MATLAB毕业设计外文翻译.docx

MATLAB毕业设计外文翻译.docx

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MATLAB毕业设计外文翻译

ComplexRidgeletsforImageDenoising

1Introduction

Wavelettransformshavebeensuccessfullyusedinmanyscientificfieldssuchasimagecompression,imagedenoising,signalprocessing,computergraphics,andpatternrecognition,tonameonlyafew.Donohoandhiscoworkerspioneeredawaveletdenoisingschemebyusingsoftthresholdingandhardthresholding.Thisapproachappearstobeagoodchoiceforanumberofapplications.Thisisbecauseawavelettransformcancompacttheenergyoftheimagetoonlyasmallnumberoflargecoefficientsandthemajorityofthewaveletcoeficientsareverysmallsothattheycanbesettozero.Thethresholdingofthewaveletcoeficientscanbedoneatonlythedetailwaveletdecompositionsubbands.Wekeepafewlowfrequencywaveletsubbandsuntouchedsothattheyarenotthresholded.ItiswellknownthatDonoho'smethodofferstheadvantagesofsmoothnessandadaptation.However,asCoifman

andDonohopointedout,thisalgorithmexhibitsvisualartifacts:

Gibbsphenomenaintheneighbourhoodofdiscontinuities.Therefore,theyproposeinatranslationinvariant（TI）denoisingschemetosuppresssuchartifactsbyaveragingoverthedenoisedsignalsofallcircularshifts.TheexperimentalresultsinconfirmthatsingleTIwaveletdenoisingperformsbetterthanthenon-TIcase.BuiandChenextendedthisTIschemetothemultiwaveletcaseandtheyfoundthatTImultiwaveletdenoisinggavebetterresultsthanTIsinglewaveletdenoising.CaiandSilvermanproposedathresholdingschemebytakingtheneighbourcoeficientsintoaccountTheirexperimentalresultsshowedapparentadvantagesoverthetraditionalterm-by-termwaveletdenoising.ChenandBuiextendedthisneighbouringwaveletthresholdingideatothemultiwaveletcase.Theyclaimedthatneighbourmultiwaveletdenoisingoutperformsneighboursinglewaveletdenoisingforsomestandardtestsignalsandreal-lifeimages.Chenetal.proposedanimagedenoisingschemebyconsideringasquareneighbourhoodinthewaveletdomain.Chenetal.alsotriedtocustomizethewavelet_lterandthethresholdforimagedenoising.Experimentalresultsshowthatthesetwomethodsproducebetterdenoisingresults.Theridgelettransformwasdevelopedoverseveralyearstobreakthelimitationsofthewavelettransform.The2Dwavelettransformofimagesproduceslargewaveletcoeficientsateveryscaleofthedecomposition.Withsomanylargecoe_cients,thedenoisingofnoisyimagesfacesalotofdiffculties.Weknowthattheridgelettransformhasbeensuccessfullyusedtoanalyzedigitalimages.Unlikewavelettransforms,theridgelettransformprocessesdatabyfirstcomputingintegralsoverdifferentorientationsandlocations.Aridgeletisconstant

alongthelinesx1cos_+x2sin_=constant.Inthedirectionorthogonaltotheseridgesitisawavelet.Ridgeletshavebeensuccessfullyappliedinimagedenoisingrecently.Inthispaper,wecombinethedual-treecomplexwaveletintheridgelettransformandapplyittoimagedenoising.Theapproximateshiftinvariancepropertyofthedual-treecomplexwaveletandthegoodpropertyoftheridgeletmakeourmethodaverygoodmethodforimagedenoising.Experimentalresultsshowthatbyusingdual-treecomplexridgelets,ouralgorithmsobtainhigherPeakSignaltoNoiseRatio（PSNR）forallthedenoisedimageswithdi_erentnoiselevels.Theorganizationofthispaperisasfollows.InSection2,weexplainhowtoincorporatethedual-tree

complexwaveletsintotheridgelettransformforimagedenoising.ExperimentalresultsareconductedinSection3.Finallywegivetheconclusionandfutureworktobedoneinsection4.

2ImageDenoisingbyusingComplex

RidgeletsDiscreteridgelettransformprovidesnear-idealsparsityofrepresentationofbothsmoothobjectsandofobjectswithedges.Itisanear-optimalmethodofdenoisingforGaussiannoise.Theridgelettransformcancompresstheenergyoftheimageintoasmallernumberofridgeletcoe_cients.Ontheotherhand,thewavelettransformproducesmanylargewaveletcoe_cientsontheedgesoneveryscaleofthe2Dwaveletdecomposition.Thismeansthatmanywaveletcoe_cientsareneededinordertoreconstructtheedgesintheimage.WeknowthatapproximateRadontransformsfordigitaldatacanbebasedondiscretefastFouriertransform.Theordinaryridgelettransformcanbeachievedasfollows:

1.Computethe2DFFToftheimage.

2.SubstitutethesampledvaluesoftheFouriertransformobtainedonthesquarelatticewithsampledvaluesonapolarlattice.

3.Computethe1DinverseFFToneachangularline.

4.Performthe1Dscalarwavelettransformontheresultingangularlinesinordertoobtaintheridgeletcoe_cients.

Itiswellknownthattheordinarydiscretewavelettransformisnotshiftinvariantbecauseofthedecimationoperationduringthetransform.Asmallshiftintheinputsignalcancauseverydi_erentoutputwaveletcoe_cients.Inordertoovercomethisproblem,Kingsburyintroducedanewkindofwavelettransform,calledthedual-treecomplexwavelettransform,thatexhibitsapproximateshiftinvariantpropertyandimprovedangularresolution.Sincethescalarwaveletisnotshiftinvariant,itisbettertoapplythedual-treecomplexwaveletintheridgelettransformsothatwecanhavewhatwecallcomplexridgelets.Thiscanbedonebyreplacingthe1Dscalarwaveletwiththe1Ddualtreecomplexwavelettransforminthelaststepoftheridgelettransform.Inthisway,wecancombinethegoodpropertyoftheridgelettransformwiththeapproximateshiftinvariantpropertyofthedual-treecomplexwavelets.

Thecomplexridgelettransformcanbeappliedtotheentireimageorwecanpartitiontheimageintoanumberofoverlappingsquaresandweapplytheridgelettransformtoeachsquare.Wedecomposetheoriginaln_nimageintosmoothlyoverlappingblocksofsidelengthRpixelssothattheoverlapbetweentwoverticallyadjacentblocksisarectangulararrayofsizeR=2_RandtheoverlapbetweentwohorizontallyadjacentblocksisarectangulararrayofsizeR_R=2.Forann_nimage,wecount2n=Rsuchblocksineachdirection.Thispartitioningintroducesaredundancyof4times.Inordertogetthedenoisedcomplexridgeletcoe_cient,weusetheaverageofthefourdenoisedcomplexridgeletcoe_cientsinthecurrentpixellocation.

Thethresholdingforthecomplexridgelettransformissimilartothecurveletthresholding[10].Onedifferenceisthatwetakethemagnitudeofthecomplexridgeletcoe_cientswhenwedothethresholding.Lety_bethenoisyridgeletcoe_cients.Weusethefollowinghardthresholdingruleforestimatingtheunknownridgeletcoe_cients.Whenjy_j>k_~_,welet^y_=y_.Otherwise,^y_=0.Here,~ItisapproximatedbyusingMonte-Carlosimulations.Theconstantkusedisdependentonthenoise.Whenthenoiseislessthan30,weusek=5forthefirstdecompositionscaleandk=4forotherdecompositionscales.Whenthenoise_isgreaterthan30,weusek=6forthe_rstdecompositionscaleandk=5forotherdecompositionscales.

Thecomplexridgeletimagedenoisingalgorithmcanbedescribedasfollows:

1.PartitiontheimageintoR*RblockswithtwoverticallyadjacentblocksoverlappingR=2*RpixelsandtwohorizontallyadjacentblocksoverlappingR_R=2pixels

2.Foreachblock,Applytheproposedcomplexridgelets,thresholdthecomplexridgeletcoefficients,andperforminversecomplexridgelettransform.

3.Taketheaverageofthedenoisingimagepixelvaluesatthesamelocation.

WecallthisalgorithmComRidgeletShrink,whilethealgorithmusingtheordinaryridgeletsRidgeletShrink.ThecomputationalcomplexityofComRidgeletShrinkissimilartothatofRidgeletShrinkbyusingthescalarwavelets.Theonlydi_erenceisthatwereplacedthe1Dwavelettransformwiththe1Ddual-treecomplexwavelettransform.Theamountofcomputationforthe1Ddual-treecomplexwaveletistwicethatofthe1Dscalarwavelettransform.However,otherstepsofthealgorithmkeepthesameamountofcomputation.OurexperimentalresultsshowthatComRidgeletShrinkoutperformsVisuShrink,RidgeletShink,andwiener2_lterforalltestingcases.Undersomecase,weobtain0.8dBimprovementinPeakSignaltoNoiseRatio（PSNR）overRidgeletShrink.TheimprovementoverVisuShinkisevenbiggerfordenoisingallimages.ThisindicatesthatComRidgeletShrinkisanexcellentchoicefordenoisingnaturalnoisyimages.

3ExperimentalResults

Weperformourexperimentsonthewell-knownimageLena.WegetthisimagefromthefreesoftwarepackageWaveLabdevelopedbyDonohoetal.atStanfordUniversity.Noisyimageswithdi_erentnoiselevelsaregeneratedbyaddingGaussianwhitenoisetotheoriginalnoise-freeimages.Forcomparison,weimplementVisuShrink,RidgeletShrink,ComRidgeletShrinkandwiener2.VisuShrinkistheuniversalsoft-thresholdingdenoisingtechnique.Thewiener2functionisavailableintheMATLABImageProcessingToolbox,andweusea5*5neighborhoodofeachpixelintheimageforit.Thewiener2functionappliesaWiener_lter（atypeoflinearfilter）toanimageadaptively,tailoringitselftothelocalimagevariance.TheexperimentalresultsinPeakSignaltoNoiseRatio（PSNR）areshowninTable1.Wefindthatthepartitionblocksizeof32*32or64*64isourbestchoice.Table1isfordenoisingimageLena,fordi_erentnoiselevelsandafixedpartitionblocksizeof32*32pixels.ThefirstcolumninthesetablesisthePSNRoftheoriginalnoisyimages,whileothercolumnsarethePSNRofthedenoisedimagesbyusingdi_erentdenoisingmethods.ThePSNRisde_nedasPSNR=􀀀10log10Pi;j（B（i;j）􀀀A（i;j））2n22552:

whereBisthedenoisedimageandAisthenoise-freeimage.FromTable1weca