# A Shearlets-Based Method for Rain Removal from Single Images
## 3 Shearlets  
Shearlets system is first given in [31] by [PERSON] et al, which not only inherits the multiscale structure of wavelets, but also possesses the directional windows in the frequency domain by introducing the shear operator, thus making it possible for the Shearlets transform to capture the directional singularities in images and high dimensional signals. This section gives a retrospection of the basic definition and construction of Shearlets, for more details refer to [28; 32].  
For the basic wavelet function \(\phi\in L^{2}(R^{2})\), it is supposed to satisfy  
\[C_{\phi}=\int_{R^{2}}\frac{|\hat{\phi}(\omega)|^{2}}{|\omega|}d\omega<\infty, \tag{2}\]  
where \(\hat{\phi}\) represents the Fourier transform of \(\phi\), and it has the following separated form:  
\[\hat{\phi}(\omega_{1},\omega_{2})=\hat{\phi}_{1}(\omega_{1})\hat{\phi}_{2}( \frac{\omega_{2}}{\omega_{1}}), \tag{3}\]  
with \(\phi_{1}\in L^{2}(R),\phi_{2}\in L^{2}(R^{2})\). Then the Shearlets multilevel system \(S\) is generated by  
\[S(\phi)=\{\phi_{j,s,t}=T_{t}D_{2}^{j}S_{s}\phi:j\in Z,s\in R,t\in R^{2}\}, \tag{4}\]  
where \(T_{t}\) is the translation operator, \(T_{t}\phi=\phi(\cdot-t)\). \(D_{s}\) is the scaling operator and \(S_{s}\) is the shear operator, their definitions are given as  
\[D_{2}^{j}=\left(\begin{array}{cc}2^{j}&0\\ 0&2^{j/2}\end{array}\right),\quad S_{s}=\left(\begin{array}{cc}1&s\\ 0&1\end{array}\right). \tag{5}\]  
where \(j\in Z\) is the scaling parameter, which decides the decomposition layer. \(s\in R\) is the shearing parameter, which decides the multi-direction property. \(t\) is the translation parameter, which decides the number of sub-bands in each scale. For the signal \(f\in L^{2}(R^{2})\), its continuous Shearlets transform  
\[f\to S(\phi)f(j,s,t)=<f,\phi_{j,s,t}>,\quad j\in Z,s\in R,t\in R^{2}, \tag{6}\]  
where \(S\) maps \(f\) into the Shearlets coefficients \(S(\phi)f(j,s,t)\) with different scale \(j\), shearing \(s\), and translation \(t\). Then the discrete Shearlets system can be given as  
**Definition 1** ([28]).: _Let \(\psi,\phi\in L^{2}(R^{2})\), \(c=(c_{1},c_{2})\in R^{2}\), the definition of the discrete Shearlets system \(S(\psi,\phi;c)\) is_  
\[S(\psi,\phi;c)=\Psi(\psi;c)\cup\Phi(\phi;c)\cup\bar{\Phi}(\bar{\phi};c), \tag{7}\]_where_  
\[\Psi(\psi;c) =\{\psi_{j,m}=T_{t}D_{2}^{j}\psi=2^{3/4}\psi(D_{2}^{j}\cdot-cm):m\in Z ^{2}\}, \tag{8}\] \[\Phi(\phi;c) =\{\phi_{j,k,m}=2^{3/4}\phi(S_{k}D_{2}^{j}\cdot-cm):j\geq 0,\,|k| \leq 2^{j/2},m\in Z^{2}\},\] \[\tilde{\Phi}(\tilde{\phi};c) =\{\tilde{\phi}_{j,k,m}=2^{3/4}\tilde{\Phi}(S_{k}D_{2}^{j}\cdot-cm ):j\geq 0,\,|k|\leq 2^{j/2},m\in Z^{2}\}.\]  
_with \(\tilde{\psi}=\hat{\phi}_{1}(\omega_{1})\hat{\phi}_{2}(\omega_{2})\), and \(\tilde{\phi}(\xi_{1},\xi_{2})=\phi(\xi_{2},\xi_{1})\)._  
The multilevel system \(\Psi(\psi;c)\) corresponds to the low frequency area in the frequency domain, while the system \(\Phi(\phi;c)\) and \(\tilde{\Phi}(\tilde{\phi};c)\) corresponds to the high frequency region in vertical and horizontal direction respectively. The sub-band of \(S(\psi,\phi;c)\) in the frequency domain is shown in Figure 3.