Fine numerical analysis of the crack-tip position for a Mumford-Shah minimizer
For researchers in image segmentation and fracture mechanics, this provides a computationally efficient method for crack-tip localization, though it is an incremental improvement over existing iterative techniques.
The paper introduces a fast iterative algorithm to locate the crack tip in Mumford-Shah minimizers, using conformal mapping and the augmented immersed interface method. Numerical results match analytic predictions and address open questions in free discontinuity problems.
A new algorithm to determine the position of the crack (discontinuity set) of certain minimizers of Mumford-Shah functional in situations when a crack-tip occurs is introduced. The conformal mapping $w=\sqrt{z}$ in the complex plane is used to transform the free discontinuity problem to a new type of free boundary problem, where the symmetry of the free boundary is an additional constraint of a non-local nature. Instead of traditional Jacobi or Newton iterative methods, we propose a simple iteration method which does not need the Jacobian but is way fast than the Jacobi iteration. In each iteration, a Laplace equation needs to be solved on an irregular domain with a Dirichlet boundary condition on the fixed part of the boundary; and a Neumann type boundary condition along the free boundary. The augmented immersed interface method is employed to solve the potential problem. The numerical results agree with the analytic analysis and provide insight into some open questions in free discontinuity problems.