A Note on Compressed Sensing of Structured Sparse Wavelet Coefficients From Subsampled Fourier Measurements
A Note on Compressed Sensing of Structured Sparse Wavelet Coefficients From Subsampled Fourier Measurements
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DOI：
10.1109/lsp.2016.2550101
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发表时间：
2016-05-01
影响因子：
3.9
通讯作者：
Roman, Bogdan
中科院分区：
文献类型：
--
作者：
Adcock, Ben;Hansen, Anders C.;Roman, Bogdan
我们考虑使用小波压缩感知（CS）从傅里叶测量中恢复信号。对于具有结构化稀疏哈尔小波系数的离散信号，我们首次证明了根据适当的可变密度采样方案从离散傅里叶样本中近乎最优恢复。关键的是，考虑到这种结构化稀疏性（称为层级稀疏性），而不仅仅是稀疏性，这一结果给出的恢复保证与在这种情况下压缩感知的经验观测恢复特性相符。这一结果补充了阿德科克等人[打破相干性障碍：压缩感知的一种新理论，arXiv预印本arXiv:1302.0561，2014]中的一个近期定理，该定理涉及连续时间信号的情况。此外，我们提供了一个明显更简短且更具解释性的论证，它清楚地说明了在这种情况下控制恢复的关键因素：即频率空间划分为与小波尺度相对应的二进频带、傅里叶/小波交叉格拉姆矩阵的近块对角性以及小波系数的结构化稀疏性。
We consider signal recovery from Fourier measurements using compressed sensing (CS) with wavelets. For discrete signals with structured sparse Haar wavelet coefficients, we give the first proof of near-optimal recovery from discrete Fourier samples taken according to an appropriate variable density sampling scheme. Crucially, in taking into account such structured sparsity-known as sparsity in levels-as opposed to just sparsity, this result yields recovery guarantees that agree with the empirically observed recovery properties of CS in this setting. This result complements a recent theorem in Adcock et al. [Breaking the coherence barrier: A new theory for compressed sensing, arXiv preprint arXiv: 1302.0561, 2014.], which addressed the case of continuous time signals. Moreover, we provide a significantly shorter and more expositional argument, which clearly illustrates the key factors governing recovery in this setting: namely the division of frequency space into dyadic bands corresponding to wavelet scales, the near-block diagonality of the Fourier/wavelet cross-Gramian matrix, and the structured sparsity of wavelet coefficients.