Wavelets with composite dilations and their MRA properties

Abstract

Affine systems are reproducing systems of the form A C = { D c T k ψ ℓ : 1 ⩽ ℓ ⩽ L , k ∈ Z n , c ∈ C } , which arise by applying lattice translation operators T k to one or more generators ψ ℓ in L 2 ( R n ) , followed by the application of dilation operators D c , associated with a countable set C of invertible matrices. In the wavelet literature, C is usually taken to be the group consisting of all integer powers of a fixed expanding matrix. In this paper, we develop the properties of much more general systems, for which C = { c = a b : a ∈ A , b ∈ B } where A and B are not necessarily commuting matrix sets. C need not contain a single expanding matrix. Nonetheless, for many choices of A and B, there are wavelet systems with multiresolution properties very similar to those of classical dyadic wavelets. Typically, A expands or contracts only in certain directions, while B acts by volume-preserving maps in transverse directions. Then the resulting wavelets exhibit the geometric properties, e.g., directionality, elongated shapes, scales, oscillations, recently advocated by many authors for multidimensional signal and image processing applications. Our method is a systematic approach to the theory of affine-like systems yielding these and more general features.