# An Operator-Splitting Method for the Gaussian Curvature Regularization Model with Applications to Surface Smoothing and Imaging

arXiv: https://arxiv.org/abs/2108.01914 · HTML: https://mlsanity.com/paper/2108.01914/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 52 references that are in the map.
- First posted on arXiv 4 Aug 2021.

## Builds on (most cited first, 7 of 7)

- [Plug-and-Play ADMM for Image Restoration: Fixed Point Convergence and Applications](https://mlsanity.com/paper/1605.01710/) (2016)
- [Curvature Regularized Surface Reconstruction from Point Cloud](https://mlsanity.com/paper/2001.07884/) (2020)
- [A Color Elastica Model for Vector-Valued Image Regularization](https://mlsanity.com/paper/2008.08255/) (2020)
- [A Discrete Scheme for Computing Image's Weighted Gaussian Curvature](https://mlsanity.com/paper/2101.07927/) (2021)
- [First order algorithms in variational image processing](https://mlsanity.com/paper/1412.4237/) (2014)
- [A novel variational model for image registration using Gaussian curvature](https://mlsanity.com/paper/1504.07643/) (2015)
- [Fast Algorithms for Surface Reconstruction from Point Cloud](https://mlsanity.com/paper/1907.01142/) (2019)

## Built on by (most cited first, 11 of 11)

- [PottsMGNet: A Mathematical Explanation of Encoder-Decoder Based Neural Networks](https://mlsanity.com/paper/2307.09039/) (2023)
- [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)
- [Euler's Elastica Based Cartoon-Smooth-Texture Image Decomposition](https://mlsanity.com/paper/2407.02794/) (2024)
- [A Mathematical Explanation of UNet](https://mlsanity.com/paper/2410.04434/) (2024)
- [Connections between Operator-splitting Methods and Deep Neural Networks with Applications in Image Segmentation](https://mlsanity.com/paper/2307.09052/) (2023)
- [A Fast Minimization Algorithm for the Euler Elastica Model Based on a Bilinear Decomposition](https://mlsanity.com/paper/2308.13471/) (2023)
- [Total Normal Curvature Regularization and its Minimization for Surface and Image Smoothing](https://mlsanity.com/paper/2512.18968/) (2025)
- [A PCA Based Model for Surface Reconstruction from Incomplete Point Clouds](https://mlsanity.com/paper/2509.15675/) (2025)
- [Bayesian methods and Markov chain Monte Carlo algorithms for curve reconstruction and point cloud data analysis](https://mlsanity.com/paper/2608.26490/) (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)

Gaussian curvature is an important geometric property of surfaces, which has been used broadly in mathematical modeling. Due to the full nonlinearity of the Gaussian curvature, efficient numerical methods for models based on it are uncommon in literature. In this article, we propose an operator-splitting method for a general Gaussian curvature model. In our method, we decouple the full nonlinearity of Gaussian curvature from differential operators by introducing two matrix- and vector-valued functions. The optimization problem is then converted into the search for the steady state solution of a time dependent PDE system. The above PDE system is well-suited to time discretization by operator splitting, the sub-problems encountered at each fractional step having either a closed form solution or being solvable by efficient algorithms. The proposed method is not sensitive to the choice of parameters, its efficiency and performances being demonstrated via systematic experiments on surface smoothing and image denoising.