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Analysis of a wavelet frame based two-scale model for enhanced edges

Bin Dong, Ting Lin, Zuowei Shen, Peichu Xie

TL;DR

This article presents a wavelet frame based image restoration model that captures potential edges and facilitates the restoration procedure by a dedicated treatment both of singularity and of cartoon, and proves that this model converges to one variant of the celebrated Mumford-Shah model when adequate asymptotic specifications are given.

Abstract

Image restoration is a class of important tasks that emerges from a wide range of scientific disciplines. It has been noticed that most practical images can be modeled as a composition from a sparse singularity set (edges) where the image contents or their gradients change drastically, and cartoon chunks in which a high degree of regularity is dominant. Enhancing edges while promoting regularity elsewhere has been an important criterion for successful restoration in many image classes. In this article, we present a wavelet frame based image restoration model that captures potential edges and facilitates the restoration procedure by a dedicated treatment both of singularity and of cartoon. Moreover, its geometric robustness is enhanced by exploiting subtle inter-scale information available in the coarse image. To substantiate our intuition, we prove that this model converges to one variant of the celebrated Mumford-Shah model when adequate asymptotic specifications are given.

Analysis of a wavelet frame based two-scale model for enhanced edges

TL;DR

This article presents a wavelet frame based image restoration model that captures potential edges and facilitates the restoration procedure by a dedicated treatment both of singularity and of cartoon, and proves that this model converges to one variant of the celebrated Mumford-Shah model when adequate asymptotic specifications are given.

Abstract

Image restoration is a class of important tasks that emerges from a wide range of scientific disciplines. It has been noticed that most practical images can be modeled as a composition from a sparse singularity set (edges) where the image contents or their gradients change drastically, and cartoon chunks in which a high degree of regularity is dominant. Enhancing edges while promoting regularity elsewhere has been an important criterion for successful restoration in many image classes. In this article, we present a wavelet frame based image restoration model that captures potential edges and facilitates the restoration procedure by a dedicated treatment both of singularity and of cartoon. Moreover, its geometric robustness is enhanced by exploiting subtle inter-scale information available in the coarse image. To substantiate our intuition, we prove that this model converges to one variant of the celebrated Mumford-Shah model when adequate asymptotic specifications are given.
Paper Structure (14 sections, 31 theorems, 160 equations, 2 figures)

This paper contains 14 sections, 31 theorems, 160 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model constructions
  3. A wavelet frame based two-scale model (discrete)
  4. A Mumford--Shah type model (continuum)
  5. Model analysis: the main theorem
  6. Toy models and heuristics
  7. Asymptotic analysis of the regularity term
  8. Finite approximation of the singularity
  9. Near-singularity estimates
  10. Off-singularity estimates
  11. Asymptotic lower-bound: the universal $\Gamma$-property
  12. Tightness argument: the existence $\Gamma$-property
  13. Counting lemmas on the lattice
  14. Technical details of the mollifier

Key Result

Theorem 2.1

If the function $u\in BV(\Omega)$, its distributional gradient has the following (unique) standard decomposition, where $\nabla^au$ is an absolutely continuous measure (with respect to the $d$-dimensional Lebesgue measure), $\nabla^su$ is a singular measure that is supported on the (approximation) jump sets (ambrosio2000functions, or evans2015measure), and $Cu$ is a measure that is perpendicular

Figures (2)

  • Figure 3.1: Deformations $\phi_{m,i}^{\pm}$, through which the function value in the shaded areas are excluded (neglected).
  • Figure A.1: Approximating the $H_n$-tubular neighbourhood by a chain of rectangles of width $H'_n$.

Theorems & Definitions (55)

  • Theorem 2.1: ambrosio2000functions
  • Definition 2.1: ambrosio2000functions
  • Proposition 2.1: evans2015measure
  • Definition 2.2
  • Theorem 2.2
  • Proposition 2.2: tv-wf
  • Proposition 2.3
  • Proposition 2.4
  • proof : Proof of Theorem \ref{['main_theorem']}
  • Lemma 2.1: tv-wf
  • ...and 45 more