Wavelets on irregular grids with arbitrary dilation matrices and frame atoms for L2(Rd)

By Akram Aldroubi, Carlos Cabrelli, Ursula Molter. 2004

126 citations recorded by the source corpus.

Abstract 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.

Subfields: Applied Mathematics, Computer Vision and Pattern Recognition, Geophysics

View Paper: https://www.semanticscholar.org/paper/a7830dab8e810e18c2c837b4ca29e41e0015153f

Citations in the corpus

• Fusion frames and distributed processing (2006, unknown): “Related approaches were undertaken by Aldroubi, Cabrelli, and Molter [1] and Fornasier [16].”
• G-frames and G-Riesz Bases (2005, unknown): “On the other hand, it is easy to check that for any f1, f2 ∈ U , 〈Sf1, f2〉 = ∑ j∈J 〈Λ∗jΛjf1, f2〉 = ∑ j∈J 〈f1,Λ ∗ jΛjf2〉 = 〈f1, Sf2〉 and therefore, ‖S‖ = sup ‖f‖=1 〈Sf, f〉 = sup ‖f‖=1 ∑ j∈J ‖Λjf‖ 2 ≤ B.”