Total normal curvature regularization and its minimization for surface and image smoothing

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.