Wavelets with composite dilations and their MRA properties

Abstract

Affine systems are reproducing systems of the form A C = {D c T k ψ : 1 L, k ∈ Z n , c ∈ C}, which arise by applying lattice translation operators T k to one or more generators ψ 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 = ab: 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.

Directional Multiscale Processing of Images Using Wavelets with Composite Dilations

“…The theory of wavelets with composite dilations extends nicely many of the standard results of wavelet theory (see [26,27,28]) and, at the same time, it allows for a much richer geometric structure. In particular, it is rather straightforward to obtain the following simple conditions for the constructions of wavelets with composite dilations in the case where the generator ψ is chosen such thatψ = χ S , where S ⊂ R 2 and χ S denotes the characteristic function of S. We state the theorem in dimension n = 2, but it holds in any dimension.…”

“…This is useful both to illustrate the variety of possible constructions which can be derived from this approach, and to set the groundwork for new discrete multiscale transforms which will be developed in Section 3. Note that Constructions 1-3 are not new (see [27,28] …”

“…For any ξ = (ξ 1 , ξ 2 ) ∈ H k , every other point ξ ′ on the same branch of the hyperbola has the unique representation ξ ′ = (ξ 1 γ −t , ξ 2 γ t ) for some t ∈ R, where γ > 1 is fixed. In particular, any ξ = (ξ 1 , ξ 2 ) in 3 Some ideas of this construction were introduced by one of the authors and collaborators in [27]. quadrant I can be parametrized by…”

“…Within the context of improved multidimensional representations, the theory of wavelets with composite dilations, originally developed in [26,27,28], is especially important. This framework is a generalization of the classical theory from which traditional wavelets are derived, and it provides a very flexible setting for the construction of many truly multidimensional variants of wavelets, such as the well known construction of shearlets 1 .…”

The Construction of Smooth Parseval Frames of Shearlets

“…In fact, shearlets were derived from the framework of wavelets with composite dilations, a method introduced to provide a truly multivariate extension of the wavelet framework through the use of affine transformations [20,21,22]. In this approach, the shearlet system is obtained by applying a countable collection of operators to a single or finite set of generators.…”