We propose a method for decomposing images into triangles. Contrary to superpixel methods, our output representation both preserves the geometric information disseminated in input images, and has an attractive storage capacity. Our method relies on the flexibility and efficiency of Delaunay point processes to address the problem. These stochastic models distribute points interacting between each other through Delaunay triangulations. The mechanism for distributing points combines several complementary ingredients including image discontinuity preservation, radiometric homogeneity inside atomic regions as well as priors on the shape of these regions. Said differently, sampled points and induced shapes work in tandem. The potential of our approach is shown through comparisons with existing oversegmentation methods and applications to vision problems.