压缩感知中的渐近结构
On asymptotic structure in compressed sensing
摘要：
本文演示了压缩感知的新原理，即渐近不相干性，渐近稀疏性和多级采样，可以用来更好地理解实际压缩感知中的潜在现象，并改善在现实世界中的应用结果。本文的贡献是四方面的：首先，它解释了如何采样策略不仅取决于信号的稀疏性，但也对它的结构，并说明如何设计有效的采样策略，利用这一点。其次，它表明，最佳的采样策略和压缩感知的效率也取决于解决问题，并显示这种现象如何显着影响压缩感知的结果，以及如何利用它。第三，由于新的框架也适合模拟（无限维）模型，在实践中管理许多逆问题，本文介绍了它如何可以用来产生实质性的改进。第四，通过使用利用信号结构的多级采样，本文解释了即使经典的$l^1$恢复算法被旨在利用诸如基于模型或贝叶斯压缩感知或近似消息传递的结构的修改算法所取代，也可以优于随机高斯/伯努利采样。这最后一点意见提出了一个问题，即即使在这种矩阵适用的情况下，普遍性是否可取。本文研究的实际应用的例子包括磁共振成像（MRI），电子显微镜（EM），压缩成像（CI）和荧光显微镜（FM）。对于后者，还提出了一种新的压缩感知方法。
英文摘要：
This paper demonstrates how new principles of compressed sensing, namely asymptotic incoherence, asymptotic sparsity and multilevel sampling, can be utilised to better understand underlying phenomena in practical compressed sensing and improve results in real-world applications. The contribution of the paper is fourfold: First, it explains how the sampling strategy depends not only on the signal sparsity but also on its structure, and shows how to design effective sampling strategies utilising this. Second, it demonstrates that the optimal sampling strategy and the efficiency of compressed sensing also depends on the resolution of the problem, and shows how this phenomenon markedly affects compressed sensing results and how to exploit it. Third, as the new framework also fits analog (infinite dimensional) models that govern many inverse problems in practice, the paper describes how it can be used to yield substantial improvements. Fourth, by using multilevel sampling, which exploits the structure of the signal, the paper explains how one can outperform random Gaussian/Bernoulli sampling even when the classical $l^1$ recovery algorithm is replaced by modified algorithms which aim to exploit structure such as model based or Bayesian compressed sensing or approximate message passaging. This final observation raises the question whether universality is desirable even when such matrices are applicable. Examples of practical applications investigated in this paper include Magnetic Resonance Imaging (MRI), Electron Microscopy (EM), Compressive Imaging (CI) and Fluorescence Microscopy (FM). For the latter, a new compressed sensing approach is also presented.
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