TY - JOUR

T1 - Life beyond bases

T2 - The advent of frames (Part I)

AU - Kovačević, Jelena

AU - Chebira, Amina

N1 - Funding Information:
We gratefully acknowledge comments from Riccardo Bernardini, Bernhard Bodmann, Helmut Bölcskei, Pete Casazza, Minh Do, Matt Fickus, Vivek Goyal, Chris Heil, Prakash Ishwar, Nick Kingsbury, Götz Pfander, Emina Sˇ oljanin, Thomas Strohmer, and Martin Vetterli; their collective wisdom improved the original manuscript significantly. In particular, Chris Heil’s and Zoran Cvetković’s help was invaluable in making the article mathematically tight and precise. This work was supported by NSF through award 0515152.

PY - 2007/7

Y1 - 2007/7

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

AN - SCOPUS:85032752112

VL - 24

SP - 86

EP - 104

JO - IEEE Acoustics, Speech, and Signal Processing Newsletter

JF - IEEE Acoustics, Speech, and Signal Processing Newsletter

SN - 1053-5888

IS - 4

ER -