发表机构
School of Mathematics, Harbin Institute of Technology（哈尔滨工业大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对相邻结构弱边界分割难题，提出基于Cahn-Hilliard方程的平滑相分离变分模型，结合区域拟合与相场正则化，引入梯度流并开发SAV方案，实验表明该方法能有效分割弱边界相邻均匀结构，提升分割精度与边界定位。
AI 中文摘要
在图像分析中，分割具有相似强度分布的相邻结构仍然是一个具有挑战性的问题，特别是当物体边界较弱或不明确时。在这种情况下，经典变分模型可能会受到退化的图像驱动力影响，导致边界泄漏或相邻区域的不期望合并。为了解决这些限制，我们提出了一种基于Cahn-Hilliard方程的平滑相分离变分模型，用于均匀外观结构的弱边界分割。所提出的框架将基于softmax的区域拟合与Cahn-Hilliard相场正则化相结合，以在弱图像驱动力下保持界面辨别力。我们进一步引入了混合$L^2-H^{-1}$梯度流，它保留了高阶界面正则化，同时允许相质量的自适应变化，建立了连续能量耗散定律，并证明了自然解类中弱解的存在性和唯一性。对于数值计算，我们开发了一种稳定的标量辅助变量（SAV）方案，该方案是线性的、基于FFT的，并且满足修改后的离散能量耗散定律。在合成图像和医学图像上的数值实验表明，与代表性的变分、相场和深度学习方法相比，该方法有效地分离了跨越弱边界的相邻均匀结构，并实现了有竞争力的分割精度和改进的边界定位。
英文摘要
Segmentation of adjacent structures with similar intensity distributions remains a challenging problem in image analysis, particularly when object boundaries are weak or ambiguous. Under such conditions, classical variational models may suffer from degenerated image-driven forces, leading to boundary leakage or undesired merging of neighboring regions. To address these limitations, we propose a smooth phase-separation variational model based on the Cahn--Hilliard equation for weak-boundary segmentation of homogeneous-appearance structures. The proposed framework integrates softmax-based region fitting with Cahn--Hilliard phase-field regularization to maintain interface discrimination under weak image-driven forces. We further introduce a mixed $L^2-H^{-1}$ gradient flow, which preserves higher-order interfacial regularization while allowing adaptive changes of phase masses, establish the continuous energy dissipation law, and prove the existence and uniqueness of weak solutions in the natural solution class. For numerical computation, we develop a stabilized scalar auxiliary variable (SAV) scheme that is linear, FFT-based, and satisfies a modified discrete energy dissipation law. Numerical experiments on synthetic and medical images demonstrate that the proposed method effectively separates adjacent homogeneous structures across weak boundaries and achieves competitive segmentation accuracy and improved boundary localization compared with representative variational, phase-field, and deep learning methods.
