# A Shearlets-Based Method for Rain Removal from Single Images
## 1 Introduction
In fact, these methods described above do remove the rain streaks from rainy images, but the recovery of background seems to be over-smoothed, that is, some details of the background layer are lost. The main reason for details missing in general methods is that the directional property of two layers is out of consideration, which is critical in edges detection. Based on this fact, the author in [21] proposed a directional global sparse model for single image rain removal, in this work, undirectional total removal (UTV) [22] is introduced to describe the basic directional property of single rainy images. The proposed optimization model is based on two directional sparse priors of rain streaks and background layer respectively and a generally sparse prior on rain streaks, that is, rain streaks are sparse in the vertical direction, while the rain-free image (background layer) is sparse in the horizontal direction; furthermore, the rain streak is approximately considered as sparse when the rain is not heavy. The directional characters are captured by rotating rainy images appropriately, which ensures that rain streaks are mainly concentrated in the vertical direction. In addition, the deep learning-based method gives another way for single image and video rain removal [23; 24; 25; 26], which is totally different from the traditional image processing. The key for this approach is the training network. By learning from lots of rainy images, the learning model can split rain streaks from rainy images after thousands of training images, which is time-consuming, but once the training model gets good generalization, it can deal with lots of rainy images simultaneously, with an accuracy of at least 80 percent. With the development of deep learning, the accuracy for splitting rain streaks can be improved further, and this approach will be widely used for other image processing and computer vision tasks. In this work, we consider removing rain streaks from single rainy images. This is different from the method in [21], which removes directional rain streaks by rotating the rainy image to the direction that rain streaks are approximately distributed in the vertical direction, then UTV is utilized to capture the'vertical' characteristics of rain streaks. This is a two-step method when the rain drops are in a non-vertical direction. In fact, directional multilevel transform, e.g., Shearlets transform, can be used to substitute the two-step transform above due to its multi-direction property, also the multilevel system is suitable for detecting rain streaks as the shape of rain streaks is not necessarily rectangular. In addition, we found that the Shearlets decomposition coefficients of rainy images in specific scale and directional frequency bands are sparse. Some statistical analysis (see Figure 1) suggests that the Shearlets coefficients of rainy images in the rain drops' direction has sparse structure, also the Shearlets coefficients of rain-free (background) layer in the rain streaks direction is sparse. Moreover, another sparse prior is the rain streaks itself, which can be approximately considered as sparse when the rain is not heavy (partly sparse when the rain is heavy). Combining these three sparse priors above, we propose a convex optimization model and the split Bregman algorithm [27] is utilized to solve the proposed model. Numerical experiments on synthetic and real rainy images demonstrate that our Shearlets-based method outperforms the recently widely used rain removal methods in [11; 13; 21]. The outline of this paper is given as follows: Section 2 gives a brief introduction of the motivation. Section 3 reviews the Shearles transform. In Section 4, a Shearlets-based rain streaks removal model, and the corresponding algorithm, are proposed. Section 5 compares our model with three state-of-art methods, some discussions of rain removal results are given in Section 6. In the end, the conclusion is given in Section 7.