Unsupervised trajectory compression

We present a method for compressing trajectories in an unsupervised manner. Given a set of trajectories sampled from a space we construct a basis for compression whose elements correspond to paths in the space which are topologically distinct. This is achieved by computing a canonical representative for each element in a generating set for the first homology group and decomposing these representatives into a set of distinct paths. Trajectory compression is subsequently accomplished through representation in terms of this basis. Robustness with respect to outliers is achieved by only considering those elements of the first homology group which exist in the super-level sets of the Kernel Density Estimation (KDE) above a threshold. Robustness with respect to small scale topological artifacts is achieved by only considering those elements of the first homology group which exist for a sufficient range in the super-level sets. We demonstrate this approach to trajectory compression in the context of a large set of crowd-sourced GPS trajectories captured in the city of Chicago. On this set, the compression method achieves a mean geometrical accuracy of 108 meters with a compression ratio of over 12

Tri-partitions and bases of an ordered complex

Generalizing the decomposition of a connected planar graph into a tree and a dual tree, we prove a combinatorial analog of the classic Helmholtz–Hodge decomposition of a smooth vector field. Specifically, we show that for every polyhedral complex, K, and every dimension, p, there is a partition of the set of p-cells into a maximal p-tree, a maximal p-cotree, and a collection of p-cells whose cardinality is the p-th reduced Betti number of K. Given an ordering of the p-cells, this tri-partition is unique, and it can be computed by a matrix reduction algorithm that also constructs canonical bases of cycle and boundary groups

Computing Linking Numbers of a Filtration

. We develop fast algorithms for computing the linking numbe