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On a cross-diffusion system arising in image denosing

Gonzalo Galiano, Julián Velasco

TL;DR

A generalization of a cross-diffusion problem deduced from a nonlinear complex-variable diffusion model for signal and image denoising is studied, and the existence of weak solutions of the time-independent problem with fidelity terms under mild conditions on the data problem is proved.

Abstract

We study a generalization of a cross-diffusion problem deduced from a nonlinear complex-variable diffusion model for signal and image denoising. We prove the existence of weak solutions of the time-independent problem with fidelity terms under mild conditions on the data problem. Then, we show that this translates on the well-posedness of a quasi-steady state approximation of the evolution problem, and also prove the existence of weak solutions of the latter under more restrictive hypothesis. We finally perform some numerical simulations for image denoising, comparing the performance of the cross-diffusion model and its corresponding scalar Perona-Malik equation.

On a cross-diffusion system arising in image denosing

TL;DR

A generalization of a cross-diffusion problem deduced from a nonlinear complex-variable diffusion model for signal and image denoising is studied, and the existence of weak solutions of the time-independent problem with fidelity terms under mild conditions on the data problem is proved.

Abstract

We study a generalization of a cross-diffusion problem deduced from a nonlinear complex-variable diffusion model for signal and image denoising. We prove the existence of weak solutions of the time-independent problem with fidelity terms under mild conditions on the data problem. Then, we show that this translates on the well-posedness of a quasi-steady state approximation of the evolution problem, and also prove the existence of weak solutions of the latter under more restrictive hypothesis. We finally perform some numerical simulations for image denoising, comparing the performance of the cross-diffusion model and its corresponding scalar Perona-Malik equation.

Paper Structure

This paper contains 7 sections, 3 theorems, 56 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main results
  3. Numerical simulations
  4. Discretization
  5. Experiment data and results
  6. Proofs
  7. Conclusions

Key Result

Theorem 1

Assume (H), and let $\gamma_i >0$, and $G_i\in L^\infty(\Omega)$, for $i=1,2$. Then, there exists a weak solution $\mathbf{u} \in H^1(\Omega)^2\cap L^\infty(\Omega)^2$, of the stationary problem (eqs:pde)-(eqs:bc).

Figures (3)

  • Figure 1: Set of test images. First and second rows are natural images: added value, boat, clock, house, tree, and test. The latter is not natural but it actually behaves as such in the denoising procedure. Third row are texture images: bark, bricks, bubbles, holes, and straw. All of them are of size $512\times512$ but clock and test, which are $256\times256$.
  • Figure 2: Detail of the natural image added value showing the performance of each method.
  • Figure 3: Contours plots for a detail of the texture image holes.

Theorems & Definitions (4)

  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Remark 1