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Empirical Voronoi Wavelets

Jerome Gilles

TL;DR

This work addresses the need for adaptable yet geometrically regular partitions in 2D empirical wavelets. It introduces Empirical Voronoid Wavelets (EVW), which detect harmonic-mode positions via scale-space analysis, construct a Voronoi partition from these seeds, and build a Voronoi-based wavelet filter bank. The EVW framework provides a Fourier-domain distance-transform-based filter construction with a dual-frame recovery, enabling stable, invertible decompositions, and demonstrates the partition's alignment with harmonic modes on synthetic data. The approach offers a practical balance between adaptability and geometric regularity, with publicly available MATLAB code to facilitate adoption in image analysis tasks.

Abstract

Recently, the construction of 2D empirical wavelets based on partitioning the Fourier domain with the watershed transform has been proposed. If such approach can build partitions of completely arbitrary shapes, for some applications, it is desirable to keep a certain level of regularity in the geometry of the obtained partitions. In this paper, we propose to build such partition using Voronoi diagrams. This solution allows us to keep a high level of adaptability while guaranteeing a minimum level of geometric regularity in the detected partition.

Empirical Voronoi Wavelets

TL;DR

This work addresses the need for adaptable yet geometrically regular partitions in 2D empirical wavelets. It introduces Empirical Voronoid Wavelets (EVW), which detect harmonic-mode positions via scale-space analysis, construct a Voronoi partition from these seeds, and build a Voronoi-based wavelet filter bank. The EVW framework provides a Fourier-domain distance-transform-based filter construction with a dual-frame recovery, enabling stable, invertible decompositions, and demonstrates the partition's alignment with harmonic modes on synthetic data. The approach offers a practical balance between adaptability and geometric regularity, with publicly available MATLAB code to facilitate adoption in image analysis tasks.

Abstract

Recently, the construction of 2D empirical wavelets based on partitioning the Fourier domain with the watershed transform has been proposed. If such approach can build partitions of completely arbitrary shapes, for some applications, it is desirable to keep a certain level of regularity in the geometry of the obtained partitions. In this paper, we propose to build such partition using Voronoi diagrams. This solution allows us to keep a high level of adaptability while guaranteeing a minimum level of geometric regularity in the detected partition.
Paper Structure (9 sections, 6 equations, 3 figures, 1 algorithm)

This paper contains 9 sections, 6 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Empirical wavelets
  3. Empirical Voronoid wavelets
  4. Detection of harmonic mode positions
  5. Voronoi partitioning
  6. Empirical Voronoi Wavelet transform
  7. Experiments
  8. Conclusion
  9. Acknowledgement

Figures (3)

  • Figure 1: Existing 2D partitions of the Fourier domain. These different types of partitions correspond to a) tensor wavelets, b) Littlewood-Paley wavelets, c) curvelet type, d) watershed wavelets.
  • Figure 2: a) existence of local maxima in the scale-space representation. Each curve correspond to one originally detected maxima. The vertical axis corresponds to the scale parameter $\sigma$. b) positions of maxima $\{\xi_n^0\}_{n\in\Lambda}$ corresponding to the meaningful modes. c) the Voronoi partition associated with $\{\xi_n^0\}_{n\in\Lambda}$.
  • Figure 3: Example of an empirical Voronoi wavelet transform. The top left image is an synthetically generated input image, in particular it contains four harmonic modes. The top right image depicted the detected Voronoi partition superimposed on the logarithm of the magnitude spectrum of the input image. The remaining images correspond to the outputs of the different empirical Voronoi wavelet filters. Note that the images that look black do actually contain some information of very small energy compared to the main harmonic modes.