The use of neural networks in safety-critical computer vision systems calls for their robustness certification against natural geometric transformations (e.g., rotation, scaling). However, current certification methods target mostly norm-based pixel perturbations and cannot certify robustness against geometric transformations. In this work, we propose a new method to compute sound and asymptotically optimal linear relaxations for any composition of transformations. Our method is based on a novel combination of sampling and optimization. We implemented the method in a system called DeepG and demonstrated that it certifies significantly more complex geometric transformations than existing methods on both defended and undefended networks while scaling to large architectures.