AgendaFoundationCh1~4ImagingSystem,Digitalization,Display,SoftwareCh5~8Histogram,PointOperations,AlgebraicOperations,GeometricOperationsTheoryCh9~12LinearSystem,FourierFrequencyTransform,FilterDesign,DiscreteSamplingCh13~15OrthogonalRadicleTransform,WaveletTime-frequencyTransform,OpticalFunctionTransformApplicationCh16~20ImageRestoration,Compression,PatternRecognitionCh21~22ColorandMulti-SpectralImageProcessing,ThreeDimensionImageProcessing

mathematicalmodelThemodelofimagedegradationandrestorationasfollow。Imagedegradationiscausedbysystemperformanceandnoise.Restorationisachievedthroughrehabilitatingfilter(inversefilter).16.2ClassicalRestoration

GeometricMeanFiltersNoticethatif=1,Eqreducestoadeconvolutionfilter.If=1/2and=1,itreducestoPSEfilter.ItisthegeometricmeanbetweenordinarydeconvolutionandWienerdeconvolution.Soitisalsocalledgeometricmeanfilter.Itiscommonpractice,however,torefertothemoregeneralfiltermentionedaboveasthegeometricmeanfilter.UnconstrainedRestoration Ifn=0orifweknownothingaboutthenoise,wecansetuptherestorationasaleastsquaresminimizationproblem.e(f)=g-Hf W(f)=(g-Hf)t(g-Hf)Settingtozerothederivativeoff,yieldsf=H-1gConstrainedLeastSquaresRestoration

Introduceintotheminimizationtheconstraintthatthenormsofeachsideofg-Hf=nbethesame，thatis,Nowwecansetuptheproblemastheminimizationof

WhereQisamatrixweselecttodefinesomelinearoperatoronfandλisaconstantcalledaLagrangemultiplier.TheabilitytospecifyQgivesusflexibilityinsettingthegoaloftherestoration.Asbefore,wesettozerothederivativeofW（f）withrespecttof:Whereisaconstantthatmustbeadjustedsothattheconstraintofaboveissatisfied.16.5Superresolution

incoherenttransferfunctionofanopticalsystemistheautocorrelationfunctionofthepupilfunction.Restorationproceduresthatseektorecoverinformationbeyondthediffractionlimitarereferredtoassuperresolutiontechnique.

Harris’Technique

Harristhoughtthatitshouldbepossibletoreconstructthatobjectininfinitedetailfromitsdiffraction-limitedimage.Thetechniqueinvolvesapplyingthesamplingtheorem,withdomainsreversed,toobtainasystemoflinearequationsthatcanbesolvedforvaluesofthesignalspectrumoutsidethediffraction-limitedpassband.

SuccessiveEnergyReduction

Itinvolvessuccessivelyenforcingspace-limitednessupontheimage,whilekeepingtheknownlow-frequencyportionofthespectrumintact.Noticethatbandlimitingthespectrumcauseg0(x)nolongertobespacelimited.Thefirststepoftherestorationisenforcingspace-limitednessupong0(x)bysettingittozerooutsidethedomainofthepulse.Thesecondstepinvolvesreplacing.Theconvergencegenerallybecomesratherslowafterthefirstfewsteps.Small-KernelConvolution

Unlesstheimageisseverelyoversampled,thesignalspectrum,andconsequentlytherestorationMTF,willnormallyextendmostofthewaytothefoldingfrequencybeforeitdiesout.FromthesimilaritytheoremoftheFouriertransform,weknowthatifthetransferfunctionisabroad,theimpulseresponsewillbenarrow.Thus,theconvolutionkernelforimplementingarestorationPSFmightwellbezero,orapproximatelyso,exceptwithinareasonablysmallradiusabouttheorigin.Inthatcase,themajorityoftheoperationsrequiredforanN-by-Nconvolutionwillcontributelittleornothingtotherestoration.Truncatingthekernel

Asimplerapproachtosmall-kernelconvolutionismerelytotruncatethePSFarraytosomeacceptablysmallsize.MultiplyingthePSFbyasquarepulseconvolvestheMTFwithasin(x)/xfunction.UnlessthePSFisspatiallybounded,thiscanalteritstransferfunctionsignificantly.KernelDecomposition

Modernimage-processingsystemoftenincorporatespecialhardwareforhigh-speedconvolutionwithasmallkernel.ThishardwarebecomesusefulwhenanM-by-Mkernelisdecomposedintoasetofsmallerkernelsthatarethenappliedsequentially.Forexample,(M-1)/2kernelsofsizethreebythreewillimplementanM-by-Mconvolution.WhilethiscannotsubstituteexactlyforanarbitraryM-by-Mkernel,theresultisoftenagoodapproximation.16.216.2ClassicalRestoration16.3LinearAlgebraicRestoration16.4restorationoflessrestricteddegradations16.5Superresolution16.6SystemIdentification16.7NoiseModeling16.8Implementation16.9SummaryAgendaFoundationCh1~4ImagingSystem,Digitalization,Display,SoftwareCh5~8Histogram,PointOperations,AlgebraicOperations,GeometricOperationsTheoryCh9~12LinearSystem,FourierFrequencyTransform,FilterDesign,DiscreteSamplingCh13~15OrthogonalRadicleTransform,WaveletTime-frequencyTransform,OpticalFunctionTransformApplicationCh16~20ImageRestoration,Compression,PatternRecognitionCh21~22ColorandMulti-SpectralImageProcessing,ThreeDimensionImageProcessingChapter17

imagecompressionSummaryLosslesscompressionLossyimagecodingImagetransformcodingImagecompressionDatapropertyredundantirrelevantDatacompressionTypeofcompressionDeletedcontentRecoveryabilityLosslesscompressionRedundantinformationExactrecoveryLossycompression1.Redundantinformation2.IrrelevantinformationApproximatereconstructionLosslesscompressiontechniquesDictionary-basedtechniquesStatistics-basedtechniqueslosslesscompression–dictionary-basedtechniques

Run-LengthEncoding:Storeacodespecifieditsgrayvalue,followedbythelengthoftherunmethod：①grayvalue+number；②grayvalue+thenumberofendrowForexample:

PCXformatlosslesscompression–dictionary-basedtechniquesLZW

coding:Whenastringoccursfirstinthetable,thestringanditsassignedcodearestoredinfull.Thereafter,whenthatstringoccursagain,onlyitscodeisstored.property：thestringtableisdynamicallybuiltduringcompression,butitneedn’tbestoredwiththecompressionfile:thedecompressionalgorithmcanreconstructitfromthecompressedfile.Bythewaytheredundancyissqueezedoutofthefile.

Forexample：GIF.Losslesscompression-statisticalmethodsTheprobabilityofsourceofmessages

are:whereKisthetotalnumbsofmessages.ameasurementoftheShannoninformation:

probabilityof

istheentropyofthemessage

redundancy。If，theformularepresentsthelowestboundonaveragewordlengthforlosslesscompression.

suchas:,thelowestboundonlosslesscompressionforbinarycodinglosslesscompression–statisticalmethods

Huffmancoding:alosslessstatisticalmethodwithavariable-lengthcodeforminimumredundancy.Statichuffmancoding:anencodingtreeconstructedinadvanceofcompressionfromatableofoccurrenceprobabilitiesofthepossibledataDynamichuffmancoding:constructstheencodingtreeduringthecompressionprocessIt’saninformation-keptcoding,orentropy-keptone,orentropyonelosslesscompression–statisticalmethods

thealgorithmofHuffmancoding:1)Arrayoriginaldatabythesizeofprobability；2)Taketwosymbolswithminimumprobabilityasleafnode

(thesmalloneisleftnode,andthebigoneisrightnode)toconstructthefathernodewiththesumofthetwoprobabilities;3)Jointhenewnodesintonodelistbyprobabilitysize;4)Repeat2-3tillallthenodesjoinintonodetable;5)Supposealltheleftnodesare0,andtherightoneis1.Fromrootnodetoleafone,pathcodeisthehuffmanoneofthenodes.exampleofhuffmancodingsetofmessagesourcea1a2a3a4a5a6probability0.40.30.10.10.060.04codeda1a2a3a4a5a6codeword10001101000101001011Codelength123455entropy:Meancodelengthlossycodingscalarquantization：Thequantizationschememinimizesthemeansquareerror.Twoproperties:EachdecisionthresholdfallsexactlyinmidvaluebetweentwoadjacentrepresentativelevelsEachrepresentativelevelfallsatthecentroidofthesectionofthePDF(ProbabilityDensityFunction)betweentwosuccessivedecisionthresholds.lossycodingdistortion：

errorbetweenreconstructionimagegandoriginalonefRatedistortionfunction：minimummeaninformationatcertaindistortion.Itgivesthelowestlimitofmeaninformation,i.ethelimitofinformationcompression（codingrateatalloweddistortion(suchasthenumberofeachpixel),forexample,when,codingrateis,Theentropyofthereconstructionerror

TheequalityholdsinthisrelationifthebalanceimagehasstatisticallyindependentpixelsandGaussianpdfLossycoding-ratedistortionfunction:ForanycertaindistortionD,arateis

arbitrarilyclosetothecodingmethodofR(D),andthemeandistortionisarbitrarilyclosetoDCodingwitharatethatislowerthanR(D),wecannotfindacodewhosedistortionisnotlessthanDtransformcoding

themeaningofminimumdistortionorthogonaltransform0DR(D)drentropycodingratedistortionfunctionThedistortionbetweenoriginalimagef(x,y)andreconstructionimage

g(x,y)isquantifiedbythemeansquareerror:doesitexistsuchtransformT,andatthesametimeDisminimumminimumdistortiontransformcodingisthemostefficientatgivendistortionratebyratedistortionfunction第8頁（共17頁）transformcodingTransformcodingisinspiredby

theanalysisoftime/frequencydomainconversioninthedigitalsignalprocessingexample-bitdistributionofimagequantizationcoding

bitdistributionofscalarYXY=TXcoordinaterotationtransformY=XXYBitofscalarquantizationdecreaseatthesamequantizationerrorsaftercoordinaterotationtransformcodingPrinciple(squeezeredundancybyfollowingmethod):Centereddistribution:reducebitofquantization;Squeezeredundancy:prepareforlossycompression;Chartoftransformcodingandencodingsystem+xW：noisecomingfrombitdistributionA：filtertransformcodingPropertyoftransformcoding:Generallyhashighcompressionrate:suchasSVD

transformcoding.becauseoftransform,greatlysqueezesinformationredundancyandstructureredundancy;Greatercalculation：suchasinversionprocessofT,sogenerallyusesnormalizationorthogonalmatrixT,here：transformcoding----K-LtransformSupposedatavectortransformmatrix,thenorthogonaltransformcanbeexpressed,theresultvectoris,inversetransformis.codingmeansquareerroris,whereiscovariancematrixofX,andfindwhenisminimum.wecangettheconditionofminimumbyLagrangemultipliermethod:,andthatiswhereischaracteristicmatrixof.wecanhavetheconclusionthat:basevectorTofoptimalorthogonaltransformisthecharacteristicmatrixofdatavectorXcovariancematrix.TheoptimalorthogonaltransformTisthefamousKanhuman-Loevetransformtransformcoding---K-LtransformcodingandencodingprocessSupposeXisN×1randomvector,eachcomponentX1ofXcanbeestimatedLvectorsamples.CovariancematrixofXis.K-Lorthogonaltransformissuchalineartransform,wheretherowofAischaracteristicmatrixof,.TransformedcovariancematrixYcanbereasonedbycovariancematrixofX

,whereischaracteristicvalueof.inversetransformofK-Lis.Forexample:One3×1randomvectorXhasfollowingcovariancematrix:Thecharacteristicvalueandcharacteristicvectorare:Foravectorwhosemeanvalueiszero,thek-LtransformresultisIfignoresthesmalltransformbaseinA,itcanreducethedimensionofY:reconstructionprocess:standarddeviationoflossycompression:(cancontroltheerror)IfCissingularity,itcanachievehighercompressionratetransformcodingcoderateallocation:generally,allocatecodelengthbythesizeofcovariancematrixsquareerroroftransformresultimage.thebigoneallocateslongcode,andthesmalloneallocatesshortcode.transformcodingevaluationandusingofminimumdistortionorthogonaltransformcoding:iscalledk-Ltransform,orprincipalcomponentanalysisLargecalculation,doesnothasfastalgorithmItisimprovedtoSVD.transformbaseTisnotbasedonoriginalimage,butonstatisticalinformationoforiginalimage,andthismakesitpossibletousethesametransformbaseforstatisticallysimilarimages;WeusuallyusefirstMarkovprocesstosimulateuniversaltransformbase,

anditisespeciallyeffectivefornaturalimageGenerally,itisusedtocomparewithnewcodingalgorithm;transformcoding----first-orderMarkov

processMarkovprocess:astationaryrandomsequenceiscalledafirst-orderMarkovsequenceiftheconditionalprobabilityofeachelementinthesequencedependsonlyuponthevalueoftheimmediatelyprecedingelement.ThepossibilitytoimproveK-LtransformcodingbyMarkovprocess:thedistributionofmostgrayimagescanbesimulatedbyfirst-orderMarkovprocess,thatiseachpixelvalueonlyhasrelationshipwithpreviouspixel,andthecorrelationiscertainknowncorrelationcoefficient,wecangetcovariancematrixoforiginaldata(normalized).SowecangetK-Ltransformbase(basisimage),andthebaseissuitableformostimages.

basicimagederivationofK-LtransformonapproximateconditionofMarkovprocess

Supposetheknowncorrelationcoefficientp(0<p≤1),Covariancematrixoftheoriginaldatais:,wecancalculatetransformbaseofK-LbyC.forexample,p=0.91,N=8thetransformmatrixisThecorrespondingcharacteristicvalue,theenergyandpercentofanteriorMdiagonalelements,areasbelow:SowecangetK-Ltransformhasgoodconcentrationofenergy,andcompletelyreducesCorrelation.K-LtransformcanapplythesametransformbasetoagroupofstatisticallysimilarobjectsMostnaturalimagesarestatisticallysimilar,andwecanexpressthesimilaritywithMarkovprocessatcorrelationcoefficient1;sowecangetageneralk-Ltransformbasefornaturalimages;Whencomparedwithotheralgorithms,weusuallyuseK-Ltransformcoding,whichissimilarwithfirst-orderMarkovprocessatcorre