Compressed Sensing & Sparse Filtering
Compressed Sensing & Sparse Filtering
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压缩感知
DOI：
10.1007/978-3-642-38398-4_2
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Blumensath T
中科院分区：
文献类型：
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作者：
Blumensath T
压缩感知的大多数发展都围绕着信号结构的开发，这些信号结构可以使用几何解释最容易地表达和理解。这种几何观点不仅是许多压缩感知理论赖以建立的初始理论发展的基础，而且还使思想能够扩展到更一般的恢复问题和结构。统一框架是非凸、低维约束集，假设要恢复的信号位于其中。传统压缩感知的稀疏信号结构转化为低维子空间的并集，每个子空间由少量坐标轴跨越。子空间解释的并集很容易推广，并且可以看出许多其他恢复问题都属于这种情况。例如，在许多问题中，数据不是矢量数据，而是更自然地以矩阵形式表示（例如，视频通常最好以时间矩阵的像素表示）。对矩阵的一个强有力的约束是对矩阵秩的约束。例如，在低秩矩阵恢复中，目标是仅在给定其条目的子集的情况下重建低秩矩阵。重要的是，低秩矩阵也存在于子空间结构的并集中，尽管现在有无限多个子空间（尽管每个子空间都是有限维的）。应用中还出现了子空间信号模型并集的许多其他示例，包括稀疏小波树结构（形成一般稀疏模型的子集）和有限创新率模型，其中我们可以拥有无限多个无限维子空间。在本章中，我将介绍这些和相关的几何概念，并展示如何使用它们来（a）开发算法来恢复具有给定结构的信号，以及（b）获得表征这些算法方法性能的理论结果。
Most developments in compressed sensing have revolved around the exploitation of signal structures that can be expressed and understood most easily using a geometrical interpretation. This geometric point of view not only underlies many of the initial theoretical developments on which much of the theory of compressed sensing is built, but has also allowed ideas to be extended to much more general recovery problems and structures. A unifying framework is that of non-convex, low-dimensional constraint sets in which the signal to be recovered is assumed to reside. The sparse signal structure of traditional compressed sensing translates into a union of low dimensional subspaces, each subspace being spanned by a small number of the coordinate axes. The union of subspaces interpretation is readily generalised and many other recovery problems can be seen to fall into this setting. For example, instead of vector data, in many problems, data is more naturally expressed in matrix form (for example a video is often best represented in a pixel by time matrix). A powerful constraint on matrices are constraints on the matrix rank. For example, in low-rank matrix recovery, the goal is to reconstruct a low-rank matrix given only a subset of its entries. Importantly, low-rank matrices also lie in a union of subspaces structure, although now, there are infinitely many subspaces (though each of these is finite dimensional). Many other examples of union of subspaces signal models appear in applications, including sparse wavelet-tree structures (which form a subset of the general sparse model) and finite rate of innovations models, where we can have infinitely many infinite dimensional subspaces. In this chapter, I will provide an introduction to these and related geometrical concepts and will show how they can be used to (a) develop algorithms to recover signals with given structures and (b) allow theoretical results that characterise the performance of these algorithmic approaches.