Analysis of gradient flow of a regularized Mumford-Shah functional for image segmentation and image inpainting

This paper studies the gradient flow of a regularized Mumford-Shah functional proposed by Ambrosio and Tortorelli (1990, 1992) for image segmentation, and adopted by Esedoglu and Shen (2002) for image inpainting. It is shown that the gradient flow with L-2 x L-infinity initial data possesses a global weak solution, and it has a unique global in time strong solution, which has at most finite number of point sinaularities in the space-time, when the initial data are in H-1 x H-1 boolean AND L-infinity. A family of fully discrete approximation schemes using low order finite elements is proposed for the gradient flow. Convergence of a subsequence (resp. the whole sequence) of the numerical solutions to a weak solution (resp. the strong solution) of the gradient flow is established as the mesh sizes tend to zero, and optimal and suboptimal order error estimates. which depend on 1/epsilon and 1/k(epsilon) only in low polynomial order, are derived for the proposed fully discrete schemes under the mesh relation k = o(h(1/2)). Numerical experiments are also presented to show effectiveness of the proposed numerical methods and to validate the theoretical analysis.