One of the main difficulties in extracting line networks from images, and in particular road networks from remote sensing images, is the existence of interruptions in the data caused, for example, by occlusions. These can lead to gaps in the extracted network that do not correspond to gaps in the real network. In this report, we describe a higher-order active contour energy that in addition to favouring network-like regions composed of thin arms joining at junctions, also includes a prior term that penalizes network configurations containing `nearby opposing extremities', and thereby makes their appearance in the extracted network less likely. If nearby opposing extremities form during the gradient descent evolution used to minimize the energy, the new energy term causes the extremities to attract one another, and hence to move towards one another and join, thus closing the gap. To minimize the energy, we develop specific techniques to handle the high-order derivatives that appear in the gradient descent equation. We present the results of automatic extraction of networks from real remote-sensing images, showing the ability of the model to overcome interruptions.