Uniform recovery in infinite-dimensional compressed sensing and applications to structured binary sampling
Uniform recovery in infinite-dimensional compressed sensing and applications to structured binary sampling
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DOI：
10.1016/j.acha.2021.04.001
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发表时间：
2019-04
期刊：
影响因子：
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通讯作者：
B. Adcock;Vegard Antun;A. Hansen
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文献类型：
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作者：
B. Adcock;Vegard Antun;A. Hansen
无限维压缩传感处理从线性测量中恢复模拟信号（函数），通常以积分变换的形式，如傅立叶变换。该框架非常适合于许多现实世界的逆问题，这些逆问题通常在无限维空间中建模，并且有限维方法的应用可能导致明显的伪影。这类问题的另一个典型特征是信号不仅在某些字典中是稀疏的，而且在层次结构中具有所谓的局部稀疏性。因此，在设计抽样方案时应考虑到这一额外结构。在本文中，我们介绍了一系列的统一恢复保证无限维压缩感知的基础上稀疏的水平和所谓的多级随机子采样。通过使用一个加权的E11-正则化，我们得到的测量条件，是尖锐的对数因子，在这个意义上说，他们同意与最知名的测量条件的甲骨文估计，其中的支持是已知的先验。这些保证也适用于有限维，并改善了现有的结果，为unweighted的NH 1-正则化。为了说明我们的结果，我们考虑了使用正交小波的沃尔什变换的二进制采样问题。二进制采样是某些成像模态的重要机制。通过仔细估计的沃尔什和小波基之间的局部相干性，我们得出了第一个已知的恢复保证这个问题。
Infinite-dimensional compressed sensing deals with the recovery of analog signals (functions) from linear measurements, often in the form of integral transforms such as the Fourier transform. This framework is well-suited to many real-world inverse problems, which are typically modeled in infinite-dimensional spaces, and where the application of finite-dimensional approaches can lead to noticeable artefacts. Another typical feature of such problems is that the signals are not only sparse in some dictionary, but possess a so-called local sparsity in levels structure. Consequently, the sampling scheme should be designed so as to exploit this additional structure. In this paper, we introduce a series of uniform recovery guarantees for infinite-dimensional compressed sensing based on sparsity in levels and so-called multilevel random subsampling. By using a weighted ℓ 1-regularizer we derive measurement conditions that are sharp up to log factors, in the sense that they agree with the best known measurement conditions for oracle estimators in which the support is known a priori. These guarantees also apply in finite dimensions, and improve existing results for unweighted ℓ 1-regularization. To illustrate our results, we consider the problem of binary sampling with the Walsh transform using orthogonal wavelets. Binary sampling is an important mechanism for certain imaging modalities. Through carefully estimating the local coherence between the Walsh and wavelet bases, we derive the first known recovery guarantees for this problem.