Elliptic and Pseudo-Parabolic PDE System with Orientation-Adaptive Anisotropy
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866914570492182528 |
|---|---|
| author | Ukai, Naotaka |
| author_facet | Ukai, Naotaka |
| contents | In this paper, we consider a coupled system of nonlinear elliptic and pseudo-parabolic PDEs arising in anisotropic monochrome image denoising with orientation-adaptation. The system is derived from the minimization process of a nonconvex energy functional. In particular, we focus on the problem of determining the initial data for the orientation variable. In previous studies, a natural procedure for determining such initial data has not been sufficiently clarified. To address this issue, we introduce a formulation in which the time derivative of the orientation variable is removed. This formulation enables the initial orientation data to be determined implicitly within a time-discrete scheme. On the other hand, this formulation weakens the conventional energy-dissipation structure and leads to new difficulties in constructing a stable variational time-evolution process. To overcome this issue, we develop an analysis based on a time-discretization method and establish the well-posedness of the proposed system, namely existence, uniqueness, and continuous dependence, as well as an energy-inequality. We also show that the proposed time-discrete scheme determines the initial orientation data consistently with the continuous model. These results provide a mathematical framework for the initial-orientation determination problem in orientation-adaptive anisotropic models. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_16354 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Elliptic and Pseudo-Parabolic PDE System with Orientation-Adaptive Anisotropy Ukai, Naotaka Analysis of PDEs In this paper, we consider a coupled system of nonlinear elliptic and pseudo-parabolic PDEs arising in anisotropic monochrome image denoising with orientation-adaptation. The system is derived from the minimization process of a nonconvex energy functional. In particular, we focus on the problem of determining the initial data for the orientation variable. In previous studies, a natural procedure for determining such initial data has not been sufficiently clarified. To address this issue, we introduce a formulation in which the time derivative of the orientation variable is removed. This formulation enables the initial orientation data to be determined implicitly within a time-discrete scheme. On the other hand, this formulation weakens the conventional energy-dissipation structure and leads to new difficulties in constructing a stable variational time-evolution process. To overcome this issue, we develop an analysis based on a time-discretization method and establish the well-posedness of the proposed system, namely existence, uniqueness, and continuous dependence, as well as an energy-inequality. We also show that the proposed time-discrete scheme determines the initial orientation data consistently with the continuous model. These results provide a mathematical framework for the initial-orientation determination problem in orientation-adaptive anisotropic models. |
| title | Elliptic and Pseudo-Parabolic PDE System with Orientation-Adaptive Anisotropy |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2510.16354 |