Image Reconstruction by Minimizing Curvatures on Image Surface

Abstract

The curvature regularities are well-known for providing strong priors in the continuity of edges, which have been applied to a wide range of applications in image processing and computer vision. However, these models are usually non-convex, non-smooth, and highly nonlinear, the first-order optimality condition of which are high-order partial differential equations. Thus, numerical computation is extremely challenging. In this paper, we estimate the discrete mean curvature and Gaussian curvature on the local $$3\times 3$$ stencil, based on the fundamental forms in differential geometry. By minimizing certain functions of curvatures over the image surface, it yields a kind of weighted image surface minimization problem, which can be efficiently solved by the alternating direction method of multipliers. Numerical experiments on image restoration and inpainting are implemented to demonstrate the effectiveness and superiority of the proposed curvature-based model compared to state-of-the-art variational approches.