拋物線縮放和剪切的幾何效果，使用一些不同參數a和 s.
連續剪切小波系統的架構是基於拋物線縮放矩陣
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}
為一個改變解析度的方法。剪切矩陣
S
s
=
[
1
s
0
1
]
,
s
∈
R
{\displaystyle S_{s}={\begin{bmatrix}1&s\\0&1\end{bmatrix}},\quad s\in \mathbb {R} }
為一個改變方向的方法。最後再用平移去改變位置。相較於曲波变换，剪切小波利用剪切的方法取代旋轉的方法，其優點在於如果
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]
給定一個
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}\},}
其對應的連續剪切小波轉換：
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}.}
離散的剪切小波系統可以直接從
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}.}
從這個式子，與剪切運算子有關的離散剪切小波系統被定義為：
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}\},}
其相關的離散剪切小波轉換被定義為：
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}.}
設
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}}
傅立葉變換 。例如，可以選擇
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].}
通常會選擇一個沖擊函數 作為
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},}
這被稱作一個典型的剪切小波。其對應的離散剪切小波系統
SH
(
ψ
)
{\displaystyle \operatorname {SH} (\psi )}
L
2
(
R
2
)
{\displaystyle L^{2}(\mathbb {R} ^{2})}
[ 5]
另外一個例子是緊支撐的剪切小波系統，其中要選定緊支撐函數
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})}
上述所定義的剪切小波有其缺陷，那就是剪切小波元素的方向性偏差與大的剪切參數有關聯。在典型剪切小波的頻率拼接(在#範例 中的圖可見)中可以看到這個影響，當剪切參數
s
{\displaystyle s}
ξ
2
{\displaystyle \xi _{2}}
ξ
2
{\displaystyle \xi _{2}}
將頻域分解成錐形和低頻區域 為了解決這個問題，頻域被分成一個低頻部分和兩個錐形部分(如圖所示)：
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適應性剪切小波系統的頻率拼接 這個自適應性剪切小波系統是由三個部分組成，每個部分都對應到這些頻域之一，這個系統是由三個函數
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}}}
式子中的一些變數定義如下；
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}}}
系統
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)
