Hyperspectral cameras have recently been miniaturized for operation on lightweight airborne platforms such as UAV or small aircraft. Unlike frame cameras (RGB or Multispectral), many hyperspectral sensors use a linear array or 'push-broom' scanning design. This design presents significant challenges for image rectification and the calibration of the intrinsic and extrinsic camera parameters. Typically, methods employed to address such tasks rely on a precise GPS/INS estimate of the airborne platform trajectory and a detailed terrain model. However, inaccuracies in the trajectory or surface model information can introduce systematic errors and complicate geometric modeling which ultimately degrade the quality of the rectification. To overcome these challenges, we propose a method for tie point extraction and camera calibration for 'push-broom' hyperspectral sensors using only the raw spectral imagery and raw, possibly low quality, GPS/INS trajectory. We demonstrate that our approach allows for the automatic calibration of airborne systems with hyperspectral cameras, outperforms other state-of-the-art automatic rectification methods and reaches an accuracy on par with manual calibration methods.