16,739 research outputs found
Sampling and Reconstruction of Sparse Signals on Circulant Graphs - An Introduction to Graph-FRI
With the objective of employing graphs toward a more generalized theory of
signal processing, we present a novel sampling framework for (wavelet-)sparse
signals defined on circulant graphs which extends basic properties of Finite
Rate of Innovation (FRI) theory to the graph domain, and can be applied to
arbitrary graphs via suitable approximation schemes. At its core, the
introduced Graph-FRI-framework states that any K-sparse signal on the vertices
of a circulant graph can be perfectly reconstructed from its
dimensionality-reduced representation in the graph spectral domain, the Graph
Fourier Transform (GFT), of minimum size 2K. By leveraging the recently
developed theory of e-splines and e-spline wavelets on graphs, one can
decompose this graph spectral transformation into the multiresolution low-pass
filtering operation with a graph e-spline filter, and subsequent transformation
to the spectral graph domain; this allows to infer a distinct sampling pattern,
and, ultimately, the structure of an associated coarsened graph, which
preserves essential properties of the original, including circularity and,
where applicable, the graph generating set.Comment: To appear in Appl. Comput. Harmon. Anal. (2017
Construction of Parseval wavelets from redundant filter systems
We consider wavelets in L^2(R^d) which have generalized multiresolutions.
This means that the initial resolution subspace V_0 in L^2(R^d) is not singly
generated. As a result, the representation of the integer lattice Z^d
restricted to V_0 has a nontrivial multiplicity function. We show how the
corresponding analysis and synthesis for these wavelets can be understood in
terms of unitary-matrix-valued functions on a torus acting on a certain vector
bundle. Specifically, we show how the wavelet functions on R^d can be
constructed directly from the generalized wavelet filters.Comment: 34 pages, AMS-LaTeX ("amsproc" document class) v2 changes minor typos
in Sections 1 and 4, v3 adds a number of references on GMRA theory and
wavelet multiplicity analysis; v4 adds material on pages 2, 3, 5 and 10, and
two more reference
Efficient Quantum Transforms
Quantum mechanics requires the operation of quantum computers to be unitary,
and thus makes it important to have general techniques for developing fast
quantum algorithms for computing unitary transforms. A quantum routine for
computing a generalized Kronecker product is given. Applications include
re-development of the networks for computing the Walsh-Hadamard and the quantum
Fourier transform. New networks for two wavelet transforms are given. Quantum
computation of Fourier transforms for non-Abelian groups is defined. A slightly
relaxed definition is shown to simplify the analysis and the networks that
computes the transforms. Efficient networks for computing such transforms for a
class of metacyclic groups are introduced. A novel network for computing a
Fourier transform for a group used in quantum error-correction is also given.Comment: 30 pages, LaTeX2e, 7 figures include
Recent Progress in Shearlet Theory: Systematic Construction of Shearlet Dilation Groups, Characterization of Wavefront Sets, and New Embeddings
The class of generalized shearlet dilation groups has recently been developed
to allow the unified treatment of various shearlet groups and associated
shearlet transforms that had previously been studied on a case-by-case basis.
We consider several aspects of these groups: First, their systematic
construction from associative algebras, secondly, their suitability for the
characterization of wavefront sets, and finally, the question of constructing
embeddings into the symplectic group in a way that intertwines the
quasi-regular representation with the metaplectic one. For all questions, it is
possible to treat the full class of generalized shearlet groups in a
comprehensive and unified way, thus generalizing known results to an infinity
of new cases. Our presentation emphasizes the interplay between the algebraic
structure underlying the construction of the shearlet dilation groups, the
geometric properties of the dual action, and the analytic properties of the
associated shearlet transforms.Comment: 28 page
Wavelets and Fast Numerical Algorithms
Wavelet based algorithms in numerical analysis are similar to other transform
methods in that vectors and operators are expanded into a basis and the
computations take place in this new system of coordinates. However, due to the
recursive definition of wavelets, their controllable localization in both space
and wave number (time and frequency) domains, and the vanishing moments
property, wavelet based algorithms exhibit new and important properties.
For example, the multiresolution structure of the wavelet expansions brings
about an efficient organization of transformations on a given scale and of
interactions between different neighbouring scales. Moreover, wide classes of
operators which naively would require a full (dense) matrix for their numerical
description, have sparse representations in wavelet bases. For these operators
sparse representations lead to fast numerical algorithms, and thus address a
critical numerical issue.
We note that wavelet based algorithms provide a systematic generalization of
the Fast Multipole Method (FMM) and its descendents.
These topics will be the subject of the lecture. Starting from the notion of
multiresolution analysis, we will consider the so-called non-standard form
(which achieves decoupling among the scales) and the associated fast numerical
algorithms. Examples of non-standard forms of several basic operators (e.g.
derivatives) will be computed explicitly.Comment: 32 pages, uuencoded tar-compressed LaTeX file. Uses epsf.sty (see
`macros'
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
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