We present a framework to super-resolve planar regions found in urban scenes and other man-made environments by taking into account their 3D geometry. Such regions have highly structured straight edges, but this prior is challenging to exploit due to deformations induced by the projection onto the imaging plane. Our method factors out such deformations by using recently developed tools based on convex optimization to learn a transform that maps the image to a domain where its gradient has a simple group-sparse structure. This allows to obtain a novel convex regularizer that enforces global consistency constraints between the edges of the image. Computational experiments with real images show that this data-driven approach to the design of regularizers promoting transform-invariant group sparsity is very effective at high super-resolution factors. We view our approach as complementary to most recent super-resolution methods, which tend to focus on hallucinating high-frequency textures.