Unified Theory for Recovery of Sparse Signals in a General Transform Domain
Unified Theory for Recovery of Sparse Signals in a General Transform Domain
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DOI：
10.1109/tit.2018.2846643
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发表时间：
2016-12
影响因子：
2.5
通讯作者：
Kiryung Lee;Yanjun Li;Kyong Hwan Jin;J. C. Ye
中科院分区：
文献类型：
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作者：
Kiryung Lee;Yanjun Li;Kyong Hwan Jin;J. C. Ye
压缩感知为稀疏信号的数据采集提供了一个范例。值得注意的是，它已被证明，实用的算法提供了强大的恢复，从噪声的线性测量，在接近最佳的采样率。在许多实际应用中，感兴趣的信号通常不是在正则基中而是在某个变换域（例如小波或有限差分）中是稀疏的。压缩感知理论被扩展到分析稀疏模型，但已知的扩展仅限于传感矩阵和稀疏化变换的具体选择。本文利用凸规划方法给出了一般变换域上稀疏信号鲁棒恢复的统一理论。特别是，我们的研究结果适用于一般的收购和稀疏模型，并显示如何恢复的测量数量取决于测量和稀疏化变换的属性。此外，我们还提供了我们的结果的情况下，在变换中的原子具有不同的不相干参数和未知的信号表现出结构化的稀疏模式的扩展。特别是，部分傅立叶恢复稀疏信号的循环变换，我们的主要结果表明，均匀随机采样。数值结果表明，变密度随机抽样的主要结果提供了一个上级恢复性能比已知的抽样策略。
Compressed sensing is provided a data-acquisition paradigm for sparse signals. Remarkably, it has been shown that the practical algorithms provide robust recovery from noisy linear measurements acquired at a near optimal sampling rate. In many real-world applications, a signal of interest is typically sparse not in the canonical basis but in a certain transform domain, such as wavelets or the finite difference. The theory of compressed sensing was extended to the analysis sparsity model, but known extensions are limited to the specific choices of sensing matrix and sparsifying transform. In this paper, we propose a unified theory for robust recovery of sparse signals in a general transform domain by convex programming. In particular, our results apply to the general acquisition and sparsity models and show how the number of measurements for recovery depends on properties of measurement and sparsifying transforms. Moreover, we also provide extensions of our results to the scenarios where the atoms in the transform have varying incoherence parameters and the unknown signal exhibits a structured sparsity pattern. In particular, for the partial Fourier recovery of sparse signals over a circulant transform, our main results suggest a uniformly random sampling. Numerical results demonstrate that the variable density random sampling by our main results provides a superior recovery performance over the known sampling strategies.