AI 中文总结
针对显微镜图像中接触和分裂细胞核难以分割的问题，本文提出带单调反应项的正则化主观曲面流模型，证明其全局经典适定性与稳定性，为细胞核图像分析提供数学基础。
AI 中文摘要
在显微镜图像中，接触和分裂的细胞核可能呈现为连通结构，这使得分割过程中难以区分相邻细胞核。本文针对该场景引入并分析一种正则化主观曲面模型：在光滑有界区域Ω⊂ℝⁿ上，给定齐次Dirichlet边界条件，模型为u_t = νΔu + (ε²+|∇u|²)^(1/2) div(G(x)∇u/(ε²+|∇u|²)^(1/2)) - μΛ(x)H_η(u-q)，其中ε>0、ν>0为固定正则化参数，G为严格正的光滑边缘系数，非负相互作用权重Λ包含相邻细胞核候选的固定信息。本研究核心目标是建立该模型的存在性理论：对取值于[0,1]的相容C^(2+α)初始数据，证明其存在唯一全局经典解，该解在每个有限时间区间上为Schauder类，且保持物理范围、满足有限时间Schauder估计，对初始数据的依赖具有L^∞非扩张性。主要分析步骤是结合∂Ω附近的梯度估计与内部梯度估计，得到全局空间梯度界。这些结果为将该模型应用于3D及3D+时间显微镜图像数据中接触和分裂细胞核的分析提供了数学基础。
英文摘要
Touching and dividing cell nuclei may appear as connected structures in microscopy images, making it difficult to distinguish neighboring nuclei during segmentation. We introduce and analyze a regularized subjective-surface model designed for this setting. On a smooth bounded domain $Ω\subset\mathbb{R}^n$ with homogeneous Dirichlet boundary conditions, the model is \[ u_t = νΔu + \left(\varepsilon^2+|\nabla u|^2\right)^{1/2} \operatorname{div}\!\left( G(x)\frac{\nabla u} {\left(\varepsilon^2+|\nabla u|^2\right)^{1/2}} \right) - μΛ(x)H_η(u-q). \] Here, $\varepsilon>0$ and $ν>0$ are fixed regularization parameters, $G$ is a strictly positive smooth edge coefficient, and the nonnegative interaction weight $Λ$ incorporates fixed information about neighboring nucleus candidates. The principal objective of this work is to establish an existence theory for the proposed model. For compatible $C^{2+α}$ initial data taking values in $[0,1]$, we prove the existence and uniqueness of a global classical solution whose restriction to every finite time interval is Schauder-classical, together with preservation of the physical range, finite-time Schauder estimates, and $L^\infty$-nonexpansive dependence on the initial data. The main analytical step is a global spatial-gradient bound, obtained by combining gradient estimates near $\partialΩ$ with interior gradient estimates. These results provide a mathematical foundation for applying the model to the analysis of touching and dividing nuclei in 3D and 3D+time microscopy image data.
