定义
连续剪切小波系统
抛物线缩放和剪切的几何效果,使用一些不同参数a和 s.
连续剪切小波系统的架构是基于抛物线缩放矩阵
A
a
=
[
a
0
0
a
1
/
2
]
,
a
>
0
{\displaystyle A_{a}={\begin{bmatrix}a&0\\0&a^{1/2}\end{bmatrix}},\quad a>0}
为一个改变分辨率的方法。剪切矩阵
S
s
=
[
1
s
0
1
]
,
s
∈
R
{\displaystyle S_{s}={\begin{bmatrix}1&s\\0&1\end{bmatrix}},\quad s\in \mathbb {R} }
为一个改变方向的方法。最后再用平移去改变位置。相较于曲波变换,剪切小波利用剪切的方法取代旋转的方法,其优点在于如果
s
∈
Z
{\displaystyle s\in \mathbb {Z} }
,剪切运算子
S
s
{\displaystyle S_{s}}
会让整数格不改变。例如二维情况下,当
s
∈
Z
{\displaystyle s\in \mathbb {Z} }
,对坐标
x
=
[
x
y
]
,
x
,
y
∈
Z
{\displaystyle \mathbf {x} ={\begin{bmatrix}x\\y\end{bmatrix}},x,y\in \mathbb {Z} }
进行剪切操作:
S
s
x
=
[
x
−
s
y
y
]
∈
Z
2
{\displaystyle S_{s}\mathbf {x} ={\begin{bmatrix}x-sy\\y\end{bmatrix}}\in \mathbb {Z} ^{2}}
结果依然在整数采样点上。[5]
给定一个
ψ
∈
L
2
(
R
2
)
{\displaystyle \psi \in L^{2}(\mathbb {R} ^{2})}
,由
ψ
∈
L
2
(
R
2
)
{\displaystyle \psi \in L^{2}(\mathbb {R} ^{2})}
产生的连续剪切小波系统被定义成:
SH
c
o
n
t
(
ψ
)
=
{
ψ
a
,
s
,
t
=
a
3
/
4
ψ
(
S
s
A
a
(
⋅
−
t
)
)
∣
a
>
0
,
s
∈
R
,
t
∈
R
2
}
,
{\displaystyle \operatorname {SH} _{\mathrm {cont} }(\psi )=\{\psi _{a,s,t}=a^{3/4}\psi (S_{s}A_{a}(\cdot -t))\mid a>0,s\in \mathbb {R} ,t\in \mathbb {R} ^{2}\},}
其对应的连续剪切小波转换:
f
↦
S
H
ψ
f
(
a
,
s
,
t
)
=
⟨
f
,
ψ
a
,
s
,
t
⟩
,
f
∈
L
2
(
R
2
)
,
(
a
,
s
,
t
)
∈
R
>
0
×
R
×
R
2
.
{\displaystyle f\mapsto {\mathcal {SH}}_{\psi }f(a,s,t)=\langle f,\psi _{a,s,t}\rangle ,\quad f\in L^{2}(\mathbb {R} ^{2}),\quad (a,s,t)\in \mathbb {R} _{>0}\times \mathbb {R} \times \mathbb {R} ^{2}.}
离散剪切小波系统
离散的剪切小波系统可以直接从
SH
c
o
n
t
(
ψ
)
{\displaystyle \operatorname {SH} _{\mathrm {cont} }(\psi )}
并借由将参数集合
R
>
0
×
R
×
R
2
.
{\displaystyle \mathbb {R} _{>0}\times \mathbb {R} \times \mathbb {R} ^{2}.}
离散化导出。有很多方法可以实现,但最常见是由下式导出:
{
(
2
j
,
k
,
A
2
j
−
1
S
k
−
1
m
)
∣
j
∈
Z
,
k
∈
Z
,
m
∈
Z
2
}
⊆
R
>
0
×
R
×
R
2
.
{\displaystyle \{(2^{j},k,A_{2^{j}}^{-1}S_{k}^{-1}m)\mid j\in \mathbb {Z} ,k\in \mathbb {Z} ,m\in \mathbb {Z} ^{2}\}\subseteq \mathbb {R} _{>0}\times \mathbb {R} \times \mathbb {R} ^{2}.}
从这个式子,与剪切运算子有关的离散剪切小波系统被定义为:
SH
(
ψ
)
=
{
ψ
j
,
k
,
m
=
2
3
j
/
4
ψ
(
S
k
A
2
j
⋅
−
m
)
∣
j
∈
Z
,
k
∈
Z
,
m
∈
Z
2
}
,
{\displaystyle \operatorname {SH} (\psi )=\{\psi _{j,k,m}=2^{3j/4}\psi (S_{k}A_{2^{j}}\cdot {}-m)\mid j\in \mathbb {Z} ,k\in \mathbb {Z} ,m\in \mathbb {Z} ^{2}\},}
其相关的离散剪切小波转换被定义为:
f
↦
S
H
ψ
f
(
j
,
k
,
m
)
=
⟨
f
,
ψ
j
,
k
,
m
⟩
,
f
∈
L
2
(
R
2
)
,
(
j
,
k
,
m
)
∈
Z
×
Z
×
Z
2
.
{\displaystyle f\mapsto {\mathcal {SH}}_{\psi }f(j,k,m)=\langle f,\psi _{j,k,m}\rangle ,\quad f\in L^{2}(\mathbb {R} ^{2}),\quad (j,k,m)\in \mathbb {Z} \times \mathbb {Z} \times \mathbb {Z} ^{2}.}
范例
设
ψ
1
∈
L
2
(
R
)
{\displaystyle \psi _{1}\in L^{2}(\mathbb {R} )}
为一个满足离散卡尔德龙条件(discrete Calderón condition)的函数,即:
∑
j
∈
Z
|
ψ
^
1
(
2
−
j
ξ
)
|
2
=
1
,
for a.e.
ξ
∈
R
,
{\displaystyle \sum _{j\in \mathbb {Z} }|{\hat {\psi }}_{1}(2^{-j}\xi )|^{2}=1,{\text{for a.e. }}\xi \in \mathbb {R} ,}
ψ
^
1
∈
C
∞
(
R
)
{\displaystyle {\hat {\psi }}_{1}\in C^{\infty }(\mathbb {R} )}
,
supp
ψ
^
1
⊆
[
−
1
2
,
−
1
16
]
∪
[
1
16
,
1
2
]
{\displaystyle \operatorname {supp} {\hat {\psi }}_{1}\subseteq [-{\tfrac {1}{2}},-{\tfrac {1}{16}}]\cup [{\tfrac {1}{16}},{\tfrac {1}{2}}]}
,其中
ψ
^
1
{\displaystyle {\hat {\psi }}_{1}}
为
ψ
1
{\displaystyle \psi _{1}}
的 傅立叶变换。例如,可以选择
ψ
1
{\displaystyle \psi _{1}}
为一个梅尔小波。此外,设
ψ
2
∈
L
2
(
R
)
{\displaystyle \psi _{2}\in L^{2}(\mathbb {R} )}
而且
ψ
^
2
∈
C
∞
(
R
)
,
{\displaystyle {\hat {\psi }}_{2}\in C^{\infty }(\mathbb {R} ),}
supp
ψ
^
2
⊆
[
−
1
,
1
]
{\displaystyle \operatorname {supp} {\hat {\psi }}_{2}\subseteq [-1,1]}
∑
k
=
−
1
1
|
ψ
^
2
(
ξ
+
k
)
|
2
=
1
,
for a.e.
ξ
∈
[
−
1
,
1
]
.
{\displaystyle \sum _{k=-1}^{1}|{\hat {\psi }}_{2}(\xi +k)|^{2}=1,{\text{for a.e. }}\xi \in \left[-1,1\right].}
通常会选择一个冲击函数 作为
ψ
^
2
{\displaystyle {\hat {\psi }}_{2}}
,然后
ψ
∈
L
2
(
R
2
)
{\displaystyle \psi \in L^{2}(\mathbb {R} ^{2})}
就会是:
ψ
^
(
ξ
)
=
ψ
^
1
(
ξ
1
)
ψ
^
2
(
ξ
2
ξ
1
)
,
ξ
=
(
ξ
1
,
ξ
2
)
∈
R
2
,
{\displaystyle {\hat {\psi }}(\xi )={\hat {\psi }}_{1}(\xi _{1}){\hat {\psi }}_{2}\left({\tfrac {\xi _{2}}{\xi _{1}}}\right),\quad \xi =(\xi _{1},\xi _{2})\in \mathbb {R} ^{2},}
这被称作一个典型的剪切小波。其对应的离散剪切小波系统
SH
(
ψ
)
{\displaystyle \operatorname {SH} (\psi )}
在
L
2
(
R
2
)
{\displaystyle L^{2}(\mathbb {R} ^{2})}
空间中构成一个紧框架,且其中包含频带限制的函数。[5]
另外一个例子是紧支撑的剪切小波系统,其中要选定紧支撑函数
ψ
∈
L
2
(
R
2
)
{\displaystyle \psi \in L^{2}(\mathbb {R} ^{2})}
让
SH
(
ψ
)
{\displaystyle \operatorname {SH} (\psi )}
形成一个
L
2
(
R
2
)
{\displaystyle L^{2}(\mathbb {R} ^{2})}
的框架。[3] [6] [7] [8]
既然这样,在
SH
(
ψ
)
{\displaystyle \operatorname {SH} (\psi )}
中所有剪切小波的元素是紧支撑且相较于频带限制的典型剪切小波有优越的空间定位。虽然紧支撑的剪切小波系统没有形成一个Parseval框架,但任意一个
f
∈
L
2
(
R
2
)
{\displaystyle f\in L^{2}(\mathbb {R} ^{2})}
的函数可以被剪切小波展开。
自适应锥形剪切小波
上述所定义的剪切小波有其缺陷,那就是剪切小波元素的方向性偏差与大的剪切参数有关联。在典型剪切小波的频率拼接(在#范例 中的图可见)中可以看到这个影响,当剪切参数
s
{\displaystyle s}
趋近无限大时,剪切小波的频率支撑越来越贴近
ξ
2
{\displaystyle \xi _{2}}
轴,这在分析傅立叶变换集中分布在
ξ
2
{\displaystyle \xi _{2}}
轴的函数时造成很严重的问题。
为了解决这个问题,频域被分成一个低频部分和两个锥形部分(如图所示):
R
=
{
(
ξ
1
,
ξ
2
)
∈
R
2
∣
|
ξ
1
|
,
|
ξ
2
|
≤
1
}
,
C
h
=
{
(
ξ
1
,
ξ
2
)
∈
R
2
∣
|
ξ
2
/
ξ
1
|
≤
1
,
|
ξ
1
|
>
1
}
,
C
v
=
{
(
ξ
1
,
ξ
2
)
∈
R
2
∣
|
ξ
1
/
ξ
2
|
≤
1
,
|
ξ
2
|
>
1
}
.
{\displaystyle {\begin{aligned}{\mathcal {R}}&=\left\{(\xi _{1},\xi _{2})\in \mathbb {R} ^{2}\mid |\xi _{1}|,|\xi _{2}|\leq 1\right\},\\{\mathcal {C}}_{\mathrm {h} }&=\left\{(\xi _{1},\xi _{2})\in \mathbb {R} ^{2}\mid |\xi _{2}/\xi _{1}|\leq 1,|\xi _{1}|>1\right\},\\{\mathcal {C}}_{\mathrm {v} }&=\left\{(\xi _{1},\xi _{2})\in \mathbb {R} ^{2}\mid |\xi _{1}/\xi _{2}|\leq 1,|\xi _{2}|>1\right\}.\end{aligned}}}
由典型剪切小波生成的自适应性剪切小波系统的频率拼接
这个自适应性剪切小波系统是由三个部分组成,每个部分都对应到这些频域之一,这个系统是由三个函数
ϕ
,
ψ
,
ψ
~
∈
L
2
(
R
2
)
{\displaystyle \phi ,\psi ,{\tilde {\psi }}\in L^{2}(\mathbb {R} ^{2})}
和晶格取样因子
c
=
(
c
1
,
c
2
)
∈
(
R
>
0
)
2
{\displaystyle c=(c_{1},c_{2})\in (\mathbb {R} _{>0})^{2}}
所产生:
SH
(
ϕ
,
ψ
,
ψ
~
;
c
)
=
Φ
(
ϕ
;
c
1
)
∪
Ψ
(
ψ
;
c
)
∪
Ψ
~
(
ψ
~
;
c
)
,
{\displaystyle \operatorname {SH} (\phi ,\psi ,{\tilde {\psi }};c)=\Phi (\phi ;c_{1})\cup \Psi (\psi ;c)\cup {\tilde {\Psi }}({\tilde {\psi }};c),}
其中:
Φ
(
ϕ
;
c
1
)
=
{
ϕ
m
=
ϕ
(
⋅
−
c
1
m
)
∣
m
∈
Z
2
}
,
Ψ
(
ψ
;
c
)
=
{
ψ
j
,
k
,
m
=
2
3
j
/
4
ψ
(
S
k
A
2
j
⋅
−
M
c
m
)
∣
j
≥
0
,
|
k
|
≤
⌈
2
j
/
2
⌉
,
m
∈
Z
2
}
,
Ψ
~
(
ψ
~
;
c
)
=
{
ψ
~
j
,
k
,
m
=
2
3
j
/
4
ψ
(
S
~
k
A
~
2
j
⋅
−
M
~
c
m
)
∣
j
≥
0
,
|
k
|
≤
⌈
2
j
/
2
⌉
,
m
∈
Z
2
}
,
{\displaystyle {\begin{aligned}\Phi (\phi ;c_{1})&=\{\phi _{m}=\phi (\cdot {}-c_{1}m)\mid m\in \mathbb {Z} ^{2}\},\\\Psi (\psi ;c)&=\{\psi _{j,k,m}=2^{3j/4}\psi (S_{k}A_{2^{j}}\cdot {}-M_{c}m)\mid j\geq 0,|k|\leq \lceil 2^{j/2}\rceil ,m\in \mathbb {Z} ^{2}\},\\{\tilde {\Psi }}({\tilde {\psi }};c)&=\{{\tilde {\psi }}_{j,k,m}=2^{3j/4}\psi ({\tilde {S}}_{k}{\tilde {A}}_{2^{j}}\cdot {}-{\tilde {M}}_{c}m)\mid j\geq 0,|k|\leq \lceil 2^{j/2}\rceil ,m\in \mathbb {Z} ^{2}\},\end{aligned}}}
式子中的一些变数定义如下;
A
~
a
=
[
a
1
/
2
0
0
a
]
,
a
>
0
,
S
~
s
=
[
1
0
s
1
]
,
s
∈
R
,
M
c
=
[
c
1
0
0
c
2
]
,
and
M
~
c
=
[
c
2
0
0
c
1
]
.
{\displaystyle {\begin{aligned}&{\tilde {A}}_{a}={\begin{bmatrix}a^{1/2}&0\\0&a\end{bmatrix}},\;a>0,\quad {\tilde {S}}_{s}={\begin{bmatrix}1&0\\s&1\end{bmatrix}},\;s\in \mathbb {R} ,\quad M_{c}={\begin{bmatrix}c_{1}&0\\0&c_{2}\end{bmatrix}},\quad {\text{and}}\quad {\tilde {M}}_{c}={\begin{bmatrix}c_{2}&0\\0&c_{1}\end{bmatrix}}.\end{aligned}}}
系统
Ψ
(
ψ
)
{\displaystyle \Psi (\psi )}
和
Ψ
~
(
ψ
~
)
{\displaystyle {\tilde {\Psi }}({\tilde {\psi }})}
基本上不同点在于
x
1
{\displaystyle x_{1}}
和
x
2
{\displaystyle x_{2}}
的角色互换。因此,它们分别对应到锥形区域
C
h
{\displaystyle {\mathcal {C}}_{\mathrm {h} }}
和
C
v
{\displaystyle {\mathcal {C}}_{\mathrm {v} }}
,而缩放函数
ϕ
{\displaystyle \phi }
则对应到低频区域
R
{\displaystyle {\mathcal {R}}}
。
应用 相关条目 参考
^ Guo, Kanghui, Gitta Kutyniok, and Demetrio Labate. "Sparse multidimensional representations using anisotropic dilation and shear operators." Wavelets and Splines (Athens, GA, 2005), G. Chen and MJ Lai, eds., Nashboro Press, Nashville, TN (2006): 189–201.
PDF PDF
^ Guo, Kanghui, and Demetrio Labate. "Optimally sparse multidimensional representation using shearlets." SIAM Journal on Mathematical Analysis 39.1 (2007): 298–318.
PDF PDF
^ 3.0 3.1 Kutyniok, Gitta, and Wang-Q Lim. "Compactly supported shearlets are optimally sparse." Journal of Approximation Theory 163.11 (2011): 1564–1589.
PDF PDF
^ Donoho, David Leigh. "Sparse components of images and optimal atomic decompositions." Constructive Approximation 17.3 (2001): 353–382.
PDF PDF
^ 5.0 5.1 5.2 5.3 5.4 Kutyniok, Gitta, and Demetrio Labate, eds. Shearlets: Multiscale analysis for multivariate data . Springer, 2012, 编辑