This thesis introduces a graph cut optimization for the Mumford Shah functional. Mumford Shah model is considered one of the basic frameworks in image segmentation. Chan and Vese [6] introduced a numerical implementation of the Mumford-Shah model in a level set framework. Level set methods suffer from being very computationally expensive and they is not guaranteed to converge to the global solution as they depend mainly on the gradient descent method in the optimization step. The thesis introduces a discrete formulation of the Chan-Vese model, prove that it can mapped to a graph and minimized using graph cuts. This discrete formulation overcomes the aforementioned drawbacks, of the level set framework, by using graph cuts to optimize the formulated energy function. Graph cuts also converges to the global minimum in a polynomial time and hence it improves the speed and accuracy. The new algorithm has been applied to some of the images used by Chan and Vese in [6]. A comparison between the output of both algorithms is introduced and it shows that our algorithm outperformed the classical curve evolution method introduced in [6]. The algorithm has also been applied to brain MRI images to solve the brain extraction problem and it has shown very promising results.