From cardinal spline wavelet bases to highly coherent dictionaries

Miroslav Andrle, Laura Rebollo-neira

Research output: Contribution to journalArticle

Abstract

Wavelet families arise by scaling and translations of a prototype function, called the 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 dictionary, spanning the same spaces as a wavelet basis but with wavelets of broader support. We characterize 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.
Original languageEnglish
Article number172001
JournalJournal of Physics A: Mathematical and Theoretical
Volume41
Issue number17
DOIs
Publication statusPublished - 2 May 2008

Bibliographical note

©2008 IOP Publishing Ltd. After the Embargo Period, the full text of the Accepted Manuscript may be made available on the non-commercial repository for anyone with an internet connection to read and download. After the Embargo Period a CC BY-NC-ND 3.0 licence applies to the Accepted Manuscript, in which case it may then only be posted under that CC BY-NC-ND licence provided that all the terms of the licence are adhered to, and any copyright notice and any cover sheet applied by IOP is not deleted or modified.

Keywords

  • wavelet families
  • prototype function
  • mother wavelet
  • cardinal spline spaces
  • multi-resolution analysis scheme

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