有关小波的几个术语及常见的小波基介绍

小波变换不同于傅里叶变换，根据小波母函数的不同，小波变换的结果也不尽相同。现实中到底选择使用哪一种小波的标准一般有以下几点：

小波函数Ψ(t)、Ψ(ω)、尺度函数φ(t)和φ(ω)的支撑区间，是当时间或频率趋向于无穷大时，Ψ(t)、Ψ(ω)、φ(t)和φ(ω)从一个有限值收敛到0的长度。支撑长度越长，一般需要耗费更多的计算时间，且产生更多高幅值的小波系数。大部分应用选择支撑长度为5~9之间的小波，因为支撑长度太长会产生边界问题，支撑长度太短消失矩太低，不利于信号能量的集中。

这里常常见到“紧支撑”的概念，通俗来讲，对于函数f(x)，如果自变量x在0附近的取值范围内，f(x)能取到值;而在此之外，f(x)取值为0，那么这个函数f(x)就是紧支撑函数，而这个0附近的取值范围就叫做紧支撑集。总结为一句话就是“除在一个很小的区域外，函数为零，即函数有速降性”。

具有对称性的小波，在图像处理中可以很有效地避免相位畸变，因为该小波对应的滤波器具有线性相位的特点。

在实际中，对基本小波往往不仅要求满足容许条件，对还要施加所谓的消失矩(Vanishing Moments)条件，使尽量多的小波系数为零或者产生尽量少的非零小波系数，这样有利于数据压缩和消除噪声。消失矩越大，就使更多的小波系数为零。但在一般情况下，消失矩越高，支撑长度也越长。所以在支撑长度和消失矩上，我们必须要折衷处理。

小波的消失矩的定义为，若

其中，Ψ(t)为基本小波，0<=p<N。则称小波函数具有N阶消失矩。从上式还可以得出，同任意n-1阶多项式正交。在频域内表示就是Ψ(ω)在ω=0处有高阶零点（一阶零点就是容许条件）。

在量化或者舍入小波系数时，为了减小重构误差对人眼的影响，我们必须尽量增大小波的光滑性或者连续可微性。因为人眼对“不规则”(irregular)误差比“平滑”误差更加敏感。换句话说，我们需要强加“正则性”(regularity)条件。也就是说正则性好的小波，能在信号或图像的重构中获得较好的平滑效果，减小量化或舍入误差的视觉影响。但在一般情况下，正则性好，支撑长度就长，计算时间也就越大。因此正则性和支撑长度上，我们也要有所权衡。

消失矩和正则性之间有很大关系，对很多重要的小波(比如，样条小波，Daubechies小波等)来说，随着消失矩的增加，小波的正则性变大，但是，并不能说随着小波消失矩的增加，小波的正则性一定增加，有的反而变小。

选择和信号波形相似的小波，这对于压缩和消噪是有参考价值的。

以下列出的15种小波基是Matlab中支持的15种。

Haar函数是小波分析中最早用到的一个具有紧支撑的正交小波函数，也是最简单的一个小波函数，它是支撑域在t∈[0,1]范围内的单个矩形波。

Haar小波在时域上是不连续的，所以作为基本小波性能不是特别好。

在Matlab中输入命令waveinfo('haar')可得到如下信息：

General characteristics: Compactlysupported

wavelet, the oldest and the simplestwavelet.

scaling function phi = 1 on [0 1] and 0otherwise.

wavelet function psi = 1 on [0 0.5], = -1on [0.5 1] and 0 otherwise.

Family Haar

Short name haar

Examples haar is the same as db1

Orthogonal yes

Biorthogonal yes

Compact support yes

DWT possible

CWT possible

Support width 1

Filters length 2

Regularity haar is not continuous

Symmetry yes

Number of vanishing

moments for psi 1

Daubechies小波(紧支集正交小波)是由世界著明的小波分析学者Ingrid Daubechies(一般音译为英格丽·多贝西)构造的小波函数，我们一般简写成dbN，N是小波的阶数。小波函数Ψ(t)和尺度函数φ(t)中的支撑区为2N-1，Ψ(t)的消失矩为N。dbN小波具有较好的正则性，即该小波作为稀疏基所引入的光滑误差不容易被察觉，使得信号重构过程比较光滑。

dbN小波的特点是随着阶次(序列N)的增大消失矩阶数越大，其中消失矩越高光滑性就越好，频域的局部化能力就越强，频带的划分效果越好，但是会使时域紧支撑性减弱，同时计算量大大增加，实时性变差。另外，除N=1外，dbN小波不具有对称性(即非线性相位)，即在对信号进行分析和重构时会产生一定的相位失真。dbN没有明确的表达式(除了N=1外，N=1时即为Haar小波)。

在Matlab中输入命令waveinfo('db')可得到如下信息：

General characteristics: Compactlysupported

wavelets with extremal phase and highest

number of vanishing moments for a given

support width. Associated scaling filtersare

minimum-phase filters.

Family Daubechies

Short name db

Order N N strictly positive integer

Examples db1 or haar, db4, db15

Orthogonal yes

Biorthogonal yes

Compact support yes

DWT possible

CWT possible

Support width 2N-1

Filters length 2N

Regularity about 0.2 N for large N

Symmetry far from

Number of vanishing

moments for psi N

在Matlab中输入命令waveinfo('sym')可得到如下信息：

General characteristics: Compactlysupported wavelets with

least asymmetry and highest number ofvanishing moments

for a given support width.

Associated scaling filters are nearlinear-phase filters.

Family Symlets

Short name sym

Order N N = 2, 3, ...

Examples sym2, sym8

Orthogonal yes

Biorthogonal yes

Compact support yes

DWT possible

CWT possible

Support width 2N-1

Filters length 2N

Regularity

Symmetry near from

Number of vanishing

moments for psi N

在Matlab中输入命令waveinfo('coif')可得到如下信息：

General characteristics: Compactlysupported

wavelets with highest number of vanishing

moments for both phi and psi for a given

support width.

Family Coiflets

Short name coif

Order N N = 1, 2, ..., 5

Examples coif2, coif4

Orthogonal yes

Biorthogonal yes

Compact support yes

DWT possible

CWT possible

Support width 6N-1

Filters length 6N

Regularity

Symmetry near from

Number of vanishing

moments for psi 2N

Number of vanishing

moments for phi 2N-1

双正交小波与正交小波的区别在于正交小波满足<Ψj,k ,Ψl,m>=δj,k δl,m，也就是对小波函数的伸缩和平移构成的基函数完全正交，而双正交小波满足的正交性为<Ψj,k ,Ψl,m>=δj,k，也就是对不同尺度伸缩下的小波函数之间有正交性，而同尺度之间通过平移得到的小波函数系之间没有正交性，所以用于分解与重构的小波不是同一个函数，相应的滤波器也不能由同一个小波生成。

该小波虽然不是正交小波，但却是双正交小波，具备正则性，同时也是紧支撑的，其重构支撑范围为2Nr+1，分解支撑范围为2Nd+1。biorNr.Nd小波的主要特征表现在具有线性相位特性。一般来说为了获得线性相位，需要降低对于正交性的局限，为此该双正交小波降低了对于正交性的要求，保留了正交小波的一部分正交性，使小波攻得了线性相位和较短支集的特性。

在Matlab中输入命令waveinfo('bior')可得到如下信息：

General characteristics: Compactly supported

biorthogonal spline wavelets for which

symmetry and exact reconstruction are possible

withFIR filters (in orthogonal case it is

impossible except for Haar).

Family Biorthogonal

Shortname bior

OrderNr,Nd Nr = 1 , Nd = 1, 3, 5

r forreconstruction Nr = 2 , Nd = 2, 4, 6,8

d fordecomposition Nr = 3 , Nd = 1, 3, 5,7, 9

Nr = 4 , Nd = 4

Nr = 5 , Nd = 5

Nr = 6 , Nd = 8

Examples bior3.1,bior5.5

Orthogonal no

Biorthogonal yes

Compact support yes

DWT possible

CWT possible

Support width 2Nr+1 forrec., 2Nd+1 for dec.

Filters length max(2Nr,2Nd)+2 but essentially

biorNr.Nd ld lr

effective length effective length

of Lo_D of Hi_D

bior1.1 2 2

bior1.3 6 2

bior1.5 10 2

bior2.2 5 3

bior2.4 9 3

bior2.6 13 3

bior2.8 17 3

bior3.1 4 4

bior3.3 8 4

bior3.5 12 4

bior3.7 16 4

bior3.9 20 4

bior 4.4 9 7

bior5.5 9 11

bior6.8 17 11

Regularity for

psirec. Nr-1 and Nr-2 at theknots

Symmetry yes

Numberof vanishing

moments for psi dec. Nr

Remark: bior 4.4 , 5.5 and 6.8 are such that reconstruction and

decomposition functions and filters are close in value.

在Matlab中输入命令waveinfo('bior')可得到如下信息：

General characteristics: Compactly supported

biorthogonal spline wavelets for which

symmetry and exact reconstruction are possible

withFIR filters (in orthogonal case it is

impossible except for Haar).

Family Biorthogonal

Shortname rbio

OrderNd,Nr Nd = 1 , Nr = 1, 3, 5

r forreconstruction Nd = 2 , Nr = 2, 4, 6,8

d fordecomposition Nd = 3 , Nr = 1, 3, 5,7, 9

Nd = 4 , Nr = 4

Nd = 5 , Nr = 5

Nd = 6 , Nr = 8

Examples rbio3.1,rbio5.5

Orthogonal no

Biorthogonal yes

Compact support yes

DWT possible

CWT possible

Support width 2Nd+1 forrec., 2Nr+1 for dec.

Filters length max(2Nd,2Nr)+2 but essentially

rbioNd.Nr lr ld

effective length effective length

of Hi_D of Lo_D

rbio1.1 2 2

rbio1.3 6 2

rbio1.5 10 2

rbio2.2 5 3

rbio2.4 9 3

rbio2.6 13 3

rbio2.8 17 3

rbio3.1 4 4

rbio3.3 8 4

rbio3.5 12 4

rbio3.7 16 4

rbio3.9 20 4

rbio4.4 9 7

rbio5.5 9 11

rbio6.8 17 11

Regularity for

psirec. Nd-1 and Nd-2 at theknots

Symmetry yes

Numberof vanishing

moments for psi dec. Nd

Remark: rbio 4.4 , 5.5 and 6.8 are such that reconstruction and

decomposition functions and filters are close in value.

在Matlab中输入命令waveinfo('meyr')可得到如下信息：

General characteristics: Infinitely regular orthogonal wavelet.

Family Meyer

Shortname meyr

Orthogonal yes

Biorthogonal yes

Compact support no

DWT possiblebut without FWT

FIR based approximation provides FWT

CWT possible

Support width infinite

Effective support [-8 8]

Regularity indefinitely derivable

Symmetry yes

在Matlab中输入命令waveinfo('dmey')可得到如下信息：

Definition: FIR based approximation of theMeyer Wavelet.

Family DMeyer

Short name dmey

Orthogonal yes

Biorthogonal yes

Compact support yes

DWT possible

CWT possible

在Matlab中输入命令waveinfo('gaus')可得到如下信息：

Definition: derivatives of the Gaussian

probability density function.

gaus(x,n) = Cn * diff(exp(-x^2),n) wherediff denotes

the symbolic derivative and where Cn issuch that

the 2-norm of gaus(x,n) = 1.

Family Gaussian

Short name gaus

Wavelet name gaus"n"

Orthogonal no

Biorthogonal no

Compact support no

DWT no

CWT possible

Support width infinite

Effective support [-5 5]

Symmetry yes

n even ==> Symmetry

n odd ==> Anti-Symmetry

在Matlab中输入命令waveinfo('mexh')可得到如下信息：

Definition: second derivative of theGaussian

probability density function

mexh(x) = c * exp(-x^2/2) * (1-x^2)

where c = 2/(sqrt(3)*pi^{1/4})

Family Mexican hat

Short name mexh

Orthogonal no

Biorthogonal no

Compact support no

DWT no

CWT possible

Support width infinite

Effective support [-5 5]

Symmetry yes

在Matlab中输入命令waveinfo('morl')可得到如下信息：

Definition:

morl(x) = exp(-x^2/2) * cos(5x)

Family Morlet

Short name morl

Orthogonal no

Biorthogonal no

Compact support no

DWT no

CWT possible

Support width infinite

Effective support [-4 4]

Symmetry yes

在Matlab中输入命令waveinfo('cgau')可得到如下信息：

Definition: derivatives of the complexGaussian

function

cgau(x) = Cn * diff(exp(-i*x)*exp(-x^2),n)where diff denotes

the symbolic derivative and where Cn is aconstant

Family Complex Gaussian

Short name cgau

Wavelet name cgau"n"

Orthogonal no

Biorthogonal no

Compact support no

DWT no

Complex CWT possible

Support width infinite

Symmetry yes

n even ==> Symmetry

n odd ==> Anti-Symmetry

Definition: a complex Shannon wavelet is

shan(x) =Fb^{0.5}*sinc(Fb*x)*exp(2*i*pi*Fc*x)

depending on two parameters:

Fb is a bandwidth parameter

Fc is a wavelet center frequency

The condition Fc > Fb/2 is sufficient toensure that

zero is not in the frequency supportinterval.

Family Complex Shannon

Short name shan

Wavelet name shan"Fb"-"Fc"

Orthogonal no

Biorthogonal no

Compact support no

DWT no

complex CWT possible

Support width infinite

在Matlab中输入命令waveinfo('fbsp')可得到如下信息：

Definition: a complex Frequency B-Splinewavelet is

fbsp(x) = Fb^{0.5}*(sinc(Fb*x/M))^M*exp(2*i*pi*Fc*x)

depending on three parameters:

M is an integer order parameter(>=1)

Fb is a bandwidth parameter

Fc is a wavelet center frequency

For M = 1, the condition Fc > Fb/2 issufficient to ensure

that zero is not in the frequency supportinterval.

Family Complex Frequency B-Spline

Short name fbsp

Wavelet name fbsp"M"-"Fb"-"Fc"

Orthogonal no

Biorthogonal no

Compact support no

DWT no

complex CWT possible

Support width infinite

在Matlab中输入命令waveinfo('cmor')可得到如下信息：

Definition: a complex Morlet wavelet is

cmor(x) =(pi*Fb)^{-0.5}*exp(2*i*pi*Fc*x)*exp(-(x^2)/Fb)

depending on two parameters:

Fb is a bandwidth parameter

Fc is a wavelet center frequency

Family Complex Morlet

Short name cmor

Wavelet name cmor"Fb"-"Fc"

Orthogonal no

Biorthogonal no

Compact support no

DWT no

complex CWT possible

Support width infinite

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【5】wcrzq，microwest.紧支集是什么意思，求详细解释，谢谢.百度知道

【6】zhaodong584584. 消失矩阶数，百度百科

【7】well3216. 对消失矩的理解(转载自matwav)，CSDN博客

【8】小米. 小波消失矩，新浪博客

【9】muchi1234. 正则性，百度百科

【10】洋务大臣. 样条函数，百度百科

【11】其它网络资源，Thanks!!!