From cardinal spline wavelet bases to highly coherent dictionaries

TL;DR

This paper introduces a method to generate highly coherent, redundant wavelet dictionaries on compact intervals by reducing translation parameters, enhancing sparse signal representation capabilities.

Contribution

It presents a novel approach to constructing wavelet dictionaries with high coherence without changing the wavelet scale, expanding the tools for sparse signal processing.

Findings

Generated highly coherent wavelet dictionaries on compact intervals.

Demonstrated the relevance of coherence in sparse signal representation.

Provided examples illustrating improved sparse coding with these dictionaries.

Abstract

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.