In this paper, we propose a semi-sparsity smoothing method based on a new sparsity-induced minimization scheme. The model is derived from the observations that semi-sparsity prior knowledge is universally applicable in situations where sparsity is not fully admitted such as in the polynomial-smoothing surfaces. We illustrate that such priors can be identified into a generalized L0-norm minimization problem in higher-order gradient domains, giving rise to a new "feature-aware" filter with a powerful simultaneous-fitting ability in both sparse singularities (corners and salient edges) and polynomial-smoothing surfaces. Notice that a direct solver to the proposed model is not available due to the non-convexity and combinatorial nature of L0-norm minimization. Instead, we propose to solve it approximately based on an efficient half-quadratic splitting technique. We demonstrate its versatility and many benefits to a series of signal/image processing and computer vision applications.