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2D Empirical Transforms. Wavelets, Ridgelets and Curvelets revisited

Jerome Gilles, Giang Tran, Stanley Osher

TL;DR

This paper revisits some well-known transforms of wavelet transform and shows that it is possible to build their empirical counterparts and proves that such constructions lead to different adaptive frames which show some promising properties for image analysis and processing.

Abstract

A recently developed new approach, called ``Empirical Wavelet Transform'', aims to build 1D adaptive wavelet frames accordingly to the analyzed signal. In this paper, we present several extensions of this approach to 2D signals (images). We revisit some well-known transforms (tensor wavelets, Littlewood-Paley wavelets, ridgelets and curvelets) and show that it is possible to build their empirical counterpart. We prove that such constructions lead to different adaptive frames which show some promising properties for image analysis and processing.

2D Empirical Transforms. Wavelets, Ridgelets and Curvelets revisited

TL;DR

This paper revisits some well-known transforms of wavelet transform and shows that it is possible to build their empirical counterparts and proves that such constructions lead to different adaptive frames which show some promising properties for image analysis and processing.

Abstract

A recently developed new approach, called ``Empirical Wavelet Transform'', aims to build 1D adaptive wavelet frames accordingly to the analyzed signal. In this paper, we present several extensions of this approach to 2D signals (images). We revisit some well-known transforms (tensor wavelets, Littlewood-Paley wavelets, ridgelets and curvelets) and show that it is possible to build their empirical counterpart. We prove that such constructions lead to different adaptive frames which show some promising properties for image analysis and processing.

Paper Structure

This paper contains 22 sections, 1 theorem, 35 equations, 24 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Notation
  4. The 1D EWT
  5. Fourier boundaries detection
  6. Original detection of local maxima method
  7. Finding the lowest minima
  8. Global trend removing
  9. Fine to Coarse histogram segmentation algorithm
  10. Comparison of the different approaches
  11. Tensor 2D EWT
  12. 2D Littlewood-Paley EWT
  13. Empirical Ridgelet Transform
  14. Empirical Curvelet Transform
  15. Frame properties
  16. ...and 7 more sections

Key Result

Theorem 1

Let $\mathcal{B}^{\mathcal{E}\mathcal{T}},\mathcal{B}^{\mathcal{E}\mathcal{LP}},\mathcal{B}^{\mathcal{E}\mathcal{R}},\mathcal{B}^{\mathcal{E}\mathcal{C}}_{I}$ and $\mathcal{B}^{\mathcal{E}\mathcal{C}}_{II}$ be the families of wavelets described in the previous sections. If their corresponding parame

Figures (24)

  • Figure 1: Fourier line decomposition principle and EWT basis construction.
  • Figure 2: Fourier supports definition based on local maxima detection and two of its main issues. See the text for full explanation.
  • Figure 3: Example of a profile extracted from the spectrum of a natural image.
  • Figure 4: Fourier support detection on a spectrum with a global trend (without any logarithm and $N=5$). Each row correspond to different preprocessing: none, plaw, poly and morpho, respectively. On the left column the detection based on the middle point between consecutive maxima is shown while the right column show the results when we keep the lowest minimum between consecutive maxima.
  • Figure 5: Fourier support detection on a spectrum with a global trend (with applying the logarithm and $N=5$). Each row correspond to different preprocessing: none, plaw, poly and morpho, respectively. On the left column the detection based on the middle point between consecutive maxima is shown while the right column show the results when we keep the lowest minimum between consecutive maxima.
  • ...and 19 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Proof 1