Wavelets on irregular grids with arbitrary dilation matrices and frame atoms for L 2 ( R d )

In this article, we develop a general method for constructing wavelets { | det A j | 1 / 2 ψ ( A j x − x j , k ) : j ∈ J , k ∈ K } on irregular lattices of the form X = { x j , k ∈ R d : j ∈ J , k ∈ K } , and with an arbitrary countable family of invertible d × d matrices { A j ∈ G L d ( R ) : j ∈ J...

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Published inApplied and computational harmonic analysis Vol. 17; no. 2; pp. 119 - 140
Main Authors Aldroubi, Akram, Cabrelli, Carlos, Molter, Ursula M.
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.09.2004
Subjects
Frames
Irregular sampling
Wavelet sets
Wavelets
Frames
Wavelets
Irregular sampling
42C40
Wavelet sets
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Summary:In this article, we develop a general method for constructing wavelets { | det A j | 1 / 2 ψ ( A j x − x j , k ) : j ∈ J , k ∈ K } on irregular lattices of the form X = { x j , k ∈ R d : j ∈ J , k ∈ K } , and with an arbitrary countable family of invertible d × d matrices { A j ∈ G L d ( R ) : j ∈ J } that do not necessarily have a group structure. This wavelet construction is a particular case of general atomic frame decompositions of L 2 ( R d ) developed in this article, that allow other time frequency decompositions such as nonharmonic Gabor frames with nonuniform covering of the Euclidean space R d . Possible applications include image and video compression, speech coding, image and digital data transmission, image analysis, estimations and detection, and seismology.
ISSN:1063-5203
1096-603X
DOI:10.1016/j.acha.2004.03.005