Identifying Common Portions between Two Trajectories

1 Introduction Trajectories are functions from a time domain—an interval on the real line—to R d with d > 1, and observed as sequences of points sampled from them. A fundamental problem in analyzing this data is that of identifying common patterns between pairs or among groups of trajecto-ries observed as sequences of sampled points. be two sequences of points in R d , sampled from two trajectories γ 1 and γ 2 defined over the time interval [0, 1]. For simplicity , we assume that P and Q are points sampled from the images of the trajectories and we ignore the temporal component. 1 Since, in practice, the underlying continuous trajectories γ 1 and γ 2 are not known but we observe only the sampled points P and Q, we will work in the discrete setting where we are only concerned with these sample points. In this abstract, we will refer to the discrete sample points P , Q as the input trajectories. We wish to compute correspondences between points belonging to similar portions of these trajectories while distinguishing these portions from the dissimilar ones. The following issues with trajectory sampling must be taken into account when identifying similarity: (i) significantly different sampling rates, (ii) presence of noise/outliers which must be distinguished from dissim-ilarities, and (iii) presence of significant unobserved portions on the trajectories with no sample points. Background. A common choice for measuring trajectory similarity is the Fréchet distance [1] defined as follows. A reparameterization is a continuous non-decreasing surjec-tion α : [0, 1] → [0, 1], such that α(0) = 0 and α(1) = 1. The Fréchet distance Fr(γ 1 , γ 2) is given by: