontrastofaregionisverypoor,thenvaluesofdifferentedgepointsareveryclosetoeachother.Ambiguityinlocatingcurvaturepointsintheseregionsmayincreaseduetocloseproximityofvaluesofdifferentpoints,asseeninthebottomrectangleofFig.10(b).Ontheotherhand,themembershipvaluesofdifferentpointsarewidelyseparatedabovethecrossoverpoints(5),wherethelocalcontrastisbetterresultinginlessambiguity.Inthetransformedset,pointshaving(5)willincludeedgepointswithhigherandmediumstrength.Whereasthosehavingvalues(0)mayselectlotofspuriousedgepointsalongwithhighandmediumtypeofcurvature Fig.7.Fuzzyedgemap:(a)(4).(b)(4)aftermembershiptransformation.(c)( Fig.8.Imageofhouse:(a)underexposed,(b)overexposed. Fig.9.Histogramplotsofimageofhouse:(a)underexposed,(b)overexposed.M.Banerjee,M.K.Kundu/AppliedSoftComputing8(2008)1680–1691 Thusaproperchoiceofthreshold()selectionisnecessary,belowwhichthevariationsareconsideredtobenoise.4.2.SelectionofthresholdonmembershipvalueThegradientmembershipvalueusedforthresholdingtheedgemap,isdecidedfromthepixelcontrastratiohistogram.Thehistogramofcontrastratiogivesanestimateofglobaldescriptionoftheappearanceofanimage.Ingeneral,thechoiceofthresholdismadeasfollows:Ahigherthresholdvalue,typically8ischosen,toreducethefalseacceptancerate,ifthenatureofthecontrasthistogramisasfollows:(i)Thecontrasthistogramoccupiesmostofthehistogramlevels,whichareincontiguouslocations.(ii)Thenumberofoccurrencesforeach()valueisquitecloseandcoversthemajorityofthetotaldynamicrange.ThisisseenfromthehistogramplotsofFigs.5(a)and8(a),(b)Asweareconcernedwiththedynamicrange,andnottheabsolutegrayscalevalues,suchthresholdingcanbeappliedforalmostallnaturalimages,evenundergonevaryingimagingconditionslikeoverexposed,underexposed,blurredetc.ThecontrasthistogramplotforFig.8(a)and(b)areshowninFig.9(a)and(b).Ontheotherhandalowerthresholdvalueoftypically0ischosen,ifthehistogramhasthefollowingproperties.(i)Sparselydistributedcontrastlevels.(ii)Havingwidelydifferentoccurrencesfordifferent()values(iii)Doesnotcovermajorityofthedynamicrange.Suchcasesmayarisefornearlybinaryimagesasseenin,Fig.5(b)andFig.19.Insuchcasestransformationof,doesnotaffecttheresultsmuch,asthecandidateweakedgesarelessinnumber.Thishasbeentestedovernumberofimagesandthestrategydescribedisfoundtobesatisfactory.4.3.EstimationoflocalshapeparametersOncethesuitablethresholdvalueofischosen,thenexttaskistocategorizetheedgepixelsbasedonthelocalpropertiesestimatedfrom.Theselectededgecandidatesconstitutethepointsofforwhichthemember-shipvaluesarecomputed.Thepropertiesofareusedtoexaminelocalshapeparameters,whicharedeÞnedasstraightnessandcornerness.Propertiesofforanyoftheselectedpoints()ontheedgemapisshowninTable1Thispropertyisdeterminedbycomparingpixelstranslatedalongthedirectionofedge.Itisexpectedthatapixeltranslatedinthedirectionofstraightedgewillbeconnectedtopixelsofsamedirection.HenceCornerness:Thispropertyisdeterminedfromcomparingpixelshavingreßexivesymmetry.Thepixelsareexpectedtobereßectedfromonearmtotheotheronbothsidesofthecurvaturejunctionwithintheregionofevaluation,asshowninFig.4ThepointsofasshowninEq.havingbothequaltozerocanbeÞlteredoutasthenoncornerpixels.Asaresulttheinterestingregionsconstitutingagroupofcurvaturepointsofthefuzzyedgeimagecanbeseparated.WeattempttoapproximatethisregionwithaquantitativemeasurebyexploitingthepropertiesofThepixelsintheproximityofthecurvaturejunctionasshowninFig.4canbecategorizedfromthefollowingrules,(i)Thepointswithhighandlowconstitutethepointsontheleftsideofthejunctionpoint.Wedesignatethesepoints,(asshowninFig.4)onforwardarmandassignmembership.Thisdifferenceisexpectedtovarywiththesharpnessofcurvature.Thepointsofrepresentafuzzysubsetas(ii)Thepointswithhighandlowconstitutethepointsontherightsideofthejunctionpoint.Wedesignatethesepointsas,(asshowninFig.4)onbackwardarmandassignmembership.Thepointsofrepresentafuzzyset Fig.10.(a)Originalimage.(b)Edgeimagefor(0).(c)(6).(d)(9).Pointsabovethresholdareplottedascrispedgepoints. Table1FuzzycornernessmeasureCornernessStraightnessLocationHighLowHighLowForwardarmHighHighHighLowNearcurvaturejunctionLowHighHighLowBackwardarmLowLowLowHighStraightedgeM.Banerjee,M.K.Kundu/AppliedSoftComputing8(2008)1680–1691 (iii)ThepointsveryneartothejunctionisexpectedtohavehighormediumvaluesofHavingobtainedthetwofuzzysets,,athirdfuzzysubsetwhichissurroundedbybothlieapproximatelyontheaxisofsymmetry.Thisregionconstitutetheambiguouscorners.Theclusterofsuchpoints(*)representedasareshowninFigs.11(a),12(a),13(a)andrespectively.Thepointsbelongingtoarethosepoints,havingotherpointswith0andintheneighborhoodofÞxedwindowsize.Theextractedcurvaturepointsmaybeofdifferentsharpnesstype(sharp,medium,weak).ThecharacteristicsofsharpcurvaturepointswillbeconÞnedwithinasmallregionbutforthatofmediumandweaktypetheregionwillbelarger.Inviewoftheabovefacts,weuseameasurethatcontrolstheshapeandsizeoftheextracted Fig.11.(a)Curvaturepoints(*),(0and1).(b)Representativepointofeachcluster. Fig.12.(a)Curvaturepoints(*),(0and=0.2).(b)Representativepointsofeachcluster. Fig.13.(a)Curvaturepoints(*),(=0.3).(b)Representativepointsofeachcluster.M.Banerjee,M.K.Kundu/AppliedSoftComputing8(2008)1680–1691 InordertodeÞneaquantitativemeasureoftheregion,constitutingofpointsof,wecomputethesumtotalofdifferencesforallthepairsofwhichfallwithintheregionofevaluation,anxnwindow.Thisvalueissubtractedfromalargevalue(keptÞxedat2.0,foundexperimentallybetter)tomakeitincreasewithsharpness.Therepresentativepointsi.e.,theclustercenterofeachlocalizedregion()isrepresentedbywhosecoordinateisequaltotheaveragevalueoftheco-ordinatesofthenpointsofeachclusterasshowninFigs.11(b),12(b),13(b)and14(b)AtÞxedvalueofthevalueofisexperimentallyvariedfrom(0.1-0.3)togeneratecornersofdifferentsharpness.Computationalcomplexity:Theanalysisofthecomputa-tionalcomplexity(worstcase)involvedindifferentoperationsforanimageofsizeandwithwindowneighborhood,isexplainedasfollows:(1)IdentiÞcationofborderregionsrequireoperations.(2)Computationofpixelcontrastratioinvolvesoperations(where0(3)Assignmentofinvolvesoperations.(4)Membershiptransformationinvolvesoperations.(5) Fig.14.(a)Curvaturepoints(*),(6and=0.1).(b)Representativepointofeachcluster. Fig.15.Cornerpoints(a)Ourdetector(=0.3).(b)Harrisdetector(c)SUSAN. Fig.16.Cornerpoints(a)Ourdetector(9and=0.2).(b)Harrisdetector(c)SUSAN.M.Banerjee,M.K.Kundu/AppliedSoftComputing8(2008)1680–1691 Thresholdinginvolvesoperations.(6)Computationofinvolvesoperations.(7)Groupingofpixelsbasedonfuzzyrulesinvolvesoperations.(8)Computationinvolvesoperations.Thetotaloperationisoftheorderof5.ExperimentalresultsWehaveexaminedtheperformanceofourdetectoronvarioustypeofimagesincludingimageswhichhaveundergoneimagealterationslikeblurring,illuminationchange,noiseetc.ImageasshowninFig.10(a)containsobjectsofdifferentshapeswithvaryingillumination.Theprocedureinvolvesextractionofedgemap.TheedgemapofFig.10(a)arethresholdedabovedifferentmembershipvaluesasshowninFig.10(b)Ð(d)for(0),(6),(respectively.Thepointsabovethethresholdvaluesarerepresentedasdarkedgepixels.Itistobenotedthattheinternalstructureoftherectangle(representedasdarkregion)showninFig.10(b)couldnotbeextractedproperlyduetopoor Fig.17.Cornerpointsfromourdetector(a)blurredimage(9and=0.2.(b)noisyimage. Fig.18.Cornerpointsunderilluminationchange:(a)Overexposed(9and=0.3).(b)Underexposedcase(9and Fig.19.Cornerpoints(a)Ourdetector(0and=0.2).(b)Harrisdetector.(c)SUSAN.M.Banerjee,M.K.Kundu/AppliedSoftComputing8(2008)1680–1691 contrastbetweenthetworegions.Insuchcases,theextractionofthestructurecouldnotbedoneproperlyasthegradientvalueoftheedgepixelsareverylowandthethresholdis0).Atahigherthresholdvalueof,strongeredgepointsmainlyrepresentingtheboundarypointscaneasilybeseparatedasshowninFig.10(d).ThecurvaturepointsofvariousregionsaredepictedbysymbolÔ*Õasshownin11(a),12(a),13(a)and14(a).TherepresentativepointfromeachclusterisshowninFigs.11(b),12(b),13(b)and14(b).ThedifferenttypeofcurvaturepointsareobtainedbyvaryingandshowninFigs.11(b),12(b)and13(b).TheresultsofouralgorithmarecomparabletothatofmostpopularlyusedcornerdetectorslikeHarrisandSUSANdetectorandareshowninFigs.15and16.Theperformanceofdifferentcornerdetectorsvarieswiththetypeoftheimageandtoobtainthebestresults,severalparametersneedtobeadjustedforalmostalldetectors.Wehavetriedtocompareourresultswiththebestresultsobtainedfromeachdetectorwiththeparametervaluesassuggestedbyauthors.Inouralgorithmbetterresultsareobtainedbykeepingthethresholdofalowervaluewhentherearelessernumberofgraylevelvariationse.g.,Fig.5(b).OntheotherhandwhentherearelargevariationsofdistinctgrayvaluesasthatofFig.5(a),higherthresholdvalueofischosentoreducethenumberweakandnoisyedgepoints.SuchresultsareshowninFigs.16Ð18.ItisseenfromFig.15(a)Ð(c)thatthecornerpointsobtainedbyourmethodshowninFig.15(a)isquitecomparabletothatofHarrisshowninFig.15(b)andSUSANdetectorinFig.15(c).HoweverSUSANisabletoextractcornersfromverylowcontrastarea.Theresultsonthehouseimagewiththresholdvalue(9and=0.2)forouralgorithm,forHarrisandSUSANmethodareshowninFig.16(a)Ð(c),respectively.ItisseenfromFig.16thatthecornerpointsobtainedbyourmethodshowninFig.16(a)iscomparabletothatofHarrisinFig.16(b)andSUSANinFig.16(c).OurresultisclosertothatofSUSANwithsomemoredetailsofcurvatureinformationthatexistsindifferentregionsofthehouseimage.TheresultsobtainedunderdifferentimagingconditionsareshownfromFigs.17and18.ItistobenotedthatourproposeddetectorisabletoextractmostofsigniÞcantstructuralcornerpointsundervaryingimagingconditions.Thisisduetothefactthattheslopeofthefuzzypropertyplaneisdeterminedfromthedynamicrange.AlthoughthegraylevelcontrastinformationisreducedintheoverexposedcaseinFig.18,butduetoadditionalcontrastintensiÞcation,signiÞcantedgepixelsareselectedabovethresholdforcornernessdetection.EvenfornearlybinaryimagesouralgorithmworkssatisfactorilyasseenfromFig.19(a)Ð(c).6.ConclusionAfuzzysettheoreticapproachfordetectionofcornersisproposedinthispaper.Theproposedalgorithmdoesnotrequirecomputationofchaincodesorcomplexdifferentialgeometricoperators.ExperimentshavebeenperformedonvarioustypesofimagestoillustratetheefÞciencyofouralgorithm.Thealgorithmperformsreasonablywellunderdifferentimagingconditions.Howeverweintendtoimprovethealgorithm,sothattheparametersmaybeselectedadaptivelyforthresholding.SigniÞcantfeaturescomputedfromthesedominanthighcurvaturefuzzypointscanbeuseddirectlyforindexinganimageforimageretrievalpurpose.[1]D.G.Lowe,PerceptualOrganizationandVisualRecognition,KluwerAcademicPublishers,USA,1985.[2]H.Freeman,L.S.Davis,Acorner-Þndingalgorithmforchain-codedcurves,IEEETrans.Comput.C-26(1977)297Ð303.[3]L.Kitchen,A.Rosenfeld,Gray-levelcornerdetection,PatternRecogn.Lett.1(1982)95Ð102.[4]Z.Zheng,H.Wang,E.Teoh,Analysisofgraylevelcornerdetection,PatternRecogn.Lett.20(2)(1999)149Ð162.[5]A.Rattarangsi,R.T.Chin,Scale-baseddetectionofcornersofplanarcurves,IEEETrans.PatternAnal.Mach.Intell.14(4)(1992)430Ð449.[6]C.Teh,R.T.Chin,Onthedetectionofdominantpointsondigitalcurves,IEEETrans.PatternAnal.Mach.Intell.11(8)(1989)859Ð[7]A.Rosenfeld,E.Johnston,Angledetectionondigitalcurves,IEEETransactiononComputersC-22(1973)858Ð875.[8]S.C.Bae,I.S.Kweon,C.D.Yoo,Cop:anewcornerdetector,PatternRecogn.Lett.20(2002)1349Ð1360.[9]H.Moravec,Towardsautomaticvisualobstacleavoidance,in:Proceed-ingsofthe5thInternationalJointConferenceonArtiÞcialIntelligence,1997,p.584.[10]C.Harris,M.Stephens,Acombinedcornerandedgedetector,in:Proceedingsofthe4thAlveyVisionConference,1988,pp.147Ð151.[11]S.Smith,M.Brady,Anewapproachtolowlevelimageprocessing,Int.J.Comput.Vision23(1)(1997)45Ð78.[12]E.Loupias,N.sebe,Wavelet-basedsalientpoints:applicationstoimageretrievalusingcolorandtexturefeatures,inAdvancesinvisualInforma-tionSystems,in:Proceedingsofthe4thIntenationalConference,VISUAL2000,(2000),pp.223Ð232.[13]M.Fischler,H.C.Wolf,Locatingperceptuallysalientpointsonplanarcurves,IEEETrans.PatternAnal.Mach.Intell.16(2)(1994)113Ð129.[14]M.Banerjee,M.K.Kundu,P.Mitra,Cornerdectectionusingsupportvectormachine,in:17thInternationalConferenceonPatternRecognitionICPR(2004),UK,vol.2,(2004),pp.819Ð822.[15]K.J.Lee,Z.Bien,Agray-levelcornerdetectorusingfuzzylogic,PatternRecogn.Lett.17(1996)939Ð950.[16]L.Li,W.Chen,Cornerdetectionandinterpretationonplanarcurvesusingfuzzyreasoning,IEEETrans.PatternAnal.Mach.Intell.14(4)(1999)[17]T.Law,H.Itoh,H.Seki,ImageÞltering,edgedetectionandedgetracingusingfuzzyreasoning,IEEETrans.PatternAnal.Mach.Intell.18(5)(1996)481Ð491.[18]J.Weijer,T.Gevers,J.Geusebroek,Edgeandcornerdetectionbyphotometricquasi-invariants,IEEETrans.PatternAnal.Mach.Intell.27(4)(2005)625Ð629.[19]L.A.Zadeh,Fuzzysets,InformationandControl8(1965)338Ð353.[20]S.K.Pal,A.Ghosh,M.K.Kundu,SoftComputingforImageProcessing,Physica-Verlag,2000,pp.44Ð78(Chapter1).[21]D.Yua,Q.Hu,C.Wua,Uncertaintymeasuresforfuzzyrelationsandtheirapplications,Appl.SoftComput.7(3)(2007)1135Ð1143.[22]S.K.Pal,D.D.Majumder,FuzzymathematicalApproachtoPatternRecognition,JohnWilley,NewYork,1985.[23]A.Rosenfeld,Fuzzydigitaltopology,in:J.C.Bezdek,S.K.Pal(Eds.),FuzzyModelsForPatternRecognition,IEEEPress,1991,pp.331Ð339.M.Banerjee,M.K.Kundu/AppliedSoftComputing8(2008)1680–1691 [24]B.B.Chaudhury,B.U.Shankar,AnefÞcientalgorithmforextremadetectionindigitalimages,PatternRecogn.Lett.10(1989)81Ð85.[25]L.Wang,T.Pavlidis,Dirextgray-scaleextractionoffeaturesforcharacterrecognition,IEEETrans.PatternAnal.Mach.Intell.15(10)(1993)1053Ð[26]R.M.Haralick,Ridgesandvalleysondigitalimages,ComputerVision,GraphicsImageProcess.22(1983)28Ð38.[27]M.Banerjee,M.K.Kundu,Edgebasedfeaturesforcontentbasedimageretrieval,PatternRecogn.36(11)(2003)2649Ð2661.[28]R.C.Gonzalez,R.E.Woods,DigitalImageProcessing,AddisonWiley,NewYork,1985.M.Banerjee,M.K.Kundu/AppliedSoftComputing8(2008)1680–1691