Generalized Sampling and Infinite-Dimensional Compressed Sensing
Generalized Sampling and Infinite-Dimensional Compressed Sensing
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DOI：
10.1007/s10208-015-9276-6
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发表时间：
2016-10-01
影响因子：
3
通讯作者：
Hansen, Anders C.
中科院分区：
文献类型：
--
作者：
Adcock, Ben;Hansen, Anders C.
我们引入并分析了一种用于无穷维压缩感知的框架及相应方法。这将现有的理论从有限维向量空间扩展到了可分希尔伯特空间的情形。我们通过证明现有的有限维技术不适合解决一些关键问题，解释了为什么需要这样一种新理论。这项工作源于经典（奈奎斯特速率）采样的广义采样定理的最新进展，该定理允许在任意基下进行重构。本文的一个结论是，可以扩展这些思想，以对稀疏或可压缩信号进行大幅欠采样。这项工作的核心是在采样理论中引入了两个新概念，即稳定采样率和平衡性质，它们规定了如何恰当地离散化一个无穷维问题。
We introduce and analyze a framework and corresponding method for compressed sensing in infinite dimensions. This extends the existing theory from finite-dimensional vector spaces to the case of separable Hilbert spaces. We explain why such a new theory is necessary by demonstrating that existing finite-dimensional techniques are ill suited for solving a number of key problems. This work stems from recent developments in generalized sampling theorems for classical (Nyquist rate) sampling that allows for reconstructions in arbitrary bases. A conclusion of this paper is that one can extend these ideas to allow for significant subsampling of sparse or compressible signals. Central to this work is the introduction of two novel concepts in sampling theory, the stable sampling rate and the balancing property, which specify how to appropriately discretize an infinite-dimensional problem.