From cardinal spline wavelet bases to highly coherent dictionaries

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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.

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Original languageEnglish
Article number172001
Pages (from-to)172001
Number of pages1
JournalJournal of Physics A: Mathematical and Theoretical
Volume41
Issue17
DOIs
StatePublished - 2 May 2008

Keywords

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

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