Edge-preserving image smoothing is a fundamental tool in computational photography and graphics. It aims to suppress insignificant details while maintaining salient structures. The classical L₀ filter provides an elegant framework for a variety of applications. However, it inclines to sharpen the salient edges, thus suffering from gradient reversals and color deviations. We propose a solution toward the objective of optimizing summed squared error regularized by the L₀-norm of the gradients. The proposed solution explores an iterative strategy, where each iteration is an optimization problem based on the truncated L₁ regularization, which can be solved efficiently with the combination of alternating direction method of multipliers and Fourier domain optimization. Leveraging L₁-norm limits the aggressive modifications of gradients during the iterative process, which alleviates various artifacts in classical L₀ filter. Experiment results indicate that the proposed method achieves superior performance in various applications, including texture removal, detail enhancement, HDR tone mapping, and compression artifact removal. The proposed filter can be conveniently implemented on GPU. It takes our filter 0.51 s to process a 720P color image on a NVIDIA GTX 1080 GPU.