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 $1024\times 1024$ within $40$s, while the ALM method requires around $200$s.

翻译：几何高阶正则化方法（如平均曲率和高斯曲率）由于其在保持图像边缘、角落和对比度等几何特性方面的能力，在过去几十年中得到了深入研究。然而，恢复质量与计算效率之间的困境是高阶方法的一个关键障碍。本文提出了快速多网格算法，用于最小化平均曲率和高斯曲率能量泛函，在不牺牲精度的情况下提高效率。与现有基于算子分裂和增广拉格朗日方法（ALM）的方法不同，我们的公式不引入任何人为参数，从而保证了所提算法的鲁棒性。同时，我们采用域分解方法促进并行计算，并利用从细到粗的结构加速收敛。通过图像去噪、CT和MRI重建问题的数值实验，展示了我们的方法在保持几何结构和细节方面的优越性。所提方法在处理大规模图像处理问题方面也表现出有效性，例如恢复一张大小为$1024\times 1024$的图像仅需不到$40$秒，而ALM方法则需要约$200$秒。