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发表于 2016-3-30 09:41
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有关小波的几个术语及常见的小波基介绍 （2）
1、Haar小波 Haar，一般音译为“哈尔”。
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
2、Daubechies(dbN)小波(紧支集正交小波) Daubechies，一般音译为“多贝西”。
Daubechies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
3、Symlet(symN)小波(近似对称的紧支集正交小波) Symlet小波函数是IngridDaubechies提出的近似对称的小波函数，它是对db函数的一种改进。Symlet小波系通常表示为symN (N=2,3,…,8)。symN小波的支撑范围为2N-1，消失矩为N，同时也具备较好的正则性。该小波与dbN小波相比，在连续性、支集长度、滤波器长度等方面与dbN小波一致，但symN小波具有更好的对称性，即一定程度上能够减少对信号进行分析和重构时的相位失真。
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
4、Coiflet(coifN)小波 根据R.Coifman的要求，Daubechies构造了Coiflet小波，它具有coifN (N=1,2,3,4,5)这一系列。Coiflet的小波函数Ψ(t)的2N阶矩为零，尺度函数φ(t)的2N-1阶矩为零。Ψ(t)和φ(t)的支撑长度为6N-1。Coiflet的Ψ(t)和φ(t)具有比dbN更好的对称性。
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
5、Biorthogonal(biorNr.Nd)小波 为了解决对称性和精确信号重构的不相容性，引入了双正交小波，称为对偶的两个小波分别用于信号的分解和重构。双正交小波解决了线性相位和正交性要求的矛盾。由于它有线性相位特性，所以主要应用在信号与图像的重构中。通常的用法是采用一个函数进行分解，用另外一个小波函娄进行重构。
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,kδl,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.
6、ReverseBior小波 由Biorthogonal而来，因此两者形式很类似。
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.
转载 blog.csdn.net/jbb0523/article/details/42586749
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发表于 2016-3-30 09:38
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