小波的幾個(gè)術(shù)語(yǔ)及常見(jiàn)的小波基介紹

1、本篇是這段時(shí)間學(xué)習(xí)小波變換的一個(gè)收尾，了解一下常見(jiàn)的小波函數(shù)，混個(gè) 臉熟，知道一下常見(jiàn)的幾個(gè)術(shù)語(yǔ)，有個(gè)印象即可，這里就當(dāng)是先作一個(gè)備忘錄，以后若有需要再深入研究。一、小波基選擇標(biāo)準(zhǔn)小波變換不同于傅里葉變換，根據(jù)小波母函數(shù)的不同，小波變換的結(jié)果也不盡 相同。現(xiàn)實(shí)中到底選擇使用哪一種小波的標(biāo)準(zhǔn)一般有以下幾點(diǎn)：1、支撐長(zhǎng)度小波函數(shù) 屮、屮(3 )、尺度函數(shù) $ 和$ ( 3 )的支撐區(qū)間，是當(dāng)時(shí)間 或頻率趨向于無(wú)窮大時(shí)，屮 、屮(3 )、$ 和$ ( 3 )從一個(gè)有限值收斂到 0的長(zhǎng)度。 支撐長(zhǎng)度越長(zhǎng)，一般需要耗費(fèi)更多的計(jì)算時(shí)間，且產(chǎn)生更多高幅值的小波系數(shù)。大部分應(yīng)用 選擇支撐長(zhǎng)度為59之間的小波，

2、因?yàn)橹伍L(zhǎng)度太長(zhǎng)會(huì)產(chǎn)生邊界問(wèn)題，支撐長(zhǎng)度太短消失矩太低，不利于信號(hào)能量的集中。這里常常見(jiàn)到“緊支撐”的概念，通俗來(lái)講，對(duì)于函數(shù)f(x)，如果自變量x在0附近的取值范圍內(nèi)，f(x)能取到值；而在此之外，f(x)取值為0,那么這個(gè)函數(shù)f(x)就是緊支撐函數(shù)，而這個(gè)0附近的取值范圍就叫做緊支撐集?？偨Y(jié)為一句話就是“除在一個(gè)很小的區(qū)域外，函數(shù)為零，即函數(shù)有速降性”。2、對(duì)稱(chēng)性具有對(duì)稱(chēng)性的小波，在圖像處理中可以很有效地避免相位畸變，因?yàn)樵撔〔?對(duì)應(yīng)的濾波器具有線性相位的特點(diǎn)。3、消失矩在實(shí)際中，對(duì)基本小波往往不僅要求滿(mǎn)足容許條件，對(duì)還要施加所謂的消失 矩(Vani shi ng Mome nts)條件，

3、使盡量多的小波系數(shù)為零或者產(chǎn)生盡量少的非零小波系數(shù)， 這樣有利于數(shù)據(jù)壓縮和消除噪聲。消失矩越大，就使更多的小波系數(shù)為零。但在一般情況下， 消失矩越高，支撐長(zhǎng)度也越長(zhǎng)。所以在支撐長(zhǎng)度和消失矩上，我們必須要折衷處理。小波的消失矩的定義為，若 其中，屮(t)為基本小波，0=p= S j,k S l,m，也就是對(duì)小波函數(shù)的伸縮和平移構(gòu)成的基函數(shù)完全正交，而雙正交小波滿(mǎn)足的正交性為= S j,k，也就是對(duì)不同尺度伸縮下的小波函數(shù)之間有正交性，而同尺度之間通過(guò)平移得到的小波函數(shù)系之間沒(méi)有正交性，所以用于分解與重構(gòu)的小波不是同一個(gè)函數(shù),應(yīng)的濾波器也不能由同一個(gè)小波生成。該小波雖然不是正交小波，但卻是雙正交小

4、波，具備正則性，同時(shí)也是緊支撐的，其重構(gòu)支撐范圍為2Nr+1，分解支撐范圍為 2Nd+1。小波的主要特征表現(xiàn)在具有線性相位特性。一般來(lái)說(shuō)為了獲得線性相位，需要降低對(duì)于正交性的局限，為此該雙正交小波降 低了對(duì)于正交性的要求，保留了正交小波的一部分正交性，使小波攻得了線性相位和較短支集的特性。在Matlab中輸入命令waveinfo(bior)可得到如下信息：Gen eral characteristics: Compactly supportedbiorthog onal spli ne wavelets for whichsymmetry and exact recon struct ion

5、are possiblewithFIR filters (in orthogo nal case it isimpossible except for Haar).FamilyBiorthogo nalShort namebiorOrderNr,NdNr = 1 , Nd = 1,3,r forrec on structio nNr = 2 , Nd = 2, 4, 6,8d fordecompositi onNr = 3 , Nd = 1,3, 5,7, 95Nr = 4 , Nd = 4Nr = 5 , Nd = 5Nr = 6 , Nd = 8Examples62102Orthogo n

6、alBiorthogo nalCompact supportDWTCWTSupport widthFilters len gthlrlen gtheffective len gthLo_D2noyesyespossiblepossible2Nr+1 forrec., 2Nd+1 for dec.max(2Nr,2Nd)+2 but esse ntially ldeffectiveofof Hi_D931331734484124164204bior979111711Regularity forpsirec.Nr-1 and Nr-2 at theknotsNryesSymmetryNumbero

7、f vanishingmome nts for psi dec.Remark: bior , and are such that recon structi on and decompositi on functions and filters are close in value.6、ReverseBior 小波由Biorthogonal 而來(lái)，因此兩者形式很類(lèi)似。在Matlab中輸入命令waveinfo(bior)可得到如下信息:Gen eral characteristics: Compactly supportedbiorthog onal spli ne wavelets for w

8、hichsymmetry and exact recon struct ion are possiblewithFIR filters (in orthogo nal case it isimpossible except for Haar).FamilyShort nameOrderNd,NrBiorthogo nal rbioNd = 1 , Nr = 1,3, 5r forrec on structio nNd = 2 , Nr = 2, 4, 6,8Idlen gthd fordecompositi onExamplesNd = 3 , Nr = 1,3, 5,7, 9Nd = 4 ,

9、 Nr = 4Nd = 5 , Nr = 5Nd = 6 , Nr = 8Orthogo nalBiorthogo nalCompact supportDWTCWTSupport widthFilters len gthlreffective len gthnoyesyespossiblepossible2Nd+1 forrec., 2Nr+1 for dec.max(2Nd,2Nr)+2 but esse ntiallyeffectiveofHi Dof Lo D10131712162223333444442097111711Regularity for psirec.SymmetryNum

10、berof vanishing mome nts for psi dec.NdNd-1 and Nd-2 at thek notsyesShort namemeyrRemark: rbio , and are such that recon structi on and decompositi on functions and filters are close in value.7、Meyer 小波Meyer小波的小波函數(shù)和尺度函數(shù)都是在頻率域中進(jìn)行定義的，它不是緊支 撐的，但它的收斂速度很快。在Matlab中輸入命令waveinfo(meyr)可得到如下信息：Gen eral charac

11、teristics: Infini tely regular orthogo nal wavelet.MeyerFamilyOrthogo nalyesShort n amedmeyBiorthogo nalCompact supportDWTFWTCWTnoyespossiblebut without FWTFIR based approximati on providespossibleSupport widthEffective supportRegularitySymmetry8 Dmeyer小波infinite-8 8in defi nitely derivableyesDmeyer

12、即離散的Meyer小波，它是Meyer小波基于FIR的近似，用于快速離 散小波變換的計(jì)算。在Matlab中輸入命令waveinfo(dmey)可得到如下信息：Defin iti on: FIR based approximati on of theMeyer Wavelet.FamilyDMeyerOrthogo nalyesWavelet n amegaus nBiorthogo nalCompact supportDWTCWT9、Gaussian 小波yesyespossiblepossibleGaussian小波是高斯密度函數(shù)的微分形式, 波，沒(méi)有尺度函數(shù)。它是一種非正交與非雙正交的小在

13、 Matlab 中輸入命令 waveinfo(gaus)可得到如下信息:Definition: derivatives of the Gaussianprobability den sity fun cti on.gaus(x ,n) = Cn * diff(exp(-xA2), n) wherediff deno testhe symbolic derivative and where Cn issuch thatthe 2-norm of gaus(x ,n) = 1.FamilyGaussia nShort n amegausOrthogo nalnoBiorthogo nalnoCom

14、pact supportnoDWTnoCWTpossibleSupport widthinfiniteEffective support-5 5Symmetryyesn even = Symmetryn odd = An ti-Symmetry10、MexicanHat(mexh)小波Mexican Hat函數(shù)為Gauss函數(shù)的二階導(dǎo)數(shù)。因數(shù)它的形狀像墨西哥帽的截 面，所以我們稱(chēng)這個(gè)函數(shù)為墨西哥草帽函數(shù)。它在時(shí)域和頻率都有很好的局部化，但不存在尺度函數(shù)，所以此小波函數(shù)不具有正交性。在Matlab中輸入命令waveinfo(mexh) 可得到如下信息：Definition: second der

15、ivative of theGaussianprobability den sity fun cti onmexh(x) = c * exp(-xA2/2) * (1-xA2) where c = 2/(sqrt(3)*piA1/4)Short n amemorlFamilyMexica n hatShort n amemexhOrthogo nalnoBiorthogo nalnoCompact supportnoDWTnoCWTpossibleSupport widthinfiniteEffective support-5 511、SymmetryyesMorlet小波Morlet小波是高

16、斯包絡(luò)下的單頻率正弦函數(shù)，沒(méi)有尺度函數(shù),在Matlab中輸入命令waveinfo(morl)可得到如下信息：Defin iti on:morl(x) = exp(-xA2/2) * cos(5x)是非正交分解。FamilyMorletOrthogo nalnoBiorthogo nalnoCompact supportnoDWTnoCWTpossibleSupport widthinfin iteEffective support-4 4Symmetryyes12、ComplexGaussian 小波屬于一類(lèi)復(fù)小波，沒(méi)有尺度函數(shù)。在Matlab中輸入命令waveinfo(cgau)可得到如下信

17、息:Definition: derivatives of the complexGaussianfun ctio ncgau(x) = Cn * diff(exp(-i*x)*exp(-xA2), n) where diff deno tes the symbolic derivative and where Cn is aeon sta ntFamilyComplex Gaussia nShort n amecgauWavelet n ameOrthogo nalBiorthogo nalCompact supportDWTcgau nnonononoComplex CWTpossibleS

18、upport widthSymmetryinfiniteyesn even = Symmetryn odd = An ti-Symmetry13、ComplexShannon Wavelets : shan在Matlab中輸入命令waveinfo(shan)可得到如下信息:Defin iti on: a complex Shannon wavelet issha n(x) =FbA*si nc(Fb*x)*exp(2*i*pi*Fc*x)depe nding on two parameters:Fb is a ban dwidth parameterFc is a wavelet cen te

19、r freque ncyThe con diti on Fc Fb/2 is sufficie nt toe nsure that zero is not in the freque ncy support in terval.FamilyShort n ameComplex Shannon sha nWavelet n ameOrthogo nalBiorthogo nalCompact supportDWTcomplex CWTSupport widthnosha nFb-Fcnononopossibleinfinite14、ComplexFrequency B-Spline Wavele

20、ts （復(fù)高斯B樣條小波）樣條函數(shù)(splinefunction)指一類(lèi)分段(片)光滑、并且在各段交接處也有一定光滑性的函數(shù)，簡(jiǎn)稱(chēng)樣條。在Matlab中輸入命令wave in fo(fbsp)可得到如下信息:Defin iti on: a complex Freque ncy B-Spli newavelet isfbsp(x) = FbA*(si nc(Fb*x/M)AM*exp(2*i*pi*Fc*x)depe nding on three parameters:M is an in teger order parameter(=1)Fb is a ban dwidth parameterFc is a wavelet cen ter freque ncyFor M = 1, the con diti on Fc Fb/2 issufficie nt to