Uncertainty principles and optimally sparse wavelet transforms

Ron Levie, Nir Sochen

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper we introduce a new localization framework for wavelet transforms, such as the 1D wavelet transform and the Shearlet transform. Our goal is to design nonadaptive window functions that promote sparsity in some sense. For that, we introduce a framework for analyzing localization aspects of window functions. Our localization theory diverges from the conventional theory in two ways. First, we distinguish between the group generators, and the operators that measure localization (called observables). Second, we define the uncertainty of a signal transform as a whole, instead of defining the uncertainty of an individual window. We show that the uncertainty of a window function, in the signal space, is closely related to the localization of the reproducing kernel of the wavelet transform, in phase space. As a result, we show that using uncertainty minimizing window functions, results in representations which are optimally sparse in some sense.

Original languageEnglish
Pages (from-to)811-867
Number of pages57
JournalApplied and Computational Harmonic Analysis
Volume48
Issue number3
DOIs
StatePublished - May 2020

Keywords

  • Continuous wavelet transform
  • Group representation
  • Sparse representation
  • Uncertainty principle

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