AgendaFoundationCh1~4ImagingSystem,Digitalization,Display,SoftwareCh5~8Histogram,PointOperations,AlgebraicOperations,GeometricOperationsTheoryCh9~12LinearSystem,FourierFrequencyTransform,FilterDesign,DiscreteSamplingCh13~15OrthogonalRadicleTransform,WaveletTime-frequencyTransform,OpticalFunctionTransformApplicationCh16~20ImageRestoration,Compression,PatternRecognitionCh21~22ColorandMulti-SpectralImageProcessing,ThreeDimensionImageProcessing1、KnowledgeofComplexAlgebraicexpression：Geometricalexpression：Complexfunction：ziscomplex

so2、TheFourierTransform§2.1definitionTheFouriertransformofaone-dimensionalfunctionf(t)is：

TheinverseFouriertransformofF(s)isdefinedas：§2.2TheDiscreteFourierTransformIfwediscretizebothtimeandfrequency：Inversetransform：where：then:§

2.3TheFouriertransformintwodimensionsThedirectFouriertransform：TheinverseFouriertransform：3、SometypicalFouriertransformRectangularpulsetransform：Figure3-12transformFouriertransformoffinitewavetrainFigure3-15Fouriertransformoffinitewavetrain

FouriertransformoftriangleBothoftheinputsarerectangularpulseConvolveoutputistrianglepulse

TheHankelTransformTwo-dimensionalFouriertransform:Two-dimensionalfunctionwithcircularsymmetrymaybetreatedasone-dimensionalfunctionsofasingleradialvariable：Dovariablesubstitution:

Thus,theFouriertransformoff(x,y)is：

where4、CorrelationandthepowerspectrumSelf-convolution：Autocorrelation：Thepowerspectrum，theFouriertransformoftheautocorrelationfunctionis：Andiscalledthepowerspectraldensityfunctionorpowerspectrumoff(t).AgendaFoundationCh1~4ImagingSystem,Digitalization,Display,SoftwareCh5~8Histogram,PointOperations,AlgebraicOperations,GeometricOperationsTheoryCh9~12LinearSystem,FourierFrequencyTransform,FilterDesign,DiscreteSamplingCh13~15OrthogonalRadicleTransform,WaveletTime-frequencyTransform,OpticalFunctionTransformApplicationCh16~20ImageRestoration,Compression,PatternRecognitionCh21~22ColorandMulti-SpectralImageProcessing,ThreeDimensionImageProcessingChapter11

FilterDesignAgendaFoundationCh1~4ImagingSystem,Digitalization,Display,SoftwareCh5~8Histogram,PointOperations,AlgebraicOperations,GeometricOperationsTheoryCh9~12LinearSystem,FourierFrequencyTransform,FilterDesign,DiscreteSamplingCh13~15OrthogonalRadicleTransform,WaveletTime-frequencyTransform,OpticalFunctionTransformApplicationCh16~20ImageRestoration,Compression,PatternRecognitionCh21~22ColorandMulti-SpectralImageProcessing,ThreeDimensionImageProcessingChapter11

FilterDesign11.1FilterWhatisafilter?Whyshouldweintroducethefilter?Duetosomenoiseinsignalorimages,thefiltercanbeusedasatoolofnoisereductioninsomefrequencybands.It’susuallyinterestedincertaincomponentsofthesignalorimages,soweusefilterstoeliminateuselesscomponentsandenhanceusefulones.LOWPASSFilter

1、ideallowpass

Sinc(t)s(t)

1

-S00

S0

stt

Remainthepartof|S|≤S0，removethepartof|S|≥S0，thetime-frequencyisafunctionofSinc：Sin(t)/t；disadvantages：“overshoot”willbeproducedinbothendsofgradationzoneoforiginalsignals.1)TheBoxFilterConvolvingthesignalf(x)withtherectangularpulsetoreducehigh-frequencynoisewithlocalaveraging.

Thewidthofthetransferfunctionisinverselyproportionaltothewidthoftheimpulseresponse；Aslongastheboxfilterisnomorethantwopixelswide，thefirstzero-crossingofitstransferfunctionfallsatorabovethehighestfrequencyfo

；Iftheboxfilterismorethantwopixelswide,however,thereisthedangerofpolarityreversalforsmallstructuresintheimage,andthefirstzero-crossingofitstransferfunctionfallsbelowthehighestfrequencyfo

TheBoxFilterIfthetimedomainis∏(t)：

ThefrequencydomainisSinc(s)：

disadvantages：polarityreversalts2)TheTriangularFilter

Wecanusethetriangularpulse,asalowpassfilterimpulseresponse.Λ(t)=>Sinc2(s)；wouldn’tfearofpolarityreversal；canbesimulatedbyrectangularfilter.

ts3)TheGaussianLowpassFilter：GaussianFunctionyieldsalowpassfilter.

11.3BANDPASSANDBANDSTOPFILTERSThistypeoffilterspassesorstopsparticularfrequenciestopickupusefulcomponents.

Weselectanonnegativeunimodalfunctionandconvolveitwithanevenimpulsepairatfrequencytoconstructabandpassfilter.

Bandstopfiltersinfactbelongtobandpassfilters,

Itstransferfunctioncanbegotwith1minusbandpassfilter.1)TheIdealBandpassFilterconsiderThattransferfunctionisgivenby：Thentheimpulseresponsebecomes：2)TheIdealBandstopFilterThetransferfunctionis：Fromthetransferfunctiontheimpulseresponseis：1、TheIdealBandpassFilter：

Movethefrequencyof∏(s)；0s

Theformula：2·Sinc(t)·cos(2πS0t)

2、TheIdealBandstopFilter：1-thenidealbandpassfilter。

s3)TheGeneralBandpassFilterWeselectanonnegativeunimodalfunctionK(s)andconvolveitwithanevenimpulsepairatfrequencyS0.Thisyieldsabandpasstransferfunction：Theimpulseresponseisgivenby：

s11.4HIGH-FREQUENCYENHANCEMENTFILTERSHigh-frequencyenhancementfilter(highpassfilter)Itisgenerallytakentodescribeatransferfunctionthatisunityatzerofrequencyandincreaseswithincreasingfrequency.Suchatransferfunctionmayeitherleveloffatsomevaluegreaterthanunityor,morecommonly,fallbacktowardzeroathigherfrequencies.

Inpractice,itissometimesdesiredtohavelessthanunitygainatzerofrequency,soastoreducethecontrastoflarge,slowlyvaryingcomponentsoftheimage.Ifthetransferfunctionpassesthroughtheorigin,itmaybecalledaLaplacianfilter.highpass/high-frequencyenhancement

Ideal：

sGeneral：

s1)TheDifference-of-Gaussians(DOG)Filter（1）Principle：thedifferenceoftwoGaussiansFigure11-8TheGaussianhigh-frequencyenhancementtransferfunctionTheimpulseresponseofthefilteris：Theimpulseresponseofsuchafilteris：isfiniteRulesofThumbforHighpassFilterDesign：（1）MaximumValue（2）Low-FrequencyResponseInlargeareawithconstantgraylevel：s=0,thenG(s)=1.Thefilterdoesnotchangeitscontrast.

11.5.1classificationoflinearfilterLow-passFiltersReduceoreliminatehigh-frequencycomponentsandpasslow-frequencycomponentsMaketheimagefuzzy,eliminatenoiseHigh-passFiltersReduceoreliminatelow-frequencycomponentsandpasshigh-frequencycomponentsMaketheimageshapening,enhancethebordersBand-passFiltersPassorstopparticularfrequenciesDefinition：asystemcanfilterssomefrequencies.Fouriertransformtofrequencydomainisoftenusedtoanalyzevariousfilters.1Lowpass1Highpass1BandpassFrequencyDomain000SpatialDomain11.5.2optimallinearfilterdesign1)RandomvariablesRandomnoise：describesanunknowncontaminatingsignal.Randomvariables:consideranensembleofinfinitelymanymemberfunctions.Whenwemakeourrecording,oneofthosememberfunctionsemergestocontaminateourrecord,butwehavenowayofknowingwhichone.Wecan,however,makegeneralstatementsabouttheensembleasagroup.Inthisway,wecanexpressourpartialknowledgeofthecontaminatingsignal.ErgodicRandomVariablesTherearetwowaysbywhichonecancomputeaveragesofarandomvariable.

(1)wecancomputeatimeaveragebyintegratingaparticularmemberfunctionoveralltime;(2)averagetogetherthevaluesofallmemberfunctionsevaluatedatsomeparticularpointintime.

Arandomvariableisergodicifandonlyif

(1)thetimeaveragesofallmemberfunctionsareequal

(2)theensembleaverageisconstantwithtime

(3)thetimeaverageandtheensembleaveragearenumericallyequal.Thus,forergodicrandomvariables,timeaveragesandensembleaverageareinterchangeable.Expectationoperatordenotestheensembleaverageoftherandomvariablexcomputedattimet.Undertheergodicityproperty,italsodenotesthevalueobtainedwhenanyparticularsampleoftherandomvariableisaveragedovertime;thatis：Aergodicrandomvariablesisanunknownfunctionthathasaknownautocorrelationfunction.Maincontents：ThepurposeandfunctionofWienerFilter;Randomvariables;DerivationofWienerfilter’stransferfunction;ExamplesoftheWienerFilterUncorrelatedsignalandnoiseWienerDeconvolutionSummarypurposeandfunction:purpose：RecoveranunknowncontaminatedsignalFunction:Estimateuncontaminatedoriginalsignal.Estimatetheformandshapeoforiginalsignal.Recoverunknownsignalfromadditivesignal,thatistheoptimallinearfilter.MaincontentsThemodelofWienerfilterTheoptimalitycriterionofWienerfilterExamplesoftheWienerfilterTransferfunctionunderuncorrelatedsignalandnoiseWienerdeconvolution1.Wienerfiltermodel——theclassiclinearfilterUsefulsignalObservedsignalImpulseresponseTheoutputofthefilterAdditivenoisesignalFigure1.theconfigurationofWienerEstimatorAim：

selecttheappropriateh(t)tomakey(t)ascloseaspossibletos(t)Points：

discussonedimensionsignal;Bothoriginalsignals(t)andadditivenoisen(t)areergodicrandomvariables;Theirpowerspectraisknown.2.PropertiesofrandomvariablesandergodicrandomvariablesNecessaryandsufficientconditionsofergodicrandomvariables：(1)thetimeaveragesofallmemberfunctionsareequal

(2)theensembleaverageisconstantwithtime

(3)thetimeaverageandtheensembleaveragearenumericallyequal.Randomvariables：weonlyhavesomegeneralknowledgeofacertainsignal,butlackspecificdetails,wedescribethesignalasrandom。Thepropertiesofergodicrandomvariables：Weintroducedtheexpectationoperatortodenotetheensembleaverageoftherandomvariablexcomputedattimet,italsodenotesthevalueobtainedwhenanyparticularsampleoftherandomvariableisaveragedovertime,thatisexpectationcanbewrittenastheformofintegral。Forergodicrandomvariables,theautocorrelationfunctionisthesameforallmemberfunction,anditthuscharacterizestheensemble.Ifn(t)isergodicrandomvariable,thenitsautocorrelationfunctionisknown：Itspowerspectrum3.OptimalitycriterionofWienerfilterUseminimizedmeansquareerrorasacriterionofoptimalityErrorsignal：meansquareerror：Points：Giventhepowerspectraofs(t)andn(t),wemustdeterminetheimpulseresponseh(t)thatminimizesthemeansquareerror;Themeansquareerrorisafunctionalofh(t),theimpulseresponse,sinceafunction,h(t),mapsintoarealnumber,MSE.

ispositiveforbothpositiveandnegativeerrors.Squaringtheerrorcauseslargeerrorstobe“penalized”moreseverelythansmallerrors。4.ObtainthefunctionalexpressionforMSEintermsofh(t)y(t)istheconvolutionofx(t)andh(t)Alsodevelopedintheformofconvolutioncross-correlationfunctionofs(t)andx(t)

5.SeekingtominimizeMSEMSEisthefunctionofh(t)NecessaryandsufficientconditionsoftheoptimalWienerestimatorTakingthefouriertransformofbothside：Substitutesuboptimalimpulseresponsefunctionintoformula(1)Points：(1)Foranylinearsystem,thecross-correlationbetweeninputandoutputisgivenbyTheautocorrelationfunctionofinputsignalWienerfiltermakesthecross-correlationfunctionofinputandoutputequalsthecross-correlationfunctionofsignaland(signalplusnoise)TransferfunctionofWienerfilter：FrequencydomainspecificationoftheWienerestimator(2)6.TransferfunctionandMSEofWienerfilterundertheconditionofuncorrelatedsignalandnoiseUncorrelatedsignalandnoisemeansthat:MSEofWienerfilter’soutput：Basedonformula(2),wecanderiveandprovetheconclusionsbelow：ThetransferfunctionofWienerfilteris：Figure2TheWienerfiltertransferfunctionAtthelowfrequnencies，thesignalpowerismuchlargerthanthatofthenoise,andthetransferfunctiontakesonvaluesnear1;Itthendecreasestoavalueof0.5atthepointwherethesignalandnoisepowerareequal;Atthehighfrequencies，thetransferfunctiondeclinestowardzero,whicharedominatedbythenoise;MSE：MSEisnonzerowhereboththesignalandnoisepowerspectraarenonzero.Figure3SeparablesignalandnoiseTheWienerestimatorpassesthesignalinitsentiretyanddiscriminatescompletelyagainstthenoise.Figure4Bandlimitedsignal7.WienerDeconvolutionFigure5WienerdeconvolutionThetransferfunctionoftheoptimaldeconvolutionfilterinthemeansquaresenseis:S(t)w(t)x(t)n(t)y(t)z(t)F(s)1/F(s)+G(s)Figure6WienerdeconvolutionexamplesThesignalhasaGaussianpowerspectrum,andthenoiseiswhite.Theblurringfunctionistheopticaltransferfunctionofaperfectlens.Atlowfrequencies,G(s)increaseswithfrequency,tocompensateforF(s);byyhemidrange，G(s)begingstorollofftowardzerotoblockthenoise.LimitationofWienerdeconvolutionThoughWienerfilteroffersaoptimalmethodtoderivedeconvolutionfunctioninnoisecases,therearethreequestionslimititsavailability.MSEcriteriongivesthesameweight

toalloftheerrors(nomatterinwhichpositionoftheimage),

buthumaneyes

aremoretolerantindarkandhighgradientareasthaninotherareas.MinimizingMSE,Wienerfilterssmoothimagesusingthemethodnotmostsuitableforhumaneyes.

ClassicWienerfiltercannotdealwiththespacevariablesituation，e.g.astigmatism,fieldbendandmotionblurincludingrotation.

Can'tdealwiththegeneralsituationwithnon-steadysignalandnoise.Mostimagesarehighlynon-steady,thathavelargesmoothareasseparatedbytheprecipitousedge,inaddition,alotofimportantnoisesourcesarerelatedtolocalgreylevel.ItisaquiteconservativecourseofWienerdeconvelution,thatemphasizesonnoiseinhibitionbutnotreconstructsthesignal.TheMatchedDetector(filter)Throughtransferfunction

maximizingthesignal-to-noisepowerratioinoutputendofthefilter,detectwhethertheprescribedsignalexistsinthepresenceofnoise(location).

Figure11-21AmodelequivalenttothatofFigure11-20Thematcheddetectordetectthepresenceorabsenceofaparticularknowninputsignal,ratherthantoestimateitsnoise-freeshape.Figure11-23InputandoutputcomponentsignalsThefigurebelowillustratesboththeWienerestimatorandthematcheddetectorwhenthesignalisaGaussianpulseembeddedinwhiterandomnoise.Inthiscase,thesignal-to-noiseratioisontheorderofunity.SignalWienerfilteroutputSignal+noiseFigure11-26ComparisonoftheWienerandmatchedfiltersMatchedfilteroutput

SpectrafunctionsofWienerestimatorandthematcheddetectorWienerestimatorwantstheareaunderMSEintegrandtobesmall,thematcheddetectorwantthesignalpowerspectraismuchlargerthanthenoisepowerspectraatsomefrequency.ForuncorrelatedsignalandnoisetheWienerestimatortransferfunctionis：TheMSEis:IfweletC=1,thematcheddetectortransferfunctionbecomes:Thesignal-to-noisepowerratioatitsoutputis:ComparisonoftheWienerestimatorandthematcheddetectorWienerfilterMatchedfilteraimRecovertheoriginalsignalfromnoiseLocatetheknownsignalinanoisybackgroundfunction“estimate”theformandshapeofthenon-contaminatedsignal“detect”theoccurrenceofasignalofprescribedforminthepresenceofnoise.Justuseittojudgewhetherthesignaloccurs.Transferfunctionandevaluation

realandeven,between0-1

c=1,HermiteisarbitraryNoisediscrim