发表机构
University of Pavia; IMATI–C.N.R. Pavia（帕维亚大学; 帕维亚 IMATI-国家研究委员会研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对基于共识的图像分割动力学模型，通过三级正则化方案解决了有界域上的适定性难题，证明了原模型的弱解全局存在性。
AI 中文摘要
本文分析了一个近期提出的基于Hegselmann-Krause型交互动力学的动力学模型，该模型描述了平均场机制下的基于共识的图像分割，其中粒子密度依赖于空间坐标和归一化灰度（特征）值。该模型被表述为一个演化型非局部偏微分方程，包含一个涉及有界置信交互核的非局部空间漂移项，该核依赖于空间和特征的接近程度，以及一个在特征变量中退化的空间扩散项，表示随机不确定性。在有界域上建立适定性（这对于实际图像处理至关重要）带来了显著的分析挑战：由于具有跳跃间断的核和退化空间扩散导致缺乏空间导数的均匀控制，在齐次Neumann条件下出现复杂的非标准边界项，以及由于缺乏关于特征变量的导数而产生的紧性缺失。为了解决这些困难，我们引入了一个三级正则化方案。具体来说，我们平滑交互核，对非局部交互项应用结构边界正则化以简化边界条件，并在空间和特征变量中引入人工扩散以解决退化问题。我们通过有限差分格式证明了正则化系统的适定性和正则性，并研究了正则化参数消失时的渐近极限，证明了原始模型的弱/分布解的全局存在性。该方法通过将分布解框架扩展到有界域和具有跳跃间断的非光滑交互核，推进了图像分割动力学模型的数学理论。
英文摘要
In this article, we analyze a recently introduced kinetic model based on Hegselmann-Krause-type interaction dynamics describing consensus-based image segmentation in the mean-field regime, where particle density depends on spatial coordinates and the normalized gray level (feature) value. The model is formulated as an evolutive nonlocal partial differential equation featuring a non-local spatial drift term involving a bounded confidence interaction kernel, depending on spatial and feature proximity, and a spatial diffusion term degenerating in the feature variable representing aleatoric uncertainties. Establishing well-posedness on a bounded domain, essential for practical image processing, presents significant analytical challenges: the lack of uniform control of space derivatives due to kernels with jump discontinuities and degenerate spatial diffusion, complex non-standard boundary terms under homogeneous Neumann conditions, and a lack of compactness arising from the absence of derivatives with respect to the feature variable. To address these difficulties, we introduce a three-level regularization scheme. Specifically, we smooth the interaction kernel, apply structural boundary regularization to the non-local interaction term to simplify boundary conditions, and incorporate artificial diffusion in spatial and feature variables to resolve degeneracy. We prove the well-posedness and regularity of the regularized system via a finite difference scheme and study the asymptotic limit as regularization parameters vanish, proving the existence of weak/distributional solutions of the original model. This approach advances the mathematical theory of kinetic models for image segmentation by extending the framework of distributional solutions to bounded domains and non-smooth interaction kernels with jump discontinuities.
