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

Abstract

In this article, we develop a general method for constructing wavelets {| det Aj|1/2ψ(Ajx-xj,k): j ∈ J, k ∈ K} on irregular lattices of the form X = {xj,k ∈ ℝd: j ∈ J, k ∈ K}, and with an arbitrary countable family of invertible d × d matrices {Aj ∈ GL 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 L2(ℝd) 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.