Complex Wavelet Based image Analysis and Synthesis.0027

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1、242AppendixC.10A-PDFSplitDEMO:Purchasefromwww.A-PDF.comtoremovethewatermarkusedthecostfunction2f(x)=λi

2、[Wx]i

3、iwhereWrepresentsaforwardrealwavelettransform(Daubechies’fifthordercom-pactlysupportedwavelet[30])and{λi}arescalingparameterschosenusingtheoutput

4、oftheedgedetector.2.StarckandPantinhaveproposed[93]amultiscalemaximumentropymethodthatusesthecostfunction2

5、[Wx]

6、f(x)=−λ[Wx]−m−

7、[Wx]

8、logiiiiimiiwhereWrepresentsanondecimatedforwardtransform(aformofLaplacianpyramidisusedwherethelowpassfilteringisperforme

9、dwitha5×5maskbasedonacubicBspline),{mi}aresomeconstantreferencevaluesgivenbymi=σ/100,and{λi}areweightingfactorsthatalterthedegreeofregularisation.Thesefactorsarechosenbasedonthesizeofthewaveletcoefficientsoftheobservedimage.Ifthecoefficientisbelowathreshold

10、of3σthenitisdeemedtobeinthemultiresolutionsupportandthefactorissettosomeconstantvalueσjthatdependsonlyonthescalejofthewaveletcoefficient.Thesecoefficientsareallowedtovaryinordertomatchtheobservations.However,ifthecoefficientislargethenthefactorissettozeroandi

11、tisnotallowedtovary.Aconstraintisaddedtotheproblemthatrequiresallcoefficientsnotinthemultiresolutionsupporttobeequaltothevalueintheobservedimage.3.BanhamandKatsaggelos[8]useanautoregressivepriormodelwhichevolvesfromcoarsetofinescales.Theparametersofthemode

12、larebasedontheoutputofanedgedetectorappliedtoaprefilteredversionofthenoisydata.AmultiscaleKalmanfilteristhenusedtoestimatetheoriginalimage.4.Belgeetal[11]useanon-Gaussianrandomprocessmodelforwhichthewaveletcoefficientsaremodelledasbeingindependentlydistribu

13、tedaccordingtogeneralisedGaussiandistributionlaws.Theresultingenergyfunctionisminimisedinadoubly2Thiscostfunctionappearsstrangebecauseitisnotasymmetricfunctionofthewaveletcoefficients.Thisisprobablyamistakebutwehavechosentokeeptheformasgiveninthereference

14、.AppendixC.10243iterativealgorithm.Inotherwords,theiterativeminimisationalgorithmisbaseduponanotheriterativealgorithm,whichitselfrequireswaveletandFouriertransformstoevaluate.Thisalgorithmcorrespondstothecostfunctionpf(x)=λi

15、[Wx]