From cardinal spline wavelet bases to highly coherent dictionaries

Wavelet families arise by scaling and translations of a prototype function, called the {\em {mother wavelet}}. The construction of wavelet bases for cardinal spline spaces is generally carried out within the multi-resolution analysis scheme. Thus, the usual way of increasing the dimension of the mul...

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Bibliographic Details
Published in:arXiv.org 2008-03
Main Authors: Andrle, Miroslav, Rebollo-Neira, Laura
Format: Article
Language:English
Subjects:
Coherence
Dictionaries
Scaling factors
Subspaces
Translations
Wavelet analysis
Online Access:Get full text
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Summary:Wavelet families arise by scaling and translations of a prototype function, called the {\em {mother wavelet}}. The construction of wavelet bases for cardinal spline spaces is generally carried out within the multi-resolution analysis scheme. Thus, the usual way of increasing the dimension of the multi-resolution subspaces is by augmenting the scaling factor. We show here that, when working on a compact interval, the identical effect can be achieved without changing the wavelet scale but reducing the translation parameter. By such a procedure we generate a redundant frame, called a {\em{dictionary}}, spanning the same spaces as a wavelet basis but with wavelets of broader support. We characterise the correlation of the dictionary elements by measuring their `coherence' and produce examples illustrating the relevance of highly coherent dictionaries to problems of sparse signal representation.
ISSN:2331-8422
DOI:10.48550/arxiv.0803.3741