Variational formulations have been proposed for a number of tasks in early vision. Discrete versions of these problems are closely related to Markov random field models and are typically used in implementing such methods. In particular, discrete and continuous versions for the problem of image segmentation have received considerable attention from both theoretical and algorithmic perspectives. It has been previously pointed out that the usual discrete version of the segmentation problem does not properly approximate the continuous formulation in the sense that the discrete solutions may not converge to a solution of the continuous problem as the lattice spacing tends to zero. One method for modifying the discrete formulations to ensure such convergence has been previously discussed. Here we consider two other partially discrete formulations which also satisfy desirable convergence properties in the continuum limit, and we discuss some general ideas about digitized versions of the variational formulation of the segmentation problem. *This work was supported by the U.S. Army Research Office under Contract DAAL03-86-K-0171, the Air Force Office of Scientific Research under grant AFOSR 89-0276 and by the Department of the Navy under Air Force Contract F19628-90-C-0002.