5,918 research outputs found
Reproducing formulas for generalized translation invariant systems on locally compact abelian groups
In this paper we connect the well established discrete frame theory of
generalized shift invariant systems to a continuous frame theory. To do so, we
let , , be a countable family of closed, co-compact
subgroups of a second countable locally compact abelian group and study
systems of the form with generators in and with each
being a countable or an uncountable index set. We refer to systems of this form
as generalized translation invariant (GTI) systems. Many of the familiar
transforms, e.g., the wavelet, shearlet and Gabor transform, both their
discrete and continuous variants, are GTI systems. Under a technical
local integrability condition (-LIC) we characterize when GTI systems
constitute tight and dual frames that yield reproducing formulas for .
This generalizes results on generalized shift invariant systems, where each
is assumed to be countable and each is a uniform lattice in
, to the case of uncountably many generators and (not necessarily discrete)
closed, co-compact subgroups. Furthermore, even in the case of uniform lattices
, our characterizations improve known results since the class of GTI
systems satisfying the -LIC is strictly larger than the class of GTI
systems satisfying the previously used local integrability condition. As an
application of our characterization results, we obtain new characterizations of
translation invariant continuous frames and Gabor frames for . In
addition, we will see that the admissibility conditions for the continuous and
discrete wavelet and Gabor transform in are special cases
of the same general characterizing equations.Comment: Minor changes (v2). To appear in Trans. Amer. Math. So
Invertible Orientation Scores of 3D Images
The enhancement and detection of elongated structures in noisy image data is
relevant for many biomedical applications. To handle complex crossing
structures in 2D images, 2D orientation scores were introduced, which already
showed their use in a variety of applications. Here we extend this work to 3D
orientation scores. First, we construct the orientation score from a given
dataset, which is achieved by an invertible coherent state type of transform.
For this transformation we introduce 3D versions of the 2D cake-wavelets, which
are complex wavelets that can simultaneously detect oriented structures and
oriented edges. For efficient implementation of the different steps in the
wavelet creation we use a spherical harmonic transform. Finally, we show some
first results of practical applications of 3D orientation scores.Comment: ssvm 2015 published version in LNCS contains a mistake (a switch
notation spherical angles) that is corrected in this arxiv versio
Interpolation in Wavelet Spaces and the HRT-Conjecture
We investigate the wavelet spaces arising from square integrable representations of a locally compact group . We show that
the wavelet spaces are rigid in the sense that non-trivial intersection between
them imposes strong conditions. Moreover, we use this to derive consequences
for wavelet transforms related to convexity and functions of positive type.
Motivated by the reproducing kernel Hilbert space structure of wavelet spaces
we examine an interpolation problem. In the setting of time-frequency analysis,
this problem turns out to be equivalent to the HRT-Conjecture. Finally, we
consider the problem of whether all the wavelet spaces
of a locally compact group
collectively exhaust the ambient space . We show that the answer is
affirmative for compact groups, while negative for the reduced Heisenberg
group.Comment: Added a relevant citation and made minor modifications to the
expositio
The wavelet transforms in Gelfand-Shilov spaces
We describe local and global properties of wavelet transforms of
ultradifferentiable functions. The results are given in the form of continuity
properties of the wavelet transform on Gelfand-Shilov type spaces and their
duals. In particular, we introduce a new family of highly time-scale localized
spaces on the upper half-space. We study the wavelet synthesis operator (the
left-inverse of the wavelet transform) and obtain the resolution of identity
(Calder\'{o}n reproducing formula) in the context of ultradistributions
Coxeter Groups and Wavelet Sets
A traditional wavelet is a special case of a vector in a separable Hilbert
space that generates a basis under the action of a system of unitary operators
defined in terms of translation and dilation operations. A
Coxeter/fractal-surface wavelet is obtained by defining fractal surfaces on
foldable figures, which tesselate the embedding space by reflections in their
bounding hyperplanes instead of by translations along a lattice. Although both
theories look different at their onset, there exist connections and
communalities which are exhibited in this semi-expository paper. In particular,
there is a natural notion of a dilation-reflection wavelet set. We prove that
dilation-reflection wavelet sets exist for arbitrary expansive matrix
dilations, paralleling the traditional dilation-translation wavelet theory.
There are certain measurable sets which can serve simultaneously as
dilation-translation wavelet sets and dilation-reflection wavelet sets,
although the orthonormal structures generated in the two theories are
considerably different
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