The curvature regularities have received growing attention with the advantage of providing strong priors in the continuity of edges in image processing applications. However, owing to the non-convex and non-smooth properties of the high-order regularizer, the numerical solution becomes challenging in real-time tasks. In this paper, we propose a novel curvature regularity, the total curvature (TC), by minimizing the normal curvatures along different directions. We estimate the normal curvatures discretely in the local neighborhood according to differential geometry theory. The resulting curvature regularity can be regarded as a re-weighted total variation (TV) minimization problem, which can be efficiently solved by the alternating direction method of multipliers (ADMM) based algorithm. By comparing with TV and Euler's elastica energy, we demonstrate the effectiveness and superiority of the total curvature regularity for various image processing applications.

The Mumford-Shah model is an important tool for image labeling and segmentation, which pursues a piecewise smooth approximation of the original image and the boundaries with the shortest length. In contrast to previous efforts, which use the total variation regularization to measure the total length of the boundaries, we build up a novel piecewise smooth Mumford-Shah model by utilizing a non-convex ℓpregularity term for p ∈ (0,1), which can well preserve sharp edges and eliminate geometric staircasing effects. We present optimization algorithms with convergence verification, where all subproblems can be solved by either the closed-form solution or fast Fourier transform (FFT). The method is compared to piecewise constant labeling algorithm and several state-of-the-art piecewise smooth Mumford-Shah models based on image decomposition approximations. Both labeling and segmentation results on synthetic and real images confirm the robustness and efficiency of the proposed method.

Exploiting sparsity in the image gradient magnitude has proved effective in preserving sharp edges and reducing noises for many image processing tasks. Based on observation, we build up a novel piecewise smooth segmentation model by utilizing a generalized total variation (TV) prior with p-th power for and a l1data fidelity. We present an efficient algorithm based on the alternating direction method of multipliers (ADMM), where all subproblems can be solved by either one-step Gauss-Seidel iteration or the closed-form solution. Numerical experiments show that the proposed model can achieve more accurate segmentation results than the classical TV based segmentation model.