Parameterized $L_{0}$ Image Smoothing With Unsupervised Learning

The traditional $L_{0}$ filter shows exquisite smoothing quality, but it suffers from high computational cost. In this paper, we propose an efficient solution to the $L_{0}$-regularized optimization problem based on deep unsupervised learning. The $L_{0}$-norm involves a discrete counting scheme, which can not be directly optimized with gradient descent. Therefore, in this paper, we propose to decompose the problem into a series of optimization problems based on a truncated $L_{1}$-norm with varying parameters. Compared with the truncated $L_{2}$-norm explored in traditional $L_{0}$ filter, the truncated $L_{1}$-norm promotes the capabilities in structure- and edge-preserving smoothing, reduces the number of iterations, and facilitates the deep learning-based optimization. Furthermore, we propose a deep learning-based parameterized approach to solve the truncated $L_{1}$-regularized problems so that we only need to train a single fully convolutional network to support varying smoothing parameters. We are not trying to reproduce the traditional $L_{0}$ filter in this paper. Instead, we show that the proposed deep $L_{0}$ filter provides a better smoothing quality. Experimental results indicate that the proposed filter outperforms the state-of-the-art on various applications, including edge-preserving smoothing, non-photorealistic rendering, texture removal, edge extraction, image composition, and compression artifact removal. Moreover, our filter is efficient, it is able to process 720P color images at interactive rates on a modern GPU.