  In this paper, we propose an interesting semi-sparsity smoothing algorithm
based on a novel sparsity-inducing optimization framework. This method is
derived from the multiple observations that semi-sparsity prior knowledge is
more universally applicable, especially in areas where sparsity is not fully
admitted, such as polynomial-smoothing surfaces. We illustrate that this
semi-sparsity can be identified into a generalized $L_0$-norm minimization in
higher-order gradient domains, thereby giving rise to a new "feature-aware"
filtering method with a powerful simultaneous-fitting ability in both sparse
features (singularities and sharpening edges) and non-sparse regions
(polynomial-smoothing surfaces). Notice that a direct solver is always
unavailable due to the non-convexity and combinatorial nature of $L_0$-norm
minimization. Instead, we solve the model based on an efficient half-quadratic
splitting minimization with fast Fourier transforms (FFTs) for acceleration. We
finally demonstrate its versatility and many benefits to a series of
signal/image processing and computer vision applications.
