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We present a novel representation of images based on a decomposition into atomic patches which we call medial visual fragments and which is particularly suited for structural grouping. Specifically, we show that the medial axis/shock graph of a contour map partitions the image domain into non-overlapping regions, which together with the image information define the visual fragments. The main advantage of such a generic representation is that both contour and regional information are explicitly available so that in the presence of partial evidence and ambiguity in maps indicating edges and regional homogeneity, both aspects can be simultaneously used for perceptual grouping of fragments into a coherent whole. Grouping of visual fragments is represented as a set of canonical transformations of visual fragments, namely, the gap and loop transforms. The advantage of this representation in comparison to perceptual grouping using only contour continuity or region grouping is demonstrated on synthetic and realistic examples.