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

2、ingandhardthresholding.Thisapproachappearstobeagoodchoiceforanumberofapplications.Thisisbecauseawavelettransformcancompacttheenergyoftheimagetoonlyasmallnumberoflargecoefficientsandthemajorityofthewaveletcoeficientsareverysmallsothattheycanbesettozero.Thethresholdingofthewaveletcoeficientsc

3、anbedoneatonlythedetailwaveletdecompositionsubbands.Wekeepafewlowfrequencywaveletsubbandsuntouchedsothattheyarenotthresholded.ItiswellknownthatDonoho'smethodofferstheadvantagesofsmoothnessandadaptation.However,asCoifmanandDonohopointedout,thisalgorithmexhibitsvisualartifacts:Gibbsphenomenai

4、ntheneighbourhoodofdiscontinuities.Therefore,theyproposeinatranslationinvariant(TI)denoisingschemetosuppresssuchartifactsbyaveragingoverthedenoisedsignalsofallcircularshifts.TheexperimentalresultsinconfirmthatsingleTIwaveletdenoisingperformsbetterthanthenon-TIcase.BuiandChenextendedthisTIsc

5、hemetothemultiwaveletcaseandtheyfoundthatTImultiwaveletdenoisinggavebetterresultsthanTIsinglewaveletdenoising.CaiandSilvermanproposedathresholdingschemebytakingtheneighbourcoeficientsintoaccountTheirexperimentalresultsshowedapparentadvantagesoverthetraditionalterm-by-termwaveletdenoising.Ch

6、enandBuiextendedthisneighbouringwaveletthresholdingideatothemultiwaveletcase.Theyclaimedthatneighbourmultiwaveletdenoisingoutperformsneighboursinglewaveletdenoisingforsomestandardtestsignalsandreal-lifeimages.Chenetal.proposedanimagedenoisingschemebyconsideringasquareneighbourhoodinthewavel

7、etdomain.Chenetal.alsotriedtocustomizethewavelet_lterandthethresholdforimagedenoising.Experimentalresultsshowthatthesetwomethodsproducebetterdenoisingresults.Theridgelettransformwasdevelopedoverseveralyearstobreakthelimitationsofthewavelettransform.The2D