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# Orientation-Preserving Vectorized Distance Between Curves

*Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference*, PMLR 145:472-496, 2022.

#### Abstract

We introduce an orientation-preserving landmark-based distance for continuous curves, which can be viewed as an alternative to the Fre ́chet or Dynamic Time Warping distances. This measure re- tains many of the properties of those measures, and we prove some relations, but can be interpreted as a Euclidean distance in a particular vector space. Hence it is significantly easier to use, faster for general nearest neighbor queries, and allows easier access to classification results than those measures. It is based on the signed distance function to the curves or other objects from a fixed set of landmark points. We also prove new stability properties with respect to the choice of landmark points, and along the way introduce a concept called signed local feature size (slfs) which param- eterizes these notions. Slfs explains the complexity of shapes such as non-closed curves where the notion of local orientation is in dispute – but is more general than the well-known concept of (unsigned) local feature size, and is for instance infinite for closed simple curves. Altogether, this work provides a novel, simple, and powerful method for oriented shape similarity and analysis.