脉冲图像去噪的BregmanDC算法

The BregmanDC Algorithm for Impulse Image Denoising

摘要: 针对脉冲图像去噪的非凸模型求解困难的问题，本文提出了一种基于Bregman距离的DC算法，简称BregmanDC算法。非凸模型虽能更准确地刻画图像先验，但在求解时面临计算复杂度高、收敛缓慢且极易陷入局部极小值等挑战。为此，本文在Difference of Convex (DC)规划框架下巧妙引入Bregman距离，有效改善了子问题的适定性，将其转化为更易处理的凸优化问题，同时指出，经典的DCA算法仅为本文提出的BregmanDC算法的一个特例。进一步地，针对求解过程中的非光滑子问题，本文利用凸共轭的性质，设计了基于对偶的增广拉格朗日算法以实现快速且稳定的求解。数值实验验证了算法的有效性，结果表明在不同强度的椒盐噪声影响下，BregmanDC算法的表现优于基准算法。

Abstract: To address the difficulty of solving non-convex models for impulse image denoising, this paper proposes a Difference-of-Convex (DC) algorithm based on the Bregman distance, abbreviated as the BregmanDC algorithm. Although non-convex models can more accurately characterize image priors, optimizing them presents significant challenges, including high computational complexity, slow convergence, and a high susceptibility to getting trapped in local minima. To overcome these, we ingeniously introduce the Bregman distance into the DC programming framework. This effectively improves the well-posedness of the subproblems, transforming them into more tractable convex optimization problems. Furthermore, we demonstrate that the classical DCA is merely a special case of our proposed BregmanDC algorithm. Additionally, to address the non-smooth subproblems encountered during the solution process, we leverage the properties of convex conjugation to design a dual-based augmented Lagrangian algorithm, achieving fast and stable resolutions. Numerical experiments validate the effectiveness of the proposed algorithm. The results demonstrate that, under varying intensities of salt-and-pepper noise, the BregmanDC algorithm consistently outperforms the baseline algorithms.