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.

众所周知，曲率规则性在边缘连续性方面具有很强的先验性，已被广泛应用于图像处理和计算机视觉中。但是，这些模型通常是非凸，非平滑和高度非线性的，其一阶最优性条件是高阶偏微分方程。因此，数值计算极具挑战性。在本文中，我们估计局部\（3 \ times 3 \）上的离散平均曲率和高斯曲率模具，基于微分几何的基本形式。通过最小化图像表面上的某些曲率函数，它产生了一种加权的图像表面最小化问题，可以通过乘数的交替方向方法有效地解决该问题。进行了图像恢复和修复的数值实验，以证明与现有的变分方法相比，所提出的基于曲率的模型的有效性和优越性。