Self-Similarity Based Time Warping

TL;DR

This paper introduces IBDTW, a novel time warping method that uses self-similarity matrices to align time-ordered point clouds without spatial alignment, applicable to cross-modal data and partial alignments.

Contribution

The work proposes a new isometry-blind dynamic time warping algorithm leveraging self-similarity matrices, avoiding explicit spatial alignment and enabling partial alignments.

Findings

IBDTW lower bounds the L1 Gromov-Hausdorff distance.

It effectively aligns spatially transformed point clouds.

Partial alignment extension improves robustness to cropping.

Abstract

In this work, we explore the problem of aligning two time-ordered point clouds which are spatially transformed and re-parameterized versions of each other. This has a diverse array of applications such as cross modal time series synchronization (e.g. MOCAP to video) and alignment of discretized curves in images. Most other works that address this problem attempt to jointly uncover a spatial alignment and correspondences between the two point clouds, or to derive local invariants to spatial transformations such as curvature before computing correspondences. By contrast, we sidestep spatial alignment completely by using self-similarity matrices (SSMs) as a proxy to the time-ordered point clouds, since self-similarity matrices are blind to isometries and respect global geometry. Our algorithm, dubbed "Isometry Blind Dynamic Time Warping" (IBDTW), is simple and general, and we show that its associated dissimilarity measure lower bounds the L1 Gromov-Hausdorff distance between the two point sets when restricted to warping paths. We also present a local, partial alignment extension of IBDTW based on the Smith Waterman algorithm. This eliminates the need for tedious manual cropping of time series, which is ordinarily necessary for global alignment algorithms to function properly.