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首页 数字图像处理答案(冈萨雷斯)

数字图像处理答案(冈萨雷斯)

数字图像处理答案(冈萨雷斯)

痴爱
2009-04-07 0人阅读 举报 0 0 暂无简介

简介:本文档为《数字图像处理答案(冈萨雷斯)pdf》,可适用于工程科技领域

DigitalImageProcessingSecondEditionProblemSolutionswStudentSetRafaelCGonzalezRichardEWoodsPrenticeHallUpperSaddleRiver,NJwwwprenhallcomgonzalezwoodsorwwwimageprocessingbookcomiiRevisionhistoryCopyrightc°byRafaelCGonzalezandRichardEWoodsPrefaceThisabbreviatedmanualcontainsdetailedsolutionstoallproblemsmarkedwithastarinDigitalImageProcessing,ndEditionThesesolutionscanalsobedownloadedfromthebookwebsite(wwwimageprocessingbookcom)Solutions(Students)ProblemThediameter,x,oftheretinalimagecorrespondingtothedotisobtainedfromsimilartriangles,asshowninFigPThatis,(d=):=(x=):whichgivesx=:dFromthediscussioninSection,andtakingsomelibertiesofinterpretation,wecanthinkofthefoveaasasquaresensorarrayhavingontheorderof,elements,whichtranslatesintoanarrayofsize£elementsAssumingequalspacingbetweenelements,thisgiveselementsandspacesonalinemmlongThesizeofeachelementandeachspaceisthens=(:mm)==:£¡mIfthesize(onthefovea)oftheimageddotislessthanthesizeofasingleresolutionelement,weassumethatthedotwillbeinvisibletotheeyeInotherwords,theeyewillnotdetectadotifitsdiameter,d,issuchthat:(d)<:£¡m,ord<:£¡mFigurePChapterSolutions(Students)Problem¸=c=v=:£(ms)=(s)=:£m=KmProblemOnepossiblesolutionistoequipamonochromecamerawithamechanicaldevicethatsequentiallyplacesared,agreen,andabluepass®lterinfrontofthelensThestrongestcameraresponsedeterminesthecolorIfallthreeresponsesareapproximatelyequal,theobjectiswhiteAfastersystemwouldutilizethreedifferentcameras,eachequippedwithanindividual®lterTheanalysiswouldbethenbasedonpollingtheresponseofeachcameraThissystemwouldbealittlemoreexpensive,butitwouldbefasterandmorereliableNotethatbothsolutionsassumethatthe®eldofviewofthecamera(s)issuchthatitiscompletely®lledbyauniformcolorie,thecamera(s)is(are)focusedonapartofthevehiclewhereonlyitscolorisseenOtherwisefurtheranalysiswouldberequiredtoisolatetheregionofuniformcolor,whichisallthatisofinterestinsolvingthisproblemProblem(a)Thetotalamountofdata(includingthestartandstopbit)inanbit,£image,is()£bitsThetotaltimerequiredtotransmitthisimageoveraAtKbaudlinkis()£==:secoraboutmin(b)AtKthistimegoesdowntoaboutsecProblemLetpandqbeasshowninFigPThen,(a)SandSarenotconnectedbecauseqisnotinthesetN(p)u(b)SandSareconnectedbecauseqisinthesetN(p)u(c)SandSaremconnectedbecause(i)qisinND(p),and(ii)thesetN(p)N(q)isemptyProblemFigurePProblemThesolutiontothisproblemconsistsofde®ningallpossibleneighborhoodshapestogofromadiagonalsegmenttoacorrespondingconnectedsegment,asshowninFigPThealgorithmthensimplylooksfortheappropriatematcheverytimeadiagonalsegmentisencounteredintheboundaryFigurePProblem(a)WhenV=fg,pathdoesnotexistbetweenpandqbecauseitisimpossibletoChapterSolutions(Students)getfromptoqbytravelingalongpointsthatarebothadjacentandalsohavevaluesfromVFigureP(a)showsthisconditionuitisnotpossibletogettoqTheshortestpathisshowninFigP(b)uitslengthisThelengthofshortestmpath(showndashed)isBothoftheseshortestpathsareuniqueinthiscase(b)OnepossibilityfortheshortestpathwhenV=fgisshowninFigP(c)uitslengthisItiseasilyveri®edthatanotherpathofthesamelengthexistsbetweenpandqOnepossibilityfortheshortestpath(itisnotunique)isshowninFigP(d)uitslengthisThelengthofashortestmpath(shoendashed)isThispathisnotuniqueFigurePProblem(a)Ashortestpathbetweenapointpwithcoordinates(xy)andapointqwithcoordinates(st)isshowninFigP,wheretheassumptionisthatallpointsalongthepatharefromVThelengthofthesegmentsofthepatharejx¡sjandjy¡tj,respectivelyThetotalpathlengthisjx¡sjjy¡tj,whichwerecognizeasthede®nitionoftheDdistance,asgiveninEq()(Recallthatthisdistanceisindependentofanypathsthatmayexistbetweenthepoints)TheDdistanceobviouslyisequaltothelengthoftheshortestpathwhenthelengthofthepathisjx¡sjjy¡tjThisoccurswheneverwecangetfromptoqbyfollowingapathwhoseelements()arefromVand()arearrangedinsuchawaythatwecantraversethepathfromptoqbymakingturnsinatmosttwodirections(eg,rightandup)(b)Thepathmayofmaynotbeunique,dependingonVandthevaluesofthepointsalongthewayProblemFigurePProblemWithreferencetoEq(),letHdenotetheneighborhoodsumoperator,letSandSdenotetwodifferentsmallsubimageareasofthesamesize,andletSSdenotethecorrespondingpixelbypixelsumoftheelementsinSandS,asexplainedinSectionNotethatthesizeoftheneighborhood(ie,numberofpixels)isnotchangedbythispixelbypixelsumTheoperatorHcomputesthesumofpixelvaluesisagivenneighborhoodThen,H(aSbS)means:()multiplyingthepixelsineachofthesubimageareasbytheconstantsshown,()addingthepixelbypixelvaluesfromSandS(whichproducesasinglesubimagearea),and()computingthesumofthevaluesofallthepixelsinthatsinglesubimageareaLetapandbpdenotetwoarbitrary(butcorresponding)pixelsfromaSbSThenwecanwriteH(aSbS)=XpSandpSapbp=XpSapXpSbp=aXpSpbXpSp=aH(S)bH(S)which,accordingtoEq(),indicatesthatHisalinearoperatorSolutions(Students)Problem(a)s=T(r)=(m=r)E:Problem(a)Thenumberofpixelshavingdifferentgraylevelvalueswoulddecrease,thuscausingthenumberofcomponentsinthehistogramtodecreaseSincethenumberofpixelswouldnotchange,thiswouldcausetheheightsomeoftheremaininghistogrampeakstoincreaseingeneralTypically,lessvariabilityingraylevelvalueswillreducecontrastProblemAllthathistogramequalizationdoesisremaphistogramcomponentsontheintensityscaleToobtainauniform(­at)histogramwouldrequireingeneralthatpixelintensitiesbeactuallyredistributedsothatthereareLgroupsofn=Lpixelswiththesameintensity,whereListhenumberofalloweddiscreteintensitylevelsandnisthetotalnumberofpixelsintheinputimageThehistogramequalizationmethodhasnoprovisionsforthistypeof(arti®cial)redistributionprocessProblemWeareinterestedinjustoneexampleinordertosatisfythestatementoftheproblemConsidertheprobabilitydensityfunctionshowninFigP(a)AplotofthetransformationT(r)inEq()usingthisparticulardensityfunctionisshowninFigP(b)Becausepr(r)isaprobabilitydensityfunctionweknowfromthediscussionChapterSolutions(Students)inSectionthatthetransformationT(r)satis®esconditions(a)and(b)statedinthatsectionHowever,weseefromFigP(b)thattheinversetransformationfromsbacktorisnotsinglevalued,asthereareanin®nitenumberofpossiblemappingsfroms==backtorItisimportanttonotethatthereasontheinversetransformationfunctionturnedoutnottobesinglevaluedisthegapinpr(r)intheinterval==FigurePProblem(c)Ifnoneofthegraylevelsrkk=:::L¡are,thenT(rk)willbestrictlymonotonicThisimpliesthattheinversetransformationwillbeof®niteslopeandthiswillbesinglevaluedProblemThevalueofthehistogramcomponentcorrespondingtothekthintensitylevelinaneighborhoodispr(rk)=nknProblemfork=:::K¡wherenkisthenumberofpixelshavinggraylevelvaluerk,nisthetotalnumberofpixelsintheneighborhood,andKisthetotalnumberofpossiblegraylevelsSupposethattheneighborhoodismovedonepixeltotherightThisdeletestheleftmostcolumnandintroducesanewcolumnontherightTheupdatedhistogramthenbecomespr(rk)=nnk¡nLknRkfork=:::K¡,wherenLkisthenumberofoccurrencesoflevelrkontheleftcolumnandnRkisthesimilarquantityontherightcolumnTheprecedingequationcanbewrittenalsoaspr(rk)=pr(rk)nnRk¡nLkfork=:::K¡:Thesameconceptappliestoothermodesofneighborhoodmotion:pr(rk)=pr(rk)nbk¡akfork=:::K¡,whereakisthenumberofpixelswithvaluerkintheneighborhoodareadeletedbythemove,andbkisthecorrespondingnumberintroducedbythemove¾g=¾fK¾´¾´¢¢¢¾´KThe®rsttermontherightsideisbecausetheelementsoffareconstantsThevarious¾´iaresimplysamplesofthenoise,whichishasvariance¾´Thus,¾´i=¾´andwehave¾g=KK¾´=K¾´whichprovesthevalidityofEq()ProblemLetg(xy)denotethegoldenimage,andletf(xy)denoteanyinputimageacquiredduringroutineoperationofthesystemChangedetectionviasubtractionisbasedoncomputingthesimpledifferenced(xy)=g(xy)¡f(xy)Theresultingimaged(xy)canbeusedintwofundamentalwaysforchangedetectionOnewayisuseapixelbypixelanalysisInthiscasewesaythatf(xy)is}closeenough}tothegoldenimageifallthepixelsind(xy)fallwithinaspeci®edthresholdbandTminTmaxwhereTminisnegativeandTmaxispositiveUsually,thesamevalueofthresholdisChapterSolutions(Students)usedforbothnegativeandpositivedifferences,inwhichcasewehaveaband¡TTinwhichallpixelsofd(xy)mustfallinorderforf(xy)tobedeclaredacceptableThesecondmajorapproachissimplytosumallthepixelsinjd(xy)jandcomparethesumagainstathresholdSNotethattheabsolutevalueneedstobeusedtoavoiderrorscancellingoutThisisamuchcrudertest,sowewillconcentrateonthe®rstapproachTherearethreefundamentalfactorsthatneedtightcontrolfordifferencebasedinspectiontowork:()properregistration,()controlledillumination,and()noiselevelsthatarelowenoughsothatdifferencevaluesarenotaffectedappreciablybyvariationsduetonoiseThe®rstconditionbasicallyaddressestherequirementthatcomparisonsbemadebetweencorrespondingpixelsTwoimagescanbeidentical,butiftheyaredisplacedwithrespecttoeachother,comparingthedifferencesbetweenthemmakesnosenseOften,specialmarkingsaremanufacturedintotheproductformechanicalorimagebasedalignmentControlledillumination(notethat|illumination}isnotlimitedtovisiblelight)obviouslyisimportantbecausechangesinilluminationcanaffectdramaticallythevaluesinadifferenceimageOneapproachoftenusedinconjunctionwithilluminationcontrolisintensityscalingbasedonactualconditionsForexample,theproductscouldhaveoneormoresmallpatchesofatightlycontrolledcolor,andtheintensity(andperhapsevencolor)ofeachpixelsintheentireimagewouldbemodi®edbasedontheactualversusexpectedintensityandorcolorofthepatchesintheimagebeingprocessedFinally,thenoisecontentofadifferenceimageneedstobelowenoughsothatitdoesnotmateriallyaffectcomparisonsbetweenthegoldenandinputimagesGoodsignalstrengthgoesalongwaytowardreducingtheeffectsofnoiseAnother(sometimescomplementary)approachistoimplementimageprocessingtechniques(eg,imageaveraging)toreducenoiseObviouslythereareanumberifvariationsofthebasicthemejustdescribedForexample,additionalintelligenceintheformofteststhataremoresophisticatedthanpixelbypixelthresholdcomparisonscanbeimplementedAtechniqueoftenusedinthisregardistosubdividethegoldenimageintodifferentregionsandperformdifferent(usuallymorethanone)testsineachoftheregions,basedonexpectedregioncontentProblem(a)Considera£mask®rstSinceallthecoef®cientsare(weareignoringtheProblemscalefactor),theneteffectofthelowpass®lteroperationistoaddallthegraylevelsofpixelsunderthemaskInitially,ittakesadditionstoproducetheresponseofthemaskHowever,whenthemaskmovesonepixellocationtotheright,itpicksuponlyonenewcolumnThenewresponsecanbecomputedasRnew=Rold¡CCwhereCisthesumofpixelsunderthe®rstcolumnofthemaskbeforeitwasmoved,andCisthesimilarsuminthecolumnitpickedupafteritmovedThisisthebasicbox®lterormovingaverageequationFora£maskittakesadditionstogetC(Cwasalreadycomputed)TothisweaddonesubtractionandoneadditiontogetRnewThus,atotalofarithmeticoperationsareneededtoupdatetheresponseafteronemoveThisisarecursiveprocedureformovingfromlefttorightalongonerowoftheimageWhenwegettotheendofarow,wemovedownonepixel(thenatureofthecomputationisthesame)andcontinuethescanintheoppositedirectionForamaskofsizen£n,(n¡)additionsareneededtoobtainC,plusthesinglesubtractionandadditionneededtoobtainRnew,whichgivesatotalof(n)arithmeticoperationsaftereachmoveAbruteforceimplementationwouldrequiren¡additionsaftereachmoveProblem(a)Therearenpointsinann£nmedian®ltermaskSincenisodd,themedianvalue,³,issuchthatthereare(n¡)=pointswithvalueslessthanorequalto³andthesamenumberwithvaluesgreaterthanorequalto³However,sincetheareaA(numberofpoints)intheclusterislessthanonehalfn,andAandnareintegers,itfollowsthatAisalwayslessthanorequalto(n¡)=Thus,evenintheextremecasewhenallclusterpointsareencompassedbythe®ltermask,therearenotenoughpointsintheclusterforanyofthemtobeequaltothevalueofthemedian(remember,weareassumingthatallclusterpointsarelighterordarkerthanthebackgroundpoints)Therefore,ifthecenterpointinthemaskisaclusterpoint,itwillbesettothemedianvalue,whichisabackgroundshade,andthusitwillbe|eliminated}fromtheclusterThisconclusionobviouslyappliestothelessextremecasewhenthenumberofclusterpointsencompassedbythemaskislessthanthemaximumsizeoftheclusterChapterSolutions(Students)Problem(a)NumericallysortthenvaluesThemedianis³=(n)=thlargestvalue(b)Oncethevalueshavebeensortedonetime,wesimplydeletethevaluesinthetrailingedgeoftheneighborhoodandinsertthevaluesintheleadingedgeintheappropriatelocationsinthesortedarrayProblemFromFig,theverticalbarsarepixelswide,pixelshigh,andtheirseparationispixelsThephenomenoninquestionisrelatedtothehorizontalseparationbetweenbars,sowecansimplifytheproblembyconsideringasinglescanlinethroughthebarsintheimageThekeytoansweringthisquestionliesinthefactthatthedistance(inpixels)betweentheonsetofonebarandtheonsetofthenextone(say,toitsright)ispixelsConsiderthescanlineshowninFigPAlsoshownisacrosssectionofa£maskTheresponseofthemaskistheaverageofthepixelsthatitencompassesWenotethatwhenthemaskmovesonepixeltotheright,itlosesonvalueoftheverticalbarontheleft,butitpicksupanidenticaloneontheright,sotheresponsedoesnztchangeInfact,thenumberofpixelsbelongingtotheverticalbarsandcontainedwithinthemaskdoesnotchange,regardlessofwherethemaskislocated(aslongasitiscontainedwithinthebars,andnotneartheedgesofthesetofbars)ThefactthatthenumberofbarpixelsunderthemaskdoesnotchangeisduetothepeculiarseparationbetweenbarsandthewidthofthelinesinrelationtothepixelwidthofthemaskThisconstantresponseisthereasonnowhitegapsisseenintheimageshownintheproblemstatementNotethatthisconstantresponsedoesnothappenwiththe£orthe£masksbecausetheyarenot}synchronized}withthewidthofthebarsandtheirseparationFigurePProblemProblemTheLaplacianoperatorisde®nedasrf=fxfyfortheunrotatedcoordinatesandasrf=fxfy:forrotatedcoordinatesItisgiventhatx=xcosµ¡ysinµandy=xsinµycosµwhereµistheangleofrotationWewanttoshowthattherightsidesofthe®rsttwoequationsareequalWestartwithfx=fxxxfyyx=fxcosµfysinµTakingthepartialderivativeofthisexpressionagainwithrespecttoxyieldsfx=fxcosµxµfy¶sinµcosµyµfx¶cosµsinµfysinµNext,wecomputefy=fxxyfyyy=¡fxsinµfycosµTakingthederivativeofthisexpressionagainwithrespecttoygivesfy=fxsinµ¡xµfy¶cosµsinµ¡yµfx¶sinµcosµfycosµAddingthetwoexpressionsforthesecondderivativesyieldsfxfy=fxfywhichprovesthattheLaplacianoperatorisindependentofrotationProblemConsiderthefollowingequation:ChapterSolutions(Students)f(xy)¡rf(xy)=f(xy)¡f(xy)f(x¡y)f(xy)f(xy¡)¡f(xy)=f(xy)¡f(xy)f(x¡y)f(xy)f(xy¡)f(xy)=f:f(xy)¡f(xy)f(x¡y)f(xy)f(xy¡)f(xy)g=£:f(xy)¡f(xy)¤wheref(xy)denotestheaverageoff(xy)inaprede®nedneighborhoodthatiscenteredat(xy)andincludesthecenterpixelanditsfourimmediateneighborsTreatingtheconstantsinthelastlineoftheaboveequationasproportionalityfactors,wemaywritef(xy)¡rf(xy)sf(xy)¡f(xy):Therightsideofthisequationisrecognizedasthede®nitionofunsharpmaskinggiveninEq()Thus,ithasbeendemonstratedthatsubtractingtheLaplacianfromanimageisproportionaltounsharpmaskingSolutions(Students)ProblemBydirectsubstitutionoff(x)Eq()intoF(u)Eq():F(u)=MM¡Xx="M¡Xr=F(r)ej¼rx=M#e¡j¼ux=M=MM¡Xr=F(r)M¡Xx=ej¼rx=Me¡j¼ux=M=MF(u)M=F(u)wherethethirdstepfollowsfromtheorthogonalityconditiongivenintheproblemstatementSubstitutionofF(u)intof(x)ishandledinasimilarmannerProblemAnimportantaspectofthisproblemistorecognizethatthequantity(uv)canb

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数字图像处理答案(冈萨雷斯)

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