Orientation-Preserving Vectorized Distance Between Curves

TL;DR

This paper introduces an orientation-preserving landmark-based distance for continuous curves that is easier to compute and interpret than traditional measures like Frechet or DTW, enabling more efficient shape analysis.

Contribution

It proposes a new Euclidean-like distance measure for curves based on signed distances to landmarks, with proven stability and a novel shape complexity parameter called signed local feature size.

Findings

The measure retains key properties of existing distances.

It is computationally faster for nearest neighbor queries.

Introduces the concept of signed local feature size for shape complexity.

Abstract

We introduce an orientation-preserving landmark-based distance for continuous curves, which can be viewed as an alternative to the \Frechet or Dynamic Time Warping distances. This measure retains 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 \emph{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 parameterizes 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.