On the Absence of Uniform Recovery in Many Real-World Applications of Compressed Sensing and the Restricted Isometry Property and Nullspace Property in Levels
On the Absence of Uniform Recovery in Many Real-World Applications of Compressed Sensing and the Restricted Isometry Property and Nullspace Property in Levels
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DOI：
10.1137/15m1043972
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发表时间：
2017-01-01
影响因子：
2.1
通讯作者：
Hansen, Anders C.
中科院分区：
文献类型：
--
作者：
Bastounis, Alexander;Hansen, Anders C.
本文件有两个目的。第一个是指出，均匀恢复的属性，这意味着所有稀疏向量被恢复，并不适用于许多成功使用压缩感知的应用。这包括磁共振成像（MRI），核磁共振计算机断层扫描，电子断层扫描，无线电干涉测量，氦原子散射和荧光显微镜等领域。我们证明，对于涉及基于水平的重建基础的自然压缩传感矩阵（例如，小波），对于合理的s，恢复所有s-稀疏信号所需的测量次数过多。特别是，所有的s-稀疏信号的统一恢复是非常不现实的。这一认识解释了为什么限制等距属性（RIP）不足以解释压缩感知在各种实际应用中的成功。本文的第二个目的是介绍一个新的框架的基础上，广义RIP和广义零空间属性，适合的应用程序中使用压缩感知。我们证明，以前用来证明统一恢复是不合理的缺点不再适用，如果我们而是要求结构化的恢复，是统一的，只有在每个级别。为了研究这一现象，一个新的工具，被称为“限制等距性质的水平”（RIPL）的描述和分析。此外，我们表明，在一定条件下的RIPL，一种形式的统一恢复在每个级别是可能的。幸运的是，Li和Adcock最近的理论进展证明了满足RIPL的大类矩阵的存在。此外，这种矩阵广泛用于诸如MRI的应用中。最后，我们总结了本文提供的例子，证明所获得的结果的最优性。
The purpose of this paper is twofold. The first is to point out that the property of uniform recovery, meaning that all sparse vectors are recovered, does not hold in many applications where compressed sensing is successfully used. This includes fields like magnetic resonance imaging (MRI), nuclear magnetic resonance computerized tomography, electron tomography, radio interferometry, helium atom scattering, and fluorescence microscopy. We demonstrate that for natural compressed sensing matrices involving a level based reconstruction basis (e.g., wavelets), the number of measurements required to recover all s-sparse signals for reasonable s is excessive. In particular, uniform recovery of all s-sparse signals is quite unrealistic. This realization explains why the restricted isometry property (RIP) is insufficient for explaining the success of compressed sensing in various practical applications. The second purpose of the paper is to introduce a new framework based on a generalized RIP and a generalized nullspace property that fit the applications where compressed sensing is used. We demonstrate that the shortcomings previously used to prove that uniform recovery is unreasonable no longer apply if we instead ask for structured recovery that is uniform only within each of the levels. To examine this phenomenon, a new tool, termed the "restricted isometry property in levels" (RIPL) is described and analyzed. Furthermore, we show that with certain conditions on the RIPL, a form of uniform recovery within each level is possible. Fortunately, recent theoretical advances made by Li and Adcock demonstrate the existence of large classes of matrices that satisfy the RIPL. Moreover, such matrices are used extensively in applications such as MRI. Finally, we conclude the paper by providing examples that demonstrate the optimality of the results obtained.