带单调反应项的正则化主观表面流：适定性与收敛性
Regularized Subjective-Surface Flow with Monotone Reaction: Viscosity Well-Posedness and Convergence
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中文总结 AI 辅助
该研究针对用于细胞核分割的正则化主观表面模型，分析其粘性极限的适定性与收敛性，确定了该模型在显微图像分割中的几何极限。
中文摘要 AI 辅助
我们研究了为接触和分裂细胞核引入的正则化主观表面模型的消失正则化极限，该模型此前针对固定正正则化参数进行过分析。在光滑有界域Ω⊂ℝᵈ上，该模型为：∂ₜu^ε,ν = νΔu^ε,ν + A_ε(∇u^ε,ν) div( G(x)∇u^ε,ν / A_ε(∇u^ε,ν) ) - μR_η(x,u^ε,ν)，其中A_ε(p)=√(ε²+|p|²)，R_η(x,r)=Λ(x)H_η(r−q)，该模型满足齐次狄利克雷边界条件和给定初始轮廓。我们将极限加权水平集平均曲率方程表述为粘性初边值问题，并建立其全局存在性、唯一性、[0,1]区间的保性以及对初始数据的非扩张依赖性。我们确定了正则化主算子在零梯度处的直接上下极限值，并确定了恢复几何方向值所需的额外自变量。对于任意T>0，当ε,ν→0时，正则化解在[0,T]×Ω̄上一致收敛到唯一的粘性解，且与两者的衰减相对速率无关。这确定了用于3D及3D+时间显微镜图像分割的正则化模型的几何极限。
英文摘要
We study the vanishing-regularization limit of the regularized subjective-surface model introduced for touching and dividing cell nuclei and previously analyzed for fixed positive regularization parameters. On a smooth bounded domain $Ω\subset\mathbb R^d$, the model is \[ \begin{aligned} \partial_tu^{\eps,ν} ={}&νΔu^{\eps,ν} +A_\eps(\nabla u^{\eps,ν}) \operatorname{div}\!\left( G(x)\frac{\nabla u^{\eps,ν}} {A_\eps(\nabla u^{\eps,ν})} \right) -μR_η(x,u^{\eps,ν}),\\ &A_\eps(p)=\sqrt{\eps^2+|p|^2}, \qquad R_η(x,r)=Λ(x)H_η(r-q), \end{aligned} \] subject to homogeneous Dirichlet data and a prescribed initial profile. We formulate the limiting weighted level-set mean-curvature equation as a viscosity initial-boundary value problem and establish global existence, uniqueness, preservation of $[0,1]$, and nonexpansive dependence on the initial data. We determine the direct upper and lower limiting values of the regularized principal operators at zero gradient and identify the additional argument needed to recover the geometric directional values. For every $T>0$, the regularized solutions converge uniformly to the unique viscosity solution on $[0,T]\times\overlineΩ$ as $\eps,ν\to0$, independently of their relative rates of decay. This determines the geometric limit of the regularized model used in 3D and 3D+time microscopy segmentation.
