5,749 research outputs found
Intertwining wavelets or Multiresolution analysis on graphs through random forests
We propose a new method for performing multiscale analysis of functions
defined on the vertices of a finite connected weighted graph. Our approach
relies on a random spanning forest to downsample the set of vertices, and on
approximate solutions of Markov intertwining relation to provide a subgraph
structure and a filter bank leading to a wavelet basis of the set of functions.
Our construction involves two parameters q and q'. The first one controls the
mean number of kept vertices in the downsampling, while the second one is a
tuning parameter between space localization and frequency localization. We
provide an explicit reconstruction formula, bounds on the reconstruction
operator norm and on the error in the intertwining relation, and a Jackson-like
inequality. These bounds lead to recommend a way to choose the parameters q and
q'. We illustrate the method by numerical experiments.Comment: 39 pages, 12 figure
Astronomical Data Analysis and Sparsity: from Wavelets to Compressed Sensing
Wavelets have been used extensively for several years now in astronomy for
many purposes, ranging from data filtering and deconvolution, to star and
galaxy detection or cosmic ray removal. More recent sparse representations such
ridgelets or curvelets have also been proposed for the detection of anisotropic
features such cosmic strings in the cosmic microwave background.
We review in this paper a range of methods based on sparsity that have been
proposed for astronomical data analysis. We also discuss what is the impact of
Compressed Sensing, the new sampling theory, in astronomy for collecting the
data, transferring them to the earth or reconstructing an image from incomplete
measurements.Comment: Submitted. Full paper will figures available at
http://jstarck.free.fr/IEEE09_SparseAstro.pd
Orthogonal Wavelets via Filter Banks: Theory and Applications
Wavelets are used in many applications, including image processing, signal analysis and seismology. The critical problem is the representation of a signal using a small number of computable functions, such that it is represented in a concise and computationally efficient form. It is shown that wavelets are closely related to filter banks (sub band filtering) and that there is a direct analogy between multiresolution analysis in continuous time and a filter bank in discrete time. This provides a clear physical interpretation of the approximation and detail spaces of multiresolution analysis in terms of the frequency bands of a signal. Only orthogonal wavelets, which are derived from orthogonal filter banks, are discussed. Several examples and applications are considered
Compactly Supported Wavelets Derived From Legendre Polynomials: Spherical Harmonic Wavelets
A new family of wavelets is introduced, which is associated with Legendre
polynomials. These wavelets, termed spherical harmonic or Legendre wavelets,
possess compact support. The method for the wavelet construction is derived
from the association of ordinary second order differential equations with
multiresolution filters. The low-pass filter associated with Legendre
multiresolution analysis is a linear phase finite impulse response filter
(FIR).Comment: 6 pages, 6 figures, 1 table In: Computational Methods in Circuits and
Systems Applications, WSEAS press, pp.211-215, 2003. ISBN: 960-8052-88-
A Multiscale Pyramid Transform for Graph Signals
Multiscale transforms designed to process analog and discrete-time signals
and images cannot be directly applied to analyze high-dimensional data residing
on the vertices of a weighted graph, as they do not capture the intrinsic
geometric structure of the underlying graph data domain. In this paper, we
adapt the Laplacian pyramid transform for signals on Euclidean domains so that
it can be used to analyze high-dimensional data residing on the vertices of a
weighted graph. Our approach is to study existing methods and develop new
methods for the four fundamental operations of graph downsampling, graph
reduction, and filtering and interpolation of signals on graphs. Equipped with
appropriate notions of these operations, we leverage the basic multiscale
constructs and intuitions from classical signal processing to generate a
transform that yields both a multiresolution of graphs and an associated
multiresolution of a graph signal on the underlying sequence of graphs.Comment: 16 pages, 13 figure
Optimal multiple description and multiresolution scalar quantizer design
The author presents new algorithms for fixed-rate multiple description and multiresolution scalar quantizer design. The algorithms both run in time polynomial in the size of the source alphabet and guarantee globally optimal solutions. To the author's knowledge, these are the first globally optimal design algorithms for multiple description and multiresolution quantizers
Wavelets and their use
This review paper is intended to give a useful guide for those who want to
apply discrete wavelets in their practice. The notion of wavelets and their use
in practical computing and various applications are briefly described, but
rigorous proofs of mathematical statements are omitted, and the reader is just
referred to corresponding literature. The multiresolution analysis and fast
wavelet transform became a standard procedure for dealing with discrete
wavelets. The proper choice of a wavelet and use of nonstandard matrix
multiplication are often crucial for achievement of a goal. Analysis of various
functions with the help of wavelets allows to reveal fractal structures,
singularities etc. Wavelet transform of operator expressions helps solve some
equations. In practical applications one deals often with the discretized
functions, and the problem of stability of wavelet transform and corresponding
numerical algorithms becomes important. After discussing all these topics we
turn to practical applications of the wavelet machinery. They are so numerous
that we have to limit ourselves by some examples only. The authors would be
grateful for any comments which improve this review paper and move us closer to
the goal proclaimed in the first phrase of the abstract.Comment: 63 pages with 22 ps-figures, to be published in Physics-Uspekh
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