### Abstract

The redundant counterpart of a basis is called a frame. Frames are more general that the typical belief that they are often associated with wavelet frames. Frames are redundant representations that only need to represent signals in a given space with a certain amount of redundancy. Frames are used anywhere where redundancy is needed. Frames are used in place of bases because certain signal characteristics become obvious in that other domain facilitating various signal processing tasks. Tackling frames involves the assumption of vectors in a vector space. Functions within the space turn the vector space into an inner product space. A precise measurement tool is obtained by introducing the distance between two vectors and turn the inner product space into a metric space. Bases in finite-dimensional spaces implies that the number of representative vectors is the same as the dimension of the space. If the number is larger, a representative set of vectors can be obtained except that the vectors are no longer linearly independent and the resulting set is no longer called a basis but a frame. Tight frames can also mimic ONBs. As such, frames are becoming a standard tool in the signal processing field especially when redundancy is required.

Original language | English (US) |
---|---|

Pages (from-to) | 86-104 |

Number of pages | 19 |

Journal | IEEE Signal Processing Magazine |

Volume | 24 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 2007 |

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### ASJC Scopus subject areas

- Signal Processing
- Electrical and Electronic Engineering
- Applied Mathematics

### Cite this

*IEEE Signal Processing Magazine*,

*24*(4), 86-104. https://doi.org/10.1109/MSP.2007.4286567

**Life beyond bases : The advent of frames (Part I).** / Kovacevic, Jelena; Chebira, Amina.

Research output: Contribution to journal › Article

*IEEE Signal Processing Magazine*, vol. 24, no. 4, pp. 86-104. https://doi.org/10.1109/MSP.2007.4286567

}

TY - JOUR

T1 - Life beyond bases

T2 - The advent of frames (Part I)

AU - Kovacevic, Jelena

AU - Chebira, Amina

PY - 2007/1/1

Y1 - 2007/1/1

N2 - The redundant counterpart of a basis is called a frame. Frames are more general that the typical belief that they are often associated with wavelet frames. Frames are redundant representations that only need to represent signals in a given space with a certain amount of redundancy. Frames are used anywhere where redundancy is needed. Frames are used in place of bases because certain signal characteristics become obvious in that other domain facilitating various signal processing tasks. Tackling frames involves the assumption of vectors in a vector space. Functions within the space turn the vector space into an inner product space. A precise measurement tool is obtained by introducing the distance between two vectors and turn the inner product space into a metric space. Bases in finite-dimensional spaces implies that the number of representative vectors is the same as the dimension of the space. If the number is larger, a representative set of vectors can be obtained except that the vectors are no longer linearly independent and the resulting set is no longer called a basis but a frame. Tight frames can also mimic ONBs. As such, frames are becoming a standard tool in the signal processing field especially when redundancy is required.

AB - The redundant counterpart of a basis is called a frame. Frames are more general that the typical belief that they are often associated with wavelet frames. Frames are redundant representations that only need to represent signals in a given space with a certain amount of redundancy. Frames are used anywhere where redundancy is needed. Frames are used in place of bases because certain signal characteristics become obvious in that other domain facilitating various signal processing tasks. Tackling frames involves the assumption of vectors in a vector space. Functions within the space turn the vector space into an inner product space. A precise measurement tool is obtained by introducing the distance between two vectors and turn the inner product space into a metric space. Bases in finite-dimensional spaces implies that the number of representative vectors is the same as the dimension of the space. If the number is larger, a representative set of vectors can be obtained except that the vectors are no longer linearly independent and the resulting set is no longer called a basis but a frame. Tight frames can also mimic ONBs. As such, frames are becoming a standard tool in the signal processing field especially when redundancy is required.

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U2 - 10.1109/MSP.2007.4286567

DO - 10.1109/MSP.2007.4286567

M3 - Article

VL - 24

SP - 86

EP - 104

JO - IEEE Signal Processing Magazine

JF - IEEE Signal Processing Magazine

SN - 1053-5888

IS - 4

ER -