小波与小波变换

13()2001-9-25110(JosephFourier)3.13.1.120(AlfredHaar)1909(Haarwavelets)2070JeanMorletWT(wavelettransform)2080Y.MeyerMeyer1986(dilations)(translations)j2(j≥0)2L(R)StephaneMallat1988[1]Mallat[1]InridDaubechiesRonaldCoifmanVictorWickerhauserInridDaubechies1988(filterbanks)[2]S.MallatInridDaubechies()23.1.23-01(b)(a)(b)(b)MoretGrossmannMorlet1984db6db6DaubechiesMeyerMeyerMeyerSym6sym6symletsDaubechiescoif2coif2DaubechiesR.Coifman(a)(b)(b)3-0133.1.3(motherwavelet)()1.3-023-02()()jtFftedtww+∞--∞=∫()ftjtew-(cossinjtetjt=+)()Fww(continuouswavelettransformCWT)(,)()(,,)Cscalepositionftscalepositiontdty+∞-∞=∫()ftyCWTC(scale)(position)CWT51:()ty()ft2:CCC43:k()tky-12()tky-212()ft4:()ty(/)ty25:1~4CWT3-033-033-04(a)MATLAB(a)xy-xyC(b)z5(a)(b)3-04[4]ww2.j2(j.0)(dyadicwavelettransform)(discretewavelettransformDWT)3-05(a)2040Gabor(shorttimeFouriertransformSTFT)-(b)2080Morlet-()3-05Mallat1988Mallat[1]3-06SADA(approximations)D(detail)63-063-07(waveletdecompositiontree)3-07(waveletpacketdecompositiontree)3-08SS=A1+AAD3+DAD3+DD273-08100010002000,(Nyquist)(downsampling)(coefficient)cDcA3-093-093.(waveletreconstruction)(synthesis)(inversediscretewavelettransformIDWT)(upsampling)3-103-1003-113-118(aliasing)(L)(H)(L'H')(quadraturemirrorfiltersQMF)3-123-123.1.4()xy()xy(motherwavelet){},()abxy()xy,()abxbxaayy-⎛⎞=⎜⎟⎝⎠1a()bxja=2bia=/2,()2(2)jjijxxiyy--=-/2,()2(2)jjijxxiyy=-/2()2(2)ijjjxxiyy=-ij()fx()xy()fx,()abxy,(,),()fabxbWabffxdxaayy+∞-∞-⎛⎞==⎜⎟⎝⎠∫11984A.GrossmanJ.Morlet,,(),()ababfxfxadadbCyyy+∞+∞--∞-∞=∫∫22Cy()xy(admissiblecondition)9ˆ()Cdyy∞-∞=∞∫ˆ()yw()xy()xy2L(R)3.23.2.1(Haarbasisfunction)1909(piecewise-constantfunction)[0,1)1002=1[0,1)00()xf0V0V00101()0xxf≤⎧=⎨⎩3-133-1300()xf(boxfunction),0V12=2[0,1)[0,1/2)[1/2,1)11101()()xxff1V21V10100.5()0xxf≤⎧=⎨⎩1110.51()0xxf≤⎧=⎨⎩3-14103-141101()()xxff21V2224j==[0,1)2224j==[0,1/4),[1/4,1/2),[1/2,3/4)[3/4,1)20()xf21()xf22()xf23()xf2V4012322221,01/41,1/41/2()()0,0,1,1/23/41,3/41()()0,0,xxxxxxxxffff≤≤⎧⎧==⎨⎨⎩⎩≤≤⎧⎧==⎨⎨⎩⎩3-153-1521()xf22()xf23()xf42VjV(basis)jV(scalingfunction)()jixf101()0xxf≤⎧=⎨⎩(3-1)()jixf()(2),0,1,,(21)jjjixxiiff=-=⋅⋅⋅-(3-2)jjiixjV{}()0,,21jjjiVspxif==⋅⋅⋅-(3-3)11sp(linearspan)2jjV[0,1)jV1jV+jV011jjVVVV+⊂⊂⋅⋅⋅⊂⊂jV1jjVV+⊆3.2.2()ijxy(Haarwaveletfunctions)101/2()11/210xxxy≤⎧⎪=-≤⎨⎪⎩(3-4)()ijxy()(2),0,,(21)ijjjxxiiyy=-=⋅⋅⋅-(3-5)jW{}()0,1,,21jjjiWspxiy==⋅⋅⋅-(3-6)spjjiix0W1W2W0W00101/2()11/210xxxy≤⎧⎪=-≤⎨⎪⎩3-16123-1600()xy1W10101/4()11/41/20xxxy≤⎧⎪=-≤⎨⎪⎩1111/23/4()13/41/20xxxy≤⎧⎪=-≤⎨⎪⎩3-173-171101()()xxyy2W22012223101/812/83/8()11/82/8()13/84/80014/85/816/87/8()15/86/8()17/8100xxxxxxxxxxxxyyyy⎧≤≤⎧⎪⎪=-≤=-≤⎨⎨⎪⎪⎩⎩⎧≤≤⎧⎪⎪=-≤=-≤⎨⎨⎪⎪⎩⎩3-183-1820()xy21()xy22()xy23()xy()jixyjW1jV+131jjWV+⊆3.2.3(normalization)[0,1)120(),()()d1jjjiiixxxxfff==∫(3-7)120(),()()d1jjjiiixxxxyyy==∫(3-8)()()jjiixxfy/2()2(2)ijjjxxiff=-(3-9)/2()2(2)ijjjxxiyy=-(3-10)/22j(innerproduct)1[0,1)3.2.4()jixf()jixyjVjW1jjjVVW+=⊕(3-11)⊕1jV+jWjVjWjV(orthogonalcomplement)(3-11)1jV+jWjVjW()ijxyØjW()jixyjV()jixf1jV+14ØjW()jixyjV()jixf3.31.[3.1]40123,,,pppp[9735]1(averaging)21/2[84]2(differencing)224(detailcoefficient)2(9-8)=18,9171(3-4)=-12[841-1]31262,1-1[621-1]3-13-14[9735]2[84][1-1]1[6][2]4(Haarwavelettransform)(Haarwaveletdecomposition)1221542.3.1(1)2V()Ix[9735]2224j==2V2222222200112233()()()()()Ixcxcxcxcxffff=+++22220123,,cccc4[9735]22220123()9()7()3()5()Ixxxxxffff=+++3-19()Ix2V(2)1V1W1V01()xf11()xf1W10()xy11()xy(3-11)211VVW=⊕()Ix1111111100110011()()()()()Ixcxcxdxdxffyy=+++10c11c210d11d10()xf11()xf1V10()xy11()xy1W()Ix163-20()Ix1V1W(3)00,VW1W0V00()xf0W00()xy1W10()xy11()xy(3-11)2001VVWW=⊕⊕()Ix0000111100000011()()()()()Ixcxdxdxdxfyyy=+++3-21()Ix00,VW1W400c00d10d11d400()xf,00()xy,10()xy11()xy2V3.4[3.2]()[2,5,8,9,7,4,1,1]fx=--17()fx3V3j=33-213-211(25,89,74,11,25,89,74,11)/2(7,17,11,0,3,1,3,2)/2f=+++-+-----=---2717110717110(,,,,3,1,3,2)/22222241110113,1,3,2)/22222f++--=-------3:()()22241124111011(,,3,1,3,2)/2222235131011(,,3,1,3,2)/22222(12.37444.5962-5.00005.5000-2.1213-0.70712.1213-1.4142)f+--=----=---=Chh2******************************************************************************procDecomposeArray(C:array[0...1]h-ofcolor):whileh1do:/2hh←for0i←to1h-do:'[]([2][21])/2CiCiCi←++18'[]([2][21])/2ChiCiCi+←-+endfor'CC←endwhileendproc******************************************************************************3.588(DCT)JPEGJPEG3.5.1A64236160675795554121351501617474620214342244026273736303133323435292838392541232244451918484915145253111056858595462631A⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥=⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦3-223-22A191.(averaging)(differencing)AR0:[642361606757]1R0N044ROR0:[642361606757]642332+=361322+=606332+=757322+=N0:[3332333231-2927-25]6433313-32-2960-33277-32-252N04N144N0N1N0:[3332333231-2927-25]333232.52+=333232.52+=N1:[32.532.50.50.531-2927-25]33-32.50.533-32.50.5312N1:[32.532.50.50.531-2927-25]32.532.532.52+=N1:[32.500.50.531-2927-25]32.5-32.502.RA32.500.50.531-2927-2532.50-0.5-0.5-2321-191732.50-0.5-0.5-1513-11932.500.50.57-53-132.500.50.5-13-5732.50-0.5-0.59-1113-1532.50-0.5-0.517-1921-2332.500.50.5-2527-2931⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥