Compressed Sensing with Nonlinear Observations
Compressed Sensing with Nonlinear Observations
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发表时间：
2010
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通讯作者：
T. Blumensath
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作者：
T. Blumensath
压缩感知是近年来发展起来的一种信号捕获技术。与传统的采样方法相比，只要信号采用稀疏表示，则需要的样本明显更少。重要的是，可以构建允许使用有效算法从少量测量中重建稀疏信号的采样方法。我们最近从两个重要方面概括了这些想法。我们已经发展了方法和理论结果，允许对信号施加更一般的约束，我们还将该方法扩展到更一般的希尔伯特空间。在这篇文章中，我们介绍了对压缩感知的进一步推广，并允许使用非线性采样方法。这是通过使用最近引入的对传统上强加于压缩传感系统的受限等距性质(或双Lipschitz条件)的推广来实现的。我们证明，如果这个更一般的条件对非线性采样系统成立，那么我们可以从非线性压缩测量中重构信号。
Compressed sensing is a recently developed signal acquisition technique. In contrast to traditional sampling methods, significantly fewer samples are required whenever the signals admit a sparse representation. Crucially, sampling methods can be constructed that allow the reconstruction of sparse signals from a small number of measurements using efficient algorithms. We have recently generalised these ideas in two important ways. We have developed methods and theoretical results that allow much more general constraints to be imposed on the signal and we have also extended the approach to more general Hilbert spaces. In this paper we introduce a further generalisation to compressed sensing and allow for non-linear sampling methods. This is achieved by using a recently introduced generalisation of the Restricted Isometry Property (or the bi-Lipschitz condition) traditionally imposed on the compressed sensing system. We show that, if this more general condition holds for the nonlinear sampling system, then we can reconstruct signals from non-linear compressive measurements.