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} }
S
s
{\displaystyle S_{s}}
s
∈
Z
{\displaystyle s\in \mathbb {Z} }
x
=
[
x
y
]
,
x
,
y
∈
Z
{\displaystyle \mathbf {x} ={\begin{bmatrix}x\\y\end{bmatrix}},x,y\in \mathbb {Z} }
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})}
ψ
∈
L
2
(
R
2
)
{\displaystyle \psi \in L^{2}(\mathbb {R} ^{2})}
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}.}
离散的剪切小波系统可以直接从
SH
c
o
n
t
(
ψ
)
{\displaystyle \operatorname {SH} _{\mathrm {cont} }(\psi )}
R
>
0
×
R
×
R
2
.
{\displaystyle \mathbb {R} _{>0}\times \mathbb {R} \times \mathbb {R} ^{2}.}
{
(
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}.}
设
ψ
1
∈
L
2
(
R
)
{\displaystyle \psi _{1}\in L^{2}(\mathbb {R} )}
∑
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}}]}
ψ
^
1
{\displaystyle {\hat {\psi }}_{1}}
ψ
1
{\displaystyle \psi _{1}}
傅立叶变换 。例如，可以选择
ψ
1
{\displaystyle \psi _{1}}
ψ
2
∈
L
2
(
R
)
{\displaystyle \psi _{2}\in L^{2}(\mathbb {R} )}
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}}
ψ
∈
L
2
(
R
2
)
{\displaystyle \psi \in L^{2}(\mathbb {R} ^{2})}
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 )}
L
2
(
R
2
)
{\displaystyle L^{2}(\mathbb {R} ^{2})}
[ 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})}
SH
(
ψ
)
{\displaystyle \operatorname {SH} (\psi )}
L
2
(
R
2
)
{\displaystyle L^{2}(\mathbb {R} ^{2})}
[ 3] [ 6] [ 7] [ 8]
SH
(
ψ
)
{\displaystyle \operatorname {SH} (\psi )}
f
∈
L
2
(
R
2
)
{\displaystyle f\in L^{2}(\mathbb {R} ^{2})}
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}
ξ
2
{\displaystyle \xi _{2}}
ξ
2
{\displaystyle \xi _{2}}
将频域分解成锥形和低频区域 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})}
c
=
(
c
1
,
c
2
)
∈
(
R
>
0
)
2
{\displaystyle c=(c_{1},c_{2})\in (\mathbb {R} _{>0})^{2}}
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 )}
Ψ
~
(
ψ
~
)
{\displaystyle {\tilde {\Psi }}({\tilde {\psi }})}
x
1
{\displaystyle x_{1}}
x
2
{\displaystyle x_{2}}
C
h
{\displaystyle {\mathcal {C}}_{\mathrm {h} }}
C
v
{\displaystyle {\mathcal {C}}_{\mathrm {v} }}
ϕ
{\displaystyle \phi }
R
{\displaystyle {\mathcal {R}}}
^ 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, ISBN 0-8176-8315-1
^ Kittipoom, Pisamai, Gitta Kutyniok, and Wang-Q Lim. "Construction of compactly supported shearlet frames." Constructive Approximation 35.1 (2012): 21–72.
PDF PDF
^ Kutyniok, Gitta, Jakob Lemvig, and Wang-Q Lim. "Optimally sparse approximations of 3D functions by compactly supported shearlet frames." SIAM Journal on Mathematical Analysis 44.4 (2012): 2962–3017.
PDF PDF
^ Purnendu Banerjee and B. B. Chaudhuri, “Video Text Localization using Wavelet and Shearlet Transforms”, In Proc. SPIE 9021, Document Recognition and Retrieval XXI, 2014 (doi:10.1117/12.2036077).PDF PDF
![{\displaystyle \operatorname {supp} {\hat {\psi }}_{1}\subseteq [-{\tfrac {1}{2}},-{\tfrac {1}{16}}]\cup [{\tfrac {1}{16}},{\tfrac {1}{2}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9781e9cfcce51978a270b8e09fc4b00106a90cc)
![{\displaystyle \operatorname {supp} {\hat {\psi }}_{2}\subseteq [-1,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4be9d7048ed3022be72a06f098288a848f9feef6)
![{\displaystyle \sum _{k=-1}^{1}|{\hat {\psi }}_{2}(\xi +k)|^{2}=1,{\text{for a.e. }}\xi \in \left[-1,1\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fa08b5c6527adfc7cc1f2651b4b413109730f1e)
