脊波和曲波变换

1JointLaboratoryofShantouandXiamenUniversity,2008,,,Candes——Ridgelet(RidgeletTransform)2JointLaboratoryofShantouandXiamenUniversity,20083.1Ridgelet3.1.1Ridgelet(3-1)dFourier(3-2)(){}1,,;,,0,daubabRauSγ−Γ==∈∈()()ˆ,ixdfefxdxxRξξ−⋅=∈∫3JointLaboratoryofShantouandXiamenUniversity,20083.1()()()()2,,,,fabRCRTabXfXdXθθψ=∫:RRψ→()0tdtψ=∫4JointLaboratoryofShantouandXiamenUniversity,2008(3-3)Ridgelet[2]RidgeletRidgelet()1/2uxbxaaγψψ−⋅−⎛⎞=⋅⎜⎟⎝⎠γψψaubγψ5JointLaboratoryofShantouandXiamenUniversity,20083-1RidgeletMarrabaψ()()221exp2tttψ⎛⎞=−−⎜⎟⎝⎠6JointLaboratoryofShantouandXiamenUniversity,2008cbb3-1RidgeletRidgeletθθ7JointLaboratoryofShantouandXiamenUniversity,20083.1.2RidgeletRidgeletContinuousRidgeletTransformCRT[2](3-4)(3-5)()()()()2,,,,fabRCRTabXfXdXθθψ=∫()()()2,,300,,4fabdadfXCRTabXdbaπθθθψπ∞∞−∞=∫∫∫8JointLaboratoryofShantouandXiamenUniversity,20083-2Ridgelet,,12(,)abxxθψ9JointLaboratoryofShantouandXiamenUniversity,2008(3-5)RidgeletRidgelet()()()2,,300,,4fabdadfXCRTabXdbaπθθθψπ∞∞−∞=∫∫∫10JointLaboratoryofShantouandXiamenUniversity,2008aRadonbRadonFourierRidgelet3-3RadonFourierRidgelet11JointLaboratoryofShantouandXiamenUniversity,2008Radon[3](3-6)RidgeletRadon(3-7)()fX()()()21,21212,cossinfRRtfxxxxtdxdxθδθθ=+−∫()()(),,,,fabfRCRTabtRtdtθψθ=∫12JointLaboratoryofShantouandXiamenUniversity,20083.2RidgeletDonohoRidgelet[4]Ridgelet“”Ridgelet(3-8)()22LR()(),:,jktjZkZψ∈∈13JointLaboratoryofShantouandXiamenUniversity,2008MeyerRidgelet(3-10)Meyer“”2[0,2)Lπ(),ˆjkψω(),jktψ()(),,;,,xjkilλρλε=()()()()()()12,,,,ˆˆˆ:/2jkiljkilεελρξξψξωθψξωθπ−=+−+14JointLaboratoryofShantouandXiamenUniversity,2008(3-11)(3-12)Meyer(4-12)DaubechiesRidgeletRidgelet[6]()(),,1jkjkttψψ−−=()1,,2iililεεωθπω+++=15JointLaboratoryofShantouandXiamenUniversity,20083.3Ridgelet3.3.1RidgeletRidgelet,,,,,,,,,,3-4ΓΓ2[0,1]16JointLaboratoryofShantouandXiamenUniversity,20083-4Ridgelet17JointLaboratoryofShantouandXiamenUniversity,2008Ridgelet,Radon,RadonRadon,RadonRadon,Ridgelet,RidgeletRidgelet,[8]18JointLaboratoryofShantouandXiamenUniversity,2008RidgeletRidgeletRidgelet19JointLaboratoryofShantouandXiamenUniversity,20083.3.2RidgeletRidgeletRidgeletRidgeletE.J.CandesD.L.DonohoCurveletsRidgeletRidgelet“”20JointLaboratoryofShantouandXiamenUniversity,20083.4RidgeletRidgeletRidgeletRidgeletRidgeletCurveletRidgeletRidgelet21JointLaboratoryofShantouandXiamenUniversity,20083.4.1RidgeletFRITRidgelet22JointLaboratoryofShantouandXiamenUniversity,2008„1„2FRITRidgelet„3„4FRIT„5Wiener“”23JointLaboratoryofShantouandXiamenUniversity,20083.13.2RidgeletRidgeletRidgelet“”Wiener24JointLaboratoryofShantouandXiamenUniversity,2008(a)(b)25JointLaboratoryofShantouandXiamenUniversity,2008(c)(d)(e)26JointLaboratoryofShantouandXiamenUniversity,20083-5Lena(a)(b)PSNR20.57dB(c)PSNR22.18dB(d)RidgeletPSNR24.88dB(e)RidgeletWienerPSNR24.93dB27JointLaboratoryofShantouandXiamenUniversity,2008(a)(b)28JointLaboratoryofShantouandXiamenUniversity,2008(c)(d)(e)29JointLaboratoryofShantouandXiamenUniversity,20083-6Goldhill(a)(b)PSNR20.29dB(c)PSNR21.90dB(d)RidgeletPSNR24.70dB(e)RidgeletWienerPSNR25.53dB30JointLaboratoryofShantouandXiamenUniversity,20083.4.2RidgeletRidgeletRadon,Ridgelet,,[10],,31JointLaboratoryofShantouandXiamenUniversity,2008„1„2„3Ridgelet„4SPIHTRidgelet32JointLaboratoryofShantouandXiamenUniversity,2008Ridgelet,4-64-7Ridgelet33JointLaboratoryofShantouandXiamenUniversity,2008(a)(b)34JointLaboratoryofShantouandXiamenUniversity,2008(c)(d)35JointLaboratoryofShantouandXiamenUniversity,20083-7Lena(a)(b)1.25PSNR=52.28dB(c)1.67PSNR=42.9dB(d)5PSNR=31.89dB36JointLaboratoryofShantouandXiamenUniversity,2008(a)(b)37JointLaboratoryofShantouandXiamenUniversity,2008(c)(d)38JointLaboratoryofShantouandXiamenUniversity,20083-8Baboon(a)(b)1.25PSNR=45.41dB(c)1.67PSNR=36.09dB(d)5PSNR=25.94dB39JointLaboratoryofShantouandXiamenUniversity,20083.4.3RidgeletRidgeletRidgelet,,,40JointLaboratoryofShantouandXiamenUniversity,2008„RidgeletRidgeletRidgeletRidgelet„1Ridgelet„2„3Ridgelet41JointLaboratoryofShantouandXiamenUniversity,2008„Ridgelet„RidgeletRidgelet“”[12]42JointLaboratoryofShantouandXiamenUniversity,2008„3.5Curvelet„3.5.1Curvelet„RidgeletRidgelet43JointLaboratoryofShantouandXiamenUniversity,2008CandesDonohoRidgelet1999CurveletCurvelet[13]Curvelet2-D“”44JointLaboratoryofShantouandXiamenUniversity,20083.5.2CurveletCurvelet:SARCurveletRidgeletCurvelet45JointLaboratoryofShantouandXiamenUniversity,2008Candes2002CurveletCurvelet[14]Curvelet2005Curvelet[15]46JointLaboratoryofShantouandXiamenUniversity,20083.5.3CurveletCurveletRidgeletRidgeletRidgeletCurvelet“Ridgelet”RidgeletCurvelet47JointLaboratoryofShantouandXiamenUniversity,2008CurveletRidgeletMultiscaleRidgeletTransformRidgeletMultiscaleRidgeletDictionaryRidgelet(3-13){},:,0,,QssQµαψψα=≥∈Ω∈Γ48JointLaboratoryofShantouandXiamenUniversity,2008„Curvelet„1„2„30Φ2(0,1,2,)ssΨ=…1ξ≤2222,2ssξ+⎡⎤∈⎣⎦22020()()1ssξξ≥Φ+Φ=∑49JointLaboratoryofShantouandXiamenUniversity,2008(3-14)Curvelet(3-15)00002(,*,*,)ssfPfffff=Φ∗∆=Ψ∆=Ψ……22202220*ssfPff≥=+∆∑,,,,sQsfQµααψα=∆∈Ω∈Γ50JointLaboratoryofShantouan