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Image smoothing is a fundamental procedure in applications of both computer vision and graphics. The required smoothing properties can be different or even contradictive among different tasks. Nevertheless, the inherent smoothing nature of one smoothing operator is usually fixed and thus cannot meet the various requirements of different applications. In this paper, we first introduce the truncated Huber penalty function which shows strong flexibility under different parameter settings. A generalized framework is then proposed with the introduced truncated Huber penalty function. When combined with its strong flexibility, our framework is able to achieve diverse smoothing natures where contradictive smoothing behaviors can even be achieved. It can also yield the smoothing behavior that can seldom be achieved by previous methods, and superior performance is thus achieved in challenging cases. These together enable our framework capable of a range of applications and able to outperform the state-of-the-art approaches in several tasks, such as image detail enhancement, clip-art compression artifacts removal, guided depth map restoration, image texture removal, etc. In addition, an efficient numerical solution is provided and its convergence is theoretically guaranteed even the optimization framework is non-convex and non-smooth. A simple yet effective approach is further proposed to reduce the computational cost of our method while maintaining its performance. The effectiveness and superior performance of our approach are validated through comprehensive experiments in a range of applications. Our code is available at https://github.com/wliusjtu/Generalized-Smoothing-Framework . [ABSTRACT FROM AUTHOR]