圖像處理與控制系統(tǒng)課件

圖像處理與控制系統(tǒng)授課教師：祝海江電子郵箱：辦公地點：科技樓502室簡介2004年6月畢業(yè)于中國科學院自動化研究所模式識別國家重點實驗室模式識別與智能系統(tǒng)專業(yè)，獲得工學博士學位；同年7月進入北京化工大學信息科學與技術(shù)學院工作；2006.6-2007.6在日本巖手大學工學部任客座研究員；主要從事機器視覺、圖像處理、信號處理與檢測等方向的教學與科研工作。承擔國家自然科學基金、教育部留學回國人員科研啟動基金、中央高?；究蒲袠I(yè)務(wù)費等國家和省部級課題?？萍紭?02研究室常規(guī)計算機控制系統(tǒng)u計算機保持器廣義對象測量變送器e(kT)u(kT)y單回路計算機控制系統(tǒng)示意圖ym采樣器A/DD/A計算機系統(tǒng)設(shè)定值r基于圖像處理的控制系統(tǒng)系統(tǒng)框圖：控制平臺/控制策略執(zhí)行機構(gòu)被控對象攝像頭圖像處理模塊數(shù)模轉(zhuǎn)換數(shù)字圖像處理概述圖像獲取圖像增強與濾波圖像分割圖像特征提取本田公司最新開發(fā)的新型機器人“阿西莫”

世界第一個機器人藝人“Ever-2Muse”ThegoalofDigitalImageProcessingistoenabletheprocessofrecognition.TheultimategoalofDIPistoenableacomputingmachinetorecognizeatleastgeometricalsizes,shapesandotherobjectsasinhumanvisionDIPisabranchofArtificialIntelligence(AI).AnattempttoemulatehumanvisioniscalledweakAI.ToexactlyproduceahumanreplicaelectronicallyiscalledstrongAI.什么是數(shù)字圖像處理（DigitalImageProcessing）?原始圖像噪聲圖像低通濾波后的圖像原始圖像直方圖增強圖像邊緣檢測（Robertoperator）邊緣檢測（Sobeloperator)數(shù)字圖像處理的應(yīng)用1.InFlexibleManufacturingSystems:ProductInspection（產(chǎn)品檢測）Assembly（裝配）VehicleGuidance（車輛導航）2.InBiomedicalEngineering:AnalyzingChromosome（染色體分析）Tomography（斷層攝影術(shù)）X-rayAnalysis（X射線分析）醫(yī)療產(chǎn)品檢測3.InMilitaryAreas:BombDisposal（炸彈處理）Infra-redNightVision（紅外線夜視）RadarImageProcessing（雷達圖像處理）TargetIdentification（目標識別）4.InCivilianAreas:Telecommunications（可視化通訊）Firefighting（消防）Fingerprintdetection(指紋識別)IntelligentVehicleHighwaySystem（智能交通系統(tǒng)）FingerprintDetectionSystem5.InCommercialAreas:BarCodeReader（條形碼閱讀器）TextReader（文本閱讀器）Multimedia（多媒體）6.InScientificExperiments:FingerprintDetection（指紋識別）SpaceExploration（太空探索）GeographicStudies（地理學）Archaeology（考古學）Physics（物理）簡要歷史回顧1920圖像在倫敦與紐約之間經(jīng)由海底電纜傳輸1921照相復制技術(shù)產(chǎn)生1929圖像亮度級別從5增加到15，圖像復制技術(shù)改進1964計算機首次應(yīng)用到處理圖像中JetPropulsionLab(JPL)Now數(shù)字圖像處理及模式識別在許多領(lǐng)域廣泛應(yīng)用數(shù)字圖像處理系統(tǒng)回顧1.ImageCapturingSystem（圖像獲取系統(tǒng)）2.ImageEnhancementSystem（圖像增強系統(tǒng)）3.FeatureExtractionSystem（特征提取系統(tǒng)）4.FeatureRepresentationandDescriptionSystem（特征表示與描述系統(tǒng)）5.ObjectClassificationSystem（目標分類系統(tǒng)）圖像獲取400800BlueGreenRedInfrared紅外線Ultraviolet紫外線VisiblelightX-raysWavelength(nanometers)成像方式Radiance光輝Irradiance發(fā)光點光源相機目標傳感器NZOpticalaxisSurfacenormalPixelPixelPixelDigitalimage196Graylevel92圖像坐標系統(tǒng)

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

光柵導向usesrowandcolumncoordinatesstartingat[0,0]fromthetopleftxyF[0,0]F[M-1,N-1]Cartesian笛卡爾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]圖像類型1:模擬圖像

Ananalogimageisa2DimageF(x,y)which -hasinfinite

precisioninspatialparameters

xandy,and -infinite

precisioninintensityateachspatialpoint(x,y).yxf(xi,yi)=Realnumberxi=Realnumberyi=Realnumber2:數(shù)字圖像

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:單色灰度圖像Agray?scaleimage

isamonochromedigitalimagef(x,y)withone

intensityvalue

perpixel.f(x,y)=0f(x,y)=89f(x,y)=2185:彩色圖像Amultispectralimage

isa2DimageM[x,y],whichhasavectorofvaluesateachspatialpointorpixel.Iftheimageisactuallyacolorimage,thenthevectorhas3elements.I=0.11R+0.59G+0.3B6:二值圖像Abinaryimageisadigitalimagewithallpixelvalues0or1.圖BA像xyf(x,y)=1f(x,y)=07:分類圖像Alabeledimage

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

ThenominalresolutionofaCCDsensoristhesizeofthesceneelementthatimagestoasinglepixelontheimageplane.Eachpixelofadigitalimagerepresentsasampleofsomeelementalregionoftherealimage.圖像度量與量化（數(shù)字化）pixel3DSceneLensImagePlaneSizeofsceneelementpixelIfthepixel

isprojected

fromtheimageplane

backouttothesourcematerialinthescene,thenthesizeofthatsceneelementisthenominalresolution標稱分辨度

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

referstotheprecisionofthesensorinmakingmeasurements,butisformallydefinedindifferentways.Ifdefinedinrealworldterms,itmayjustbethenominalresolution,asin“theresolutionofthisscannerisonemeterontheground”O(jiān)ritmaybeinthenumberoflinepairspermillimeterthatcanberesolvedordistinguishedinthesensedimage.3:視野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圖像量化（數(shù)字化）Agraphicalrepresentation(staircasemap)ofthequantizationfunctionisasbelow:Usually,L=2B

(B-bitrepresentation).D=tL–t0

=umax-uminiscalledthedynamicrange.ErrorofquantizationclearlydependsonLandD,aswellasonthechoiceofreconstructionlevelsandtransitionlevels.uu*Quantizerrktkuu*rLr0t0tL圖像量化中的一些問題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數(shù)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)數(shù)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.圖像格式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.圖像增強與濾波Animageneedsimprovement

Low?levelfeaturesmustbedetected

圖像增強

例1:圖像中的劃痕被去掉。Scratches例2:亮度增強例3:機器零件邊緣增強Left

-Originalsensedfingerprint;

Center

-Imageenhancedbydetectionandthinningofridges;

Right

-Identificationofspecialfeaturescalledminutia,whichcanbeusedformatchingtomillionsoffingerprintrepresentationsinadatabase.Example圖像增強操作(1)點操作ContraststretchingNoiseclippingWindowslicingHistogrammodeling(2)掩膜操作NoisesmoothingMedianfilteringSharpingmaskingZooming對比度增強(a)Original(b)Enhanced(b)Enhanced(a)Original(a)Original(b)EnhancedClipingandthresholdingClipingandthresholding反色反色反色直方圖增強

Histogramafterequalization

Originalimage

OriginalhistogramModifiedimage

Originalimage

OriginalhistogramModifiedimage

Histogramafterequalization

(a)Inputimage

(b)Processedimage(c)Inputimage

(d)Processedimage(e)Inputimage

(f)Processedimage

圖像濾波

Often,animageiscomposedofsomeunderlyingidealstructure,whichwewanttodetectanddescribe,togetherwithsomerandom

noiseorartifact,whichwewouldliketoremove.ImagecontainsbothGaussiannoiseandbrightringartifactsImagewithrandomnoiseScratchesImagecontainsartifacts方框濾波器（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鄰閾平均法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高斯濾波（GaussianFilter）disthedistanceoftheneighborhoodpixel[x,y]fromthecenterpixel[xc,yc]oftheoutputimagewherethefilterisbeingapplied.[xc]g(x)[x]d[x] Ratherthanweightallinputpixelsequally,itisbetter

toreducetheweight

oftheinputpixelswithincreasingdistancefromthecenterpixelI[xc,yc].TheGaussianfilterdoesthisandisperhapsthemostcommonlyusedofallfilters.[xc]xg(x)高斯函數(shù) 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圖像分割1.基于掩膜窗口的分割I(lǐng)magepointsofhighcontrastcanbedetectedbycomputingintensitydifferencesinlocalimageregions.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三種掩膜窗口: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.基于閾值的分割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圖像特征提取漢字識別的困難

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?在模式識別中，特征指的是把兩類或多類目標區(qū)別開來的一種描述方法。

Featuresarefunctionsofthemeasurementsperformedonaclassofobjectsthatenablethatclasstobedistinguishedfromotherclassesinthesamegeneralcategory.ExampleForpolygonsthenumberofverticesthenumberofsidesThelengthsofsidesthevaluesoftheanglesverticesideqPurposeoffeatureselection:

reducedimensionalityofrepresentation

minimizemeasurementextractioncosts

assessthepotentialperformanceofthepatternrecognitionsystemimprovesystem'sperformanceThreeingredientsinfeatureselection/extractionfeatureevaluationcriteriadimensionalityofthefeaturespaceoptimizationprocedure如何選擇特征?(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