# Total Normal Curvature Regularization and its Minimization for Surface and Image Smoothing

arXiv: https://arxiv.org/abs/2512.18968 · HTML: https://mlsanity.com/paper/2512.18968/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 51 references that are in the map.
- First posted on arXiv 22 Dec 2025.

## Builds on (most cited first, 6 of 6)

- [A Color Elastica Model for Vector-Valued Image Regularization](https://mlsanity.com/paper/2008.08255/) (2020)
- [An Operator-Splitting Method for the Gaussian Curvature Regularization Model with Applications to Surface Smoothing and Imaging](https://mlsanity.com/paper/2108.01914/) (2021)
- [Elastica Models for Color Image Regularization](https://mlsanity.com/paper/2203.09995/) (2022)
- [CurvPnP: Plug-and-play Blind Image Restoration with Deep Curvature Denoiser](https://mlsanity.com/paper/2211.07286/) (2022)
- [Fast Multi-grid Methods for Minimizing Curvature Energy](https://mlsanity.com/paper/2204.07921/) (2022)
- [A Fast Minimization Algorithm for the Euler Elastica Model Based on a Bilinear Decomposition](https://mlsanity.com/paper/2308.13471/) (2023)

## Built on by (most cited first, 2 of 2)

- [A Globally Convergent Algorithm for Total Scaled-Gradient Variation via Cone-Constrained Bilinear Decomposition](https://mlsanity.com/paper/2609.00036/) (2026)
- [Gauss-Map Variation for Image Denoising: Geometric Analysis and an Anderson--Accelerated Majorization--Minimization Method](https://mlsanity.com/paper/2610.05801/) (2026)

## Abstract (arXiv)

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.