图像处理与控制系统课件

图像处理与控制系统授课教师：祝海江电子邮箱：办公地点：科技楼502室简介2004年6月毕业于中国科学院自动化研究所模式识别国家重点实验室模式识别与智能系统专业，获得工学博士学位；同年7月进入北京化工大学信息科学与技术学院工作；2006.6-2007.6在日本岩手大学工学部任客座研究员；主要从事机器视觉、图像处理、信号处理与检测等方向的教学与科研工作。承担国家自然科学基金、教育部留学回国人员科研启动基金、中央高校基本科研业务费等国家和省部级课题。科技楼502研究室常规计算机控制系统u计算机保持器广义对象测量变送器e(kT)u(kT)y单回路计算机控制系统示意图ym采样器A/DD/A计算机系统设定值r基于图像处理的控制系统系统框图：控制平台/控制策略执行机构被控对象摄像头图像处理模块数模转换数字图像处理概述图像获取图像增强与滤波图像分割图像特征提取本田公司最新开发的新型机器人“阿西莫”

世界第一个机器人艺人“Ever-2Muse”ThegoalofDigitalImageProcessingistoenabletheprocessofrecognition.TheultimategoalofDIPistoenableacomputingmachinetorecognizeatleastgeometricalsizes,shapesandotherobjectsasinhumanvisionDIPisabranchofArtificialIntelligence(AI).AnattempttoemulatehumanvisioniscalledweakAI.ToexactlyproduceahumanreplicaelectronicallyiscalledstrongAI.什么是数字图像处理（DigitalImageProcessing）?原始图像噪声图像低通滤波后的图像原始图像直方图增强图像边缘检测（Robertoperator）边缘检测（Sobeloperator)数字图像处理的应用1.InFlexibleManufacturingSystems:ProductInspection（产品检测）Assembly（装配）VehicleGuidance（车辆导航）2.InBiomedicalEngineering:AnalyzingChromosome（染色体分析）Tomography（断层摄影术）X-rayAnalysis（X射线分析）医疗产品检测3.InMilitaryAreas:BombDisposal（炸弹处理）Infra-redNightVision（红外线夜视）RadarImageProcessing（雷达图像处理）TargetIdentification（目标识别）4.InCivilianAreas:Telecommunications（可视化通讯）Firefighting（消防）Fingerprintdetection(指纹识别)IntelligentVehicleHighwaySystem（智能交通系统）FingerprintDetectionSystem5.InCommercialAreas:BarCodeReader（条形码阅读器）TextReader（文本阅读器）Multimedia（多媒体）6.InScientificExperiments:FingerprintDetection（指纹识别）SpaceExploration（太空探索）GeographicStudies（地理学）Archaeology（考古学）Physics（物理）简要历史回顾1920图像在伦敦与纽约之间经由海底电缆传输1921照相复制技术产生1929图像亮度级别从5增加到15，图像复制技术改进1964计算机首次应用到处理图像中JetPropulsionLab(JPL)Now数字图像处理及模式识别在许多领域广泛应用数字图像处理系统回顾1.ImageCapturingSystem（图像获取系统）2.ImageEnhancementSystem（图像增强系统）3.FeatureExtractionSystem（特征提取系统）4.FeatureRepresentationandDescriptionSystem（特征表示与描述系统）5.ObjectClassificationSystem（目标分类系统）图像获取400800BlueGreenRedInfrared红外线Ultraviolet紫外线VisiblelightX-raysWavelength(nanometers)成像方式Radiance光辉Irradiance发光点光源相机目标传感器NZOpticalaxisSurfacenormalPixelPixelPixelDigitalimage196Graylevel92图像坐标系统

crI[0,0]I[M-1,0]I[M-1,N-1]rasteroriented

光栅导向usesrowandcolumncoordinatesstartingat[0,0]fromthetopleftxyF[0,0]F[M-1,N-1]Cartesian笛卡尔coordinateframewith[0,0]atthelowerleftxy[0,0][W/2,H/2][-W/2,-H/2]Cartesiancoordinateframewith[0,0]attheimagecenterRelationshipofpixelcenterpoint[x,y]toareaelementsampledinarrayelementI[i,j][x0,y0][x0+ix,y0+jy]F[i,j]F[i+1,j]图像类型1:模拟图像

Ananalogimageisa2DimageF(x,y)which -hasinfinite

precisioninspatialparameters

xandy,and -infinite

precisioninintensityateachspatialpoint(x,y).yxf(xi,yi)=Realnumberxi=Realnumberyi=Realnumber2:数字图像

Adigitalimageisa2D

imageI[r,c]representedbyadiscrete2Darrayofintensitysamples,eachofwhichisrepresentedusingalimitedprecision.Itiscommontorecordintensityasan8‑bit(1‑byte)numberwhichallowsvaluesof0to255.256differentlevelsisusuallyalltheprecisionavailable-fromthesensorand-alsoisusuallyenoughtosatisfytheconsumer.

yxf(xi,yi)=Integerxi=Integeryi=Integer3:Apicturefunction

isamathematicalrepresentation

f(x,y)ofapictureasafunctionoftwospatialvariablesxandy.

xandyarerealvaluesdefiningpointsofthepicture.

f(x,y)isusuallyalsoarealvaluedefiningtheintensityofthepictureatpoint(x,y).

4:单色灰度图像Agray‑scaleimage

isamonochromedigitalimagef(x,y)withone

intensityvalue

perpixel.f(x,y)=0f(x,y)=89f(x,y)=2185:彩色图像Amultispectralimage

isa2DimageM[x,y],whichhasavectorofvaluesateachspatialpointorpixel.Iftheimageisactuallyacolorimage,thenthevectorhas3elements.I=0.11R+0.59G+0.3B6:二值图像Abinaryimageisadigitalimagewithallpixelvalues0or1.图BA像xyf(x,y)=1f(x,y)=07:分类图像Alabeledimage

isadigitalimageL[r,c]whosepixelvaluesaresymbols.Thesymbolvalueofapixeldenotestheoutcomeofsomedecisionmadeforthatpixel.OriginalimageLabeledimageBoundariesoftheextractedfaceregionOriginalimage(tiger)Labeledimage1:标称分辨度

ThenominalresolutionofaCCDsensoristhesizeofthesceneelementthatimagestoasinglepixelontheimageplane.Eachpixelofadigitalimagerepresentsasampleofsomeelementalregionoftherealimage.图像度量与量化（数字化）pixel3DSceneLensImagePlaneSizeofsceneelementpixelIfthepixel

isprojected

fromtheimageplane

backouttothesourcematerialinthescene,thenthesizeofthatsceneelementisthenominalresolution标称分辨度

ofthesensor.pixel3DSceneLensImagePlaneForexample,ifa10inchsquaresheetofpaperisimagedtoforma500500digitalimage,thenthenominalresolutionofthesensoris0.02inches(10/500=0.02).1010inch25005002:分辨率Thetermresolution

referstotheprecisionofthesensorinmakingmeasurements,butisformallydefinedindifferentways.Ifdefinedinrealworldterms,itmayjustbethenominalresolution,asin“theresolutionofthisscannerisonemeterontheground”Oritmaybeinthenumberoflinepairspermillimeterthatcanberesolvedordistinguishedinthesensedimage.3:视野Thefieldofviewofasensor(FOV)isthesizeofthescenethatitcansense,forexample10inchesby10inches.(a)Digitalimagewith127rowsof176columns;(b)(6388)createdbyaveragingeach22neighborhoodof(a)andreplicatingtheaveragetoproducea22averageblock;(c)(3144)createdinsamemannerfrom(b);

and(d)(1522)createdinsamemannerfrom(c).Effectivenominalresolutionsare(127176),(6388),

(3144),and(1522)respectively.AquantizerisanAnalog-to-Digitaldevicewhichconvertsacontinuous

inputsignalutooneofasetofdiscretelevelscalledreconstructionlevels

rk.Supposetheuliesintherange:umin

u

umaxandwewishtoquantizeuintoL

levels.Thenwedefine

L+1

transitionlevelstk:

t0=umin<t1<……<tL-1<tL=umaxThequantizationstepinvolvesmappingutoitsquantizedvalue,u*,usingtherule: Define{tk,k=0,…,L}asasetofincreasingtransitionordecisionlevelswitht0andtLastheminimumandmaximumvalues,respectively,ofu.ImageQuantization图像量化（数字化）Agraphicalrepresentation(staircasemap)ofthequantizationfunctionisasbelow:Usually,L=2B

(B-bitrepresentation).D=tL–t0

=umax-uminiscalledthedynamicrange.ErrorofquantizationclearlydependsonLandD,aswellasonthechoiceofreconstructionlevelsandtransitionlevels.uu*Quantizerrktkuu*rLr0t0tL图像量化中的一些问题1010arrayofblack(brightness0)andwhite(brightness8)tiles;(b)

Intensitiesrecordedina55imageofpreciselythebrightnessfieldattheabove,whereeachpixelsensestheaverage

brightnessofa22

neighborhoodoftiles;(0+0+0+8)/4=2(0+0+0+8)/4=2(d)Intensitiesrecordedfromtheshiftedcamerainthesamemannerasin(b).(c)Imagesensedbyshifted

cameraonetiledownandonetiletotheright.0000000000000080880(8+0+0+0)/4=2Notethatthequantizedbrightnessvaluesdepend

onboththeactualpixelsizeandpositionrelativetothebrightnessfield;Interpretationoftheactualscenefeatureswillbeproblematicwitheitherimage(b)or(d).(b)(d)Notethatthequantizedbrightnessvaluesdependonboththeactualpixelsizeandpositionrelativetothebrightnessfield.Differentdigitalimages数字化效果ColourimageGray-scaleimage512X256Gray-scaleimage256X128Theresolutiondependsonthespatialquantization(thenumberofsamplesperinch)Gray-scaleimage64X32Gray-scaleimage32X16Gray-scaleimage128X64Theresolutiondependsonthespatialquantization(thenumberofsamplesperinch)8-Bitimage(256levels)7-Bitimage(128levels)Thequalityoftheimagedependsontheintensityquantization(thenumberofgraylevels)6-Bit(64levels)5-Bit(32levels)4-Bit(16levels)3-Bit(8levels)2-Bitimage(4levels)1-Bitimage(2levels)Thequalityoftheimagedependsontheintensityquantization(thenumberofgraylevels)数字化效果1.Run‑CodedBinaryImages

Run‑codingisanefficientcodingschemeforbinaryorlabeledimages:notonlydoesitreducememoryspace,butitcanalsospeedupimageoperations.Example:

ImageRowr

1111111111100000

Run‑codeA

8(0)5(1)12(0)3(1)7(0)9(1)5(0)Run‑codingisoftenusedforcompressionwithinstandard.图像格式2.PGM:PortableGrayMapP2#samplesmallpicture8rowsof16columns,maxgrayvalueof192#makinganimageoftheword"Hi".

168192PrintedpictureImagemadeusingalossycompressionalgorithmOneofthesimplestforstoringandexchangingimagedataisthePBMorPortableBitMapfamilyofformats(PBM/PGM,PPNI).

TheimageheaderandpixelinformationareencodedinASCII.3.GIFImageTheGraphicsInterchangeFormat(GIF)originatedfromCompuServe,Inc.IthasbeenusedtoencodeahugenumberofimagesontheWorldWideWeborincurrentdatabases.GIFfilesarerelativelyeasytoworkwith,butcannotbeusedforhigh‑precisioncolor,sinceonly8‑bitsareusedtoencodecolor.4.TIFFImageTIFForTIFis

verygeneralandverycomplex.Itisusedonallpopularplatformsandisoftentheformatusedbyscanners.Itsupportsmultipleimageswith1to24bitsofcolor

perpixel.TIFForTIFis

availableforeitherlossyorlosslesscompression.5.JPEGFormatforStillPhotosJPEG(JFIF/JFI/JPG)isamorerecentstandardfromtheJointPhotographicExpertsGroup.Themajorpurposeistoprovideforpracticalcompressionofhigh‑qualitycolorstillimages.Animagecanhaveupto64K64Kpixelsof24bitseach.6.PostScriptThefamilyofformatsBDF/PDL/EPSstoreimagedatausingprintableASCIIcharactersandareoftenusedwithX11graphicsdisplaysandprinters.PDLisapagedescriptionlanguageEPSisencapsulatedpostscript(originallyfromAdobe),whichiscommonlyusedtocontaingraphicsorimagestobeinsertedintoalargerdocument.7.MPEGFormatforVideoMPEG(MPG/MPEG‑1/MPEG‑2)isastream‑orientedencodingschemeforvideo,audio,text,andgraphics.MPEGstandsforMotionPictureExpertsGroup,aninternationalgroupofrepresentativesfromindustryandgovernments.MPEG‑1isprimarilydesignedformultimediasystemsandprovidesforadatarateof0.25Mbitspersecondofcompressedaudioand1.25Mbitsofcompressedvideo.Theseratesaresuitableformultimediaforpopularpersonalcomputers,butaretoolowforhigh‑qualityTV.MPEG‑2standardprovidesforupto15MbitsperseconddataratestohandlehighdefinitionTVrates.Thecompressionschemetakesadvantageofbothspatialredundancy,asusedinJPEG,andtemporalredundancyandgenerallyprovidesausefulcompressionratioof25to1,with200to1ratiospossible.图像增强与滤波Animageneedsimprovement

Low‑levelfeaturesmustbedetected

图像增强

例1:图像中的划痕被去掉。Scratches例2:亮度增强例3:机器零件边缘增强Left

-Originalsensedfingerprint;

Center

-Imageenhancedbydetectionandthinningofridges;

Right

-Identificationofspecialfeaturescalledminutia,whichcanbeusedformatchingtomillionsoffingerprintrepresentationsinadatabase.Example图像增强操作(1)点操作ContraststretchingNoiseclippingWindowslicingHistogrammodeling(2)掩膜操作NoisesmoothingMedianfilteringSharpingmaskingZooming对比度增强(a)Original(b)Enhanced(b)Enhanced(a)Original(a)Original(b)EnhancedClipingandthresholdingClipingandthresholding反色反色反色直方图增强

Histogramafterequalization

Originalimage

OriginalhistogramModifiedimage

Originalimage

OriginalhistogramModifiedimage

Histogramafterequalization

(a)Inputimage

(b)Processedimage(c)Inputimage

(d)Processedimage(e)Inputimage

(f)Processedimage

图像滤波

Often,animageiscomposedofsomeunderlyingidealstructure,whichwewanttodetectanddescribe,togetherwithsomerandom

noiseorartifact,whichwewouldliketoremove.ImagecontainsbothGaussiannoiseandbrightringartifactsImagewithrandomnoiseScratchesImagecontainsartifacts方框滤波器（BoxFilter）

Definition

:Smoothinganimagebyequally

weightingarectangularneighborhoodofpixelsiscalledusingaboxfilter.

Output-Image[r,c]= Averageofsomeneighborhoodof Input-Image[r,c]

Example:55NeighborhoodFilter-averages25pixelvaluesina55neighborhoodoftheinputimagepixelinordertocreateasmoothedoutputimage.Example80912308081331808030820803040340405050204000+03+08+12+03+05+40+30+09+13+40+40+80+80+00+50+30+80+80+00+20+40+20+30+1825=3080912308081331808030820803040340405050204030邻阈平均法OriginalImageNoisyImageNAF(3-by-3)NAF(5-by-5)NAF(7-by-7)UsefornoisesmoothingLPfilteringandsubsamplingofimages.AssumingwhitenoiseηwithzeromeanandvarianceThenthespatialaverage:assumingequalweightwhereisthespatialaverageof.Notethathaszeromeanandi.e.NoisepowerisreducedbyafactorofRemark:Neighborhoodaveragingintroducesadistortionintheformofblurring.

(a)Original(b)noisy(c)3×3filter(d)5×5filterSpatialaveragingfiltersforsmoothingimagescontainingGaussiannoise.

Definition

:WhenaGaussianfilterisused,pixel[x,y]isweightedaccordingtox高斯滤波（GaussianFilter）disthedistanceoftheneighborhoodpixel[x,y]fromthecenterpixel[xc,yc]oftheoutputimagewherethefilterisbeingapplied.[xc]g(x)[x]d[x] Ratherthanweightallinputpixelsequally,itisbetter

toreducetheweight

oftheinputpixelswithincreasingdistancefromthecenterpixelI[xc,yc].TheGaussianfilterdoesthisandisperhapsthemostcommonlyusedofallfilters.[xc]xg(x)高斯函数 One-DimensionalGaussianFunction Two-DimensionalGaussianFunctionExample809123080813318080308208030403404050502040001+031+081+121+031+051

+402+302

+092+131+401+402+803+802+001+501+302+802+802+001+201+401+201+301+181

25=5280912308081331808030820803040340405050204052Examples

NoisyimageIdealimagePixelvaluesinrow100ofthenoisyimagePixelvaluesinrow100ofthesmoothedimageNoiseaveragedusinga55neighborhoodExamples

NoisyimageM=32M=16M=8M=2M=128图像分割1.基于掩膜窗口的分割Imagepointsofhighcontrastcanbedetectedbycomputingintensitydifferencesinlocalimageregions.HighcontrastHighcontrastTypically,suchpointsformtheborder(oredge)betweendifferentobjectsorsceneparts.Neighborhood

templatesormaskscanbeused.Westartbyusingone‑dimensional(1D)signals.The1Dsignalscouldjustberowsorcolumnsofa2Dimage.(a)(b)BorderDifferencing2DImages(DetectingEdgesof2DImages)

Themaximumchangeofthecontrastinthe2Dpicturefunctionf(x,y)

occursalongthedirectionofthegradient

梯度

ofthefunction.(Edge)HighcontrastThedirectionofthe

gradient

梯度Mathematicformulaofthegradient:

Gradientmagnitudeor

GradientdirectionfxfyfLower/HigherintensitiesHigher/Lowerintensitiesfxfyf三种掩膜窗口:Sobelmasks210-1-2-1100Mx=0-12-10110-2My=110-1-1-11000-11-10110-1Mx=My=PrewittmasksMx=My=Robertmasks10-1-11000OriginalImage-LenaEnhancedLenabyHistogramEuqalizationEdgemapbyRobertoperatorEdgemapbySobeloperatorStep-1.Compute:MaskMx

isoverlaidonimageneighborhoodN8[x,y]

sothateachintensityNij

canbemultipliedbyweightMij;Finallyalltheseproductsaresummed.Prewittmasks644215356612143865fxfyfN8[x,y]110-1-1-1100Mx=Step-2.Compute:MaskMy

isoverlaidonimageneighborhoodN8[x,y]

sothateachintensityNij

canbemultipliedbyweightMij;Finallyalltheseproductsaresummed.644215356612143865fxfyfN8[x,y]0-11-10110-1My=Step-3.Compute:

Gradientmagnitude

Gradientdirection110-1-1-1100Mx=0-11-10110-1My=LowerintensitiesHigherintensities644215356612143865fxfyfN8[x,y]ExampleImageofJudithPrewittGradientimage

showingresultusingthePrewitt33operator(a)(b)Sobelmasks:theSobeloperatorrepresentsmany,butnotall,oftheimageedges.210-1-2-1100Mx=0-12-10110-2My= (a)Imageofnoisysquaredandrings,(b)Codingofgradientdirectioncomputedby33Sobeloperator.(a)(b)ExampleAninputimage(a)issmoothedusingGaussianfiltersofsize(b)=4,and(c)=1

beforeperformingedgedetection.Moredetailandmorenoiseisshownforthesmallerfilter.(b)(c)(a)=4=1Example: (a)ImageofthegreatarchinSt.Louis; (b)resultsofCannyoperatorwith=1; (c)resultsofCannyoperatorwith=4;(a)(b)(c)ExamplesImageofMao’sMemorial.ResultofapplyingCannyoperatorwith=1.Resultof=2.

Someobjectsaredetectedverywell,soaresomeshadows.(a)(b)(c)DifferentHistograms2.基于阈值的分割Tofindthethreshold,Letthehistogrambep(z)Find2localmaximaonp(z)thatareatleastsomeminimumdistanceapart;z1,z2sayFindz3betweenz1andz2atwhichp(z3)ismin.Checkthatp(z3)/[min(p(z1),p(z2))]tobesmallChoosez3tobethethresholdIfprobabilitydistributionofthe2regionisknown,thenwecanuseBayesiandecisiontheorytofindthethreshold. Letwi=pixelbelongstoregionI ThenchoosezsuchthatP(w1︱z)=P(w2︱z) ApplyBayesrule:

i.e.Selectzsuchthatp(z︱w2)P(w2)=p(z︱w1)P(w1)HistogramrepresentingobjectondarkbackgroundOriginalImage-LenaBinaryLenabyOtsuLenaHistogramThresholdat50Thresholdat165Thresholdat80图像特征提取汉字识别的困难

LargeinDataSetComplexinStructure1.ALargeSetofCharacters:English: 26lettersRussian: 32lettersGreek: 24lettersChinese: 3,000-7,000charactersareoftenusedStandardinP.O.China:6,763Thefirstlevel:3,755,Thesecondlevel:3,0087,000-10,000Chrs.arecollectedinsmalldictionaries70,000ChelargestcontemporarydictionaryInthelonghistoryofChina,thetotalnumberofChinesecharactersbecamelargerandlargerCellularFeatureExtractionPreprocessingNeuralTreeClassification008109212597191832302Input:Chinesecharacter“tree”......Features(Matrix)什么是特征?在模式识别中，特征指的是把两类或多类目标区别开来的一种描述方法。

Featuresarefunctionsofthemeasurementsperformedonaclassofobjectsthatenablethatclasstobedistinguishedfromotherclassesinthesamegeneralcategory.ExampleForpolygonsthenumberofverticesthenumberofsidesThelengthsofsidesthevaluesoftheanglesverticesideqPurposeoffeatureselection:

reducedimensionalityofrepresentation

minimizemeasurementextractioncosts

assessthepotentialperformanceofthepatternrecognitionsystemimprovesystem'sperformanceThreeingredientsinfeatureselection/extractionfeatureevaluationcriteriadimensionalityofthefeaturespaceoptimizationprocedure如何选择特征?(1)Aformal,number-crunchingapproach:forstatisticalPR.(2)Designfeatureswithsemanticcontents,someintuitivewaycorrespondtohumanperceptionoftheobjects:forstructuralPRExamples:(1)ProjectionFourierTransformFeaturesof750123467(2)

00555S00555ABHKVT={0,5}VN={S,A,B,H,K}P:S0A,A0B,B5H,H5K,K5

Featuresof7DensityFeaturesApatternimageisdividedintoNNsub-images

N=4N=4WWi(CellularFeature)N=4N=4Calculatethe

densityofeachsub-image

Howmanyimagepixelsineachsub-imageProjectionsInpatternrecognition,thetermprojectionusuallyreferstomappinganimageintoawaveform

Thevaluesofthewaveformarethesumsoftheimagepointsalong

particular

directionsAccordingtodirections,3projectionshavebeendeveloped:

HorizontalandverticalprojectionRingprojectionCentralprojectionProjectionsFormulaoftheprojection:istheprojectiondirection,Risareaofimage[z]isafunctionsuchthatxytProjectionsofsomeparticulardirections=0°,45°,90°and135°:tf(t)tf(t)Horizontalandvertical

Projections2-Dobjectisconvertedintotwo1-Dsignals

tf(t)tf(t)Orthogonaltransform(Fouriertransform)isusedtoobtainnumericalfeaturesFeatureVectorsVa

=

{va1,va2,...,van,}Vb

=

{vb1,vb2,...,vbn,}Vc

=

{vc1,vc2,...,vcn,}………………..Vz

=

{vz1,vz2,...,vzn,}Theprojectionisrotationsensitive779881047riCenterofgravityrk4911p(r)rp(ri)ri0rkRing-ProjectionAlgorithmThe1-DpatternobtainedfromtheRing-ProjectionalgorithmisinvarianttorotationsExtractionofrotation-invariantfeatures Ring-projection

Cumulativeangularfunction- Fourierdescriptor(CAF-FD) MomentInvariantCumulativeAngularFunction-

FourierDescriptors

(CAF-FD)Cumulativeangularfunction-Fourierdescriptor(CAF-FD)canproducerotation-invariantfeaturesWhentheobjectisrotatedwithdifferentangles,thefeaturearesameStep-1:Representapatternbyaboundary(closedcurve)Step-2:TracethecurveclockwiseovertheentireboundaryStep-3:Findtheangulardirection(t)ateachkeypointsStep-4:Findcumulativeangularfunction(t)Step-5:NormalizethecumulativeangularfunctionandproduceNCAF*(t)Step-6:ExpandNCAFintoFourierseriesAlgorithmofCAF-FDStep-1:

RepresentapatternbyaboundaryApatternisrepresentedbyaboundaryCumulativeangularfunction-Fourierdescriptor(CAF-FD)requireclosedcurvesAnypattern,thatcanbeapproximatedbyaboundarycurve,canbedescribedbyCAF-FDThecurveistracedclockwiseovertheentireboundaryAlgorithm:

(Boundarytracing)StartingpointThestartingpointofthecurveisarbitrarilychosenStep-2:

TracetheboundaryAlgorithm:(Boun