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This paper presents a method for computing the visual hull that is based on two novel representations: the rim mesh, which describes the connectivity of contour generators on the object surface; and the visual hull mesh, which describes the exact structure of the surface of the solid formed by intersecting a finite number of visual cones. We describe the topological features of these meshes and show how they can be identified in the image using epipolar constraints. These constraints are used to derive an image-based practical reconstruction algorithm that works with weakly calibrated cameras. Experiments on synthetic and real data validate the proposed approach.