Finite frames for sparse signal processing

Waheed U. Bajwa, Ali Pezeshki

Research output: Chapter in Book/Report/Conference proceedingChapter

3 Scopus citations

Abstract

Over the last decade, considerable progress has been made toward developing new signal processing methods to manage the deluge of data caused by advances in sensing, imaging, storage, and computing technologies. Most of these methods are based on a simple but fundamental observation: high-dimensional data sets are typically highly redundant and live on low-dimensional manifolds or subspaces. This means that the collected data can often be represented in a sparse or parsimonious way in a suitably selected finite frame. This observation has also led to the development of a new sensing paradigm, called compressed sensing, which shows that high-dimensional data sets can often be reconstructed, with high fidelity, from only a small number of measurements. Finite frames play a central role in the design and analysis of both sparse representations and compressed sensing methods. In this chapter, we highlight this role primarily in the context of compressed sensing for estimation, recovery, support detection, regression, and detection of sparse signals. The recurring theme is that frames with small spectral norm and/or small worst-case coherence, average coherence, or sum coherence are well suited for making measurements of sparse signals.

Original languageEnglish (US)
Title of host publicationApplied and Numerical Harmonic Analysis
PublisherSpringer International Publishing
Pages303-335
Number of pages33
Edition9780817683726
DOIs
Publication statusPublished - Jan 1 2013

Publication series

NameApplied and Numerical Harmonic Analysis
Number9780817683726
ISSN (Print)2296-5009
ISSN (Electronic)2296-5017

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All Science Journal Classification (ASJC) codes

  • Applied Mathematics

Keywords

  • Approximation theory
  • Coherence property
  • Compressed sensing
  • Detection
  • Estimation
  • Grassmannian frames
  • Model selection
  • Regression
  • Restricted isometry property
  • Typical guarantees
  • Uniform guarantees
  • Welch bound

Cite this

Bajwa, W. U., & Pezeshki, A. (2013). Finite frames for sparse signal processing. In Applied and Numerical Harmonic Analysis (9780817683726 ed., pp. 303-335). (Applied and Numerical Harmonic Analysis; No. 9780817683726). Springer International Publishing. https://doi.org/10.1007/978-0-8176-8373-3_9