In this work, the wavelet based sparse optimization approach is introduced to process rain removal problem from a single image by considering the direction and shape of rain streaks. Firstly, a new kind of wavelet, the uni-directional multilevel system is construct to describe the singularities of noisy in particular direction and scales, like rain streaks and stripes in radar images. Compared with total variation, the uni-directional multilevel transform of rainy images gives the sparse representation of singularities in different scales and frequency bands due to its multiscale structure, which includes more rain details. Secondly, a convex optimization rain removal model is proposed by considering the intrinsic directional and structure information of the rain streak and the background image. The model involves three sparse priors, including the sparse regularizer on rain streaks and two sparse regularizers on the uni-directional multilevel transform of background layer in the rain drop’s direction and the multilevel transform of rain streaks across the rain direction. The split Bregman algorithm is utilized to solve the proposed convex optimization model which ensures the global optimal solution. Thirdly, comparison tests with four stat-of-the-art methods are implemented on synthetic and real rainy images, which suggests that the proposed method is efficient both in rain removal and details preservation of the background layer.

在这项工作中，引入了基于小波的稀疏优化方法，通过考虑雨条纹的方向和形状来处理单个图像中的雨去除问题。首先，构造了一种新型的小波单向多级系统，用于描述特定方向和尺度上的噪声奇异性，例如雷达图像中的雨条纹和条纹。与总变化量相比，多雨图像的单向多级变换由于其多尺度结构而包含了更多的雨细节，因此可以稀疏表示不同尺度和频带的奇异点。其次，考虑雨条和背景图像的内在方向和结构信息，提出了凸优化除雨模型。该模型涉及三个稀疏先验，包括在雨条纹上的稀疏正则器和在雨滴方向上背景层的单向多级变换和在雨方向上的雨条纹的多级变换的两个稀疏正则器。利用分裂的Bregman算法来求解所提出的凸优化模型，从而保证了全局最优解。第三，在合成和真实的雨天图像上进行了四种最新技术的比较测试，这表明所提出的方法在去除雨水和保留背景层的细节方面都是有效的。利用分裂的Bregman算法来求解所提出的凸优化模型，从而保证了全局最优解。第三，在合成和真实的雨天图像上进行了四种最新技术的比较测试，这表明所提出的方法在去除雨水和保留背景层的细节方面都是有效的。利用分裂的Bregman算法来求解所提出的凸优化模型，从而保证了全局最优解。第三，在合成和真实的雨天图像上进行了四种最新技术的比较测试，这表明所提出的方法在去除雨水和保留背景层的细节方面都是有效的。