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

摘要

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 x d matrices [A_j ∈ 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 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.