Wavelets on Irregular Grids with Arbitrary Dilation Matrices, and Frame Atoms for L2(Rd)

ALDROUBI, AKRAM; CABRELLI, CARLOS A.; MOLTER, URSULA M.

APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS

Elsevier

Lugar: San Diego, CA, USA; Año: 2004 vol. 17 p. 119 - 140

1063-5203

In this article, we develop a general method for con\-struct\-ingwa\-ve\-lets $\{|\det A_j|^{1/2}\psi(A_jx-x_{j,k}): \,j\in J, k \in K\}$  on irregular lattices of the form$X=\{x_{j,k} \in \R^d: \; j \in J, k\in K\}$, and with an arbitrary countablefamily of invertible $d\times d$matrices $\{A_j \in GL_d(\R): \; j \in J\}$ that do not necessarilyhave a groupstructure. This waveletconstruction is a particular case of  general atomic framedecompositions 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 $\R^d$. Possibleapplications include image and video compression, speech coding,image and digital data transmission,image analysis, estimations and detection, and seismology.