Department of Electrical and Computer Engineering, Rice University, Houston, TX 77005, USA.
IEEE Trans Image Process. 2012 Feb;21(2):494-504. doi: 10.1109/TIP.2011.2165289. Epub 2011 Aug 18.
Compressive sensing (CS) is an emerging approach for the acquisition of signals having a sparse or compressible representation in some basis. While the CS literature has mostly focused on problems involving 1-D signals and 2-D images, many important applications involve multidimensional signals; the construction of sparsifying bases and measurement systems for such signals is complicated by their higher dimensionality. In this paper, we propose the use of Kronecker product matrices in CS for two purposes. First, such matrices can act as sparsifying bases that jointly model the structure present in all of the signal dimensions. Second, such matrices can represent the measurement protocols used in distributed settings. Our formulation enables the derivation of analytical bounds for the sparse approximation of multidimensional signals and CS recovery performance, as well as a means of evaluating novel distributed measurement schemes.
压缩感知（CS）是一种新兴的信号获取方法，适用于在某些基下具有稀疏或可压缩表示的信号。虽然 CS 文献主要集中在涉及一维信号和二维图像的问题上，但许多重要的应用涉及多维信号；由于其更高的维度，此类信号的稀疏基和测量系统的构建变得复杂。在本文中，我们提出将 Kronecker 积矩阵用于 CS 的两个目的。首先，此类矩阵可以作为稀疏基，共同模拟所有信号维度中存在的结构。其次，此类矩阵可以表示分布式设置中使用的测量协议。我们的公式可用于推导多维信号的稀疏逼近和 CS 恢复性能的分析界，以及评估新型分布式测量方案的一种手段。
