Semi-analytical penalized threshold dynamics method for binary image segmentation

Binary image segmentation is a fundamental task in image analysis, often requiring methods that ensure both stability and interface continuity. In this paper, inspired by the Allen–Cahn equation, we propose a semi-analytical penalized threshold dynamics method to improve the efficiency and stability of binary image segmentation. The method employs a spectral approach in conjunction with operator splitting techniques to effectively address different components of the problem. First, the penalization term is solved analytically, allowing for accurate treatment of intensity differences. Next, the spectral method is utilized to solve the heat equation, providing exact solutions for the dynamics of interface evolution. Finally, a thresholding step is applied to achieve a clear demarcation of the interface. It is shown that the maximum principle is preserved throughout the whole process. The method can also be extended to three-dimensional (3D) segmentation, allowing for the analysis of volumetric data. This framework provides a robust approach to stable segmentation, preserving interface continuity and accurate region differentiation in both 2D and 3D contexts. Visual results demonstrate the effectiveness of the method across various image segmentation tasks, highlighting its potential for practical applications in binary image analysis. The basic 2D code implementation is provided in the appendix for reproducibility and further exploration.