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Multi-camera systems are an important sensor platform for intelligent systems such as self-driving cars. Pattern-based calibration techniques can be used to calibrate the intrinsics of the cameras individually. However, extrinsic calibration of systems with little to no visual overlap between the cameras is a challenge. Given the camera intrinsics, infrastucture-based calibration techniques are able to estimate the extrinsics using 3D maps pre-built via SLAM or Structure-from-Motion. In this paper, we propose to fully calibrate a multi-camera system from scratch using an infrastructure-based approach. Assuming that the distortion is mainly radial, we introduce a two-stage approach. We first estimate the camera-rig extrinsics up to a single unknown translation component per camera. Next, we solve for both the intrinsic parameters and the missing translation components. Extensive experiments on multiple indoor and outdoor scenes with multiple multi-camera systems show that our calibration method achieves high accuracy and robustness. In particular, our approach is more robust than the naive approach of first estimating intrinsic parameters and pose per camera before refining the extrinsic parameters of the system. The implementation is available at https://github.com/youkely/InfrasCal.