Images of the continuous wavelet transform (original) (raw)
Wavelet transforms via generalized quasi-regular representations
Applied and Computational Harmonic Analysis, 2009
The construction of the well-known continuous wavelet transform has been extended before to higher dimensions. Then it was generalized to a group which is topologically isomorphic to a homogeneous space of the semidirect product of an abelian locally compact group and a locally compact group. In this paper, we consider a more general case. We introduce a class of continuous wavelet transforms obtained from the generalized quasi-regular representations. To define such a representation of a group G, we need a homogeneous space with a relatively invariant Radon measure and a character of G.
A functional Hilbert space approach to the theory of wavelets
Autom Constr, 2004
We approach the theory of wavelets from the theory of functional Hilbert spaces. Starting with a Hilbert space H, we consider a subset V of H, for which the span is dense in H. We define a function of positive type on the index set I which labels the elements of V. This function of positive type induces uniquely a functional Hilbert space, which is a subspace of C I and there exists a unitary mapping from H onto this functional Hilbert space. Such functional Hilbert spaces, however, are not easily characterized. Next we consider a group G for the index set I and create the set V using a representation R of the group on H. The unitary mapping between H and the functional Hilbert space is easily recognized as the wavelet transform. We do not insist the representation to be irreducible and derive a generalization of the wavelet theorem as formulated by Grossmann, Morlet and Paul. The functional Hilbert space can in general not be identified with a closed subspace of L 2 (G), in contrast to the case of unitary, irreducible and square integrable representations. Secondly, we take for G a semi-direct product of two locally compact groups S T , where S is abelian. In this case we give a more tangible description for the functional Hilbert space, which is easier to grasp. Finally, we provide an example where we take H = L 2 (R 2) and the Euclidean motion group for G. This example is inspired by an application of biomedical imaging, namely orientation bundle theory, which was the motivation for this report.
On the structure of the space of wavelet transforms
Comptes Rendus Mathematique, 2008
Let G be the "ax + b"-group with the left invariant Haar measure dν and ψ be a fixed real-valued admissible wavelet on L2(R). The complete decomposition of L2(G, dν) onto the space of wavelet transforms W ψ (L2(R)) is obtained after identifying the group G with the upper half-plane Π in C. To cite this article: O. Hutník, C. R. Acad. Sci. Paris, Ser. I 346 (2008).
Journal of Fourier Analysis and Applications, 2002
Continuous wavelet transforms arising from the quasiregular representation of a semidirect product group G = R k ⋊ H have been studied by various authors. Recently the attention has shifted from the irreducible case to include more general dilation groups H, for instance cyclic (more generally: discrete) or one-parameter groups. These groups do not give rise to irreducible square-integrable representations, yet it is possible (and quite simple) to give admissibility conditions for a large class of them. We put these results in a theoretical context by establishing a connection to the Plancherel theory of the semidirect products, and show how the admissibility conditions relate to abstract admissibility conditions which use Plancherel theory. *
Wavelet Transforms for Semidirect Product Groups with Not Necessarily Commutative Normal Subgroups
Journal of Fourier Analysis and Applications, 2006
Let G be the semidirect product group of a separable locally compact unimodular group N of type I with a closed subgroup H of Aut(N). The group N is not necessarily commutative. We consider irreducible subrepresentations of the unitary representation of G realized naturally on L 2 (N), and investigate the wavelet transforms associated to them. Furthermore, the irreducible subspaces are characterized by certain singular integrals on N analogous to the Cauchy-Szegö integral.
Wavelet Packets on Locally Compact Abelian Groups
2010
The objective of this paper is to construct wavelet packets associated with multiresolution analysis on locally compact Abelian groups. Moreover, from the collection of dilations and translations of the wavelet packets, we characterize the subcollections which form an orthonormal basis for L 2 (G).
Orthogonal wavelets on locally compact Abelian groups
Functional Analysis and Its Applications, 1997
We extend and improve the results of W. Lang (1998) on the wavelet analysis on the Cantor dyadic group C. Our construction is realized on a locally compact abelian group G which is defined for an integer p 2 and coincides with C when p = 2. For any integers p, n 2 we determine a function ϕ in L 2 (G) which 1) is the sum of a lacunary series by generalized Walsh functions, 2) has orthonormal "integer" shifts in L 2 (G), 3) satisfies "the scaling equation" with p n numerical coefficients, 4) has compact support whose Haar measure is proportional to p n , 5) generates a multiresolution analysis in L 2 (G). Orthogonal wavelets ψ with compact supports on G are defined by such functions ϕ. The family of these functions ϕ is in many respects analogous to the well-known family of Daubechies' scaling functions. We give a method for estimating the moduli of continuity of the functions ϕ, which leads to sharp estimates for small p and n. We also show that the notion of adapted multiresolution analysis recently suggested by Sendov is applicable in this situation.
Wavelets invariant under finite reflection groups
Mathematical Methods in The Applied Sciences, 2009
In this paper we use approximate identities in the Dunkl setting in order to construct spherical Dunkl wavelets, which do not involve the knowledge of the intertwining operator, the Dunkl translation or of the Dunkl transform. The practicality of the proposed approach will be shown with the example of Abel-Poisson wavelets. Copyright © 2009 John Wiley & Sons, Ltd.
Wavelet subspaces invariant under groups of translation operators
Proceedings Mathematical Sciences, 2003
We study the action of translation operators on wavelet subspaces. This action gives rise to an equivalence relation on the set of all wavelets. We show by explicit construction that each of the associated equivalence classes is non-empty.
Wavelet filters and infinite-dimensional unitary groups
2000
In this paper, we study wavelet filters and their dependence on two numbers, the scale N and the genus g. We show that the wavelet filters, in the quadrature mirror case, have a harmonic analysis which is based on representations of the C^*-algebra O_N. A main tool in our analysis is the infinite-dimensional group of all maps T -> U(N) (where U(N) is the group of all unitary N-by-N matrices), and we study the extension problem from low-pass filter to multiresolution filter using this group.