Image Segmentation using Chan-Vese Active Contours

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.