In this paper, we propose a new class of imaging denoising models by using the \(L^p\)-norm of mean curvature of image graphs as regularizers with \(p\in (1,2]\). The motivation of introducing such models is to add stronger regularizations than that of the original mean curvature based image denoising model (Zhu and Chan in SIAM J Imaging Sci 5(1):1–32, 2012) in order to remove noise more efficiently. To minimize these variational models, we develop a novel augmented Lagrangian method, and one thus just needs to solve two linear elliptic equations to find saddle points of the associated augmented Lagrangian functionals. Specifically, we linearize the nonlinear term in one of the two subproblems and minimize a proximal-like functional that can be easily treated. We prove that the minimizer of the substitute functional does reduce the value of the original subproblem under certain conditions. Numerical results are presented to illustrate the features of the proposed models and also the efficiency of the designed algorithm.

在本文中，我们通过使用图像图的平均曲率的\（L ^ p \）-范数作为\（p \ in（1,2] \）的正则化器，提出了一类新的成像降噪模型。引入此类模型的动机是要添加比原始的基于平均曲率的图像去噪模型（SIAM J Imaging Sci 5（1）：1-32，2012年的Zhu和Chan）更强的正则化，以便更有效地去除噪声。为了最小化这些变分模型，我们开发了一种新颖的增强拉格朗日方法，因此只需解决两个线性椭圆方程，即可找到相关的增强拉格朗日函数的鞍点。具体来说，我们将两个子问题之一中的非线性项线性化，并最小化可以轻松处理的类似近端的函数。我们证明了在某些条件下，替代函数的极小值确实会减小原始子问题的值。