We propose a novel graphical model for probabilistic image segmentation that contributes both to aspects of perceptual grouping in connection with image segmentation, and to globally optimal inference with higher-order graphical models. We represent image partitions in terms of cellular complexes in order to make the duality between connected regions and their contours explicit. This allows us to formulate a graphical model with higher-order factors that represent the requirement that all contours must be closed. The model induces a probability measure on the space of all partitions, concentrated on perceptually meaningful segmentations. We give a complete polyhedral characterization of the resulting global inference problem in terms of the multicut polytope and efficiently compute global optima by a cutting plane method. Competitive results for the Berkeley segmentation benchmark confirm the consistency of our approach.