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Image Decomposition: Theory, Numerical Schemes, and Performance Evaluation

Jerome Gilles

TL;DR

A method to evaluate the performance of algorithms that allow separate structures and textures or structures, textures, and noise to enhance understanding of the behavior of these models is proposed.

Abstract

This paper describes the many image decomposition models that allow to separate structures and textures or structures, textures, and noise. These models combined a total variation approach with different adapted functional spaces such as Besov or Contourlet spaces or a special oscillating function space based on the work of Yves Meyer. We propose a method to evaluate the performance of such algorithms to enhance understanding of the behavior of these models.

Image Decomposition: Theory, Numerical Schemes, and Performance Evaluation

TL;DR

A method to evaluate the performance of algorithms that allow separate structures and textures or structures, textures, and noise to enhance understanding of the behavior of these models is proposed.

Abstract

This paper describes the many image decomposition models that allow to separate structures and textures or structures, textures, and noise. These models combined a total variation approach with different adapted functional spaces such as Besov or Contourlet spaces or a special oscillating function space based on the work of Yves Meyer. We propose a method to evaluate the performance of such algorithms to enhance understanding of the behavior of these models.

Paper Structure

This paper contains 31 sections, 11 theorems, 126 equations, 29 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Wavelets
  4. Continuous Case
  5. Discrete Case
  6. Multiresolution Analysis
  7. Directional Multiresolution Analysis
  8. Ridgelets
  9. Curvelets
  10. Contourlets
  11. Function Spaces
  12. Sobolev Spaces
  13. Besov Spaces
  14. Ridgelet Spaces
  15. Contourlet Spaces
  16. ...and 16 more sections

Key Result

Theorem 1

Let $\psi\in L^2({\mathbb{R}})$ be a real wavelet that respects the following admissibility condition: where $\hat{\psi}$ is the Fourier transform of $\psi$. Then, all functions $f\in L^2({\mathbb{R}})$ verify and (Parseval relation)

Figures (29)

  • Figure 1: Contourlet transform principle.
  • Figure 2: Original Barbara, House, and Leopard images.
  • Figure 3: $BV$-$G$ structures $+$ textures image decomposition of Barbara image.
  • Figure 4: $BV$-$G$ structures $+$ textures image decomposition of House image.
  • Figure 5: $BV$-$G$ structures $+$ textures image decomposition of Leopard image.
  • ...and 24 more figures

Theorems & Definitions (22)

  • Theorem 1
  • Definition 1
  • Definition 2
  • Proposition 1
  • Proposition 2
  • Theorem 2
  • Corollary 1
  • Definition 3
  • Definition 4
  • Definition 5
  • ...and 12 more