A Smooth Phase-Separation Model for Weak-Boundary Segmentation of Homogeneous Structures

2026-07-24

AI Summary

This work addresses the challenge of segmenting homogeneous structures in images characterized by similar intensities and ambiguous boundaries. A novel variational model is proposed that integrates softmax-based region fitting with Cahn–Hilliard phase-field regularization, establishing a smooth phase-separation framework endowed with a continuous energy dissipation law. This is the first formulation to unify these two components, and the existence and uniqueness of its solution are rigorously proven. A linear numerical scheme preserving a discrete energy dissipation law is further developed, leveraging a mixed L²–H⁻¹ gradient flow and FFT acceleration for efficient computation. Experimental results on both synthetic and medical images demonstrate that the method significantly outperforms state-of-the-art variational, phase-field, and deep learning approaches under weak boundary conditions, achieving superior segmentation accuracy and boundary localization.

Abstract

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.

Problem

• weak-boundary segmentation
• homogeneous structures
• phase separation
• image segmentation
• boundary ambiguity

Innovation

• Cahn–Hilliard equation
• phase-field segmentation
• weak-boundary segmentation
• energy dissipation
• scalar auxiliary variable (SAV)