Sampling Theorems for Signals From the Union of Finite-Dimensional Linear Subspaces
Sampling Theorems for Signals From the Union of Finite-Dimensional Linear Subspaces
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DOI：
10.1109/tit.2009.2013003
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发表时间：
2009-04
影响因子：
2.5
通讯作者：
T. Blumensath;M. Davies
中科院分区：
文献类型：
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作者：
T. Blumensath;M. Davies
压缩感知是一种新兴的信号捕获技术，它使信号的采样能够远低于奈奎斯特速率，因为信号在正交化的基础上具有稀疏表示。事实上，正交基中的稀疏性只是允许采样策略低于奈奎斯特比率的一种可能的信号模型。在这篇文章中，我们考虑一个更一般的信号模型，并假设信号位于或接近低维线性子空间的并。我们给出了这个模型的抽样定理，它们与Nyquist-Shannon抽样定理的精神相同，因为它们将所需样本的数量与某些模型参数联系起来。与Nyquist-Shannon采样定理给出了所需样本数的充要条件以及信号重构的简单线性算法相反，本文所研究的模型更为复杂。因此，我们集中在信号模型的两个方面，一对一映射到低维观测空间的存在性和逆映射的光滑性。我们证明了当观测空间至少与模型中任意两个子空间并的凸壳的最大维度相同时，几乎所有的线性映射都是一对一的。然而，我们也证明了为了使逆映射具有某些光滑性，例如给定有限的Lipschitz常数，所需的观测维度必然对数地依赖于信号模型中的子空间的数目。换句话说，虽然唯一的线性抽样方案只需要少量的样本，但为了具有稳定的抽样方法，样本的数量必然对数地依赖于模型中的子空间的数量。然后将这些结果应用于两个例子，其中信号具有以正交基表示的稀疏表示的标准压缩传感信号模型和具有附加树结构的稀疏信号模型。
Compressed sensing is an emerging signal acquisition technique that enables signals to be sampled well below the Nyquist rate, given that the signal has a sparse representation in an orthonormal basis. In fact, sparsity in an orthonormal basis is only one possible signal model that allows for sampling strategies below the Nyquist rate. In this paper, we consider a more general signal model and assume signals that live on or close to the union of linear subspaces of low dimension. We present sampling theorems for this model that are in the same spirit as the Nyquist-Shannon sampling theorem in that they connect the number of required samples to certain model parameters. Contrary to the Nyquist-Shannon sampling theorem, which gives a necessary and sufficient condition for the number of required samples as well as a simple linear algorithm for signal reconstruction, the model studied here is more complex. We therefore concentrate on two aspects of the signal model, the existence of one to one maps to lower dimensional observation spaces and the smoothness of the inverse map. We show that almost all linear maps are one to one when the observation space is at least of the same dimension as the largest dimension of the convex hull of the union of any two subspaces in the model. However, we also show that in order for the inverse map to have certain smoothness properties such as a given finite Lipschitz constant, the required observation dimension necessarily depends logarithmically on the number of subspaces in the signal model. In other words, while unique linear sampling schemes require a small number of samples depending only on the dimension of the subspaces involved, in order to have stable sampling methods, the number of samples depends necessarily logarithmically on the number of subspaces in the model. These results are then applied to two examples, the standard compressed sensing signal model in which the signal has a sparse representation in an orthonormal basis and to a sparse signal model with additional tree structure.