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

¹ Department of Mathematics, The University of Tennessee, Knoxville, TN 37996, USA.
² Department of Mathematics, ETHZ, 8092 Zürich, Switzerland, apr@math.ethz.ch.

Received: 10 June 2003
Revised: 5 September 2003

Abstract

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 L² x L^∞ 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 singularities in the space-time, when the initial data are in H¹ x H¹ ∩ L^∞. 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 and only in low polynomial order, are derived for the proposed fully discrete schemes under the mesh relation . Numerical experiments are also presented to show effectiveness of the proposed numerical methods and to validate the theoretical analysis.