Geometric Separation in $$\mathbb {R}^3$$ R 3
Geometric Separation in $$\mathbb {R}^3$$ R 3
复制标题
$$mathbb {R}^3$$ R 3 中的几何分离
DOI：
10.1007/s00041-017-9569-z
复制
发表时间：
2018
影响因子：
1.2
通讯作者：
Labate, Demetrio
中科院分区：
文献类型：
--
作者：
Guo, Kanghui;Labate, Demetrio
几何分离问题最初由Donoho和Kutyniok提出(Commun Pure Appl Math 66：1-47,2013)，其目的是将包含点和曲线奇点的非平凡叠加的分布分离成其不同的几何组成部分。Donoho和Kutyniok(2013)提出的解决方案考虑了关于组合小波-曲线字典的展开，并对展开系数应用a-范数最小化，以实现在精细尺度上的渐近分离。然而，这一结果的原始证明使用了依赖于傅里叶积分算子的稀疏表示的重型机械，这并不直接扩展到3D设置。在本文中，我们使用一个新的和更简单的论点将几何分离结果推广到3D环境中，该论点部分依赖于作者开发的基于Shearlet的曲线边缘分析技术。我们的新结果还给出了原始2D几何分离问题的一个简单得多的证明，并推广了作者先前的结果，该结果仅限于分段线性奇点。
The geometric separation problem, initially posed by Donoho and Kutyniok (Commun Pure Appl Math 66:1–47, 2013), aims to separate a distribution containing a non-trivial superposition of point and curvilinear singularities into its distinct geometric constituents. The solution proposed in Donoho and Kutyniok (2013) considers expansions with respect to a combined wavelet-curvelet dictionary and applies an-norm minimization over the expansion coefficients to achieve separation asymptotically at fine scales. However, the original proof of this result uses a heavy machinery relying on sparse representations of Fourier integral operators which does not extend directly to the 3D setting. In this paper, we extend the geometric separation result to the 3D setting using a novel and simpler argument which relies in part on techniques developed by the authors for the shearlet-based analysis of curvilinear edges. Our new result also yields a significantly simpler proof of the original 2D geometric separation problem and extends a prior result by the authors which was limited to piecewise linear singularities.