A new augmented Lagrangian primal dual algorithm for elastica regularization
A new augmented Lagrangian primal dual algorithm for elastica regularization
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DOI：
10.1177/1748301816668044
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发表时间：
2016-09
影响因子：
0.9
通讯作者：
Jianping Zhang;Ke-long Chen
中科院分区：
文献类型：
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作者：
Jianping Zhang;Ke-long Chen
正则化是图像处理中变分模型的关键要素。为了克服基于全变分模型的弱点，最近提出并研究了各种高阶（通常是二阶）正则化模型。其中，欧拉基于弹性能量的正则化器在数学和物理论证方面可能是最有趣的。更重要的是它的成功已经在应用中得到了证明；然而，开发快速有效的算法一直是一个重大挑战。在本文中，我们提出了一种基于勒让德-芬切尔变换的原始对偶算法的新想法，用于表示弹性正则化器。结合增强拉格朗日公式，我们能够导出等效的无约束优化，与之前基于分裂方法的工作相比，需要使用的变量更少。我们将介绍图像恢复问题和图像分割模型的算法。这个想法适用于需要弹性正则化器的其他模型。数值实验表明，该方法能够以更高的效率产生极具竞争力的结果。
Regularization is a key element of variational models in image processing. To overcome the weakness of models based on total variation, various high order (typically second order) regularization models have been proposed and studied recently. Among these, Euler's elastica energy based regularizer is perhaps the most interesting in terms of both mathematical and physical justifications. More importantly its success has been proven in applications; however it has been a major challenge to develop fast and effective algorithms. In this paper we propose a new idea for deriving a primal dual algorithm, based on Legendre–Fenchel transformations, for representing the elastica regularizer. Combined with an augmented Lagrangian for-mulation, we are able to derive an equivalent unconstrained optimization that has fewer variables to work with than previous works based on splitting methods. We shall present our algorithms for both the image restoration problem and the image segmentation model. The idea applies to other models where the elastica regularizer is required. Numerical experiments show that the proposed method can produce highly competitive results with better efficiency.