Crystallographic Haar-Type Composite Dilation Wavelets

Matricial filters and crystallographic composite dilation wavelets

“…Here we present the matricial filter equations for the group of full symmetries of the hexagon and show that these filters satisfy the necessary and sufficient condition (3.7). Blanchard and Steffen [3] have cataloged the Haar-type composite dilation wavelets when the semi-direct product B Γ forms a crystallographic group on R 2 (also called a plane symmetry group). Independently, MacArthur and Taylor have utilized the representation theoretic framework to produce Haar-type wavelets from the plane crystallographic groups [17] and MacArthur also cataloged these wavelets in R 2 [16].…”

“…(In the plane symmetry (crystallographic) group notation from [3], B Γ = p6m.) Choose the ordering of B to map the set R 0 counterclockwise around the octagon (see Figure 2(a)).…”

“…However, it is possible to find orthogonal, compactly supported, MRA CDW in the case of a finite group B (e.g. [3,13]). The goal of this paper is to develop a theoretical, matricial filter base for such examples and present some new compactly supported CDW.…”

Crystallographic Haar Wavelets

“…Results on particular cases of CDW were obtained in [4,5,18] for A = {a j : j ∈ Z} and B a group satisfying the crystallographic condition B L = L. Under this hypothesis, in [5], it is established that for any Haar-type (a, B, L) CDW, B must be finite. Moreover B L is a crystallographic splitting group (see Sect.…”

“…Sufficient conditions for the existence of such CDW are also proved. It is noticed in [5,23], and we will show in Sect. 2.1 that not all crystallographic wavelets can be obtained as CDW.…”