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1、ComplexRidgeletsforImageDenoising1IntroductionWavelettransformshavebeensuccessfullyusedinmanyscientificfieldssuchasimagecompression,imagedenoising,signalprocessing,computergraphics,andpatternrecognition,tonameonlyafew.Donohoandhiscoworkerspioneeredawaveletdenoisings

2、chemebyusingsoftthresholdingandhardthresholding.Thisapproachappearstobeagoodchoiceforanumberofapplications.Thisisbecauseawavelettransformcancompacttheenergyoftheimagetoonlyasmallnumberoflargecoefficientsandthemajorityofthewaveletcoeficientsareverysmallsothattheycanb

3、esettozero.Thethresholdingofthewaveletcoeficientscanbedoneatonlythedetailwaveletdecompositionsubbands.Wekeepafewlowfrequencywaveletsubbandsuntouchedsothattheyarenotthresholded.ItiswellknownthatDonoho'smethodofferstheadvantagesofsmoothnessandadaptation.However,asCoif

4、manandDonohopointedout,thisalgorithmexhibitsvisualartifacts:Gibbsphenomenaintheneighbourhoodofdiscontinuities.Therefore,theyproposeinatranslationinvariant(TI)denoisingschemetosuppresssuchartifactsbyaveragingoverthedenoisedsignalsofallcircularshifts.Theexperimentalre

5、sultsinconfirmthatsingleTIwaveletdenoisingperformsbetterthanthenon-TIcase.BuiandChenextendedthisTIschemetothemultiwaveletcaseandtheyfoundthatTImultiwaveletdenoisinggavebetterresultsthanTIsinglewaveletdenoising.CaiandSilvermanproposedathresholdingschemebytakingthenei

6、ghbourcoeficientsintoaccountTheirexperimentalresultsshowedapparentadvantagesoverthetraditionalterm-by-termwaveletdenoising.ChenandBuiextendedthisneighbouringwaveletthresholdingideatothemultiwaveletcase.Theyclaimedthatneighbourmultiwaveletdenoisingoutperformsneighbou

7、rsinglewaveletdenoisingforsomestandardtestsignalsandreal-lifeimages.Chenetal.proposedanimagedenoisingschemebyconsideringasquareneighbourhoodinthewaveletdomain.Chenetal.alsotriedtocustomizethewavelet_lterandthethresholdforimagedenoising.Experimentalresultsshowthatthe

8、setwomethodsproducebetterdenoisingresults.Theridgelettransformwasdevelopedoverseveralyearstobreakthelimitationsofthewavelettransform.The2D