A variational method for the reconstruction and segmentation of images was recently proposed by Mumford and Shah [15]. In this paper we treat two aspects of the problem. The first concerns existence of solutions to the problem; the second concerns representations suitable for computation. Discrete versions of this problem have been proposed and studied in [5,12,14,15]. However, it seems that these discrete versions do not properly approximate the continuous problem in the sense that their solutions may not converge to a solution of the continuous problem as the lattice spacing tends to zero.Here we consider the use of an alternate lattice approximation for the boundaries of the image and Minkowski content as a cost term for the boundaries. Several properties of Minkowski content are derived. These are used to show that partially discrete versions of the variational problem possess some desirable convergence properties. Specifically, under suitable conditions, solutions to the discrete problem converge in the continuum limit to a solution of the continuous problem.The existence result included here is applicable to both discrete and continuous versions of the problem.