Linear time offline tracking and lower envelope algorithms

Offline tracking of visual objects is particularly helpful in the presence of significant occlusions, when a frame-by-frame, causal tracker is likely to lose sight of the target. In addition, the trajectories found by offline tracking are typically smoother and more stable because of the global optimization this approach entails. In contrast with previous work, we show that this global optimization can be performed in O(MNT) time for T frames of video at M × N resolution, with the help of the generalized distance transform developed by Felzenszwalb and Huttenlocher [13]. Recognizing the importance of this distance transform, we extend the computation to a more general lower envelope algorithm in certain heterogeneous l₁-distance metric spaces. The generalized lower envelope algorithm is of complexity O(MN(M+N)) and is useful for a more challenging offline tracking problem. Experiments show that trajectories found by offline tracking are superior to those computed by online tracking methods, and are computed at 100 frames per second.