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Continuous empirical wavelets systems

Jerome Gilles

Abstract

The recently proposed empirical wavelet transform was based on a particular type of filter. In this paper, we aim to propose a general framework for the construction of empirical wavelet systems in the continuous case. We define a well-suited formalism and then investigate some general properties of empirical wavelet systems. In particular, we provide some sufficient conditions to the existence of a reconstruction formula. In the second part of the paper, we propose the construction of empirical wavelet systems based on some classic mother wavelets.

Continuous empirical wavelets systems

Abstract

The recently proposed empirical wavelet transform was based on a particular type of filter. In this paper, we aim to propose a general framework for the construction of empirical wavelet systems in the continuous case. We define a well-suited formalism and then investigate some general properties of empirical wavelet systems. In particular, we provide some sufficient conditions to the existence of a reconstruction formula. In the second part of the paper, we propose the construction of empirical wavelet systems based on some classic mother wavelets.

Paper Structure

This paper contains 11 sections, 7 theorems, 39 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Generalities - Notations
  3. Partitioning of the Fourier line
  4. Empirical Wavelet Systems
  5. Construction of empirical wavelet systems
  6. Empirical Littlewood-Paley wavelets
  7. Empirical Meyer wavelets
  8. Empirical Shannon wavelets
  9. Empirical Gabor wavelets
  10. Numerical implementation
  11. Conclusion

Key Result

Proposition 1

The Continuous Empirical Wavelet Transform ${\mathcal{E}}_{\psi}^f(b,n)$ is equivalent to the convolution of $f$ with the function $\overline{\psi^*_n}(t)$, i.e

Figures (8)

  • Figure 1: Example of partitions ${\mathcal{V}}$ (top) and ${\mathcal{V}}^*$ (bottom) of the Fourier line.
  • Figure 2: Example of a ${\mathcal{V}}$ partition with infinite rays with its associated support centers.
  • Figure 3: Construction of empirical Littlewood-Paley wavelets
  • Figure 4: Example of an empirical Littlewood-Paley wavelet filter bank in the Fourier domain corresponding to a ${\mathcal{V}}^*$ partition with left and right rays.
  • Figure 5: Example of an empirical Meyer wavelet filter bank in the Fourier domain corresponding to a ${\mathcal{V}}^*$ partition with left and right rays.
  • ...and 3 more figures

Theorems & Definitions (9)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Proposition 2
  • Corollary 1
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6