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 image 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 the 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 the convergence. Numerical experiments are presented on both image denoising and CT reconstruction problem to demonstrate the ability to recover image texture and the efficiency of the proposed method.
翻译:在过去几十年中,由于具有保存几何特性的能力,包括图像边缘、角和图像对比,对诸如平均曲线和高斯曲线等几何高阶正规化方法进行了深入研究。然而,恢复质量和计算效率之间的两难是高阶方法的基本障碍。在本文中,我们提出了快速多格算法,以尽量减少平均曲线和高斯曲线的能量功能,同时又不牺牲效率的准确性。与基于操作员分裂的现有方法和Augmented Lagrangian方法(ALM)不同的是,在我们的配方中没有人为参数,这保证了拟议算法的稳健性。与此同时,我们采用了域分解法,以促进平行计算,并使用微缩至细结构加速汇合。在图像分解和CT重建问题上都进行了数字实验,以显示恢复图像纹度的能力和拟议方法的效率。