🤖 AI Summary
Conventional curvature-based regularization methods struggle to simultaneously preserve sharp edges and maintain isotropic fidelity in surface and image smoothing. Method: This paper proposes a total curvature regularization model incorporating multi-directional normal curvature penalization. We introduce a novel total normal curvature regularizer that adaptively weights principal curvature directions, enabling isotropic modeling for the first time. The resulting high-order nonconvex optimization problem is reformulated as a steady-state partial differential equation (PDE), solved efficiently via operator-splitting time discretization and closed-form subproblem solvers. Contribution/Results: Experiments demonstrate superior robustness across diverse noise types and geometric structures, markedly reduced parameter sensitivity, and significant improvements in both edge preservation accuracy and isotropic fidelity compared to classical curvature regularization approaches.
📝 Abstract
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.