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Various types of video can be captured with fisheye lenses; their wide field of view is particularly suited to surveillance video. However, fisheye lenses introduce distortion, and this changes as objects in the scene move, making fisheye video difficult to interpret. Current still fisheye image correction methods are either limited to small angles of view, or are strongly content dependent, and therefore unsuitable for processing video streams. We present an efficient and robust scheme for fisheye video correction, which minimizes time-varying distortion and preserves salient content in a coherent manner. Our optimization process is controlled by user annotation, and takes into account a wide set of measures addressing different aspects of natural scene appearance. Each is represented as a quadratic term in an energy minimization problem, leading to a closed-form solution via a sparse linear system. We illustrate our method with a range of examples, demonstrating coherent natural-looking video output. The visual quality of individual frames is comparable to those produced by state-of-the-art methods for fisheye still photograph correction.