The Euler Elastica (EE) model with surface curvature can generate artifact-free results compared with the traditional total variation regularization model in image processing. However, strong nonlinearity and singularity due to the curvature term in the EE model pose a great challenge for one to design fast and stable algorithms for the EE model. In this paper, we propose a new, fast, hybrid alternating minimization (HALM) algorithm for the EE model based on a bilinear decomposition of the gradient of the underlying image and prove the global convergence of the minimizing sequence generated by the algorithm under mild conditions. The HALM algorithm comprises three sub-minimization problems and each is either solved in the closed form or approximated by fast solvers making the new algorithm highly accurate and efficient. We also discuss the extension of the HALM strategy to deal with general curvature-based variational models, especially with a Lipschitz smooth functional of the curvature. A host of numerical experiments are conducted to show that the new algorithm produces good results with much-improved efficiency compared to other state-of-the-art algorithms for the EE model. As one of the benchmarks, we show that the average running time of the HALM algorithm is at most one-quarter of that of the fast operator-splitting-based Deng-Glowinski-Tai algorithm.

翻译：含曲面曲率的欧拉弹性（EE）模型在图像处理中相比传统全变分正则化模型可产生无伪影结果。然而，EE模型中曲率项带来的强非线性和奇异性给设计快速稳定的算法带来巨大挑战。本文基于底层图像梯度的双线性分解，提出一种新颖快速的混合交替最小化（HALM）算法，并在温和条件下证明了该算法生成的最小化序列的全局收敛性。HALM算法包含三个子最小化问题，每个问题要么通过闭式求解，要么采用快速求解器近似求解，使新算法兼具高精度和高效率。我们还讨论了将HALM策略推广至处理基于曲率的通用变分模型，特别是具有Lipschitz光滑曲率泛函的模型。大量数值实验表明，与现有先进的EE模型算法相比，新算法在显著提升效率的同时仍能获得良好结果。作为基准测试之一，我们证明HALM算法的平均运行时间不超过基于快速算子分裂的Deng-Glowinski-Tai算法的四分之一。