# A Shearlets-Based Method for Rain Removal from Single Images
## 2 Motivations of the Proposed Method  
In general, the rain streak removal model can be represented as follows:  
\[o=r+b; \tag{1}\]  
where \(o\in R^{M\times N}\) is the rainy image, \(r\in R^{M\times N}\) represents rain streaks layer, and \(b\in R^{M\times N}\) is the background layer (rain-free layer).  
Figure 1: The sparse structure for rain streaks and the Shearlets coefficients of rain streaks and the background layer in and across the rain directions.  
Motivated by the work in [14], which decomposed rainy images into high- and low-frequency, then the high-frequency parts can be represented by the addition of rain streaks layer and rain-free layer by performing dictionary learning and sparse coding. This method does remove rain streaks from the rainy image, but the rain-free layer estimated tends to be blurry, which implies that the high-frequency decomposition leads to the loss of details in high frequency not only from rain streaks, but also from the details of the background layer. This makes us consider splitting rain streaks in specific frequency bands, so that more details can be preserved. Therefore, the directional multilevel transform, such as Shearlets transform, is a good choice as it can be used to separate the frequency bands of rain streaks only.  
In fact, the Shearlets transform is a powerful tool in detecting directional singularities in frequency band [28; 29; 30]. Different from UTV, which is used to enforce the sparse prior of the gradient along vertical and horizontal directions, the Shearlets can be utilized to describe the variation along different directions in various scales (see Figure 2). Since the rain streaks can be viewed as a particular directional singularity in different scales, which makes it possible to remove from the rainy image without additional details missing. However, loss of the details from the rain-free layer in corresponding frequency bands cannot be avoided. In order to minimize this kind of details loss, the multi-direction Shearlets is used to preserve most original image details.  
Compared with the recent state-of-the-art directional UTV rotation-based rain removal model in [21], the proposed model has the following advantages:  
* The directional multilevel transform, Shearlets transform, is utilized to describe the sparse structure of rain streaks and the background layer. Different from the rotation UTV method,  
Figure 2: The visual results of Shearlets decomposition coefficients in different scales (high frequency) and different directions are displayed (the testing image is “Sydney Opera House”, can be downloaded in [[http://homepages.lboro.ac.uk/cgs/datasets/ucid/ucid.html](http://homepages.lboro.ac.uk/cgs/datasets/ucid/ucid.html)]([http://homepages.lboro.ac.uk/cgs/datasets/ucid/ucid.html](http://homepages.lboro.ac.uk/cgs/datasets/ucid/ucid.html))).  
the Shearlets-based method can obtain the gradient variation in different directions and different scales efficiently, thus the recovery keeps more directional singularities details.
* For different directions of rain streaks, the Shearlets transform can capture the rain streak layer details due to its multi-direction property. Moreover, the fast algorithm for solving the proposed model can be obtained as the discrete Shearlets transform and inverse are available.
* The split Bregman algorithm is utilized to solve the proposed convex optimization model, which guarantees that the solver is global optimal. The computation of algorithm includes three soft-thresholding processes and two Shearlets transforms, and the total computing complexity is \(O(N\log N)+O(N)\), where \(N\) is the total number of pixels.