Shearlet

Trapezoidal frequency support of the classical shearlet.

Frequency tiling of the (discrete) classical shearlet system.

Let ${\displaystyle \psi _{1}\in L^{2}(\mathbb {R} )}$ be a function satisfying the discrete Calderón condition, i.e.,

${\displaystyle \sum _{j\in \mathbb {Z} }|{\hat {\psi }}_{1}(2^{-j}\xi )|^{2}=1,\quad \xi \in \mathbb {R} ,}$

with ${\displaystyle {\hat {\psi }}_{1}\in C^{\infty }(\mathbb {R} )}$ and ${\displaystyle \operatorname {supp} {\hat {\psi }}_{1}\subseteq [-{\tfrac {1}{2}},-{\tfrac {1}{16}}]\cup [{\tfrac {1}{16}},{\tfrac {1}{2}}],}$ where ${\displaystyle {\hat {\psi }}_{1}}$ denotes the Fourier transform of ${\displaystyle \psi _{1}.}$ For instance, one can choose ${\displaystyle \psi _{1}}$ to be a Meyer wavelet. Furthermore, let ${\displaystyle \psi _{2}\in L^{2}(\mathbb {R} )}$ be such that ${\displaystyle {\hat {\psi }}_{2}\in C^{\infty }(\mathbb {R} ),}$ ${\displaystyle \operatorname {supp} {\hat {\psi }}_{2}\subseteq [-1,1]}$ and

${\displaystyle \sum _{k\in Z}|{\hat {\psi }}_{2}(\xi +k)|^{2}=1,\quad \xi \in \mathbb {R} .}$

One typically chooses ${\displaystyle {\hat {\psi }}_{2}}$ to be a smooth bump function. Then ${\displaystyle \psi \in L^{2}(\mathbb {R} ^{2})}$ given by

${\displaystyle {\hat {\psi }}(\xi )={\hat {\psi }}_{1}(\xi _{1}){\hat {\psi }}_{2}\left({\tfrac {\xi _{2}}{\xi _{1}}}\right),\quad \xi =(\xi _{1},\xi _{2})\in \mathbb {R} ^{2},}$

is called a classical shearlet. It can be shown that the corresponding discrete shearlet system ${\displaystyle \operatorname {SH} (\psi )}$ constitutes a Parseval frame for ${\displaystyle L^{2}(\mathbb {R} ^{2})}$ consisting of bandlimited functions.[5]

Another example are compactly supported shearlet systems, where a compactly supported function ${\displaystyle \psi \in L^{2}(\mathbb {R} ^{2})}$ can be chosen so that ${\displaystyle \operatorname {SH} (\psi )}$ forms a frame for ${\displaystyle L^{2}(\mathbb {R} ^{2})}$.[4][6][7][8] In this case, all shearlet elements in ${\displaystyle \operatorname {SH} (\psi )}$ are compactly supported providing superior spatial localization compared to the classical shearlets, which are bandlimited. Although a compactly supported shearlet system does not generally form a Parseval frame, any function ${\displaystyle f\in L^{2}(\mathbb {R} ^{2})}$ can be represented by the shearlet expansion due to its frame property.

One drawback of shearlets defined as above is the directional bias of shearlet elements associated with large shearing parameters. This effect is already recognizable in the frequency tiling of classical shearlets (see Figure in Section #Examples), where the frequency support of a shearlet increasingly aligns along the ${\displaystyle \xi _{2}}$-axis as the shearing parameter ${\displaystyle s}$ goes to infinity. This causes serious problems when analyzing a function whose Fourier transform is concentrated around the ${\displaystyle \xi _{2}}$-axis.

Decomposition of the frequency domain into cones.

To deal with this problem, the frequency domain is divided into a low-frequency part and two conic regions (see Figure):

${\displaystyle {\begin{aligned}{\mathcal {R}}&=\left\{(\xi _{1},\xi _{2})\in \mathbb {R} ^{2}\mid |\xi _{1}|,|\xi _{2}|\leq 1\right\},\\{\mathcal {C}}_{\mathrm {h} }&=\left\{(\xi _{1},\xi _{2})\in \mathbb {R} ^{2}\mid |\xi _{2}/\xi _{1}|\leq 1,|\xi _{1}|>1\right\},\\{\mathcal {C}}_{\mathrm {v} }&=\left\{(\xi _{1},\xi _{2})\in \mathbb {R} ^{2}\mid |\xi _{1}/\xi _{2}|\leq 1,|\xi _{2}|>1\right\}.\end{aligned}}}$

Frequency tiling of the cone-adapted shearlet system generated by the classical shearlet.

The associated cone-adapted discrete shearlet system consists of three parts, each one corresponding to one of these frequency domains. It is generated by three functions ${\displaystyle \phi ,\psi ,{\tilde {\psi }}\in L^{2}(\mathbb {R} ^{2})}$ and a lattice sampling factor ${\displaystyle c=(c_{1},c_{2})\in (\mathbb {R} _{>0})^{2}:}$

${\displaystyle \operatorname {SH} (\phi ,\psi ,{\tilde {\psi }};c)=\Phi (\phi ;c_{1})\cup \Psi (\psi ;c)\cup {\tilde {\Psi }}({\tilde {\psi }};c),}$

where

${\displaystyle {\begin{aligned}\Phi (\phi ;c_{1})&=\{\phi _{m}=\phi (\cdot {}-c_{1}m)\mid m\in \mathbb {Z} ^{2}\},\\\Psi (\psi ;c)&=\{\psi _{j,k,m}=2^{3j/4}\psi (S_{k}A_{2^{j}}\cdot {}-M_{c}m)\mid j\geq 0,|k|\leq \lceil 2^{j/2}\rceil ,m\in \mathbb {Z} ^{2}\},\\{\tilde {\Psi }}({\tilde {\psi }};c)&=\{{\tilde {\psi }}_{j,k,m}=2^{3j/4}\psi ({\tilde {S}}_{k}{\tilde {A}}_{2^{j}}\cdot {}-{\tilde {M}}_{c}m)\mid j\geq 0,|k|\leq \lceil 2^{j/2}\rceil ,m\in \mathbb {Z} ^{2}\},\end{aligned}}}$

with

${\displaystyle {\begin{aligned}&{\tilde {A}}_{a}={\begin{bmatrix}a^{1/2}&0\\0&a\end{bmatrix}},\;a>0,\quad {\tilde {S}}_{s}={\begin{bmatrix}1&0\\s&1\end{bmatrix}},\;s\in \mathbb {R} ,\quad M_{c}={\begin{bmatrix}c_{1}&0\\0&c_{2}\end{bmatrix}},\quad {\text{and}}\quad {\tilde {M}}_{c}={\begin{bmatrix}c_{2}&0\\0&c_{1}\end{bmatrix}}.\end{aligned}}}$

The systems ${\displaystyle \Psi (\psi )}$ and ${\displaystyle {\tilde {\Psi }}({\tilde {\psi }})}$ basically differ in the reversed roles of ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$. Thus, they correspond to the conic regions ${\displaystyle {\mathcal {C}}_{\mathrm {h} }}$ and ${\displaystyle {\mathcal {C}}_{\mathrm {v} }}$, respectively. Finally, the scaling function ${\displaystyle \phi }$ is associated with the low-frequency part ${\displaystyle {\mathcal {R}}}$.

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• 3D-Shearlets [7][9]
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