The active contour/snake model is a successful variational model for image segmentation, evolving a contour toward object boundaries. Its success relies on mathematical properties and efficient numerical schemes, but it suffers from local minima in the energy function, making initial conditions critical. This paper proposes solving this issue by finding a global minimum of the active contour model. The approach unifies image segmentation and denoising into a global minimization framework, combining the snake model, ROF denoising model, and Mumford–Shah segmentation model. Theorems are established to prove the existence of a global minimum. A dual formulation is proposed for efficient numerical solution, avoiding the need for re-initialization in level set methods. The method is applied to synthetic and real-world images, showing improved performance compared to other models. The paper also discusses the limitations of the standard snake/GAC model, which is sensitive to initial conditions due to non-convexity and local minima. The goal is to find a global minimum of a convex functional for reliable segmentation. The paper introduces new active contour energies based on the GAC model, aiming for the correct segmentation result regardless of initial conditions. Chan, Esedoglu, and Nikolova proposed a method to address global minima by relating segmentation to denoising. This work develops three theoretical models for global minimization, combining the snake/GAC model and the ROF denoising model. Image denoising aims to remove noise while preserving features like edges.The active contour/snake model is a successful variational model for image segmentation, evolving a contour toward object boundaries. Its success relies on mathematical properties and efficient numerical schemes, but it suffers from local minima in the energy function, making initial conditions critical. This paper proposes solving this issue by finding a global minimum of the active contour model. The approach unifies image segmentation and denoising into a global minimization framework, combining the snake model, ROF denoising model, and Mumford–Shah segmentation model. Theorems are established to prove the existence of a global minimum. A dual formulation is proposed for efficient numerical solution, avoiding the need for re-initialization in level set methods. The method is applied to synthetic and real-world images, showing improved performance compared to other models. The paper also discusses the limitations of the standard snake/GAC model, which is sensitive to initial conditions due to non-convexity and local minima. The goal is to find a global minimum of a convex functional for reliable segmentation. The paper introduces new active contour energies based on the GAC model, aiming for the correct segmentation result regardless of initial conditions. Chan, Esedoglu, and Nikolova proposed a method to address global minima by relating segmentation to denoising. This work develops three theoretical models for global minimization, combining the snake/GAC model and the ROF denoising model. Image denoising aims to remove noise while preserving features like edges.