We introduce a novel formulation for curvature regularization by penalizing normal curvatures from multiple directions. This total normal curvature regularization is capable of producing solutions with sharp edges and precise isotropic properties. To tackle the resulting high-order nonlinear optimization problem, we reformulate it as the task of finding the steady-state solution of a time-dependent partial differential equation (PDE) system. Time discretization is achieved through operator splitting, where each subproblem at the fractional steps either has a closed-form solution or can be efficiently solved using advanced algorithms. Our method circumvents the need for complex parameter tuning and demonstrates robustness to parameter choices. The efficiency and effectiveness of our approach have been rigorously validated in the context of surface and image smoothing problems.

翻译：本文提出了一种新颖的曲率正则化方法，通过从多个方向惩罚法曲率来实现。这种总法曲率正则化能够产生具有锐利边缘和精确各向同性特性的解。为解决由此产生的高阶非线性优化问题，我们将其重新表述为寻找时间依赖偏微分方程系统稳态解的任务。时间离散化通过算子分裂实现，其中每个分数步子问题要么具有闭式解，要么可以利用先进算法高效求解。我们的方法避免了复杂参数调整的需求，并展现出对参数选择的鲁棒性。该方法的效率与有效性已在曲面与图像平滑问题中得到严格验证。