A multigrid finite-difference method for the solution of Euler equations of the variational image segmentation Original Research Article

This paper deals with multigrid finite-difference approximations of Euler equations (Eqs) arising in the variational formulation of image segmentation problems. We illustrate that the Eqs can be obtained by the definition of the minimization problem for the Mumford–Shah functional (MSf), along with a sequence of functionals Γ-convergent to the MSf, and the multigrid finite-difference solution of the Eqs, associated to the kth functional of the sequence, can be carried out. We assume finite-difference approximations of the Euler equations, we define the related multigrid solution algorithm and we evaluate algorithmic performance by application to segmentation of synthetic images. We analyze computed discontinuity contours and convergence histories of method executions.