# Fast Multi-grid Methods for Minimizing Curvature Energy

arXiv: https://arxiv.org/abs/2204.07921 · HTML: https://mlsanity.com/paper/2204.07921/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 64 references that are in the map.
- First posted on arXiv 17 Apr 2022.

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

- [Robustness via curvature regularization, and vice versa](https://mlsanity.com/paper/1811.09716/) (2018)
- [Low Curvature Activations Reduce Overfitting in Adversarial Training](https://mlsanity.com/paper/2102.07861/) (2021)
- [Weighted Mean Curvature](https://mlsanity.com/paper/1903.07189/) (2019)
- [Curvature Regularization to Prevent Distortion in Graph Embedding](https://mlsanity.com/paper/2011.14211/) (2020)
- [CURE: Curvature Regularization For Missing Data Recovery](https://mlsanity.com/paper/1901.09548/) (2019)
- [On Connections between Regularizations for Improving DNN Robustness](https://mlsanity.com/paper/2007.02209/) (2020)

## Built on by (most cited first, 3 of 3)

- [Total Normal Curvature Regularization and its Minimization for Surface and Image Smoothing](https://mlsanity.com/paper/2512.18968/) (2025)
- [Learning Dynamic Local Context Representations for Infrared Small Target Detection](https://mlsanity.com/paper/2412.17401/) (2024)
- [Effective Receptive Field Ordering Matters for Infrared Small Target Detection](https://mlsanity.com/paper/2607.23994/) (2026)

## Abstract (arXiv)

The geometric high-order regularization methods such as mean curvature and Gaussian curvature, have been intensively studied during the last decades due to their abilities in preserving geometric properties including image edges, corners, and contrast. However, the dilemma between restoration quality and computational efficiency is an essential roadblock for high-order methods. In this paper, we propose fast multi-grid algorithms for minimizing both mean curvature and Gaussian curvature energy functionals without sacrificing accuracy for efficiency. Unlike the existing approaches based on operator splitting and the Augmented Lagrangian method (ALM), no artificial parameters are introduced in our formulation, which guarantees the robustness of the proposed algorithm. Meanwhile, we adopt the domain decomposition method to promote parallel computing and use the fine-to-coarse structure to accelerate convergence. Numerical experiments are presented on image denoising, CT, and MRI reconstruction problems to demonstrate the superiority of our method in preserving geometric structures and fine details. The proposed method is also shown effective in dealing with large-scale image processing problems by recovering an image of size 1024× 1024 within 40s, while the ALM method requires around 200s.