ImageEnhancementintheSpatialImageEnhancementintheSpatialImageenhancementisakindprocessingofimagequalitytomakeoriginalimagesuitsomepredefinedsense;ImageenhancementissubjectiveImageenhancementisoneofthemostandvisuallyappealingareasofimageImageEnhancementapproachesfallintobroadcategories:SpatialdomainmethodsandFrequencydomainmethods;SomeBasicGrayLevelTransformationsHistogramProcessingEnhancementUsingArithmetic/LogicOperationsBasicsofSpatialFilteringSmoothingSpatialFiltersSharpeningSpatialCombiningSpatialEnhancementSomeBasicGrayLevelTransformationsHistogramProcessingEnhancementUsingArithmetic/LogicOperationsBasicsofSpatialFilteringSmoothingSpatialFiltersSharpeningSpatialCombiningSpatialEnhancementImageenhancementinspatialdomainistheImageenhancementinspatialdomainistheprocedurethatoperatesdirectlyonimagepixels.Spatialdomainprocesseswillbedenotedbytheexpression??,=??,where??(??,??)istheinputimage，??(??,??)istheprocessedimage,and??isoperatoron??,definedoversomeneighborhoodof(??,??)Defininganeighborhoodofpixel(??,y點(??,??)的鄰域主要是指以(??,??)為xDefininganeighborhoodofpixel(??,y點(??,??)的鄰域主要是指以(??,??)為xToperationofsingleTheneighborhoodisofsizesInputOutput??=當灰度變化的??操作針對單個像素時，輸出圖像的??????,??的值，??操作變成了灰度級變換函數(shù)（強度映射）ToperationofsingleTheneighborhoodisofsizesInputOutput??=當灰度變化的??操作針對單個像素時，輸出圖像的??????,??的值，??操作變成了灰度級變換函數(shù)（強度映射）ToperationofconvolutionTheneighborhoodisofsize??×??(e.g.3×ConvolutionInputOutput??(??,??)=??1??1+??2??2+??3??3+??4??4+??5??5+??6??6??7??7+??8??8+ToperationofconvolutionTheneighborhoodisofsize??×??(e.g.3×ConvolutionInputOutput??(??,??)=??1??1+??2??2+??3??3+??4??4+??5??5+??6??6??7??7+??8??8+SomeBasicGrayLevelTransformationsHistogramProcessingEnhancementUsingArithmetic/LogicOperationsBasicsofSpatialFilteringSmoothingSpatialFiltersSharpeningSpatialCombiningSpatialEnhancementSomeBasicGrayLevelImage灰度級范圍為[0,?1]獲得,即??=??10OutputInput??=??(??)=???1?SomeBasicGrayLevelImage灰度級范圍為[0,?1]獲得,即??=??10OutputInput??=??(??)=???1?SomeBasicGrayLevelImageSomeBasicGrayLevelImageOutputInput??=??(??)=255?SomeBasicGrayLevelSomeBasicGrayLevelLogThegeneralformofthelogtransformation??=????????(1+Wherecisaconstant,anditisassumedthat??≥SomeBasicGrayLevelSomeBasicGrayLevelLogSomeBasicGrayLevelSomeBasicGrayLevelLogtransformationsInputOutputSomeBasicGrayLevelSomeBasicGrayLevelPower-lawThebasicformofPower–law??=??????,wherecandγarepositiveSomeBasicGrayLevelPower-lawSomeBasicGrayLevelPower-lawtransformation—GammaSomeBasicGrayLevelSomeBasicGrayLevelAdditionalremarksofgammaGammacorrectionisimportantifdisplayinganimageaccuratelyonacomputerscreenisofconcern.Imagesthatarenotcorrectedproperlycanlookeitherbleachedout,or,whatismorelikely,toodark.Gammacorrectionchangesnotonlythebrightness,butalsotheratiosofredtogreentoblue.SomeBasicGrayLevelSomeBasicGrayLevelSomeBasicGrayLevelSomeBasicGrayLevelSomeBasicGrayLevelContrastPiecewise-lineartransformationSomeBasicGrayLevelContrastPiecewise-lineartransformation點(??1,??1)和（??2,??2）的位置決定如果??2=??1,=如果=0,??2=???=??2,TypicaltransformationusedforcontrastSomeBasicGrayLevelSomeBasicGrayLevelContraststretchingSomeBasicGrayLevelSomeBasicGrayLevelSomeBasicGrayLevelSomeBasicGrayLevelPiecewise-lineartransformationfunction—Gray-levelSomeBasicGrayLevelSomeBasicGrayLeveltransformationfunction—Bit-Piecewise-SomeBasicGrayLevelSomeBasicGrayLevelBit-planeslicingSomeBasicGrayLevelSomeBasicGrayLevelBit-planeslicingSomebasicGrayLevelTransformationsHistogramProcessingEnhancementUsingArithmetic/LogicOperationsBasicsofSpatialFilteringSmoothingSpatialFiltersSharpeningSpatialCombiningSpatialEnhancementHistogramWhatisTheHistogramofadigitalHistogramWhatisTheHistogramofadigitalimagewithgrayslevelsintherange[0,???1]isadiscretefunction?(????)=where????isthe??thgrayleveland????isthenumberpixelsintheimagehavinggraylevel????.Normalizedhistogramisgivenby=;Actually,thehistogramofadigitalimagereflectsprobabilityofoccurrenceofgraylevel????HistogramHistogramsof4HistogramHistogramsof4typesofHistogramHistogramHistogramHistogramHistogramHistogram？HistogramHistogramHistogramHistogramConsiderforamomentcontinuousfunctions,Letthevariable??representthegraylevelsoftheimagetobeenhanced.ThenHistogramHistogramConsiderforamomentcontinuousfunctions,Letthevariable??representthegraylevelsoftheimagetobeenhanced.Thenhistogramequalizationcanberealizedbythefollowingfunction:??=0≤??≤???where??isthegraylevelHistogramAssumefunctionsatisfiesthefollowingHistogramAssumefunctionsatisfiesthefollowing??(??)issinglevaluedandmonotonicallyincreasingtheinterval0≤??≤??? ≤???1,0≤??≤???0≤HistogramAssumefunctionsatisfiesthefollowingHistogramAssumefunctionsatisfiesthefollowing??(??)issinglevaluedandmonotonicallyincreasingintheinterval0≤??≤???1（保證輸出灰度不改變原始相對亮度關(guān)系 ≤???1,0≤??≤???0≤（保證輸出灰度范圍與輸入灰度范圍相同HistogramThe??=HistogramThe??=0≤??≤HistogramLet’sseethefollowingtransformation??==(??HistogramLet’sseethefollowingtransformation??==(???0Where??isadummyvariableofintegration.transformationfunctionsatisfiestheprevioustwo??(??)issinglevaluedandmonotonicallyintheinterval0≤??≤??? ≤???1,0≤??≤???0≤function(CDF)HistogramPrincipleofhistogram denotetheprobabilityHistogramPrincipleofhistogram denotetheprobabilityLet andfunction(PDF)of??and??,thePDFvariablescanbeobtainedusingthesimple=HistogramLet??and??HistogramLet??and??arerandomvariableininterval[0,???Assumethat????(??)and????(??)respectivelyprobabilitydensityfunctionof??and??.HistogramHistogramWewanttoanalyzetheprobabilityfunctionof??b????????????we.=??=HistogramWewanttoanalyzetheprobabilityfunctionof??b????????????we.=??==(???0=(???1)0??==(???Histogram1(???1)????1=??=??Histogram1(???1)????1=??=??=0≤??≤???HistogramFordiscrete,)??=0,1,2,…,???AplotofHistogramFordiscrete,)??=0,1,2,…,???Aplotof versus????iscommonlyreferredtoasThediscretetransformation???=???=????=??=0,1,2,…,??? ishistogramequalizationHistogram例：假設(shè)一幅×8個灰度級，各個灰度級出現(xiàn)的概率如下表所示，請對該圖像的值方圖進行均衡化處理。==???0====Histogram例：假設(shè)一幅×8個灰度級，各個灰度級出現(xiàn)的概率如下表所示，請對該圖像的值方圖進行均衡化處理。==???0=========+7????HistogramHistogram=1.33→=4.55→=6.23→=6.86→=3.08→=5.67→=6.65→=7.00→HistogramHistogramHistogramHistogramHistogramsuitableSpecialexplanation:HistogramHistogramsuitableSpecialexplanation:Histogramprocessingisallimageenhancement.InputimageandoutputimageandHistogramHistogramMatchingHistogramHistogramMatchingThemethodusedtogenerateaprocessedimagethathasaspecifiedhistogramiscalledhistogrammatchingorhistogramspecification.HistogramHistogramMatchingLet????(??)denotetheprobabilitydensityfunctionofthegivenHistogramHistogramMatchingLet????(??)denotetheprobabilitydensityfunctionofthegiveninputimageand????(??)denotethespecifiedprobabilitydensityfunctionthatwewishtheoutputimagetohave.BasicProcedureofhistogrammatching(continuous（1）Equalizetheoriginalimage??=??=(?????0（2）Thedesiredimagehistogram??=(?????????????=0（3）Obtaintheinversetransformation??==（4）ObtaintheoutputHistogramFordiscrete（1）Equalizetheoriginalimage??,??=0,1,2,…,??? =??=???HistogramFordiscrete（1）Equalizetheoriginalimage??,??=0,1,2,…,??? =??=???=（2）Thedesiredimagehistogram???=???=????=??????theinversetransformation==??（4）ObtaintheoutputimagebytheinversetransformationHistogramHistogramHistogramStep1:obtainthescaledhistogram-equalizedvaluesusingthefollowingequation:??,??=0,1,2,…,??? =??HistogramStep1:obtainthescaledhistogram-equalizedvaluesusingthefollowingequation:??,??=0,1,2,…,??? =??=???=For7790=1.33→ =0==00Then,we??0=1,??2=5,??4=6,??6=??1=3,??3=6,??5=7,??7=HistogramStep2:computeallthevaluesofthetransformation???=???==??==0HistogramStep2:computeallthevaluesofthetransformation???=???==??==0 0 1==+??(??1)]=======HistogramConvertthesefractionalvaluesto==HistogramConvertthesefractionalvaluesto========HistogramStep3:findtheHistogramStep3:findthesmallestvalueof????sothatthevalue??(????)istheclosetHistogramStep4:mapeverypixelinthehistogramequalizedimageintoacorrespondingHistogramStep4:mapeverypixelinthehistogramequalizedimageintoacorrespondingpixelinthenewlycreatedhistogram-specifiedimage.HistogramHistogramHistogramCompareHistogramEqualizationHistogramCompareHistogramEqualizationandHistogramHistogramHistogramComparingHistogramEqualizationandHistogramComparingHistogramEqualizationandHistogramHistogramHistogramLocalLocalHistogramLocalLocalHistogramHistogramUseofHistogramStatisticsforimageHistogramUseofHistogramStatisticsforimageThemeanofan???11??????(????)=??(??,????=0Where????isthegrayleveland?? istheofgraylevel????HistogramUseofHistogramStatisticsforimageThevarianceofan???1HistogramUseofHistogramStatisticsforimageThevarianceofan???112=?????2??(??)????,?Where????isthegraylevelandofgraylevel???? istheHistogramUseofHistogramStatisticsforimagesubThemeanHistogramUseofHistogramStatisticsforimagesubThemean=??????(??)Thevarianceofsub22=?????(??Where??????denotesthesubimagecentered??,.HistogramEnhance??,??1????HistogramEnhance??,??1????≤????????≤≤??0??????(??,??)????,SomebasicGrayLevelTransformationsHistogramProcessingEnhancementUsingArithmetic/LogicOperationsBasicsofSpatialFilteringSmoothingSpatialFiltersSharpeningSpatialCombiningSpatialEnhancementEnhancementUsingArithmeticoperationsEnhancementUsingArithmeticoperationsThesubsectiondiscussadditionandLogic:AND,ORExceptfortheLogicoperationNot,Otherarithmetic/logicoperationsinvolvingimagesareperformedonapixel-by-pixelbasisbetweenormoreSomebasicGrayLevelTransformationsHistogramProcessingEnhancementUsingArithmetic/LogicOperationsBasicsofSpatialFilteringSmoothingSpatialFiltersSharpeningSpatialCombiningSpatialEnhancementBasicsSpatialSpatialFilteringBasicsSpatialSpatialFilteringareperformeddirectlyonthepixelsofanimage.BasicsSpatialofspatialBasicsSpatialofspatialSpatialfiltercanalsobecalledmask,kernel,templateandwindow.BasicsSpatialAspatialfilterconsistsofaneighborhood(typicallyasmallapredefinedoperationthatisperformedontheimagepixelsencompassedbytheneighborhood.InputOutputBasicsSpatialAspatialfilterconsistsofaneighborhood(typicallyasmallapredefinedoperationthatisperformedontheimagepixelsencompassedbytheneighborhood.InputOutputBasicsSpatialTwokindsofspatialBasicsSpatialTwokindsofspatialLinearspatialTheoperationperformedontheimagepixelsislinear,thenthefilteriscalledalinearspatialfilter.NonlinearspatialTheoperationperformedontheimagepixelsisnonlinear,thenthefilteriscalledalinearspatialBasicsSpatial1.LinearSpatialTheresponseofthefilterisgivenbyasumofproductsofthefiltercoefficientsandthecorrespondingimagepixelsintheareaspannedbythefiltermask.ThefilterInputOutput??(??,??)=???1,?1?????1,???+???1,0?????1,+??0,0????,+?+??1,0????+1,+??1,1????+1,??+??(?1,??(0,??(1,BasicsSpatial1.LinearSpatialTheresponseofthefilterisgivenbyasumofproductsofthefiltercoefficientsandthecorrespondingimagepixelsintheareaspannedbythefiltermask.ThefilterInputOutput??(??,??)=???1,?1?????1,???+???1,0?????1,+??0,0????,+?+??1,0????+1,+??1,1????+1,??+??(?1,??(0,??(1,BasicsSpatialForamasksizeofBasicsSpatialForamasksizeof??×??,weassumethat??=2??1and??=2??+1.Ingeneral,linearspatialfilteringofanimageofsize??×??withafilterofsize??×??isgivenbytheexpression:??(??,??)??,??(??+??,??+??=???BasicsSpatial2.NonlinearSpatialBasicsSpatial2.NonlinearSpatialTheresponseofnonlinearspatialfilteringcan'tbeobtainedbytheformulaoflinespatialfiltering,forexamplemedianfilter.（Fordetailofmedianrefertothe)SomebasicGrayLevelTransformationsHistogramProcessingEnhancementUsingArithmetic/LogicOperationsBasicsofSpatialFilteringSmoothingSpatialFiltersSharpeningSpatialCombiningSpatialEnhancementBasicsSpatialThekeyBasicsSpatialThekeyusefulnessofsoothspatialfilternoisereduction.isblurringBasicsSpatialSmoothingLinearTheBasicsSpatialSmoothingLinearTheresponseofthefilerissimplytheaverageofpixelscontainedintheneighborhoodoffilter3×3Weighting3×3boxSmoothDifferentfilersize,SmoothDifferentfilersize,DifferenteffectofsmoothingSmoothAnapplicationofSmoothAnapplicationofsmoothinglinearSmoothOrders-Statistics(Nonlinear)SmoothOrders-Statistics(Nonlinear)Orders-statisticsfiltersisakindfilter.Itsoutputisbasedonorderingthepixelscontainedintheimageareaencompassedbythefilter,andthenreplacingthevalueofthecenterpixelwiththevaluedeterminedbytherankingThebest-knownexampleisthemedianSmoothSpatialMedianfilterreplacesthevalueofapixelbythemedianoftheintensityvaluesinSmoothSpatialMedianfilterreplacesthevalueofapixelbythemedianoftheintensityvaluesintheneighborhoodofthatpixel.ForMedianoftheSmoothApplicationexampleSmoothApplicationexampleMedianfilterareparticularlyeffectiveinthepresenceofimpulsenoise,alsocalledsalt-and-peppernoisebecauseofitsappearanceaswhiteandblackdotssuperimposedonanimage.SomebasicGrayLevelTransformationsHistogramProcessingEnhancementUsingArithmetic/LogicOperationsBasicsofSpatialFilteringSmoothingSpatialFiltersSharpeningSpatialCombiningSpatialEnhancementSharpeningSpatialTheprincipalobjectiveofSharpeningSpatialTheprincipalobjectiveofsharpeningistohighlightfinedetailinanimageorenhanceblurreddetail.SpatialismostlyaccomplishedspatialInputOutputSharpeningSpatialAbasicdefinitionofthefirst-orderderivativeSharpeningSpatialAbasicdefinitionofthefirst-orderderivativeofaone-dimensionalfunction??(??)isthedifference=????+?Similarly，wedefineasecond-orderderivativeasthe=????++??????SharpeningSpatialExplanationofSharpeningSpatialExplanationofcomparingandsecond-orderSharpeningSpatialPropertiesSharpeningSpatialPropertiesoffirstmustbezeroinareasofconstantmustbenonzeroattheonsetofanintensitysteporrampmustbenonzeroalongSharpeningSpatialPropertiesSharpeningSpatialPropertiesofsecondmustbezeroinareasofconstantmustbenonzeroattheonsetandendofanintensitysteporrampmustbezeroalongrampsofconstantSharpeningSpatialConclusionsofcomparingfirst-SharpeningSpatialConclusionsofcomparingfirst-andsecond-orderFirst-orderderivativesgenerallyproducethickeredgesinanimage;Second-orderderivativeshaveastrongerresponsefinedetail,suchasthinlinesandisolatedpoints;First-orderderivativesgenerallyhaveastrongerresponsetoagray-levelstep.Second-orderderivativesproduceadoubleresponseatstepchangesingraylevel.Forsimilarchangesingray-levelvaluesinanimage,responseofsecond-orderderivativesisstrongertoalinethantoastep,andtoapointthantoaline.SharpeningSpatialUseofsecondderivativesforenhancement–TheTheLaplacianofafunction??(??,??)oftwoSharpeningSpatialUseofsecondderivativesforenhancement–TheTheLaplacianofafunction??(??,??)oftwodefined??2??+Fordiscrete??2??+=????+1,?2????,+?????1,+[????,??+?2????, +??(??,???=????+1,+?????1,+????,??++????,????4??(??,SharpeningSpatialLap