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

In this article, we develop a general method for constructing wavelets {\det A(j)\(1/2)psi(A(j)x - x(j,k)): j is an element of J, k is an element of K} on irregular lattices of the form X = {X-j,X-k is an element of R-d: j is an element of J, k is an element of K}, and with an arbitrary countable family of invertible d x d matrices {A(j) is an element of GL(d)(R): j is an element of 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. (C) 2004 Elsevier Inc. All rights reserved.