We present a novel approach to denoising and inpainting problems for surface meshes. The purpose of these problems is to remove noise or fill in missing parts while preserving important features such as sharp edges. A discrete variant of the total variation of the unit normal vector field serves as a regularizing functional to achieve these goals. In order to solve the resulting problem, we use a version of the split Bregman (ADMM) iteration adapted to the problem. A new formulation of the total variation regularizer, as well as the use of an inexact Newton method for the shape optimization step, bring significant speed-up compared to earlier methods. Numerical examples are included, demonstrating the performance of our algorithm with some complex 3D geometries.

翻译：针对曲面网格的去噪与修复问题，我们提出了一种新颖方法。这类问题的目标是在去除噪声或填补缺失区域的同时，保留尖锐边缘等重要特征。为实现这些目标，我们采用单位法向量场的离散化全变分作为正则化泛函。为求解该问题，我们应用了针对该问题改进的分裂布列格曼（ADMM）迭代算法。通过提出全变分正则化器的新形式，以及在形状优化步骤中引入非精确牛顿法，相较于早期方法实现了显著的加速。数值实验表明，该算法在若干复杂三维几何体上具有良好的性能表现。