In this article, we develop a general method for constructing wavelets {|detA_j|^{1/2} g(A_jx-x_{j,k}): j in J, k in K}, on irregular lattices of the formX={x_{j,k} in R^d: j in J, k in K}, and with an arbitrary countable family ofinvertible dxd matrices {A_j in GL_d(R): j in J} that do not necessarily have agroup structure. This wavelet construction is a particular case of generalatomic frame decompositions of L^2(R^d) developed in this article, that allowother time frequency decompositions such as non-harmonic Gabor frames withnon-uniform covering of the Euclidean space R^d. Possible applications includeimage and video compression, speech coding, image and digital datatransmission, image analysis, estimations and detection, and seismology.
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