In view of the exceptional ability of curvature in connecting missing edges and structures, we propose novel sparse reconstruction models via the Euler’s elastica energy. In particular, we firstly extend the Euler’s elastica regularity into the nonlocal formulation to fully take the advantages of the pattern redundancy and structural similarity in image data. Due to its non-convexity, non-smoothness and nonlinearity, we regard both local and nonlocal elastica functional as the weighted total variation for a good trade-off between the runtime complexity and performance. The splitting techniques and alternating direction method of multipliers (ADMM) are used to achieve efficient algorithms, the convergence of which is also discussed under certain assumptions. The weighting function occurred in our model can be well estimated according to the local approach. Numerical experiments demonstrate that our nonlocal elastica model achieves the state-of-the-art reconstruction results for different sampling patterns and sampling ratios, especially when the sampling rate is extremely low.

中文翻译：

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稀疏重建的非局部Elastica模型

鉴于在连接缺失的边缘和结构方面具有出色的曲率能力，我们通过欧拉弹性能提出了新颖的稀疏重建模型。特别是，我们首先将欧拉弹性规则性扩展到非局部公式中，以充分利用图像数据中图案冗余和结构相似性的优势。由于其非凸性，非平滑性和非线性性，我们将局部和非局部弹性函数视为加权总变化，以便在运行时复杂性和性能之间取得良好的平衡。使用乘法器的拆分技术和交替方向方法（ADMM）来实现有效的算法，并且在某些假设下还讨论了算法的收敛性。我们模型中发生的加权函数可以根据局部方法很好地估计。数值实验表明，我们的非局部弹性模型可以在不同的采样模式和采样率下获得最新的重建结果，尤其是在采样率极低的情况下。