SIAM Journal on Imaging Sciences, Volume 14, Issue 1, Page 389-417, January 2021.
Euler's elastica model has been widely used in image processing. Since it is a challenging nonconvex and nonsmooth optimization model, most existing algorithms do not have convergence theory for it. In this paper, we propose a penalty relaxation algorithm with mathematical guarantee to find a stationary point of Euler's elastica model. To deal with the nonsmoothness of Euler's elastica model, we first introduce a smoothing relaxation problem, and then propose an exact penalty method to solve it. We establish the relationships between Euler's elastica model, the smoothing relaxation problem, and the penalty problem in theory regarding optimal solutions and stationary points. Moreover, we propose an efficient block coordinate descent algorithm to solve the penalty problem by taking advantage of convexity of its subproblems. We prove global convergence of the algorithm to a stationary point of the penalty problem. Finally we apply the proposed algorithm to denoise the optical coherence tomography images with real data from an optometry clinic and show the efficiency of the method for image processing using Euler's elastica model.

中文翻译：

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欧拉弹性模型的惩罚松弛方法

SIAM影像科学杂志，第14卷，第1期，第389-417页，2021年1月。
欧拉的Elastica模型已广泛应用于图像处理中。由于它是一个具有挑战性的非凸且非平滑的优化模型，因此大多数现有算法都没有针对其的收敛理论。在本文中，我们提出了一种具有数学保证的惩罚松弛算法，以找到欧拉弹性模型的固定点。为了处理Euler弹性模型的不光滑性，我们首先引入了平滑松弛问题，然后提出了一种精确的罚分方法来解决该问题。我们在理论上建立了关于最优解和平稳点的欧拉弹性模型，平滑松弛问题和惩罚问题之间的关系。此外，我们提出了一种有效的块坐标下降算法，以利用子问题的凸性来解决惩罚问题。我们证明了该算法的全局收敛性到惩罚问题的一个稳定点。最后，我们将所提出的算法用于对来自光学验光诊所的真实数据进行光学相干断层扫描图像进行降噪，并证明了使用欧拉弹性模型进行图像处理的方法的效率。