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Well-posedness of a nonlinear integro-differential problem and its rearranged formulation

Gonzalo Galiano, Emanuele Schiavi, Julián Velasco

TL;DR

This work analyzes a nonlinear nonlocal integro-differential evolution with kernel $\mathcal{K}_h$ and fidelity parameter $\lambda$, posed on a bounded domain $\Omega$ with initial data $u_0\in BV(\Omega)\cap L^\infty(\Omega)$. By introducing the decreasing rearrangement, the authors reduce the problem to a one-dimensional reformulation on $\Omega_*=(0,|\Omega|)$, establishing well-posedness, stability, and time-invariant level-set structure, and proving an exact equivalence between the multi-dimensional and rearranged problems. They derive asymptotic, shock-filter–style behavior as the window parameter $h$ varies and develop a fast implicit discretization based on the 1D reformulation, demonstrated on histogram-based image segmentation tasks. The results yield a principled, efficient framework for nonlocal denoising-segmentation with potential extensions to non-homogeneous kernels via relative rearrangements.

Abstract

We study the existence and uniqueness of solutions of a nonlinear integro-differential problem which we reformulate introducing the notion of the decreasing rearrangement of the solution. A dimensional reduction of the problem is obtained and a detailed analysis of the properties of the solutions of the model is provided. Finally, a fast numerical method is devised and implemented to show the performance of the model when typical image processing tasks such as filtering and segmentation are performed.

Well-posedness of a nonlinear integro-differential problem and its rearranged formulation

TL;DR

This work analyzes a nonlinear nonlocal integro-differential evolution with kernel and fidelity parameter , posed on a bounded domain with initial data . By introducing the decreasing rearrangement, the authors reduce the problem to a one-dimensional reformulation on , establishing well-posedness, stability, and time-invariant level-set structure, and proving an exact equivalence between the multi-dimensional and rearranged problems. They derive asymptotic, shock-filter–style behavior as the window parameter varies and develop a fast implicit discretization based on the 1D reformulation, demonstrated on histogram-based image segmentation tasks. The results yield a principled, efficient framework for nonlocal denoising-segmentation with potential extensions to non-homogeneous kernels via relative rearrangements.

Abstract

We study the existence and uniqueness of solutions of a nonlinear integro-differential problem which we reformulate introducing the notion of the decreasing rearrangement of the solution. A dimensional reduction of the problem is obtained and a detailed analysis of the properties of the solutions of the model is provided. Finally, a fast numerical method is devised and implemented to show the performance of the model when typical image processing tasks such as filtering and segmentation are performed.

Paper Structure

This paper contains 7 sections, 5 theorems, 82 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. The decreasing rearrangement
  3. Main results
  4. Discretization and numerical examples
  5. Example. Histogram based image segmentation
  6. Proofs
  7. Conclusions

Key Result

Theorem 1

Assume (H). Then there exists a unique solution $u\in C^\infty([0,T];{\cal X})$ of problem P$(\Omega,u_0)$. In addition, if $u_{01},u_{02}\in {\cal X}$ and $u_1,u_2 \in C^\infty([0,T];{\cal X})$ are the corresponding solutions to problems $P(\Omega,u_{01}),~P(\Omega,u_{02})$ then for some constant $C>0$. Finally, suppose that $u_0(\mathbf{x}_1)=u_0(\mathbf{x}_2)$ for some $\mathbf{x}_1,\mathbf{x}

Figures (1)

  • Figure 1: Example. Nucleus and citoplasm segmentation process. First column corresponds to the initial image. Second column, to the background extraction, and third column to the nucleus segmentation. The citoplasm is the difference between the images shown in the third and second columns. Finally, fourth column shows the difference between the ground-truth nucleus segmentation and the obtained with our method.

Theorems & Definitions (7)

  • Theorem 1
  • Remark 1
  • Corollary 1
  • Remark 2
  • Theorem 2
  • Theorem 3
  • Lemma 1