### 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 language | English (US) |
---|---|

Title of host publication | Applied and Numerical Harmonic Analysis |

Publisher | Springer International Publishing |

Pages | 303-335 |

Number of pages | 33 |

Edition | 9780817683726 |

DOIs | |

State | Published - Jan 1 2013 |

### Publication series

Name | Applied and Numerical Harmonic Analysis |
---|---|

Number | 9780817683726 |

ISSN (Print) | 2296-5009 |

ISSN (Electronic) | 2296-5017 |

### Fingerprint

### 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

*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

}

*Applied and Numerical Harmonic Analysis.*9780817683726 edn, Applied and Numerical Harmonic Analysis, no. 9780817683726, Springer International Publishing, pp. 303-335. https://doi.org/10.1007/978-0-8176-8373-3_9

**Finite frames for sparse signal processing.** / Bajwa, Waheed Uz Zaman; Pezeshki, Ali.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

TY - CHAP

T1 - Finite frames for sparse signal processing

AU - Bajwa, Waheed Uz Zaman

AU - Pezeshki, Ali

PY - 2013/1/1

Y1 - 2013/1/1

N2 - 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.

AB - 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.

KW - Approximation theory

KW - Coherence property

KW - Compressed sensing

KW - Detection

KW - Estimation

KW - Grassmannian frames

KW - Model selection

KW - Regression

KW - Restricted isometry property

KW - Typical guarantees

KW - Uniform guarantees

KW - Welch bound

UR - http://www.scopus.com/inward/record.url?scp=85047391782&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85047391782&partnerID=8YFLogxK

U2 - 10.1007/978-0-8176-8373-3_9

DO - 10.1007/978-0-8176-8373-3_9

M3 - Chapter

T3 - Applied and Numerical Harmonic Analysis

SP - 303

EP - 335

BT - Applied and Numerical Harmonic Analysis

PB - Springer International Publishing

ER -