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Elliptic and Pseudo-Parabolic Gradient System Arising from Anisotropic Image-Denoising with Orientation Adaptation

Naotaka Ukai

TL;DR

The paper analyzes a coupled nonlinear elliptic–pseudo-parabolic PDE system for anisotropic monochrome image denoising with orientation adaptation, where the orientation variable $\alpha$ evolves without a time derivative, enabling automatic initialization. Formulated as a gradient-flow of the nonconvex energy $E(\alpha,u)$, the model demands rigorous well-posedness and energy-dissipation analysis, which the authors achieve via a time-discretization scheme that also prescribes how to initialize $\alpha$. They prove existence, uniqueness, continuous dependence, and an energy-dissipation inequality for the continuous system, and establish convergence and stability of the discrete scheme to a weak solution, along with crucial bounds $0\le u\le 1$. The results provide a robust numerical framework for orientation initialization in orientation-adaptive anisotropic image denoising and contribute to the mathematical understanding of nonconvex coupled gradient systems with automatic initial data determination.

Abstract

In this paper, we consider a coupled system of nonlinear elliptic and pseudo-parabolic PDEs arising in anisotropic monochrome image-denoising with an orientation adaptation. This system is motivated by the minimization of a nonconvex energy functional. This study focuses on the treatment of the initial data for the orientation variable. Previous models have not provided explicit procedures or clarified a natural and convincing method for its determination. To resolve this issue, we propose a formulation in which the time derivative of the orientation variable is removed. This enables the initial value to be computed automatically within the equation. This formulation weakens the energy-dissipation property and introduces new challenges in constructing a stable minimization process. Consequently, a different approach from previous works is required. In light of the above, this paper aims to establish the well-posedness and energy-dissipation for our system. The proofs are based on a time-discretization method. The proposed time-discrete scheme determines the initial orientation data and ensures consistency with the continuous model. These results guarantee the stability and effectiveness of the proposed method for numerical implementation and provide a method for determining the initial data of orientation in the time-discretization scheme.

Elliptic and Pseudo-Parabolic Gradient System Arising from Anisotropic Image-Denoising with Orientation Adaptation

TL;DR

The paper analyzes a coupled nonlinear elliptic–pseudo-parabolic PDE system for anisotropic monochrome image denoising with orientation adaptation, where the orientation variable evolves without a time derivative, enabling automatic initialization. Formulated as a gradient-flow of the nonconvex energy , the model demands rigorous well-posedness and energy-dissipation analysis, which the authors achieve via a time-discretization scheme that also prescribes how to initialize . They prove existence, uniqueness, continuous dependence, and an energy-dissipation inequality for the continuous system, and establish convergence and stability of the discrete scheme to a weak solution, along with crucial bounds . The results provide a robust numerical framework for orientation initialization in orientation-adaptive anisotropic image denoising and contribute to the mathematical understanding of nonconvex coupled gradient systems with automatic initial data determination.

Abstract

In this paper, we consider a coupled system of nonlinear elliptic and pseudo-parabolic PDEs arising in anisotropic monochrome image-denoising with an orientation adaptation. This system is motivated by the minimization of a nonconvex energy functional. This study focuses on the treatment of the initial data for the orientation variable. Previous models have not provided explicit procedures or clarified a natural and convincing method for its determination. To resolve this issue, we propose a formulation in which the time derivative of the orientation variable is removed. This enables the initial value to be computed automatically within the equation. This formulation weakens the energy-dissipation property and introduces new challenges in constructing a stable minimization process. Consequently, a different approach from previous works is required. In light of the above, this paper aims to establish the well-posedness and energy-dissipation for our system. The proofs are based on a time-discretization method. The proposed time-discrete scheme determines the initial orientation data and ensures consistency with the continuous model. These results guarantee the stability and effectiveness of the proposed method for numerical implementation and provide a method for determining the initial data of orientation in the time-discretization scheme.
Paper Structure (8 sections, 2 theorems, 97 equations)

This paper contains 8 sections, 2 theorems, 97 equations.

Table of Contents

  1. Preliminaries
  2. Main results
  3. Time-discretization scheme
  4. Proof of Main Theorem.
  5. Uniqueness and continuous dependence of system (S)
  6. Existence of a limit $[\alpha,u]$ for a subsequence of $\{[\alpha]_\tau,[u]_\tau\}$
  7. Existence and energy-dissipation for the system (S)
  8. Verification of Boundedness: $0\leq u \leq 1$ in $\overline{Q}$

Key Result

Theorem 1

(Existence, uniqueness of solution with energy-inequality) There exists a positive constant $\widehat{\kappa} > 0$, and a sufficiently small constant $\widehat{\tau}\in (0,1)$ such that for any $\kappa > \widehat{\kappa}$, and for any $\tau \in (0,\widehat{\tau})$, (AP)$^\tau$ admits a unique soluti

Theorems & Definitions (12)

  • Example 1
  • Definition 1: Mosco-convergence: cf.MR0298508
  • Definition 2: $\Gamma$-convergence; cf.MR1201152
  • Remark 1
  • Definition 3
  • Remark 2
  • Definition 4
  • Theorem 1
  • Lemma 3.1
  • proof
  • ...and 2 more