Abstract
This paper presents a comprehensive derivation and implementation of the Chan-Vese active contour model for image segmentation. The model, derived from the Mumford-Shah variational framework, evolves contours based on regional intensity differences rather than image gradients, making it highly effective for segmenting noisy images or images with weak boundaries. We provide a rigorous mathematical derivation of the level set formulation, including detailed treatment of each energy term using the divergence theorem and curve evolution theory. The resulting algorithm is implemented in Python using finite difference methods with special care to numerical stability, including an upwind entropy scheme and curvature-based regularization. Experimental results on medical and synthetic images demonstrate accurate segmentation, robustness to noise, and superior performance compared to classical edge-based methods. This study confirms the suitability of the Chan-Vese model for complex segmentation tasks and highlights its potential for use in real-world imaging applications.
Abstract (translated)
本文介绍了Chan-Vese活动轮廓模型在图像分割中的全面推导和实现。该模型基于Mumford-Shah变分框架，通过区域强度差异来演化轮廓线，而不是依赖于图像梯度，因此对于处理噪声大或边界弱的图像非常有效。我们提供了对水平集形式化的严格的数学推导，包括使用散度定理和曲线演化理论详细分析每个能量项。 算法采用Python实现，并利用有限差分方法进行数值稳定性的特别注意，其中包括迎风熵方案和基于曲率的正则化技术。通过医学图像和合成图像的实验结果表明，该模型能够实现准确的分割、对噪声具有鲁棒性，并且相较于传统的边缘基础的方法表现出更优越的性能。 本研究确认了Chan-Vese模型在复杂分割任务中的适用性，并强调其在实际成像应用中使用的潜力。
URL
https://arxiv.org/abs/2506.19344
